Research ArticleA Novel Approach for Nonstationary Time Series Analysis withTime-Invariant Correlation Coefficient

Chengrui Liu1 Zhihua Wang2 Huimin Fu2 and Yongbo Zhang2

1 Beijing Institute of Control Engineering Beijing 100190 China2 School of Aeronautic Science and Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Zhihua Wang wangzhihuabuaaeducn

Received 15 July 2013 Accepted 25 November 2013 Published 22 January 2014

Academic Editor Yingwei Zhang

Copyright copy 2014 Chengrui Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We will concentrate on the modeling and analysis of a class of nonstationary time series called correlation coefficient stationaryseries which commonly exists in practical engineering First the concept and scope of correlation coefficient stationary series arediscussed to get a better understanding Second a theorem is proposed to determine standard deviation function for correlationcoefficient stationary seriesThird we propose a moving multiple-point average method to determine the function forms for meanand standard deviation which can help to improve the analysis precision especially in the context of limited sample size Fourththe conditional likelihood approach is utilized to estimate the model parameters In addition we discuss the correlation coefficientstationarity test method which can contribute to the verification of modeling validity Monte Carlo simulation study illustrates theauthentication of the theorem and the validity of the establishedmethod Empirical study shows that the approach can satisfactorilyexplain the nonstationary behavior ofmany practical data sets including stock returnsmaximumpower load Chinamoney supplyand foreign currency exchange rate The effectiveness of these processes is addressed by forecasting performance

1 Introduction

Time series methods have been generally accepted as oneof the most important means in an increasing number ofreal-world applications including finance In the past severaldecades considerable efforts have been made for time seriesanalysis and prediction [1ndash3] Time series approaches [4]regression models [5] artificial intelligence method [6] andGrey theory [7] are the commonly used techniques [8] Manyanalyses are based on the assumption that the probabilisticproperties of the underlying process are time invariant thatis the series to be analyzed is covariance stationaryModelingthis stationary time series one frequently chooses time seriesmethods because of their high performance and robustnesswhich mainly include autoregressive (AR) moving average(MA) autoregressive moving average (ARMA) autoregres-sive integrated moving average (ARIMA) and Box-Jenkinsmodels

Although the stationary assumption is very useful for theconstruction of simple models it does not seem to be thebest strategy in practice and sometimes such stationarity

assumptions are often questionable [9] because time serieswith time-varying means and variances are commonly seenin economic forecast [10] fault diagnosis [11] quality control[12] signal processing [13] performance test [14] automaticcontrol [15] biopharmaceutical [16] and other fields Whenthe heteroscedasticity time series is processed by existingcovariance stationary time series analysis method the modelparameters will lose the minimum variance property andthe variance estimator is no longer the unbiased estimation[17] Referring to time series approaches and regressionanalysis reasonable analysis and accurate prediction cannotbe achieved for the nonstationary time series Consideringartificial intelligence such as expert system and neuralnetwork abundant prediction rule and practical experiencefrom specific experts and large historical data banks arerequisite for precise forecast Although the Grey predictionmodel has been successfully applied in various fields and hasdemonstrated satisfactory results its prediction performancestill could be improved The reason is that the Grey forecast-ing model is constructed of exponential function and hence

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 148432 12 pageshttpdxdoiorg1011552014148432

2 Mathematical Problems in Engineering

it may have worse prediction precise in the case of more ran-dom data sets

Meanwhile great progress has been achieved related toprocess monitoring in industrial fields To solve the multi-mode problem illustrated by industrial process because ofmultiple production patterns in the same production linevarious methods including partial least squares methods[18] model library-based methods [19] the Gaussian mix-ture model [20] the localized Fisher discriminant analysisapproach [21] and the recent independent component anal-ysis (ICA) based statistical processing methods [22 23] havebeen constructedThe industrial process monitoring is of sig-nificant importance in the literature However the statisticalmethod based on time series analysis is focused on in thisstudy

In 1982 Engle [24] proposed the concept of conditionalheteroscedasticity with which they solved the conditionalheteroscedasticity estimation problem for time series withconstant unconditional variance The proposed theory hasbeen widely applied in financial risk evaluation For his sig-nificant contribution Engle gained 2003rsquos Nobel EconomicsPrize However the analysis problem for time series withunconditional time-varying variance still exists It can becommonly seen in application [25 26] In addition somehybrid models are also seen in the literature [27 28] whichcombine dissimilar models or models that disagree with eachother strongly to lower the generalization variance or errorAlthough hybrid models have shown advantages in some cir-cumstances there is no denying that they are much compli-cated for application

A simple nonstationary model contains a second-orderstationary process modulated by a deterministic time-vary-ing mean and a deterministic unconditional time-varyingvariance [29] Let 119910

119905 119905 = 1 2 3 be a stationary pro-

cess with zero mean and a simple nonstationary model canbe given by 119909

119905= 120583(119905) + 120590(119905)119910

119905 where 120583(119905) is the determin-

istic time-varying mean function and 120590(119905) is the determinis-tic unconditional time-varying standard deviation functionwhich is strictly positive We can conclude that the non-stationarity of 119909

119905is expressed by its evolving mean and

unconditional variance The efficient analysis of this nonsta-tionary process is a substantial drawback in practice and hasgradually attached great importance to the researchers

Based on systematic study of mass measured data Fuand Liu [30] found that some common characteristics areshared by certain nonstationary time series Like general non-stationary ones these series exhibit time-varying mean 120583(119905)and variance 1205902(119905) and their autocovariance function 120574(119905 119905 +120591) = Cov(119909

119905 119909119905+120591) = 119864[119909

119905minus 120583(119905)][119909

119905+120591minus 120583(119905 + 120591)] is no

longer a univariate function of time interval 120591 that is 120574(119905 119905 +120591) = 120574(120591) while their correlation coefficient function 120588(119905 119905 +120591) = Cov(119909

119905 119909119905+120591)[120590(119905)120590(119905 + 120591)] is still a univariate function

of time interval 120591 that is 120588(119905 119905 + 120591) = 120588120591 Accordingly we

can conclude that (i) they are not covariance stationary timeseries [31] whose autocovariance function 120574(119905 119905 + 120591) is a uni-variate function of time interval 120591 that is 120574(119905 119905 + 120591) =120574(120591) (ii) they are a certain class of nonstationary time seriesand different from other nonstationary time series whose

correlation coefficient function 120588(119905 119905 + 120591) varies with time119905 On this basis Fu and Liu [30] proposed the concept ofldquocorrelation coefficient stationary processrdquo and discussed theestablishment of the correlation coefficient autoregressivemoving average (CCARMA) model

In this paper we further study the nonstationary behav-ior of this correlation coefficient stationary series Firstcharacteristics of the variance function have been furtherstudied and a rigorous theorem was proposed which canhelp not only determination of the standard deviation butalso verification of the modeling process Second a rollingwindow determination scheme named moving multiple-point average method has been established to obtain themean and standard deviation functions This technology canenhance the accuracy under the same sample size and theeffect is more obvious in case of limited sample sizeThird westudied the scope of correlation coefficient stationary processin which discussion can be helpful to better understand theconcept of correlation coefficient stationary process Finallythe correlation coefficient stationary test method has beeninvestigated which can assess the validity of the modelingprocess andmake themodeling process a closed-loop system

In the next section the concept of CCARMA processand its basic properties are introduced we also discussthe CCARMA model In Section 3 we develop a methodfor determining the function forms for mean and standarddeviation Section 4 establishes the parameter determinationmethod and a correlation coefficient stationary test methodSection 5 illustrates simulation studies to assess the validity ofthe approach And Section 6 is devoted to the practical evalu-ation of the proposed method on several data sets includingdaily returns to Shanghai composite index Guangxi monthlymaximum power load China monthly money supply anddaily foreign exchange (FX) rate EURUSD A comparisonbetween our forecasting results and ARIMA variable differ-ential GARCH GM(1 1) and Modified GM(1 1) models isalso provided in this section Finally we conclude this paperwith a discussion in Section 7

2 Concept of CCARMA Process

21 Concept of CCARMA Process Generally speaking tradi-tional stationarity means covariance stationarity [31] Timeseries 119909

119905 119905 = 1 2 119899 is a covariance stationary time series

if the following two conditions are satisfied

(i) The mean function 120583(119905) = 119864(119909119905) does not evolve

through time that is 120583(119905) = 120583(ii) The autocovariance function 120574(119905 119905 + 120591) =

Cov(119909119905 119909119905+120591) = 119864[119909

119905minus 120583(119905)][119909

119905+120591minus 120583(119905 + 120591)]

is a univariate function of time interval 120591 that is120574(119905 119905 + 120591) = 120574(120591)

Let 120591 = 0 and the autocovariance Cov(119909119905 119909119905) equals

the variance Var(119909119905) at time 119905 Consequently variance of

covariance stationarity time series dose not vary with timethat is Var(119909

119905) = 1205902 However most time series encountered

in practice cannot satisfy the above two requirements Tosolve the analysis problem of a certain class of nonstationary

Mathematical Problems in Engineering 3

time series Fu and Liu [30] extended the above conceptand proposed the following two concepts of correlationcoefficient stationary time series

Concept 1 Let 119909119905 119905 = 1 2 119899 be a second-order moment

time series and let its correlation coefficient function 120588(119905 119905 +120591) = Cov(119909

119905 119909119905+120591)[120590(119905)120590(119905 + 120591)] be a univariate function of

time interval 120591 that is 120588(119905 119905 + 120591) = 120588120591 then 119909

119905 119905 = 1 2 119899

is called a correlation coefficient stationary time series

Concept 2 Correlation coefficient stationary time series 119909119905

119905 = 1 2 119899 is called CCARMA series if 119910119905= [119909119905minus

120583(119905)]120590(119905) 119905 = 1 2 119899 is an ARMA sequence where119864(119909119905) = 120583(119905) and Var(119909

119905) = 120590

2(119905) are mean and variance

functions of series 119909119905 119905 = 1 2 119899 respectively

The Gaussian CCARMA(119901 119902)model can be denoted by

Φ (119871)119909119905minus 120583 (119905)

120590 (119905)= Θ (119871) 120576

119905 120576119905sim NID [0 1205902

120576] (1)

where Φ(119871) = 1 minus 1205931119871 minus sdot sdot sdot minus 120593

119901119871119901 specifies the AR lag-

polynomial Θ(119871) = 1 minus 1205791119871 minus sdot sdot sdot minus 120579

119902119871119902 specifies the MA

polynomial and 120576119905sim iid 119873(0 1205902

120576) As special cases of

CCARMA(119901 119902) model the CCAR(119901) model and CCMA(119902)model can be given as

Φ (119871)119909119905minus 120583 (119905)

120590 (119905)= 120576119905 120576119905sim NID [0 1205902

120576]

119909119905minus 120583 (119905)

120590 (119905)= Θ (119871) 120576

119905 120576119905sim NID [0 1205902

120576]

(2)

Based on the above definitions we can conclude the fol-lowing

(i) The statistical property difference between covariancestationarity series and correlation coefficient station-ary series is whether its mean and variance varywith time By plotting the sequence this statisticalproperty difference can be detected intuitively Anda quantitative method to determine the operation isthe covariance stationary test method introduced inSection 42

(ii) The difference between the correlation coefficient sta-tionary series and other nonstationary sequence liesin whether the correlation coefficient 120588(119905 119905 + 120591) of 119909

119905

and 119909119905+120591

120591 = 1 2 varies with time 119905 meaning that120588(119905 119905+120591) is only a univariate function of time interval120591We can get a judgment by a simple way First dividethe series into several subseries which are consideredas series with constant mean and variance Secondcalculate the correlation coefficients 120588(120591) 120591 = 1 2 of each subsequence Generally speaking it is enoughto derive the first five-order correlation coefficientsFinally examine whether the correlation coefficient120588(120591) 120591 = 1 2 of each subsequence equals to eachother This can be completed by plotting or simplequantitative tests

22 Scope of CCARMA Process Basic properties show thatthe covariance stationary series is a special case of correlationcoefficient stationary sequenceWhen themean and variancedo not vary with time that is 119864(119909

119905) = 120583 Var(119909

119905) = 120590

2the correlation coefficient stationary series 119909

119905 119905 = 1 2 119899

degenerates to covariance stationary sequenceSuppose that 119910

119905is a covariance stationary series and

120583(119905) is a deterministic function then we can conclude thatseries 119909

119905= 119910119905+ 120583(119905) 119905 = 1 2 119899 is a correlation coef-

ficient stationary time series with time-varying mean 120583(119905)and constant variance Var(119909

119905) = 120590

2 For this series 119910119905and

120583(119905) represent the random part and deterministic part res-pectively Sequences of this kind are very common in practicesuch as ground movement and deformation time seriesmeteorological data and observation sequence in other fieldsWang [32] called it variance stationary sequence

Let 119910119905be a zero mean covariance stationary series and

120590(119905) is a deterministic positive function and thenwe can knowthat 119909

119905= 120590(119905) times 119910

119905 119905 = 1 2 119899 is a correlation coeffi-

cient stationary time series with zero mean and time-varyingvariance Amplitude modulation signal commonly seen inradio communication monitoring and other fields belongsto this case In these signals carrier signal is the zero meancovariance stationary series 119910

119905 and modulated signal is the

positive deterministic function 120590(119905)Considering a more composite circumstance we assume

that 119910119905is a zero mean covariance stationary series 120583(119905) is a

deterministic function and 120590(119905) is a positive deterministicfunction then it can be inferred that 119909

119905= 120590(119905) times 119910

119905+ 120583(119905)

119905 = 1 2 119899 is a correlation coefficient stationary timeseries with time-varying mean and variance Actually this isa comprehensive result of the former two cases

Furthermore correlation coefficient stationary series alsoincludes the sequences which can satisfy the correlationcoefficient stationary conditions The correlation coefficientstability test method will be discussed in Section 4

3 Function Form Determination ofMean and Standard Deviation

We know that one can hardly efficiently obtain the meanand standard deviation functions when these two functionsdiscontinuously vary with time Consequently in this studywe consider the general cases in which mean and standarddeviation functions vary with time continuously and slowlyThis is a common assumption in model constructing fortime series analysis In our theoretical study process wefound that some constraints have to be satisfied for rigorousderivation Accordingly we proposed the following theoremfor determining the mean and standard deviation functionsfor nonstationary time series

Theorem 1 Let 119909119905 119905 = 1 2 119899 be a Gaussian correlation

coefficient stationary series with time-varying mean 119864(119909119905) =

120583(119905) and variance Var(119909119905) = 120590

2(119905) and its standard deviation

function 120590(119905) has the same form with the trend item of series|nabla119909119905| = |119909

119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| 119905 = 1 2 119899 minus 1

whenΔ120590(119905)120590(119905) is a constant or [Δ120590(119905)120590(119905)]max is a negligible

4 Mathematical Problems in Engineering

small amount compared with one that is [Δ120590(119905)120590(119905)]max ≪1 where [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) andΔ120590(119905) = 120590(119905 + 1) minus 120590(119905) 119905 = 1 2 119899 then the standarddeviation function 120590(119905) can be depicted by

120590 (119905) = 1198881198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816 (3)

where 119888 is a positive real number See Appendix A for theoremproof

Generally speaking [Δ120590(119905)120590(119905)]max can be consideredas a negligible small amount when two orders smaller thanone It can be inferred from the above theorem that thestandard deviation 120590(119905) has the same function form withthe mean function of series |119909

119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)|

119905 = 1 2 119899 minus 1 Consequently in the process of mean andstandard deviation function form determination we need toconduct the following steps (1) obtain the trend estimator_120583(119905) and derive series |119909

119905+1minus

_120583(119905 + 1) minus 119909

119905+

_120583 (119905)| 119905 =

1 2 119899 minus 1 (2) determine the trend of series |119909119905+1minus

_120583(119905 +

1)minus119909119905+

_120583(119905)| 119905 = 1 2 119899minus1 whichwe take as the function

form of standard deviation function 120590(119905) (3) take the result_120590(119905) as the estimate of standard deviation function when thetheorem condition can be satisfied

Otherwise when the theorem conditions cannot be metthat is neither Δ120590(119905)120590(119905) is a constant nor [Δ120590(119905)120590(119905)]maxis a negligible small amount compared with one we have tochange the determination strategy In this case its standarddeviation function 120590(119905) has the same function form with thetrend item of series |119909

119905minus 120583(119905)| 119905 = 1 2 119899 That is for

correlation coefficient stationary series 119909119905 119905 = 1 2 119899 its

standard deviation function can be determined by

120590 (119905) = 11988811198641003816100381610038161003816119909119905 minus 120583 (119905)

1003816100381610038161003816 (4)

where 1198881is a positive real number See Appendices for proof

The function form determination of mean and variancefocuses on accessing the trend items of 119909

119905 |119909119905+1minus

_120583(119905 + 1) minus

119909119905+

_120583(119905)| or |119909

119905minus

_120583(119905)| 119905 = 1 2 119899 We consider that

the trend function contains nonperiodic part and periodicpart In this paper we propose a rolling window methodcalled ldquomovingmultiple-point averagemethodrdquo for the deter-mination of sequence trend item In order to determine thenonperiodic part in the trend item the proposed methodmovingly fits on the whole sample data length 119899 with themultiple-point average method Meanwhile we adopt thesample periodogram method to obtain the periodic partTo better address this issue the following steps can beperformed

(1) Determine the periodic part of trend itemwith sampleperiodogram method

First we suppose the existence of frequency 1205961 1205962

120596119872 and then we can express the periodic part of series

119909119905 119905 = 1 2 119899 with periodogram method [33] as

119909119905= 119909 +

119872

sum

119895=1

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)] 119905 = 1 2 119899

(5)

where 119909 is constant mean of series 119909119905 119905 = 1 2 119899 which

can be obtained by 119909 = sum119899119905=1119909119905119899119872 is the existing frequency

number which equals 1198992 when sample size 119899 is an evennumber or equals (119899 minus 1)2 when 119899 is an odd number 120596

119895=

2119895120587119899 119895 = 1 2 119872 is an existing frequency 120572119895and 120573

119895are

cosine and sine coefficients corresponding to frequency 120596119895

119895 = 1 2 119872When sample size 119899 is an odd number coefficients 120572

119895and

120573119895 119895 = 1 2 119872 can be calculated by

120572119895=2

119899

119899

sum

119905=1

119909119905cos [120596

119895(119905 minus 1)] 119895 = 1 2 119872

120573119895=2

119899

119899

sum

119905=1

119909119905sin [120596

119895(119905 minus 1)] 119895 = 1 2 119872

(6)

When sample size 119899 is an even number coefficients 120572119895and 120573

119895

119895 = 1 2 119872 minus 1 can also be worked out by (6) and

120572119872=1

119899

119899

sum

119905=1

(minus1)119905minus1119909119905

120573119872= 0

(7)

Then we introduce a parameter119860119895= radic1205722

119895+ 1205732119895depicting

the amplitude of frequency 120596119895 119895 = 1 2 119872 When

one or more 119860119895is significantly greater than the other ones

1198601 1198602 119860

119895minus1 119860119895+1 119860

119872 we can affirm that a periodic

item with frequency 120596119895exists And then the existing periodic

itemwith frequency120596119895can be expressed as 120572

119895cos[120596

119895(119905minus1)]+

120573119895sin[120596119895(119905minus1)] For the circumstance ofmultiple frequencies

the periodic item is sum of the periodic items correspondingto each crest value

sum

119895

120572119895cos [120596

119895(119905 minus 1)] + 120573

119895sin [120596

119895(119905 minus 1)] (8)

(2) Calculate nonperiodic part of trend item with movingmultiple-point average method

With the former results of step (1) nonperiodic trend partcan be obtained by

119909119905minussum

119895

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)] 119905 = 1 2 119899

(9)

Then we select the point number of each averaging segment119898 (subsequence length) and moving time interval Δ basedon the volatility of the obtained series from (9) Generallyspeaking 119898 is in the range of [11989950 1198992] where 119899 is thesample length and Δ is in the range of [119898101198982] In orderto get an accurate periodic part of trend item we shouldnote that averaging point number 119898 must be not less thanent(2120587120596min) whereas 120596min is the smallest frequency in thedetermined periodic function that is (8)

Mathematical Problems in Engineering 5

Then we implement moving 119898-point average on thewhole sample data length 119899 based on themoving time intervalΔ and obtain a group of mean value (119905

119894 119909119894) by

119905119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119905 119894 = 1 2 119897 (10)

119909119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119909119905 119894 = 1 2 119897 (11)

where 119897 = ent((119899 minus 119898)Δ) + 1 indicates the total averagingtimes

Consequently fit the obtained group of mean value(119905119894 119909119894) 119894 = 1 2 119897 to get the regression function 119891(119905)

which is the nonperiodic trend function part of series 119909119905

119905 = 1 2 119899(3) Redetermine the periodic part of trend itemBased on the nonperiodic trend function 119891(119905) obtained

above the following series can be calculated

119909119905minus 119891 (119905) 119905 = 1 2 119899 (12)

Process the obtained series from (12) and then the periodicpart function expressed by (8) can be redetermined with theperiodogram method

(4) Repeat step (2) and step (3) until each parameterresults in periodic part function sum

119895120572119895cos[120596

119895(119905 minus 1)] +

120573119895sin[120596119895(119905minus1)] and nonperiodic part function119891(119905) becomes

numerical stabilized Then we can obtain the final trend itemexpression as

_120583(119905) = 119891 (119905) +sum

119895

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)]

(13)

The mean function 120583(119905) can be directly determined byimplementing the above steps from (1) to (4) on series 119909

119905

119905 = 1 2 119899 Based on the theorem given in Section 3the standard deviation function 120590(119905) can be obtained byconducting the same steps from (1) to (4) on series |119909

119905+1minus

120583(119905 + 1) minus 119909119905+ 120583(119905)| = |119909

119905+1minus

_120583(119905 + 1) minus 119909

119905+

_120583(119905)|

119905 = 1 2 119899 minus 1 if the theorem condition can be satisfiedOtherwise the standard deviation function 120590(119905) can beobtained by conducting the same steps from (1) to (4) onseries on sequence |119909

119905minus 120583(119905)| = |119909

119905minus

_120583(119905)| 119905 = 1 2 119899

In a word the function forms of mean 120583(119905) and standarddeviation 120590(119905) can be determined In order to facilitate theparameter estimation process we depict the time-varyingfunctions of mean and standard deviation by

120583 (a 119905) =119903

sum

119894=0

119886119894120601119894(119905)

120590 (b 119905) =119904

sum

119895=0

119887119895120595119895(119905)

(14)

where 1206010(119905) = 120595

0(119905) = 1 120601

119894(119905) 119894 = 1 2 119903 and 120595

119895(119905)119895 =

1 2 119904 are functions that can be known through the above

determination process and a = (1198860 1198861 119886

119903)119879 and b = (119887

0

1198871 119887

119904)119879 are general sets of unknown parameters to be

calculated

4 Model Construction and Testing

41 CCARMA Model Parameter Estimation Let 119909119905be a

CCARMA series with mean 120583(a 119905) and standard deviation120590(b 119905) and then transformed sequence 119910

119905= [119909119905minus 120583(119905)]120590(119905)

is an ARMA series according to concept 2 in Section 21The relationship between 119909

119905and 119910

119905can be rewritten as

119909119905= 120583(119905) + 120590(119905)119910

119905 Joint probability density functions

(PDF) 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) can be derived by PDF

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) through

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus1sdotsdotsdot 1198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(15)

See Appendices for proofAccording to time series analysis theory [33] a common

approximation of the likelihood function for ARMA processconditions on initial values of both 119910rsquos and 120576rsquos Based on therecommendation given by Box and Jenkins [34] we set 120576rsquos tozero for 119896 = 0 minus1 minus119902 + 1 and 119910rsquos to their actual valuesfor 119896 = 0 minus1 minus119901 + 1 Then the sequence 120576

1 1205762 120576

119899

can be calculated from 1199101 1199102 119910

119899 and the conditional

log likelihood is then

ln 119871 = minus119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896 (16)

Considering the Gaussian CCARMA(119901 119902) process 119909119905

depicted by (1) suppose that we have a sample of 119899 obser-vations 119909

119905 119905 = 1 2 119899 Maximum likelihood estimation

with conditional likelihood function is utilized to estimatethe vector of population parameters 120579 = (119886

0 1198861

119886119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 1205791 1205792 120579

119902 120590120576)119879 Accord-

ingly a common approximation of the likelihood functionfor CCARMA process conditions on initial values of both119909rsquos and 120576rsquos We set 120576rsquos to zero for 119896 = 0 minus1 minus119902 + 1and 119909rsquos to their actual values for 119896 = 0 minus1 minus119901 + 1Then the sequence 120576

1 1205762 sdot sdot sdot 120576

119899 can be calculated from

1199091 1199092 119909

119899 by iterating on

120576119896=119909119896minus 120583 (a 119905

119896)

120590 (b 119905119896)minus

119901

sum

119894=1

120593119894

119909119896minus119894minus 120583 (a 119905

119896minus119894)

120590 (b 119905119896minus119894)+

119902

sum

119894=1

120579119894120576119896minus119894

(17)

for 119896 = 1 2 119899 Based on the joint PDF relationshipdepicted by (15) the conditional log likelihood is then

ln 119871 (120579) = minus 119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576

minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896minus

119899

sum

119896=1

ln120590 (b 119905119896)

(18)

6 Mathematical Problems in Engineering

where 120582 equals the maximum one of 119901 and 119902 that is 120582 =max(119901 119902) Model parameters can be determined by solvingthe following equations

120597 ln 119871 (120579)120597119886119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119886119897

= 0 119897 = 0 1 119903 (19)

120597 ln 119871 (120579)120597119887119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119887119897

minus

119899

sum

119896=1

1

120590 (b 119905119896)

120597120590 (b 119905119896)

120597119887119897

= 0 119897 = 0 1 119904

(20)

120597 ln 119871 (120579)120597120593119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120593119897

= 0 119897 = 0 1 119901 (21)

120597 ln 119871 (120579)120597120579119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120579119897

= 0 119897 = 0 1 119902 (22)

120597 ln 119871 (120579)120597120590120576

= minus119899 minus 120582

120590120576

+1

1205903120576

119899

sum

119896=120582+1

1205762

119896= 0 (23)

Based on the initial values for 120576rsquos given above we canconclude that 120597120576

119896120597119886119897 120597120576119896120597119887119897 120597120576119896120597120593119897 and 120597120576

119896120597120579119897equal zero

for 119896 = 0 minus1 minus119902 + 1 Then these values for 119896 = 1 2 119899can be calculated by iterating on

120597120576119896

120597119886119897

= minus1

119868 (b 119905119896)

120597120583 (a 119905119896)

120597119886119897

+

119901

sum

119894=1

120593119894

119868 (b 119905119896minus119894)

120597120583 (a 119905119896minus119894)

120597119886119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119886119897

= 0

120597120576119896

120597119887119897

= minus119909119896minus 120597120583 (a 119905

119896)

1198682 (b 119905119896)

119868 (b 119905119896)

120597119886119897

+

119901

sum

119894=1

120578119894[119909119896minus119894minus 120583 (a 119905

119896minus119894)]

1198682 (b 119905119896minus119894)

120597119868 (b 119905119896minus119894)

120597119887119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119887119897

= 0

120597120576119896

120597120578119897

= minus119909119896minus119897minus 120597120583 (a 119905

119896minus119897)

119868 (b 119905119896minus119897)

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120578119897

= 0

120597120576119896

120597120579119897

= 120576119896minus119897+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120579119897

= 0

(24)

Then (23) can be rewritten as

1205902

120576=1

119899 minus 120582

119899

sum

119894=120582+1

1205762

119894= 0 (25)

Parameters 1198860 1198861 119886

119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 and

1205791 1205792 120579

119902can be obtained by solving (19) to (22) and

parameter 120590120576can be calculated through (25)

42 CCARMA Model Testing Here we focus on the modeltesting method which can demonstrate whether the pro-posed model is appropriate to describe the observation dataFor the nonstationary series 119909

119905 119905 = 1 2 119899 based on

the parameter estimation process given above we can obtainits mean function 120583(119905) and standard deviation function 120590(119905)Then we can implement the correlation coefficient stationarytest of series119909

119905through the covariance stationary test of series

119910119905 where 119910

119905= [119909

119905minus 120583(119905)]120590(119905) Several tests have been

proposed and applied to examine the covariance stationarityin literature In this paper we take the postsample predictiontesting method presented by Pagan and Schwert [35] asan example to illustrate the testing procedure Postsampleprediction test for covariance stationarity is a nonparametricmethod and it is facilitative to implement and familiar byscholars

First obtain the sequence 119910119905 119905 = 1 2 119899 through the

transformation119910119905= [119909119905minus120583(119905)]120590(119905) based on the determined

mean function 120583(119905) and standard deviation function 120590(119905)Second split series 119910

119905 119905 = 1 2 119899 averagely into

two parts and calculate the sample variance_1205902

(1)= 119864((2

119899)sum1198992

119894=11199102

119894) and

_1205902

(2)= 119864((2119899)sum

119899

119894=1+11989921199102

119894) for eachThen the

test statistic_120591=

_1205902

(2)minus

_1205902

(1)follows

radic2

119899

_120591sim 119873[0 2(119877

0+ 2

infin

sum

119894=1

119877119894)] (26)

If 1199102119905is a covariance stationary process with autocovariances

119877119894 and let ] = 119877

0+ 2suminfin

119894=1119877119894 then it can be estimated by

_]=

_1198770 + 2

8

sum

119894=1

_119877119894 (1 minus

119894

9) (27)

where_119877119894 is the estimated serial correlation coefficients of 1199102

119905

calculated over the whole sampleFinally define null hypothesis that119867

0 119910119905 119905 = 1 2 119899

is a covariance stationarity series versus alternative hypothe-sis that119867

119886 119910119905 119905 = 1 2 119899 is not a covariance stationarity

series Construct test statistic 119861 and the rejection region canbe expressed by

119861 =

1003816100381610038161003816100381610038161003816100381610038161003816

radic119899

2

_120591

radic2]

1003816100381610038161003816100381610038161003816100381610038161003816

gt 1199061minus1205722 (28)

where 119906119901is the 100119906th percentile of the standard normal

distribution and 120572 is the selected significance level indicatingthe probability of type I error

5 Simulation Experiment

To assess the computational performance of the proposedmethod and determine whether the approach seems to givereasonable results we study the effectiveness and the per-formance of presented methods in Sections 3 and 4 from aMonte Carlo simulated example

Mathematical Problems in Engineering 7

The following zero mean CCAR(1)model

119909119905

120590 (119905)= 120593

119909119905minus1

120590 (119905 minus 1)+ 120576119905 120576119905sim NID [0 1205902

120576] (29)

is considered with four simulation function forms for thestandard deviation including linear function quadraticfunction periodic function and the combination functionof periodic function and linear function The parametersassumed in each experiment are summarized in Table 1 forconvenience The sample size is 119899 = 200 The point numberof averaging and moving time interval for each experiment isalso listed in Table 1

Series |119909119905+1minus 119909119905| and |119909

119905| can be taken to deter-

mine the standard deviation function 120590(119905) by (3) (Theoremmethod) presented in Section 3 since the mean of series119909119905equals zero that is 120583(119905) = 0 Simulation results are

analyzed by the index of average percent relative errorerr = (1119899)sum119899

119905=1|120590(119905)minus

_120590(119905)| times 100120590(119905) which is presented

in Table 2 The results are based on 20000 Monte Carlosimulations with innovations drawn from an IID Gaussiandistribution

Nonzero mean CCAR(119901) and CCARMA(119901 119902) modelwith higher order and different parameter specifications arealso considered to study the authentication of the theoremand the results were very similar to those reported in thispaper

6 Empirical Results

In this section we focus on the practical performance ofthe proposed approach Experiments are presented for fourdifferent economic data sets presented in Section 61 InSection 62 the mean and standard deviation functions aredetermined Then in Section 63 we apply the statistical testmethod discussed in Section 4 Finally Section 64 is devotedto evaluating the forecasting performance of the correlationcoefficient stationary method

61 The Data Sets

Daily CIRThe daily returns to composite index for Shanghaifrom January 5 1999 through September 30 2003 (1131observations)

Monthly MPL The monthly maximum power load forGuangxi from January 1990 through December 1999 (120observations)

Monthly M2 The monthly money supply for China fromJanuary 2000 through December 2009 (112 observations)

Daily FX Rate EURUSD Euro to the United States dollarparity from January 1 2005 throughDecember 30 2005 (260observations)

62 Determination of Mean and Variance In order to test thestability of the three data sets we apply the determinationmethod presented in Sections 3 and 4 on the four data setsAll parameter results are summarized in Tables 3 and 4 Note

that the standard deviation functions 120590(119905) are obtained by (3)for Daily CIR Monthly MPL and Daily FX rate data sets andby (4) forMonthlyM2 because the theorem condition can besatisfied for the former three sequences

63 Testing for Correlation Coefficient Stationarity As asecond step we apply the correlation coefficient stationaritytest presented in Section 4 on the four data sets All resultsare summarized in Table 5 We first derive sequence 119910

119905by

transformation 119910119905= [119909119905minus 120583(119905)]120590(119905) with the results listed

in Table 3 Let significance level 120572 = 005 and the rejectionregion is 119861 gt 119906

1minus1205722= 196

From the results we can conclude that the four datasets are correlation coefficient stationary time series and theapplication of the proposed method is reasonable

64 Prediction Based on the CCAR models constructedabove the forecast of original sequence 119909

119905can be obtained by

(1) In this section we will consider autoregressive integratedmoving average (ARIMA) model variable differential (VD)model generalized autoregressive conditional heteroscedas-ticity (GARCH)model Grey predictionmodel GM(1 1) andmodified GM(1 1) model which are considered as standardremarks to study its forecasting accuracy The future 5daily Shanghai composite index data the future 24 monthlymaximum power load data for Guangxi from January 2000through December 2001 the future 8 monthly money supplydata for China fromMay 2009 through December 2009 andthe future 10 daily FX rate data EURUSD are forecasted bythe established models The prediction results are summa-rized from Tables 6 7 8 and 9 and they are analyzed bythe index of percent relative error (Per err) 120576

119894= |119909119899+119894minus

_119909119899+119894| times 100119909

119899+119894 119894 = 1 2 119897 and mean percent relative

error (Mper err) 120576 = (1119897) sum119897119894=1120576119894 where 119897 is the prediction

step aheadFrom the results we can conclude that the application

of the proposed method for correlation coefficient stationarytime series is reasonable and effective Furthermore thepresented methodology can be considered good and showsa promise for future applications in nonstationary time seriesanalysis and forecasting

7 Discussion

In this paper we discussed the category of correlationcoefficient stationary series a nonstationary time series withtime-varying mean and variance We proposed a movingdetermination method for its time-varying mean functionand standard deviation function We also discussed thecorrelation coefficient stationary test method

The determination principle of function form and orderof mean 120583(119905) and standard deviation 120590(119905) cannot be sep-arated from primary sequence analysis It is worth notingin prediction problem that the function models of 120583(119905) and120590(119905) should also trade off between sequence volatility andaccuracy requirements

For mean function determination the proposed movingmethod can be used to establish the mean function of allkinds of nonstationary time series and can help improve the

8 Mathematical Problems in Engineering

Table 1 Model specifications by experiments

Experiment [Δ120590(119905)120590(119905)]max 119898 Δ 120593 1198881

1198882

1198883

(1) 120590(119905) = 1198881+ 1198882119905 00040 40 20

04 5 002

06 5 002

08 5 002

(2) 120590(119905) = 1198881+ 1198882119905 + 11988831199053 00087 40 20

04 4 minus001 0000306 4 minus001 0000308 4 minus001 00003

(3) 120590(119905) = 1198881+ 1198882sin (003120587119905) 00241

04 13 2

06 13 2

08 13 2

(4) 120590(119905) = 1198881+ 1198882119905 + 1198883sin (003120587119905) 00154

04 13 002 206 13 002 208 13 002 2

Note119898 and Δ are point number of each averaging and moving time interval 120593 is the autocorrelation coefficient of simulation model 1198881 1198882 and 119888

3are model

parameters in standard deviation function [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) Symbol ldquordquo indicates that the parameter does not exist

Table 2 Results of parameter and the average percent relative error for each experiment

120593 04 06 08 04 06 08

Method Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Experiment 1 Experiment 21198881

47635 43543 44568 49790 41984 66076 47635 43543 44568 49790 41984 660761198882

00190 00174 00178 00200 00168 00268 00190 00174 00178 00200 00168 00268Per err 771 1348 1157 885 1615 3371 771 1348 1157 885 1615 3371

Experiment 3 Experiment 41198881

123941 113212 116007 129740 109307 172514 124157 113302 115889 129288 109200 1716421198882

19023 17321 17246 19175 16054 25234 00188 00173 00178 00203 00169 002761198883

18880 17320 17573 19839 16753 26349Per err 1022 1449 1121 832 1594 3330 862 1382 1189 1029 1617 3458

Note Experiments 1 to 4 are defined in Table 1 Symbol ldquordquo indicates that the parameter does not exist

Table 3 Determination results of mean and variance functions

Trend function 120583(119905)Daily CIR 0Monthly MPL 105762 + 20498119905 minus 10297 sin(120587119905

6) + 15926 cos (120587119905

6)

Monthly M2 11986292 + 22406119905 + 652801199052minus 07726119905

3+ 4077 times 10

minus31199054

Daily FX rate 12489 + 51943 times 10minus3119905 minus 11693 times 10

minus41199052+ 94416 times 10

minus71199053minus 33508 times 10

minus91199054+ 43846 times 10

minus121199055

+001006 cos(312058711990565) + 001587 cos(3120587119905

130)

Standard deviation function 120590(119905)Daily CIR 05456 + 31940 times 10

minus2119905 minus 19532 times 10

minus41199052+ 43798 times 10

minus71199053minus 41174 times 10

minus101199054+ 13751 times 10

minus131199055

Monthly MPL 11607 minus 79024 sin(1205871199056) minus 40838 cos (120587119905

6)

Monthly M2 257950 minus 35024119905 + 176221199052minus 02823119905

3+ 14410 times 10

minus31199054

Daily FX rate 54449 times 10minus3

Mathematical Problems in Engineering 9

Table 4 Model parameter results

Data set Model 119910119905= sum119901

119894=1120593119894119910119905minus119894+ 120576119905

119901 1205931

1205932

Daily CIR 1 002861

Monthly MPL 1 06353

Monthly M2 2 11572 minus01401Daily FX rate 1 08616

accuracy especially in situations of small sample size Forstandard deviation function fixing the presented theoremis only suitable for the correlation coefficient stationarysequence

In addition the mean and standard deviation functionsdepicted by (14) are general expressions Take the MonthlyMPL data set in Section 6 for example we can assume that1206010(119905) = 120595

0(119905) = 1 120601

1(119905) = 119905 120601

2(119905) = 120595

1(119905) = sin(1205871199056)

1206013(119905) = 120595

2(119905) = cos(1205871199056) and a = (119886

0 1198861 1198862 1198863)1015840 and b =

(1198870 1198871 1198872)1015840

Appendices

A Proof of Theorem 1

Suppose that 119909119905 119905 = 1 2 119899 is a correlation coefficient

stationary time series119864(119909119905) = 120583(119905) is a deterministic function

representing time-varying mean and radicVar(119909119905) = 120590(119905)

is a positive deterministic function denoting time-varyingstandard deviation function According to the concept ofcorrelation coefficient stationary time series we can express119909119905as

119909119905= 120583 (119905) + 120590 (119905) 120576

119905 120576119905sim NID [0 1] (A1)

We can derive the mean of sequence |119909119905+1minus120583(119905+1)minus119909

119905+

120583(119905)| 119905 = 2 3 119899 by

1198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816

= 1198641003816100381610038161003816120590 (119905 + 1) 120576119905+1 minus 120590 (119905) 120576119905

1003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816

120590 (119905) + Δ120590 (119905)

120590 (119905)120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

(A2)

Define a new randomvariable 120585(119905) = [1+Δ120590(119905)120590(119905)]120576119905+1minus

120576119905 We can conclude that 120585

119905is still a Gaussian random variable

because it is a linear combination of two Gaussian random

variables 120576119905+1

and 120576119905 And its mean and variance can be

derived by

119864 (120585119905) = 119864[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905 = 0

Var (120585119905) = 119864[1 +

Δ120590(119905)

120590(119905)] 120576119905+1minus 120576119905

2

= (1 +Δ120590 (119905)

120590 (119905))

2

+ 1 minus 21205881(1 +

Δ120590 (119905)

120590 (119905))

(A3)

Therefore Var(120585119905) is a constant when Δ120590(119905)120590(119905) is a

constant or [Δ120590(119905)120590(119905)]max is a negligible small amountcompared with one And

11986410038161003816100381610038161205851199051003816100381610038161003816 = 119864

1003816100381610038161003816[1 + Δ120590 (119905) 120590 (119905)] 120576119905+1 minus 1205761199051003816100381610038161003816 =radic2

120587Var (120585

119905)

(A4)

Let constant 119888 = 1radic(2120587)Var(120585119905) then the standard

deviation of original series 119909119905can be expressed as 120590(119905) =

119888119864|119909119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| Theorem 1 in Section 3 has

been proved

B Proof of PDF Relationship

Let us derive the relationship between joint PDF119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) and 119891

119884119899119884119899minus11198841

(119910119899 119910119899minus1

1199101) Cumulative distribution function (CDF) can be

expressed by

119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

= 119875119883119899minus 120583 (119905

119899)

120590 (119905119899)le 119910119899119883119899minus1minus 120583 (119905

119899minus1)

120590 (119905119899minus1)

le 119910119899minus1

1198831minus 120583 (119905

1)

120590 (1199051)le 1199101

= 119875 119883119899le 120583 (119905

119899) + 120590 (119905

119899) 119910119899 119883119899minus1le 120583 (119905

119899minus1)

+ 120590 (119905119899minus1) 119910119899minus1 119883

1le 120583 (119905

1) + 120590 (119905

1) 1199101

= 119875 119883119899le 119909119899 119883119899minus1le 119909119899minus1 119883

1le 1199091

= 119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

(B1)

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

it may have worse prediction precise in the case of more ran-dom data sets

Meanwhile great progress has been achieved related toprocess monitoring in industrial fields To solve the multi-mode problem illustrated by industrial process because ofmultiple production patterns in the same production linevarious methods including partial least squares methods[18] model library-based methods [19] the Gaussian mix-ture model [20] the localized Fisher discriminant analysisapproach [21] and the recent independent component anal-ysis (ICA) based statistical processing methods [22 23] havebeen constructedThe industrial process monitoring is of sig-nificant importance in the literature However the statisticalmethod based on time series analysis is focused on in thisstudy

In 1982 Engle [24] proposed the concept of conditionalheteroscedasticity with which they solved the conditionalheteroscedasticity estimation problem for time series withconstant unconditional variance The proposed theory hasbeen widely applied in financial risk evaluation For his sig-nificant contribution Engle gained 2003rsquos Nobel EconomicsPrize However the analysis problem for time series withunconditional time-varying variance still exists It can becommonly seen in application [25 26] In addition somehybrid models are also seen in the literature [27 28] whichcombine dissimilar models or models that disagree with eachother strongly to lower the generalization variance or errorAlthough hybrid models have shown advantages in some cir-cumstances there is no denying that they are much compli-cated for application

A simple nonstationary model contains a second-orderstationary process modulated by a deterministic time-vary-ing mean and a deterministic unconditional time-varyingvariance [29] Let 119910

119905 119905 = 1 2 3 be a stationary pro-

cess with zero mean and a simple nonstationary model canbe given by 119909

119905= 120583(119905) + 120590(119905)119910

119905 where 120583(119905) is the determin-

istic time-varying mean function and 120590(119905) is the determinis-tic unconditional time-varying standard deviation functionwhich is strictly positive We can conclude that the non-stationarity of 119909

119905is expressed by its evolving mean and

unconditional variance The efficient analysis of this nonsta-tionary process is a substantial drawback in practice and hasgradually attached great importance to the researchers

Based on systematic study of mass measured data Fuand Liu [30] found that some common characteristics areshared by certain nonstationary time series Like general non-stationary ones these series exhibit time-varying mean 120583(119905)and variance 1205902(119905) and their autocovariance function 120574(119905 119905 +120591) = Cov(119909

119905 119909119905+120591) = 119864[119909

119905minus 120583(119905)][119909

119905+120591minus 120583(119905 + 120591)] is no

longer a univariate function of time interval 120591 that is 120574(119905 119905 +120591) = 120574(120591) while their correlation coefficient function 120588(119905 119905 +120591) = Cov(119909

119905 119909119905+120591)[120590(119905)120590(119905 + 120591)] is still a univariate function

of time interval 120591 that is 120588(119905 119905 + 120591) = 120588120591 Accordingly we

can conclude that (i) they are not covariance stationary timeseries [31] whose autocovariance function 120574(119905 119905 + 120591) is a uni-variate function of time interval 120591 that is 120574(119905 119905 + 120591) =120574(120591) (ii) they are a certain class of nonstationary time seriesand different from other nonstationary time series whose

correlation coefficient function 120588(119905 119905 + 120591) varies with time119905 On this basis Fu and Liu [30] proposed the concept ofldquocorrelation coefficient stationary processrdquo and discussed theestablishment of the correlation coefficient autoregressivemoving average (CCARMA) model

In this paper we further study the nonstationary behav-ior of this correlation coefficient stationary series Firstcharacteristics of the variance function have been furtherstudied and a rigorous theorem was proposed which canhelp not only determination of the standard deviation butalso verification of the modeling process Second a rollingwindow determination scheme named moving multiple-point average method has been established to obtain themean and standard deviation functions This technology canenhance the accuracy under the same sample size and theeffect is more obvious in case of limited sample sizeThird westudied the scope of correlation coefficient stationary processin which discussion can be helpful to better understand theconcept of correlation coefficient stationary process Finallythe correlation coefficient stationary test method has beeninvestigated which can assess the validity of the modelingprocess andmake themodeling process a closed-loop system

In the next section the concept of CCARMA processand its basic properties are introduced we also discussthe CCARMA model In Section 3 we develop a methodfor determining the function forms for mean and standarddeviation Section 4 establishes the parameter determinationmethod and a correlation coefficient stationary test methodSection 5 illustrates simulation studies to assess the validity ofthe approach And Section 6 is devoted to the practical evalu-ation of the proposed method on several data sets includingdaily returns to Shanghai composite index Guangxi monthlymaximum power load China monthly money supply anddaily foreign exchange (FX) rate EURUSD A comparisonbetween our forecasting results and ARIMA variable differ-ential GARCH GM(1 1) and Modified GM(1 1) models isalso provided in this section Finally we conclude this paperwith a discussion in Section 7

2 Concept of CCARMA Process

21 Concept of CCARMA Process Generally speaking tradi-tional stationarity means covariance stationarity [31] Timeseries 119909

119905 119905 = 1 2 119899 is a covariance stationary time series

if the following two conditions are satisfied

(i) The mean function 120583(119905) = 119864(119909119905) does not evolve

through time that is 120583(119905) = 120583(ii) The autocovariance function 120574(119905 119905 + 120591) =

Cov(119909119905 119909119905+120591) = 119864[119909

119905minus 120583(119905)][119909

119905+120591minus 120583(119905 + 120591)]

is a univariate function of time interval 120591 that is120574(119905 119905 + 120591) = 120574(120591)

Let 120591 = 0 and the autocovariance Cov(119909119905 119909119905) equals

the variance Var(119909119905) at time 119905 Consequently variance of

covariance stationarity time series dose not vary with timethat is Var(119909

119905) = 1205902 However most time series encountered

in practice cannot satisfy the above two requirements Tosolve the analysis problem of a certain class of nonstationary

Mathematical Problems in Engineering 3

time series Fu and Liu [30] extended the above conceptand proposed the following two concepts of correlationcoefficient stationary time series

Concept 1 Let 119909119905 119905 = 1 2 119899 be a second-order moment

time series and let its correlation coefficient function 120588(119905 119905 +120591) = Cov(119909

119905 119909119905+120591)[120590(119905)120590(119905 + 120591)] be a univariate function of

time interval 120591 that is 120588(119905 119905 + 120591) = 120588120591 then 119909

119905 119905 = 1 2 119899

is called a correlation coefficient stationary time series

Concept 2 Correlation coefficient stationary time series 119909119905

119905 = 1 2 119899 is called CCARMA series if 119910119905= [119909119905minus

120583(119905)]120590(119905) 119905 = 1 2 119899 is an ARMA sequence where119864(119909119905) = 120583(119905) and Var(119909

119905) = 120590

2(119905) are mean and variance

functions of series 119909119905 119905 = 1 2 119899 respectively

The Gaussian CCARMA(119901 119902)model can be denoted by

Φ (119871)119909119905minus 120583 (119905)

120590 (119905)= Θ (119871) 120576

119905 120576119905sim NID [0 1205902

120576] (1)

where Φ(119871) = 1 minus 1205931119871 minus sdot sdot sdot minus 120593

119901119871119901 specifies the AR lag-

polynomial Θ(119871) = 1 minus 1205791119871 minus sdot sdot sdot minus 120579

119902119871119902 specifies the MA

polynomial and 120576119905sim iid 119873(0 1205902

120576) As special cases of

CCARMA(119901 119902) model the CCAR(119901) model and CCMA(119902)model can be given as

Φ (119871)119909119905minus 120583 (119905)

120590 (119905)= 120576119905 120576119905sim NID [0 1205902

120576]

119909119905minus 120583 (119905)

120590 (119905)= Θ (119871) 120576

119905 120576119905sim NID [0 1205902

120576]

(2)

Based on the above definitions we can conclude the fol-lowing

(i) The statistical property difference between covariancestationarity series and correlation coefficient station-ary series is whether its mean and variance varywith time By plotting the sequence this statisticalproperty difference can be detected intuitively Anda quantitative method to determine the operation isthe covariance stationary test method introduced inSection 42

(ii) The difference between the correlation coefficient sta-tionary series and other nonstationary sequence liesin whether the correlation coefficient 120588(119905 119905 + 120591) of 119909

119905

and 119909119905+120591

120591 = 1 2 varies with time 119905 meaning that120588(119905 119905+120591) is only a univariate function of time interval120591We can get a judgment by a simple way First dividethe series into several subseries which are consideredas series with constant mean and variance Secondcalculate the correlation coefficients 120588(120591) 120591 = 1 2 of each subsequence Generally speaking it is enoughto derive the first five-order correlation coefficientsFinally examine whether the correlation coefficient120588(120591) 120591 = 1 2 of each subsequence equals to eachother This can be completed by plotting or simplequantitative tests

22 Scope of CCARMA Process Basic properties show thatthe covariance stationary series is a special case of correlationcoefficient stationary sequenceWhen themean and variancedo not vary with time that is 119864(119909

119905) = 120583 Var(119909

119905) = 120590

2the correlation coefficient stationary series 119909

119905 119905 = 1 2 119899

degenerates to covariance stationary sequenceSuppose that 119910

119905is a covariance stationary series and

120583(119905) is a deterministic function then we can conclude thatseries 119909

119905= 119910119905+ 120583(119905) 119905 = 1 2 119899 is a correlation coef-

ficient stationary time series with time-varying mean 120583(119905)and constant variance Var(119909

119905) = 120590

2 For this series 119910119905and

120583(119905) represent the random part and deterministic part res-pectively Sequences of this kind are very common in practicesuch as ground movement and deformation time seriesmeteorological data and observation sequence in other fieldsWang [32] called it variance stationary sequence

Let 119910119905be a zero mean covariance stationary series and

120590(119905) is a deterministic positive function and thenwe can knowthat 119909

119905= 120590(119905) times 119910

119905 119905 = 1 2 119899 is a correlation coeffi-

cient stationary time series with zero mean and time-varyingvariance Amplitude modulation signal commonly seen inradio communication monitoring and other fields belongsto this case In these signals carrier signal is the zero meancovariance stationary series 119910

119905 and modulated signal is the

positive deterministic function 120590(119905)Considering a more composite circumstance we assume

that 119910119905is a zero mean covariance stationary series 120583(119905) is a

deterministic function and 120590(119905) is a positive deterministicfunction then it can be inferred that 119909

119905= 120590(119905) times 119910

119905+ 120583(119905)

119905 = 1 2 119899 is a correlation coefficient stationary timeseries with time-varying mean and variance Actually this isa comprehensive result of the former two cases

Furthermore correlation coefficient stationary series alsoincludes the sequences which can satisfy the correlationcoefficient stationary conditions The correlation coefficientstability test method will be discussed in Section 4

3 Function Form Determination ofMean and Standard Deviation

We know that one can hardly efficiently obtain the meanand standard deviation functions when these two functionsdiscontinuously vary with time Consequently in this studywe consider the general cases in which mean and standarddeviation functions vary with time continuously and slowlyThis is a common assumption in model constructing fortime series analysis In our theoretical study process wefound that some constraints have to be satisfied for rigorousderivation Accordingly we proposed the following theoremfor determining the mean and standard deviation functionsfor nonstationary time series

Theorem 1 Let 119909119905 119905 = 1 2 119899 be a Gaussian correlation

coefficient stationary series with time-varying mean 119864(119909119905) =

120583(119905) and variance Var(119909119905) = 120590

2(119905) and its standard deviation

function 120590(119905) has the same form with the trend item of series|nabla119909119905| = |119909

119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| 119905 = 1 2 119899 minus 1

whenΔ120590(119905)120590(119905) is a constant or [Δ120590(119905)120590(119905)]max is a negligible

4 Mathematical Problems in Engineering

small amount compared with one that is [Δ120590(119905)120590(119905)]max ≪1 where [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) andΔ120590(119905) = 120590(119905 + 1) minus 120590(119905) 119905 = 1 2 119899 then the standarddeviation function 120590(119905) can be depicted by

120590 (119905) = 1198881198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816 (3)

where 119888 is a positive real number See Appendix A for theoremproof

Generally speaking [Δ120590(119905)120590(119905)]max can be consideredas a negligible small amount when two orders smaller thanone It can be inferred from the above theorem that thestandard deviation 120590(119905) has the same function form withthe mean function of series |119909

119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)|

119905 = 1 2 119899 minus 1 Consequently in the process of mean andstandard deviation function form determination we need toconduct the following steps (1) obtain the trend estimator_120583(119905) and derive series |119909

119905+1minus

_120583(119905 + 1) minus 119909

119905+

_120583 (119905)| 119905 =

1 2 119899 minus 1 (2) determine the trend of series |119909119905+1minus

_120583(119905 +

1)minus119909119905+

_120583(119905)| 119905 = 1 2 119899minus1 whichwe take as the function

form of standard deviation function 120590(119905) (3) take the result_120590(119905) as the estimate of standard deviation function when thetheorem condition can be satisfied

Otherwise when the theorem conditions cannot be metthat is neither Δ120590(119905)120590(119905) is a constant nor [Δ120590(119905)120590(119905)]maxis a negligible small amount compared with one we have tochange the determination strategy In this case its standarddeviation function 120590(119905) has the same function form with thetrend item of series |119909

119905minus 120583(119905)| 119905 = 1 2 119899 That is for

correlation coefficient stationary series 119909119905 119905 = 1 2 119899 its

standard deviation function can be determined by

120590 (119905) = 11988811198641003816100381610038161003816119909119905 minus 120583 (119905)

1003816100381610038161003816 (4)

where 1198881is a positive real number See Appendices for proof

The function form determination of mean and variancefocuses on accessing the trend items of 119909

119905 |119909119905+1minus

_120583(119905 + 1) minus

119909119905+

_120583(119905)| or |119909

119905minus

_120583(119905)| 119905 = 1 2 119899 We consider that

the trend function contains nonperiodic part and periodicpart In this paper we propose a rolling window methodcalled ldquomovingmultiple-point averagemethodrdquo for the deter-mination of sequence trend item In order to determine thenonperiodic part in the trend item the proposed methodmovingly fits on the whole sample data length 119899 with themultiple-point average method Meanwhile we adopt thesample periodogram method to obtain the periodic partTo better address this issue the following steps can beperformed

(1) Determine the periodic part of trend itemwith sampleperiodogram method

First we suppose the existence of frequency 1205961 1205962

120596119872 and then we can express the periodic part of series

119909119905 119905 = 1 2 119899 with periodogram method [33] as

119909119905= 119909 +

119872

sum

119895=1

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)] 119905 = 1 2 119899

(5)

where 119909 is constant mean of series 119909119905 119905 = 1 2 119899 which

can be obtained by 119909 = sum119899119905=1119909119905119899119872 is the existing frequency

number which equals 1198992 when sample size 119899 is an evennumber or equals (119899 minus 1)2 when 119899 is an odd number 120596

119895=

2119895120587119899 119895 = 1 2 119872 is an existing frequency 120572119895and 120573

119895are

cosine and sine coefficients corresponding to frequency 120596119895

119895 = 1 2 119872When sample size 119899 is an odd number coefficients 120572

119895and

120573119895 119895 = 1 2 119872 can be calculated by

120572119895=2

119899

119899

sum

119905=1

119909119905cos [120596

119895(119905 minus 1)] 119895 = 1 2 119872

120573119895=2

119899

119899

sum

119905=1

119909119905sin [120596

119895(119905 minus 1)] 119895 = 1 2 119872

(6)

When sample size 119899 is an even number coefficients 120572119895and 120573

119895

119895 = 1 2 119872 minus 1 can also be worked out by (6) and

120572119872=1

119899

119899

sum

119905=1

(minus1)119905minus1119909119905

120573119872= 0

(7)

Then we introduce a parameter119860119895= radic1205722

119895+ 1205732119895depicting

the amplitude of frequency 120596119895 119895 = 1 2 119872 When

one or more 119860119895is significantly greater than the other ones

1198601 1198602 119860

119895minus1 119860119895+1 119860

119872 we can affirm that a periodic

item with frequency 120596119895exists And then the existing periodic

itemwith frequency120596119895can be expressed as 120572

119895cos[120596

119895(119905minus1)]+

120573119895sin[120596119895(119905minus1)] For the circumstance ofmultiple frequencies

the periodic item is sum of the periodic items correspondingto each crest value

sum

119895

120572119895cos [120596

119895(119905 minus 1)] + 120573

119895sin [120596

119895(119905 minus 1)] (8)

(2) Calculate nonperiodic part of trend item with movingmultiple-point average method

With the former results of step (1) nonperiodic trend partcan be obtained by

119909119905minussum

119895

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)] 119905 = 1 2 119899

(9)

Then we select the point number of each averaging segment119898 (subsequence length) and moving time interval Δ basedon the volatility of the obtained series from (9) Generallyspeaking 119898 is in the range of [11989950 1198992] where 119899 is thesample length and Δ is in the range of [119898101198982] In orderto get an accurate periodic part of trend item we shouldnote that averaging point number 119898 must be not less thanent(2120587120596min) whereas 120596min is the smallest frequency in thedetermined periodic function that is (8)

Mathematical Problems in Engineering 5

Then we implement moving 119898-point average on thewhole sample data length 119899 based on themoving time intervalΔ and obtain a group of mean value (119905

119894 119909119894) by

119905119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119905 119894 = 1 2 119897 (10)

119909119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119909119905 119894 = 1 2 119897 (11)

where 119897 = ent((119899 minus 119898)Δ) + 1 indicates the total averagingtimes

Consequently fit the obtained group of mean value(119905119894 119909119894) 119894 = 1 2 119897 to get the regression function 119891(119905)

which is the nonperiodic trend function part of series 119909119905

119905 = 1 2 119899(3) Redetermine the periodic part of trend itemBased on the nonperiodic trend function 119891(119905) obtained

above the following series can be calculated

119909119905minus 119891 (119905) 119905 = 1 2 119899 (12)

Process the obtained series from (12) and then the periodicpart function expressed by (8) can be redetermined with theperiodogram method

(4) Repeat step (2) and step (3) until each parameterresults in periodic part function sum

119895120572119895cos[120596

119895(119905 minus 1)] +

120573119895sin[120596119895(119905minus1)] and nonperiodic part function119891(119905) becomes

numerical stabilized Then we can obtain the final trend itemexpression as

_120583(119905) = 119891 (119905) +sum

119895

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)]

(13)

The mean function 120583(119905) can be directly determined byimplementing the above steps from (1) to (4) on series 119909

119905

119905 = 1 2 119899 Based on the theorem given in Section 3the standard deviation function 120590(119905) can be obtained byconducting the same steps from (1) to (4) on series |119909

119905+1minus

120583(119905 + 1) minus 119909119905+ 120583(119905)| = |119909

119905+1minus

_120583(119905 + 1) minus 119909

119905+

_120583(119905)|

119905 = 1 2 119899 minus 1 if the theorem condition can be satisfiedOtherwise the standard deviation function 120590(119905) can beobtained by conducting the same steps from (1) to (4) onseries on sequence |119909

119905minus 120583(119905)| = |119909

119905minus

_120583(119905)| 119905 = 1 2 119899

In a word the function forms of mean 120583(119905) and standarddeviation 120590(119905) can be determined In order to facilitate theparameter estimation process we depict the time-varyingfunctions of mean and standard deviation by

120583 (a 119905) =119903

sum

119894=0

119886119894120601119894(119905)

120590 (b 119905) =119904

sum

119895=0

119887119895120595119895(119905)

(14)

where 1206010(119905) = 120595

0(119905) = 1 120601

119894(119905) 119894 = 1 2 119903 and 120595

119895(119905)119895 =

1 2 119904 are functions that can be known through the above

determination process and a = (1198860 1198861 119886

119903)119879 and b = (119887

0

1198871 119887

119904)119879 are general sets of unknown parameters to be

calculated

4 Model Construction and Testing

41 CCARMA Model Parameter Estimation Let 119909119905be a

CCARMA series with mean 120583(a 119905) and standard deviation120590(b 119905) and then transformed sequence 119910

119905= [119909119905minus 120583(119905)]120590(119905)

is an ARMA series according to concept 2 in Section 21The relationship between 119909

119905and 119910

119905can be rewritten as

119909119905= 120583(119905) + 120590(119905)119910

119905 Joint probability density functions

(PDF) 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) can be derived by PDF

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) through

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus1sdotsdotsdot 1198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(15)

See Appendices for proofAccording to time series analysis theory [33] a common

approximation of the likelihood function for ARMA processconditions on initial values of both 119910rsquos and 120576rsquos Based on therecommendation given by Box and Jenkins [34] we set 120576rsquos tozero for 119896 = 0 minus1 minus119902 + 1 and 119910rsquos to their actual valuesfor 119896 = 0 minus1 minus119901 + 1 Then the sequence 120576

1 1205762 120576

119899

can be calculated from 1199101 1199102 119910

119899 and the conditional

log likelihood is then

ln 119871 = minus119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896 (16)

Considering the Gaussian CCARMA(119901 119902) process 119909119905

depicted by (1) suppose that we have a sample of 119899 obser-vations 119909

119905 119905 = 1 2 119899 Maximum likelihood estimation

with conditional likelihood function is utilized to estimatethe vector of population parameters 120579 = (119886

0 1198861

119886119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 1205791 1205792 120579

119902 120590120576)119879 Accord-

ingly a common approximation of the likelihood functionfor CCARMA process conditions on initial values of both119909rsquos and 120576rsquos We set 120576rsquos to zero for 119896 = 0 minus1 minus119902 + 1and 119909rsquos to their actual values for 119896 = 0 minus1 minus119901 + 1Then the sequence 120576

1 1205762 sdot sdot sdot 120576

119899 can be calculated from

1199091 1199092 119909

119899 by iterating on

120576119896=119909119896minus 120583 (a 119905

119896)

120590 (b 119905119896)minus

119901

sum

119894=1

120593119894

119909119896minus119894minus 120583 (a 119905

119896minus119894)

120590 (b 119905119896minus119894)+

119902

sum

119894=1

120579119894120576119896minus119894

(17)

for 119896 = 1 2 119899 Based on the joint PDF relationshipdepicted by (15) the conditional log likelihood is then

ln 119871 (120579) = minus 119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576

minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896minus

119899

sum

119896=1

ln120590 (b 119905119896)

(18)

6 Mathematical Problems in Engineering

where 120582 equals the maximum one of 119901 and 119902 that is 120582 =max(119901 119902) Model parameters can be determined by solvingthe following equations

120597 ln 119871 (120579)120597119886119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119886119897

= 0 119897 = 0 1 119903 (19)

120597 ln 119871 (120579)120597119887119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119887119897

minus

119899

sum

119896=1

1

120590 (b 119905119896)

120597120590 (b 119905119896)

120597119887119897

= 0 119897 = 0 1 119904

(20)

120597 ln 119871 (120579)120597120593119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120593119897

= 0 119897 = 0 1 119901 (21)

120597 ln 119871 (120579)120597120579119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120579119897

= 0 119897 = 0 1 119902 (22)

120597 ln 119871 (120579)120597120590120576

= minus119899 minus 120582

120590120576

+1

1205903120576

119899

sum

119896=120582+1

1205762

119896= 0 (23)

Based on the initial values for 120576rsquos given above we canconclude that 120597120576

119896120597119886119897 120597120576119896120597119887119897 120597120576119896120597120593119897 and 120597120576

119896120597120579119897equal zero

for 119896 = 0 minus1 minus119902 + 1 Then these values for 119896 = 1 2 119899can be calculated by iterating on

120597120576119896

120597119886119897

= minus1

119868 (b 119905119896)

120597120583 (a 119905119896)

120597119886119897

+

119901

sum

119894=1

120593119894

119868 (b 119905119896minus119894)

120597120583 (a 119905119896minus119894)

120597119886119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119886119897

= 0

120597120576119896

120597119887119897

= minus119909119896minus 120597120583 (a 119905

119896)

1198682 (b 119905119896)

119868 (b 119905119896)

120597119886119897

+

119901

sum

119894=1

120578119894[119909119896minus119894minus 120583 (a 119905

119896minus119894)]

1198682 (b 119905119896minus119894)

120597119868 (b 119905119896minus119894)

120597119887119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119887119897

= 0

120597120576119896

120597120578119897

= minus119909119896minus119897minus 120597120583 (a 119905

119896minus119897)

119868 (b 119905119896minus119897)

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120578119897

= 0

120597120576119896

120597120579119897

= 120576119896minus119897+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120579119897

= 0

(24)

Then (23) can be rewritten as

1205902

120576=1

119899 minus 120582

119899

sum

119894=120582+1

1205762

119894= 0 (25)

Parameters 1198860 1198861 119886

119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 and

1205791 1205792 120579

119902can be obtained by solving (19) to (22) and

parameter 120590120576can be calculated through (25)

42 CCARMA Model Testing Here we focus on the modeltesting method which can demonstrate whether the pro-posed model is appropriate to describe the observation dataFor the nonstationary series 119909

119905 119905 = 1 2 119899 based on

the parameter estimation process given above we can obtainits mean function 120583(119905) and standard deviation function 120590(119905)Then we can implement the correlation coefficient stationarytest of series119909

119905through the covariance stationary test of series

119910119905 where 119910

119905= [119909

119905minus 120583(119905)]120590(119905) Several tests have been

proposed and applied to examine the covariance stationarityin literature In this paper we take the postsample predictiontesting method presented by Pagan and Schwert [35] asan example to illustrate the testing procedure Postsampleprediction test for covariance stationarity is a nonparametricmethod and it is facilitative to implement and familiar byscholars

First obtain the sequence 119910119905 119905 = 1 2 119899 through the

transformation119910119905= [119909119905minus120583(119905)]120590(119905) based on the determined

mean function 120583(119905) and standard deviation function 120590(119905)Second split series 119910

119905 119905 = 1 2 119899 averagely into

two parts and calculate the sample variance_1205902

(1)= 119864((2

119899)sum1198992

119894=11199102

119894) and

_1205902

(2)= 119864((2119899)sum

119899

119894=1+11989921199102

119894) for eachThen the

test statistic_120591=

_1205902

(2)minus

_1205902

(1)follows

radic2

119899

_120591sim 119873[0 2(119877

0+ 2

infin

sum

119894=1

119877119894)] (26)

If 1199102119905is a covariance stationary process with autocovariances

119877119894 and let ] = 119877

0+ 2suminfin

119894=1119877119894 then it can be estimated by

_]=

_1198770 + 2

8

sum

119894=1

_119877119894 (1 minus

119894

9) (27)

where_119877119894 is the estimated serial correlation coefficients of 1199102

119905

calculated over the whole sampleFinally define null hypothesis that119867

0 119910119905 119905 = 1 2 119899

is a covariance stationarity series versus alternative hypothe-sis that119867

119886 119910119905 119905 = 1 2 119899 is not a covariance stationarity

series Construct test statistic 119861 and the rejection region canbe expressed by

119861 =

1003816100381610038161003816100381610038161003816100381610038161003816

radic119899

2

_120591

radic2]

1003816100381610038161003816100381610038161003816100381610038161003816

gt 1199061minus1205722 (28)

where 119906119901is the 100119906th percentile of the standard normal

distribution and 120572 is the selected significance level indicatingthe probability of type I error

5 Simulation Experiment

To assess the computational performance of the proposedmethod and determine whether the approach seems to givereasonable results we study the effectiveness and the per-formance of presented methods in Sections 3 and 4 from aMonte Carlo simulated example

Mathematical Problems in Engineering 7

The following zero mean CCAR(1)model

119909119905

120590 (119905)= 120593

119909119905minus1

120590 (119905 minus 1)+ 120576119905 120576119905sim NID [0 1205902

120576] (29)

is considered with four simulation function forms for thestandard deviation including linear function quadraticfunction periodic function and the combination functionof periodic function and linear function The parametersassumed in each experiment are summarized in Table 1 forconvenience The sample size is 119899 = 200 The point numberof averaging and moving time interval for each experiment isalso listed in Table 1

Series |119909119905+1minus 119909119905| and |119909

119905| can be taken to deter-

mine the standard deviation function 120590(119905) by (3) (Theoremmethod) presented in Section 3 since the mean of series119909119905equals zero that is 120583(119905) = 0 Simulation results are

analyzed by the index of average percent relative errorerr = (1119899)sum119899

119905=1|120590(119905)minus

_120590(119905)| times 100120590(119905) which is presented

in Table 2 The results are based on 20000 Monte Carlosimulations with innovations drawn from an IID Gaussiandistribution

Nonzero mean CCAR(119901) and CCARMA(119901 119902) modelwith higher order and different parameter specifications arealso considered to study the authentication of the theoremand the results were very similar to those reported in thispaper

6 Empirical Results

In this section we focus on the practical performance ofthe proposed approach Experiments are presented for fourdifferent economic data sets presented in Section 61 InSection 62 the mean and standard deviation functions aredetermined Then in Section 63 we apply the statistical testmethod discussed in Section 4 Finally Section 64 is devotedto evaluating the forecasting performance of the correlationcoefficient stationary method

61 The Data Sets

Daily CIRThe daily returns to composite index for Shanghaifrom January 5 1999 through September 30 2003 (1131observations)

Monthly MPL The monthly maximum power load forGuangxi from January 1990 through December 1999 (120observations)

Monthly M2 The monthly money supply for China fromJanuary 2000 through December 2009 (112 observations)

Daily FX Rate EURUSD Euro to the United States dollarparity from January 1 2005 throughDecember 30 2005 (260observations)

62 Determination of Mean and Variance In order to test thestability of the three data sets we apply the determinationmethod presented in Sections 3 and 4 on the four data setsAll parameter results are summarized in Tables 3 and 4 Note

that the standard deviation functions 120590(119905) are obtained by (3)for Daily CIR Monthly MPL and Daily FX rate data sets andby (4) forMonthlyM2 because the theorem condition can besatisfied for the former three sequences

63 Testing for Correlation Coefficient Stationarity As asecond step we apply the correlation coefficient stationaritytest presented in Section 4 on the four data sets All resultsare summarized in Table 5 We first derive sequence 119910

119905by

transformation 119910119905= [119909119905minus 120583(119905)]120590(119905) with the results listed

in Table 3 Let significance level 120572 = 005 and the rejectionregion is 119861 gt 119906

1minus1205722= 196

From the results we can conclude that the four datasets are correlation coefficient stationary time series and theapplication of the proposed method is reasonable

64 Prediction Based on the CCAR models constructedabove the forecast of original sequence 119909

119905can be obtained by

(1) In this section we will consider autoregressive integratedmoving average (ARIMA) model variable differential (VD)model generalized autoregressive conditional heteroscedas-ticity (GARCH)model Grey predictionmodel GM(1 1) andmodified GM(1 1) model which are considered as standardremarks to study its forecasting accuracy The future 5daily Shanghai composite index data the future 24 monthlymaximum power load data for Guangxi from January 2000through December 2001 the future 8 monthly money supplydata for China fromMay 2009 through December 2009 andthe future 10 daily FX rate data EURUSD are forecasted bythe established models The prediction results are summa-rized from Tables 6 7 8 and 9 and they are analyzed bythe index of percent relative error (Per err) 120576

119894= |119909119899+119894minus

_119909119899+119894| times 100119909

119899+119894 119894 = 1 2 119897 and mean percent relative

error (Mper err) 120576 = (1119897) sum119897119894=1120576119894 where 119897 is the prediction

step aheadFrom the results we can conclude that the application

of the proposed method for correlation coefficient stationarytime series is reasonable and effective Furthermore thepresented methodology can be considered good and showsa promise for future applications in nonstationary time seriesanalysis and forecasting

7 Discussion

In this paper we discussed the category of correlationcoefficient stationary series a nonstationary time series withtime-varying mean and variance We proposed a movingdetermination method for its time-varying mean functionand standard deviation function We also discussed thecorrelation coefficient stationary test method

The determination principle of function form and orderof mean 120583(119905) and standard deviation 120590(119905) cannot be sep-arated from primary sequence analysis It is worth notingin prediction problem that the function models of 120583(119905) and120590(119905) should also trade off between sequence volatility andaccuracy requirements

For mean function determination the proposed movingmethod can be used to establish the mean function of allkinds of nonstationary time series and can help improve the

8 Mathematical Problems in Engineering

Table 1 Model specifications by experiments

Experiment [Δ120590(119905)120590(119905)]max 119898 Δ 120593 1198881

1198882

1198883

(1) 120590(119905) = 1198881+ 1198882119905 00040 40 20

04 5 002

06 5 002

08 5 002

(2) 120590(119905) = 1198881+ 1198882119905 + 11988831199053 00087 40 20

04 4 minus001 0000306 4 minus001 0000308 4 minus001 00003

(3) 120590(119905) = 1198881+ 1198882sin (003120587119905) 00241

04 13 2

06 13 2

08 13 2

(4) 120590(119905) = 1198881+ 1198882119905 + 1198883sin (003120587119905) 00154

04 13 002 206 13 002 208 13 002 2

Note119898 and Δ are point number of each averaging and moving time interval 120593 is the autocorrelation coefficient of simulation model 1198881 1198882 and 119888

3are model

parameters in standard deviation function [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) Symbol ldquordquo indicates that the parameter does not exist

Table 2 Results of parameter and the average percent relative error for each experiment

120593 04 06 08 04 06 08

Method Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Experiment 1 Experiment 21198881

47635 43543 44568 49790 41984 66076 47635 43543 44568 49790 41984 660761198882

00190 00174 00178 00200 00168 00268 00190 00174 00178 00200 00168 00268Per err 771 1348 1157 885 1615 3371 771 1348 1157 885 1615 3371

Experiment 3 Experiment 41198881

123941 113212 116007 129740 109307 172514 124157 113302 115889 129288 109200 1716421198882

19023 17321 17246 19175 16054 25234 00188 00173 00178 00203 00169 002761198883

18880 17320 17573 19839 16753 26349Per err 1022 1449 1121 832 1594 3330 862 1382 1189 1029 1617 3458

Note Experiments 1 to 4 are defined in Table 1 Symbol ldquordquo indicates that the parameter does not exist

Table 3 Determination results of mean and variance functions

Trend function 120583(119905)Daily CIR 0Monthly MPL 105762 + 20498119905 minus 10297 sin(120587119905

6) + 15926 cos (120587119905

6)

Monthly M2 11986292 + 22406119905 + 652801199052minus 07726119905

3+ 4077 times 10

minus31199054

Daily FX rate 12489 + 51943 times 10minus3119905 minus 11693 times 10

minus41199052+ 94416 times 10

minus71199053minus 33508 times 10

minus91199054+ 43846 times 10

minus121199055

+001006 cos(312058711990565) + 001587 cos(3120587119905

130)

Standard deviation function 120590(119905)Daily CIR 05456 + 31940 times 10

minus2119905 minus 19532 times 10

minus41199052+ 43798 times 10

minus71199053minus 41174 times 10

minus101199054+ 13751 times 10

minus131199055

Monthly MPL 11607 minus 79024 sin(1205871199056) minus 40838 cos (120587119905

6)

Monthly M2 257950 minus 35024119905 + 176221199052minus 02823119905

3+ 14410 times 10

minus31199054

Daily FX rate 54449 times 10minus3

Mathematical Problems in Engineering 9

Table 4 Model parameter results

Data set Model 119910119905= sum119901

119894=1120593119894119910119905minus119894+ 120576119905

119901 1205931

1205932

Daily CIR 1 002861

Monthly MPL 1 06353

Monthly M2 2 11572 minus01401Daily FX rate 1 08616

accuracy especially in situations of small sample size Forstandard deviation function fixing the presented theoremis only suitable for the correlation coefficient stationarysequence

In addition the mean and standard deviation functionsdepicted by (14) are general expressions Take the MonthlyMPL data set in Section 6 for example we can assume that1206010(119905) = 120595

0(119905) = 1 120601

1(119905) = 119905 120601

2(119905) = 120595

1(119905) = sin(1205871199056)

1206013(119905) = 120595

2(119905) = cos(1205871199056) and a = (119886

0 1198861 1198862 1198863)1015840 and b =

(1198870 1198871 1198872)1015840

Appendices

A Proof of Theorem 1

Suppose that 119909119905 119905 = 1 2 119899 is a correlation coefficient

stationary time series119864(119909119905) = 120583(119905) is a deterministic function

representing time-varying mean and radicVar(119909119905) = 120590(119905)

is a positive deterministic function denoting time-varyingstandard deviation function According to the concept ofcorrelation coefficient stationary time series we can express119909119905as

119909119905= 120583 (119905) + 120590 (119905) 120576

119905 120576119905sim NID [0 1] (A1)

We can derive the mean of sequence |119909119905+1minus120583(119905+1)minus119909

119905+

120583(119905)| 119905 = 2 3 119899 by

1198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816

= 1198641003816100381610038161003816120590 (119905 + 1) 120576119905+1 minus 120590 (119905) 120576119905

1003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816

120590 (119905) + Δ120590 (119905)

120590 (119905)120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

(A2)

Define a new randomvariable 120585(119905) = [1+Δ120590(119905)120590(119905)]120576119905+1minus

120576119905 We can conclude that 120585

119905is still a Gaussian random variable

because it is a linear combination of two Gaussian random

variables 120576119905+1

and 120576119905 And its mean and variance can be

derived by

119864 (120585119905) = 119864[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905 = 0

Var (120585119905) = 119864[1 +

Δ120590(119905)

120590(119905)] 120576119905+1minus 120576119905

2

= (1 +Δ120590 (119905)

120590 (119905))

2

+ 1 minus 21205881(1 +

Δ120590 (119905)

120590 (119905))

(A3)

Therefore Var(120585119905) is a constant when Δ120590(119905)120590(119905) is a

constant or [Δ120590(119905)120590(119905)]max is a negligible small amountcompared with one And

11986410038161003816100381610038161205851199051003816100381610038161003816 = 119864

1003816100381610038161003816[1 + Δ120590 (119905) 120590 (119905)] 120576119905+1 minus 1205761199051003816100381610038161003816 =radic2

120587Var (120585

119905)

(A4)

Let constant 119888 = 1radic(2120587)Var(120585119905) then the standard

deviation of original series 119909119905can be expressed as 120590(119905) =

119888119864|119909119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| Theorem 1 in Section 3 has

been proved

B Proof of PDF Relationship

Let us derive the relationship between joint PDF119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) and 119891

119884119899119884119899minus11198841

(119910119899 119910119899minus1

1199101) Cumulative distribution function (CDF) can be

expressed by

119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

= 119875119883119899minus 120583 (119905

119899)

120590 (119905119899)le 119910119899119883119899minus1minus 120583 (119905

119899minus1)

120590 (119905119899minus1)

le 119910119899minus1

1198831minus 120583 (119905

1)

120590 (1199051)le 1199101

= 119875 119883119899le 120583 (119905

119899) + 120590 (119905

119899) 119910119899 119883119899minus1le 120583 (119905

119899minus1)

+ 120590 (119905119899minus1) 119910119899minus1 119883

1le 120583 (119905

1) + 120590 (119905

1) 1199101

= 119875 119883119899le 119909119899 119883119899minus1le 119909119899minus1 119883

1le 1199091

= 119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

(B1)

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

time series Fu and Liu [30] extended the above conceptand proposed the following two concepts of correlationcoefficient stationary time series

Concept 1 Let 119909119905 119905 = 1 2 119899 be a second-order moment

time series and let its correlation coefficient function 120588(119905 119905 +120591) = Cov(119909

119905 119909119905+120591)[120590(119905)120590(119905 + 120591)] be a univariate function of

time interval 120591 that is 120588(119905 119905 + 120591) = 120588120591 then 119909

119905 119905 = 1 2 119899

is called a correlation coefficient stationary time series

Concept 2 Correlation coefficient stationary time series 119909119905

119905 = 1 2 119899 is called CCARMA series if 119910119905= [119909119905minus

120583(119905)]120590(119905) 119905 = 1 2 119899 is an ARMA sequence where119864(119909119905) = 120583(119905) and Var(119909

119905) = 120590

2(119905) are mean and variance

functions of series 119909119905 119905 = 1 2 119899 respectively

The Gaussian CCARMA(119901 119902)model can be denoted by

Φ (119871)119909119905minus 120583 (119905)

120590 (119905)= Θ (119871) 120576

119905 120576119905sim NID [0 1205902

120576] (1)

where Φ(119871) = 1 minus 1205931119871 minus sdot sdot sdot minus 120593

119901119871119901 specifies the AR lag-

polynomial Θ(119871) = 1 minus 1205791119871 minus sdot sdot sdot minus 120579

119902119871119902 specifies the MA

polynomial and 120576119905sim iid 119873(0 1205902

120576) As special cases of

CCARMA(119901 119902) model the CCAR(119901) model and CCMA(119902)model can be given as

Φ (119871)119909119905minus 120583 (119905)

120590 (119905)= 120576119905 120576119905sim NID [0 1205902

120576]

119909119905minus 120583 (119905)

120590 (119905)= Θ (119871) 120576

119905 120576119905sim NID [0 1205902

120576]

(2)

Based on the above definitions we can conclude the fol-lowing

(i) The statistical property difference between covariancestationarity series and correlation coefficient station-ary series is whether its mean and variance varywith time By plotting the sequence this statisticalproperty difference can be detected intuitively Anda quantitative method to determine the operation isthe covariance stationary test method introduced inSection 42

(ii) The difference between the correlation coefficient sta-tionary series and other nonstationary sequence liesin whether the correlation coefficient 120588(119905 119905 + 120591) of 119909

119905

and 119909119905+120591

120591 = 1 2 varies with time 119905 meaning that120588(119905 119905+120591) is only a univariate function of time interval120591We can get a judgment by a simple way First dividethe series into several subseries which are consideredas series with constant mean and variance Secondcalculate the correlation coefficients 120588(120591) 120591 = 1 2 of each subsequence Generally speaking it is enoughto derive the first five-order correlation coefficientsFinally examine whether the correlation coefficient120588(120591) 120591 = 1 2 of each subsequence equals to eachother This can be completed by plotting or simplequantitative tests

22 Scope of CCARMA Process Basic properties show thatthe covariance stationary series is a special case of correlationcoefficient stationary sequenceWhen themean and variancedo not vary with time that is 119864(119909

119905) = 120583 Var(119909

119905) = 120590

2the correlation coefficient stationary series 119909

119905 119905 = 1 2 119899

degenerates to covariance stationary sequenceSuppose that 119910

119905is a covariance stationary series and

120583(119905) is a deterministic function then we can conclude thatseries 119909

119905= 119910119905+ 120583(119905) 119905 = 1 2 119899 is a correlation coef-

ficient stationary time series with time-varying mean 120583(119905)and constant variance Var(119909

119905) = 120590

2 For this series 119910119905and

120583(119905) represent the random part and deterministic part res-pectively Sequences of this kind are very common in practicesuch as ground movement and deformation time seriesmeteorological data and observation sequence in other fieldsWang [32] called it variance stationary sequence

Let 119910119905be a zero mean covariance stationary series and

120590(119905) is a deterministic positive function and thenwe can knowthat 119909

119905= 120590(119905) times 119910

119905 119905 = 1 2 119899 is a correlation coeffi-

cient stationary time series with zero mean and time-varyingvariance Amplitude modulation signal commonly seen inradio communication monitoring and other fields belongsto this case In these signals carrier signal is the zero meancovariance stationary series 119910

119905 and modulated signal is the

positive deterministic function 120590(119905)Considering a more composite circumstance we assume

that 119910119905is a zero mean covariance stationary series 120583(119905) is a

deterministic function and 120590(119905) is a positive deterministicfunction then it can be inferred that 119909

119905= 120590(119905) times 119910

119905+ 120583(119905)

119905 = 1 2 119899 is a correlation coefficient stationary timeseries with time-varying mean and variance Actually this isa comprehensive result of the former two cases

Furthermore correlation coefficient stationary series alsoincludes the sequences which can satisfy the correlationcoefficient stationary conditions The correlation coefficientstability test method will be discussed in Section 4

3 Function Form Determination ofMean and Standard Deviation

We know that one can hardly efficiently obtain the meanand standard deviation functions when these two functionsdiscontinuously vary with time Consequently in this studywe consider the general cases in which mean and standarddeviation functions vary with time continuously and slowlyThis is a common assumption in model constructing fortime series analysis In our theoretical study process wefound that some constraints have to be satisfied for rigorousderivation Accordingly we proposed the following theoremfor determining the mean and standard deviation functionsfor nonstationary time series

Theorem 1 Let 119909119905 119905 = 1 2 119899 be a Gaussian correlation

coefficient stationary series with time-varying mean 119864(119909119905) =

120583(119905) and variance Var(119909119905) = 120590

2(119905) and its standard deviation

function 120590(119905) has the same form with the trend item of series|nabla119909119905| = |119909

119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| 119905 = 1 2 119899 minus 1

whenΔ120590(119905)120590(119905) is a constant or [Δ120590(119905)120590(119905)]max is a negligible

4 Mathematical Problems in Engineering

small amount compared with one that is [Δ120590(119905)120590(119905)]max ≪1 where [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) andΔ120590(119905) = 120590(119905 + 1) minus 120590(119905) 119905 = 1 2 119899 then the standarddeviation function 120590(119905) can be depicted by

120590 (119905) = 1198881198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816 (3)

where 119888 is a positive real number See Appendix A for theoremproof

Generally speaking [Δ120590(119905)120590(119905)]max can be consideredas a negligible small amount when two orders smaller thanone It can be inferred from the above theorem that thestandard deviation 120590(119905) has the same function form withthe mean function of series |119909

119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)|

119905 = 1 2 119899 minus 1 Consequently in the process of mean andstandard deviation function form determination we need toconduct the following steps (1) obtain the trend estimator_120583(119905) and derive series |119909

119905+1minus

_120583(119905 + 1) minus 119909

119905+

_120583 (119905)| 119905 =

1 2 119899 minus 1 (2) determine the trend of series |119909119905+1minus

_120583(119905 +

1)minus119909119905+

_120583(119905)| 119905 = 1 2 119899minus1 whichwe take as the function

form of standard deviation function 120590(119905) (3) take the result_120590(119905) as the estimate of standard deviation function when thetheorem condition can be satisfied

Otherwise when the theorem conditions cannot be metthat is neither Δ120590(119905)120590(119905) is a constant nor [Δ120590(119905)120590(119905)]maxis a negligible small amount compared with one we have tochange the determination strategy In this case its standarddeviation function 120590(119905) has the same function form with thetrend item of series |119909

119905minus 120583(119905)| 119905 = 1 2 119899 That is for

correlation coefficient stationary series 119909119905 119905 = 1 2 119899 its

standard deviation function can be determined by

120590 (119905) = 11988811198641003816100381610038161003816119909119905 minus 120583 (119905)

1003816100381610038161003816 (4)

where 1198881is a positive real number See Appendices for proof

The function form determination of mean and variancefocuses on accessing the trend items of 119909

119905 |119909119905+1minus

_120583(119905 + 1) minus

119909119905+

_120583(119905)| or |119909

119905minus

_120583(119905)| 119905 = 1 2 119899 We consider that

the trend function contains nonperiodic part and periodicpart In this paper we propose a rolling window methodcalled ldquomovingmultiple-point averagemethodrdquo for the deter-mination of sequence trend item In order to determine thenonperiodic part in the trend item the proposed methodmovingly fits on the whole sample data length 119899 with themultiple-point average method Meanwhile we adopt thesample periodogram method to obtain the periodic partTo better address this issue the following steps can beperformed

(1) Determine the periodic part of trend itemwith sampleperiodogram method

First we suppose the existence of frequency 1205961 1205962

120596119872 and then we can express the periodic part of series

119909119905 119905 = 1 2 119899 with periodogram method [33] as

119909119905= 119909 +

119872

sum

119895=1

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)] 119905 = 1 2 119899

(5)

where 119909 is constant mean of series 119909119905 119905 = 1 2 119899 which

can be obtained by 119909 = sum119899119905=1119909119905119899119872 is the existing frequency

number which equals 1198992 when sample size 119899 is an evennumber or equals (119899 minus 1)2 when 119899 is an odd number 120596

119895=

2119895120587119899 119895 = 1 2 119872 is an existing frequency 120572119895and 120573

119895are

cosine and sine coefficients corresponding to frequency 120596119895

119895 = 1 2 119872When sample size 119899 is an odd number coefficients 120572

119895and

120573119895 119895 = 1 2 119872 can be calculated by

120572119895=2

119899

119899

sum

119905=1

119909119905cos [120596

119895(119905 minus 1)] 119895 = 1 2 119872

120573119895=2

119899

119899

sum

119905=1

119909119905sin [120596

119895(119905 minus 1)] 119895 = 1 2 119872

(6)

When sample size 119899 is an even number coefficients 120572119895and 120573

119895

119895 = 1 2 119872 minus 1 can also be worked out by (6) and

120572119872=1

119899

119899

sum

119905=1

(minus1)119905minus1119909119905

120573119872= 0

(7)

Then we introduce a parameter119860119895= radic1205722

119895+ 1205732119895depicting

the amplitude of frequency 120596119895 119895 = 1 2 119872 When

one or more 119860119895is significantly greater than the other ones

1198601 1198602 119860

119895minus1 119860119895+1 119860

119872 we can affirm that a periodic

item with frequency 120596119895exists And then the existing periodic

itemwith frequency120596119895can be expressed as 120572

119895cos[120596

119895(119905minus1)]+

120573119895sin[120596119895(119905minus1)] For the circumstance ofmultiple frequencies

the periodic item is sum of the periodic items correspondingto each crest value

sum

119895

120572119895cos [120596

119895(119905 minus 1)] + 120573

119895sin [120596

119895(119905 minus 1)] (8)

(2) Calculate nonperiodic part of trend item with movingmultiple-point average method

With the former results of step (1) nonperiodic trend partcan be obtained by

119909119905minussum

119895

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)] 119905 = 1 2 119899

(9)

Then we select the point number of each averaging segment119898 (subsequence length) and moving time interval Δ basedon the volatility of the obtained series from (9) Generallyspeaking 119898 is in the range of [11989950 1198992] where 119899 is thesample length and Δ is in the range of [119898101198982] In orderto get an accurate periodic part of trend item we shouldnote that averaging point number 119898 must be not less thanent(2120587120596min) whereas 120596min is the smallest frequency in thedetermined periodic function that is (8)

Mathematical Problems in Engineering 5

Then we implement moving 119898-point average on thewhole sample data length 119899 based on themoving time intervalΔ and obtain a group of mean value (119905

119894 119909119894) by

119905119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119905 119894 = 1 2 119897 (10)

119909119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119909119905 119894 = 1 2 119897 (11)

where 119897 = ent((119899 minus 119898)Δ) + 1 indicates the total averagingtimes

Consequently fit the obtained group of mean value(119905119894 119909119894) 119894 = 1 2 119897 to get the regression function 119891(119905)

which is the nonperiodic trend function part of series 119909119905

119905 = 1 2 119899(3) Redetermine the periodic part of trend itemBased on the nonperiodic trend function 119891(119905) obtained

above the following series can be calculated

119909119905minus 119891 (119905) 119905 = 1 2 119899 (12)

Process the obtained series from (12) and then the periodicpart function expressed by (8) can be redetermined with theperiodogram method

(4) Repeat step (2) and step (3) until each parameterresults in periodic part function sum

119895120572119895cos[120596

119895(119905 minus 1)] +

120573119895sin[120596119895(119905minus1)] and nonperiodic part function119891(119905) becomes

numerical stabilized Then we can obtain the final trend itemexpression as

_120583(119905) = 119891 (119905) +sum

119895

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)]

(13)

The mean function 120583(119905) can be directly determined byimplementing the above steps from (1) to (4) on series 119909

119905

119905 = 1 2 119899 Based on the theorem given in Section 3the standard deviation function 120590(119905) can be obtained byconducting the same steps from (1) to (4) on series |119909

119905+1minus

120583(119905 + 1) minus 119909119905+ 120583(119905)| = |119909

119905+1minus

_120583(119905 + 1) minus 119909

119905+

_120583(119905)|

119905 = 1 2 119899 minus 1 if the theorem condition can be satisfiedOtherwise the standard deviation function 120590(119905) can beobtained by conducting the same steps from (1) to (4) onseries on sequence |119909

119905minus 120583(119905)| = |119909

119905minus

_120583(119905)| 119905 = 1 2 119899

In a word the function forms of mean 120583(119905) and standarddeviation 120590(119905) can be determined In order to facilitate theparameter estimation process we depict the time-varyingfunctions of mean and standard deviation by

120583 (a 119905) =119903

sum

119894=0

119886119894120601119894(119905)

120590 (b 119905) =119904

sum

119895=0

119887119895120595119895(119905)

(14)

where 1206010(119905) = 120595

0(119905) = 1 120601

119894(119905) 119894 = 1 2 119903 and 120595

119895(119905)119895 =

1 2 119904 are functions that can be known through the above

determination process and a = (1198860 1198861 119886

119903)119879 and b = (119887

0

1198871 119887

119904)119879 are general sets of unknown parameters to be

calculated

4 Model Construction and Testing

41 CCARMA Model Parameter Estimation Let 119909119905be a

CCARMA series with mean 120583(a 119905) and standard deviation120590(b 119905) and then transformed sequence 119910

119905= [119909119905minus 120583(119905)]120590(119905)

is an ARMA series according to concept 2 in Section 21The relationship between 119909

119905and 119910

119905can be rewritten as

119909119905= 120583(119905) + 120590(119905)119910

119905 Joint probability density functions

(PDF) 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) can be derived by PDF

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) through

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus1sdotsdotsdot 1198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(15)

See Appendices for proofAccording to time series analysis theory [33] a common

approximation of the likelihood function for ARMA processconditions on initial values of both 119910rsquos and 120576rsquos Based on therecommendation given by Box and Jenkins [34] we set 120576rsquos tozero for 119896 = 0 minus1 minus119902 + 1 and 119910rsquos to their actual valuesfor 119896 = 0 minus1 minus119901 + 1 Then the sequence 120576

1 1205762 120576

119899

can be calculated from 1199101 1199102 119910

119899 and the conditional

log likelihood is then

ln 119871 = minus119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896 (16)

Considering the Gaussian CCARMA(119901 119902) process 119909119905

depicted by (1) suppose that we have a sample of 119899 obser-vations 119909

119905 119905 = 1 2 119899 Maximum likelihood estimation

with conditional likelihood function is utilized to estimatethe vector of population parameters 120579 = (119886

0 1198861

119886119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 1205791 1205792 120579

119902 120590120576)119879 Accord-

ingly a common approximation of the likelihood functionfor CCARMA process conditions on initial values of both119909rsquos and 120576rsquos We set 120576rsquos to zero for 119896 = 0 minus1 minus119902 + 1and 119909rsquos to their actual values for 119896 = 0 minus1 minus119901 + 1Then the sequence 120576

1 1205762 sdot sdot sdot 120576

119899 can be calculated from

1199091 1199092 119909

119899 by iterating on

120576119896=119909119896minus 120583 (a 119905

119896)

120590 (b 119905119896)minus

119901

sum

119894=1

120593119894

119909119896minus119894minus 120583 (a 119905

119896minus119894)

120590 (b 119905119896minus119894)+

119902

sum

119894=1

120579119894120576119896minus119894

(17)

for 119896 = 1 2 119899 Based on the joint PDF relationshipdepicted by (15) the conditional log likelihood is then

ln 119871 (120579) = minus 119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576

minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896minus

119899

sum

119896=1

ln120590 (b 119905119896)

(18)

6 Mathematical Problems in Engineering

where 120582 equals the maximum one of 119901 and 119902 that is 120582 =max(119901 119902) Model parameters can be determined by solvingthe following equations

120597 ln 119871 (120579)120597119886119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119886119897

= 0 119897 = 0 1 119903 (19)

120597 ln 119871 (120579)120597119887119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119887119897

minus

119899

sum

119896=1

1

120590 (b 119905119896)

120597120590 (b 119905119896)

120597119887119897

= 0 119897 = 0 1 119904

(20)

120597 ln 119871 (120579)120597120593119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120593119897

= 0 119897 = 0 1 119901 (21)

120597 ln 119871 (120579)120597120579119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120579119897

= 0 119897 = 0 1 119902 (22)

120597 ln 119871 (120579)120597120590120576

= minus119899 minus 120582

120590120576

+1

1205903120576

119899

sum

119896=120582+1

1205762

119896= 0 (23)

Based on the initial values for 120576rsquos given above we canconclude that 120597120576

119896120597119886119897 120597120576119896120597119887119897 120597120576119896120597120593119897 and 120597120576

119896120597120579119897equal zero

for 119896 = 0 minus1 minus119902 + 1 Then these values for 119896 = 1 2 119899can be calculated by iterating on

120597120576119896

120597119886119897

= minus1

119868 (b 119905119896)

120597120583 (a 119905119896)

120597119886119897

+

119901

sum

119894=1

120593119894

119868 (b 119905119896minus119894)

120597120583 (a 119905119896minus119894)

120597119886119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119886119897

= 0

120597120576119896

120597119887119897

= minus119909119896minus 120597120583 (a 119905

119896)

1198682 (b 119905119896)

119868 (b 119905119896)

120597119886119897

+

119901

sum

119894=1

120578119894[119909119896minus119894minus 120583 (a 119905

119896minus119894)]

1198682 (b 119905119896minus119894)

120597119868 (b 119905119896minus119894)

120597119887119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119887119897

= 0

120597120576119896

120597120578119897

= minus119909119896minus119897minus 120597120583 (a 119905

119896minus119897)

119868 (b 119905119896minus119897)

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120578119897

= 0

120597120576119896

120597120579119897

= 120576119896minus119897+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120579119897

= 0

(24)

Then (23) can be rewritten as

1205902

120576=1

119899 minus 120582

119899

sum

119894=120582+1

1205762

119894= 0 (25)

Parameters 1198860 1198861 119886

119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 and

1205791 1205792 120579

119902can be obtained by solving (19) to (22) and

parameter 120590120576can be calculated through (25)

42 CCARMA Model Testing Here we focus on the modeltesting method which can demonstrate whether the pro-posed model is appropriate to describe the observation dataFor the nonstationary series 119909

119905 119905 = 1 2 119899 based on

the parameter estimation process given above we can obtainits mean function 120583(119905) and standard deviation function 120590(119905)Then we can implement the correlation coefficient stationarytest of series119909

119905through the covariance stationary test of series

119910119905 where 119910

119905= [119909

119905minus 120583(119905)]120590(119905) Several tests have been

proposed and applied to examine the covariance stationarityin literature In this paper we take the postsample predictiontesting method presented by Pagan and Schwert [35] asan example to illustrate the testing procedure Postsampleprediction test for covariance stationarity is a nonparametricmethod and it is facilitative to implement and familiar byscholars

First obtain the sequence 119910119905 119905 = 1 2 119899 through the

transformation119910119905= [119909119905minus120583(119905)]120590(119905) based on the determined

mean function 120583(119905) and standard deviation function 120590(119905)Second split series 119910

119905 119905 = 1 2 119899 averagely into

two parts and calculate the sample variance_1205902

(1)= 119864((2

119899)sum1198992

119894=11199102

119894) and

_1205902

(2)= 119864((2119899)sum

119899

119894=1+11989921199102

119894) for eachThen the

test statistic_120591=

_1205902

(2)minus

_1205902

(1)follows

radic2

119899

_120591sim 119873[0 2(119877

0+ 2

infin

sum

119894=1

119877119894)] (26)

If 1199102119905is a covariance stationary process with autocovariances

119877119894 and let ] = 119877

0+ 2suminfin

119894=1119877119894 then it can be estimated by

_]=

_1198770 + 2

8

sum

119894=1

_119877119894 (1 minus

119894

9) (27)

where_119877119894 is the estimated serial correlation coefficients of 1199102

119905

calculated over the whole sampleFinally define null hypothesis that119867

0 119910119905 119905 = 1 2 119899

is a covariance stationarity series versus alternative hypothe-sis that119867

119886 119910119905 119905 = 1 2 119899 is not a covariance stationarity

series Construct test statistic 119861 and the rejection region canbe expressed by

119861 =

1003816100381610038161003816100381610038161003816100381610038161003816

radic119899

2

_120591

radic2]

1003816100381610038161003816100381610038161003816100381610038161003816

gt 1199061minus1205722 (28)

where 119906119901is the 100119906th percentile of the standard normal

distribution and 120572 is the selected significance level indicatingthe probability of type I error

5 Simulation Experiment

To assess the computational performance of the proposedmethod and determine whether the approach seems to givereasonable results we study the effectiveness and the per-formance of presented methods in Sections 3 and 4 from aMonte Carlo simulated example

Mathematical Problems in Engineering 7

The following zero mean CCAR(1)model

119909119905

120590 (119905)= 120593

119909119905minus1

120590 (119905 minus 1)+ 120576119905 120576119905sim NID [0 1205902

120576] (29)

is considered with four simulation function forms for thestandard deviation including linear function quadraticfunction periodic function and the combination functionof periodic function and linear function The parametersassumed in each experiment are summarized in Table 1 forconvenience The sample size is 119899 = 200 The point numberof averaging and moving time interval for each experiment isalso listed in Table 1

Series |119909119905+1minus 119909119905| and |119909

119905| can be taken to deter-

mine the standard deviation function 120590(119905) by (3) (Theoremmethod) presented in Section 3 since the mean of series119909119905equals zero that is 120583(119905) = 0 Simulation results are

analyzed by the index of average percent relative errorerr = (1119899)sum119899

119905=1|120590(119905)minus

_120590(119905)| times 100120590(119905) which is presented

in Table 2 The results are based on 20000 Monte Carlosimulations with innovations drawn from an IID Gaussiandistribution

Nonzero mean CCAR(119901) and CCARMA(119901 119902) modelwith higher order and different parameter specifications arealso considered to study the authentication of the theoremand the results were very similar to those reported in thispaper

6 Empirical Results

In this section we focus on the practical performance ofthe proposed approach Experiments are presented for fourdifferent economic data sets presented in Section 61 InSection 62 the mean and standard deviation functions aredetermined Then in Section 63 we apply the statistical testmethod discussed in Section 4 Finally Section 64 is devotedto evaluating the forecasting performance of the correlationcoefficient stationary method

61 The Data Sets

Daily CIRThe daily returns to composite index for Shanghaifrom January 5 1999 through September 30 2003 (1131observations)

Monthly MPL The monthly maximum power load forGuangxi from January 1990 through December 1999 (120observations)

Monthly M2 The monthly money supply for China fromJanuary 2000 through December 2009 (112 observations)

Daily FX Rate EURUSD Euro to the United States dollarparity from January 1 2005 throughDecember 30 2005 (260observations)

62 Determination of Mean and Variance In order to test thestability of the three data sets we apply the determinationmethod presented in Sections 3 and 4 on the four data setsAll parameter results are summarized in Tables 3 and 4 Note

that the standard deviation functions 120590(119905) are obtained by (3)for Daily CIR Monthly MPL and Daily FX rate data sets andby (4) forMonthlyM2 because the theorem condition can besatisfied for the former three sequences

63 Testing for Correlation Coefficient Stationarity As asecond step we apply the correlation coefficient stationaritytest presented in Section 4 on the four data sets All resultsare summarized in Table 5 We first derive sequence 119910

119905by

transformation 119910119905= [119909119905minus 120583(119905)]120590(119905) with the results listed

in Table 3 Let significance level 120572 = 005 and the rejectionregion is 119861 gt 119906

1minus1205722= 196

From the results we can conclude that the four datasets are correlation coefficient stationary time series and theapplication of the proposed method is reasonable

64 Prediction Based on the CCAR models constructedabove the forecast of original sequence 119909

119905can be obtained by

(1) In this section we will consider autoregressive integratedmoving average (ARIMA) model variable differential (VD)model generalized autoregressive conditional heteroscedas-ticity (GARCH)model Grey predictionmodel GM(1 1) andmodified GM(1 1) model which are considered as standardremarks to study its forecasting accuracy The future 5daily Shanghai composite index data the future 24 monthlymaximum power load data for Guangxi from January 2000through December 2001 the future 8 monthly money supplydata for China fromMay 2009 through December 2009 andthe future 10 daily FX rate data EURUSD are forecasted bythe established models The prediction results are summa-rized from Tables 6 7 8 and 9 and they are analyzed bythe index of percent relative error (Per err) 120576

119894= |119909119899+119894minus

_119909119899+119894| times 100119909

119899+119894 119894 = 1 2 119897 and mean percent relative

error (Mper err) 120576 = (1119897) sum119897119894=1120576119894 where 119897 is the prediction

step aheadFrom the results we can conclude that the application

of the proposed method for correlation coefficient stationarytime series is reasonable and effective Furthermore thepresented methodology can be considered good and showsa promise for future applications in nonstationary time seriesanalysis and forecasting

7 Discussion

In this paper we discussed the category of correlationcoefficient stationary series a nonstationary time series withtime-varying mean and variance We proposed a movingdetermination method for its time-varying mean functionand standard deviation function We also discussed thecorrelation coefficient stationary test method

The determination principle of function form and orderof mean 120583(119905) and standard deviation 120590(119905) cannot be sep-arated from primary sequence analysis It is worth notingin prediction problem that the function models of 120583(119905) and120590(119905) should also trade off between sequence volatility andaccuracy requirements

For mean function determination the proposed movingmethod can be used to establish the mean function of allkinds of nonstationary time series and can help improve the

8 Mathematical Problems in Engineering

Table 1 Model specifications by experiments

Experiment [Δ120590(119905)120590(119905)]max 119898 Δ 120593 1198881

1198882

1198883

(1) 120590(119905) = 1198881+ 1198882119905 00040 40 20

04 5 002

06 5 002

08 5 002

(2) 120590(119905) = 1198881+ 1198882119905 + 11988831199053 00087 40 20

04 4 minus001 0000306 4 minus001 0000308 4 minus001 00003

(3) 120590(119905) = 1198881+ 1198882sin (003120587119905) 00241

04 13 2

06 13 2

08 13 2

(4) 120590(119905) = 1198881+ 1198882119905 + 1198883sin (003120587119905) 00154

04 13 002 206 13 002 208 13 002 2

Note119898 and Δ are point number of each averaging and moving time interval 120593 is the autocorrelation coefficient of simulation model 1198881 1198882 and 119888

3are model

parameters in standard deviation function [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) Symbol ldquordquo indicates that the parameter does not exist

Table 2 Results of parameter and the average percent relative error for each experiment

120593 04 06 08 04 06 08

Method Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Experiment 1 Experiment 21198881

47635 43543 44568 49790 41984 66076 47635 43543 44568 49790 41984 660761198882

00190 00174 00178 00200 00168 00268 00190 00174 00178 00200 00168 00268Per err 771 1348 1157 885 1615 3371 771 1348 1157 885 1615 3371

Experiment 3 Experiment 41198881

123941 113212 116007 129740 109307 172514 124157 113302 115889 129288 109200 1716421198882

19023 17321 17246 19175 16054 25234 00188 00173 00178 00203 00169 002761198883

18880 17320 17573 19839 16753 26349Per err 1022 1449 1121 832 1594 3330 862 1382 1189 1029 1617 3458

Note Experiments 1 to 4 are defined in Table 1 Symbol ldquordquo indicates that the parameter does not exist

Table 3 Determination results of mean and variance functions

Trend function 120583(119905)Daily CIR 0Monthly MPL 105762 + 20498119905 minus 10297 sin(120587119905

6) + 15926 cos (120587119905

6)

Monthly M2 11986292 + 22406119905 + 652801199052minus 07726119905

3+ 4077 times 10

minus31199054

Daily FX rate 12489 + 51943 times 10minus3119905 minus 11693 times 10

minus41199052+ 94416 times 10

minus71199053minus 33508 times 10

minus91199054+ 43846 times 10

minus121199055

+001006 cos(312058711990565) + 001587 cos(3120587119905

130)

Standard deviation function 120590(119905)Daily CIR 05456 + 31940 times 10

minus2119905 minus 19532 times 10

minus41199052+ 43798 times 10

minus71199053minus 41174 times 10

minus101199054+ 13751 times 10

minus131199055

Monthly MPL 11607 minus 79024 sin(1205871199056) minus 40838 cos (120587119905

6)

Monthly M2 257950 minus 35024119905 + 176221199052minus 02823119905

3+ 14410 times 10

minus31199054

Daily FX rate 54449 times 10minus3

Mathematical Problems in Engineering 9

Table 4 Model parameter results

Data set Model 119910119905= sum119901

119894=1120593119894119910119905minus119894+ 120576119905

119901 1205931

1205932

Daily CIR 1 002861

Monthly MPL 1 06353

Monthly M2 2 11572 minus01401Daily FX rate 1 08616

accuracy especially in situations of small sample size Forstandard deviation function fixing the presented theoremis only suitable for the correlation coefficient stationarysequence

In addition the mean and standard deviation functionsdepicted by (14) are general expressions Take the MonthlyMPL data set in Section 6 for example we can assume that1206010(119905) = 120595

0(119905) = 1 120601

1(119905) = 119905 120601

2(119905) = 120595

1(119905) = sin(1205871199056)

1206013(119905) = 120595

2(119905) = cos(1205871199056) and a = (119886

0 1198861 1198862 1198863)1015840 and b =

(1198870 1198871 1198872)1015840

Appendices

A Proof of Theorem 1

Suppose that 119909119905 119905 = 1 2 119899 is a correlation coefficient

stationary time series119864(119909119905) = 120583(119905) is a deterministic function

representing time-varying mean and radicVar(119909119905) = 120590(119905)

is a positive deterministic function denoting time-varyingstandard deviation function According to the concept ofcorrelation coefficient stationary time series we can express119909119905as

119909119905= 120583 (119905) + 120590 (119905) 120576

119905 120576119905sim NID [0 1] (A1)

We can derive the mean of sequence |119909119905+1minus120583(119905+1)minus119909

119905+

120583(119905)| 119905 = 2 3 119899 by

1198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816

= 1198641003816100381610038161003816120590 (119905 + 1) 120576119905+1 minus 120590 (119905) 120576119905

1003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816

120590 (119905) + Δ120590 (119905)

120590 (119905)120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

(A2)

Define a new randomvariable 120585(119905) = [1+Δ120590(119905)120590(119905)]120576119905+1minus

120576119905 We can conclude that 120585

119905is still a Gaussian random variable

because it is a linear combination of two Gaussian random

variables 120576119905+1

and 120576119905 And its mean and variance can be

derived by

119864 (120585119905) = 119864[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905 = 0

Var (120585119905) = 119864[1 +

Δ120590(119905)

120590(119905)] 120576119905+1minus 120576119905

2

= (1 +Δ120590 (119905)

120590 (119905))

2

+ 1 minus 21205881(1 +

Δ120590 (119905)

120590 (119905))

(A3)

Therefore Var(120585119905) is a constant when Δ120590(119905)120590(119905) is a

constant or [Δ120590(119905)120590(119905)]max is a negligible small amountcompared with one And

11986410038161003816100381610038161205851199051003816100381610038161003816 = 119864

1003816100381610038161003816[1 + Δ120590 (119905) 120590 (119905)] 120576119905+1 minus 1205761199051003816100381610038161003816 =radic2

120587Var (120585

119905)

(A4)

Let constant 119888 = 1radic(2120587)Var(120585119905) then the standard

deviation of original series 119909119905can be expressed as 120590(119905) =

119888119864|119909119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| Theorem 1 in Section 3 has

been proved

B Proof of PDF Relationship

Let us derive the relationship between joint PDF119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) and 119891

119884119899119884119899minus11198841

(119910119899 119910119899minus1

1199101) Cumulative distribution function (CDF) can be

expressed by

119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

= 119875119883119899minus 120583 (119905

119899)

120590 (119905119899)le 119910119899119883119899minus1minus 120583 (119905

119899minus1)

120590 (119905119899minus1)

le 119910119899minus1

1198831minus 120583 (119905

1)

120590 (1199051)le 1199101

= 119875 119883119899le 120583 (119905

119899) + 120590 (119905

119899) 119910119899 119883119899minus1le 120583 (119905

119899minus1)

+ 120590 (119905119899minus1) 119910119899minus1 119883

1le 120583 (119905

1) + 120590 (119905

1) 1199101

= 119875 119883119899le 119909119899 119883119899minus1le 119909119899minus1 119883

1le 1199091

= 119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

(B1)

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

small amount compared with one that is [Δ120590(119905)120590(119905)]max ≪1 where [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) andΔ120590(119905) = 120590(119905 + 1) minus 120590(119905) 119905 = 1 2 119899 then the standarddeviation function 120590(119905) can be depicted by

120590 (119905) = 1198881198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816 (3)

where 119888 is a positive real number See Appendix A for theoremproof

Generally speaking [Δ120590(119905)120590(119905)]max can be consideredas a negligible small amount when two orders smaller thanone It can be inferred from the above theorem that thestandard deviation 120590(119905) has the same function form withthe mean function of series |119909

119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)|

119905 = 1 2 119899 minus 1 Consequently in the process of mean andstandard deviation function form determination we need toconduct the following steps (1) obtain the trend estimator_120583(119905) and derive series |119909

119905+1minus

_120583(119905 + 1) minus 119909

119905+

_120583 (119905)| 119905 =

1 2 119899 minus 1 (2) determine the trend of series |119909119905+1minus

_120583(119905 +

1)minus119909119905+

_120583(119905)| 119905 = 1 2 119899minus1 whichwe take as the function

form of standard deviation function 120590(119905) (3) take the result_120590(119905) as the estimate of standard deviation function when thetheorem condition can be satisfied

Otherwise when the theorem conditions cannot be metthat is neither Δ120590(119905)120590(119905) is a constant nor [Δ120590(119905)120590(119905)]maxis a negligible small amount compared with one we have tochange the determination strategy In this case its standarddeviation function 120590(119905) has the same function form with thetrend item of series |119909

119905minus 120583(119905)| 119905 = 1 2 119899 That is for

correlation coefficient stationary series 119909119905 119905 = 1 2 119899 its

standard deviation function can be determined by

120590 (119905) = 11988811198641003816100381610038161003816119909119905 minus 120583 (119905)

1003816100381610038161003816 (4)

where 1198881is a positive real number See Appendices for proof

The function form determination of mean and variancefocuses on accessing the trend items of 119909

119905 |119909119905+1minus

_120583(119905 + 1) minus

119909119905+

_120583(119905)| or |119909

119905minus

_120583(119905)| 119905 = 1 2 119899 We consider that

the trend function contains nonperiodic part and periodicpart In this paper we propose a rolling window methodcalled ldquomovingmultiple-point averagemethodrdquo for the deter-mination of sequence trend item In order to determine thenonperiodic part in the trend item the proposed methodmovingly fits on the whole sample data length 119899 with themultiple-point average method Meanwhile we adopt thesample periodogram method to obtain the periodic partTo better address this issue the following steps can beperformed

(1) Determine the periodic part of trend itemwith sampleperiodogram method

First we suppose the existence of frequency 1205961 1205962

120596119872 and then we can express the periodic part of series

119909119905 119905 = 1 2 119899 with periodogram method [33] as

119909119905= 119909 +

119872

sum

119895=1

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)] 119905 = 1 2 119899

(5)

where 119909 is constant mean of series 119909119905 119905 = 1 2 119899 which

can be obtained by 119909 = sum119899119905=1119909119905119899119872 is the existing frequency

number which equals 1198992 when sample size 119899 is an evennumber or equals (119899 minus 1)2 when 119899 is an odd number 120596

119895=

2119895120587119899 119895 = 1 2 119872 is an existing frequency 120572119895and 120573

119895are

cosine and sine coefficients corresponding to frequency 120596119895

119895 = 1 2 119872When sample size 119899 is an odd number coefficients 120572

119895and

120573119895 119895 = 1 2 119872 can be calculated by

120572119895=2

119899

119899

sum

119905=1

119909119905cos [120596

119895(119905 minus 1)] 119895 = 1 2 119872

120573119895=2

119899

119899

sum

119905=1

119909119905sin [120596

119895(119905 minus 1)] 119895 = 1 2 119872

(6)

When sample size 119899 is an even number coefficients 120572119895and 120573

119895

119895 = 1 2 119872 minus 1 can also be worked out by (6) and

120572119872=1

119899

119899

sum

119905=1

(minus1)119905minus1119909119905

120573119872= 0

(7)

Then we introduce a parameter119860119895= radic1205722

119895+ 1205732119895depicting

the amplitude of frequency 120596119895 119895 = 1 2 119872 When

one or more 119860119895is significantly greater than the other ones

1198601 1198602 119860

119895minus1 119860119895+1 119860

119872 we can affirm that a periodic

item with frequency 120596119895exists And then the existing periodic

itemwith frequency120596119895can be expressed as 120572

119895cos[120596

119895(119905minus1)]+

120573119895sin[120596119895(119905minus1)] For the circumstance ofmultiple frequencies

the periodic item is sum of the periodic items correspondingto each crest value

sum

119895

120572119895cos [120596

119895(119905 minus 1)] + 120573

119895sin [120596

119895(119905 minus 1)] (8)

(2) Calculate nonperiodic part of trend item with movingmultiple-point average method

With the former results of step (1) nonperiodic trend partcan be obtained by

119909119905minussum

119895

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)] 119905 = 1 2 119899

(9)

Then we select the point number of each averaging segment119898 (subsequence length) and moving time interval Δ basedon the volatility of the obtained series from (9) Generallyspeaking 119898 is in the range of [11989950 1198992] where 119899 is thesample length and Δ is in the range of [119898101198982] In orderto get an accurate periodic part of trend item we shouldnote that averaging point number 119898 must be not less thanent(2120587120596min) whereas 120596min is the smallest frequency in thedetermined periodic function that is (8)

Mathematical Problems in Engineering 5

Then we implement moving 119898-point average on thewhole sample data length 119899 based on themoving time intervalΔ and obtain a group of mean value (119905

119894 119909119894) by

119905119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119905 119894 = 1 2 119897 (10)

119909119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119909119905 119894 = 1 2 119897 (11)

where 119897 = ent((119899 minus 119898)Δ) + 1 indicates the total averagingtimes

Consequently fit the obtained group of mean value(119905119894 119909119894) 119894 = 1 2 119897 to get the regression function 119891(119905)

which is the nonperiodic trend function part of series 119909119905

119905 = 1 2 119899(3) Redetermine the periodic part of trend itemBased on the nonperiodic trend function 119891(119905) obtained

above the following series can be calculated

119909119905minus 119891 (119905) 119905 = 1 2 119899 (12)

Process the obtained series from (12) and then the periodicpart function expressed by (8) can be redetermined with theperiodogram method

(4) Repeat step (2) and step (3) until each parameterresults in periodic part function sum

119895120572119895cos[120596

119895(119905 minus 1)] +

120573119895sin[120596119895(119905minus1)] and nonperiodic part function119891(119905) becomes

numerical stabilized Then we can obtain the final trend itemexpression as

_120583(119905) = 119891 (119905) +sum

119895

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)]

(13)

The mean function 120583(119905) can be directly determined byimplementing the above steps from (1) to (4) on series 119909

119905

119905 = 1 2 119899 Based on the theorem given in Section 3the standard deviation function 120590(119905) can be obtained byconducting the same steps from (1) to (4) on series |119909

119905+1minus

120583(119905 + 1) minus 119909119905+ 120583(119905)| = |119909

119905+1minus

_120583(119905 + 1) minus 119909

119905+

_120583(119905)|

119905 = 1 2 119899 minus 1 if the theorem condition can be satisfiedOtherwise the standard deviation function 120590(119905) can beobtained by conducting the same steps from (1) to (4) onseries on sequence |119909

119905minus 120583(119905)| = |119909

119905minus

_120583(119905)| 119905 = 1 2 119899

In a word the function forms of mean 120583(119905) and standarddeviation 120590(119905) can be determined In order to facilitate theparameter estimation process we depict the time-varyingfunctions of mean and standard deviation by

120583 (a 119905) =119903

sum

119894=0

119886119894120601119894(119905)

120590 (b 119905) =119904

sum

119895=0

119887119895120595119895(119905)

(14)

where 1206010(119905) = 120595

0(119905) = 1 120601

119894(119905) 119894 = 1 2 119903 and 120595

119895(119905)119895 =

1 2 119904 are functions that can be known through the above

determination process and a = (1198860 1198861 119886

119903)119879 and b = (119887

0

1198871 119887

119904)119879 are general sets of unknown parameters to be

calculated

4 Model Construction and Testing

41 CCARMA Model Parameter Estimation Let 119909119905be a

CCARMA series with mean 120583(a 119905) and standard deviation120590(b 119905) and then transformed sequence 119910

119905= [119909119905minus 120583(119905)]120590(119905)

is an ARMA series according to concept 2 in Section 21The relationship between 119909

119905and 119910

119905can be rewritten as

119909119905= 120583(119905) + 120590(119905)119910

119905 Joint probability density functions

(PDF) 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) can be derived by PDF

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) through

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus1sdotsdotsdot 1198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(15)

See Appendices for proofAccording to time series analysis theory [33] a common

approximation of the likelihood function for ARMA processconditions on initial values of both 119910rsquos and 120576rsquos Based on therecommendation given by Box and Jenkins [34] we set 120576rsquos tozero for 119896 = 0 minus1 minus119902 + 1 and 119910rsquos to their actual valuesfor 119896 = 0 minus1 minus119901 + 1 Then the sequence 120576

1 1205762 120576

119899

can be calculated from 1199101 1199102 119910

119899 and the conditional

log likelihood is then

ln 119871 = minus119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896 (16)

Considering the Gaussian CCARMA(119901 119902) process 119909119905

depicted by (1) suppose that we have a sample of 119899 obser-vations 119909

119905 119905 = 1 2 119899 Maximum likelihood estimation

with conditional likelihood function is utilized to estimatethe vector of population parameters 120579 = (119886

0 1198861

119886119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 1205791 1205792 120579

119902 120590120576)119879 Accord-

ingly a common approximation of the likelihood functionfor CCARMA process conditions on initial values of both119909rsquos and 120576rsquos We set 120576rsquos to zero for 119896 = 0 minus1 minus119902 + 1and 119909rsquos to their actual values for 119896 = 0 minus1 minus119901 + 1Then the sequence 120576

1 1205762 sdot sdot sdot 120576

119899 can be calculated from

1199091 1199092 119909

119899 by iterating on

120576119896=119909119896minus 120583 (a 119905

119896)

120590 (b 119905119896)minus

119901

sum

119894=1

120593119894

119909119896minus119894minus 120583 (a 119905

119896minus119894)

120590 (b 119905119896minus119894)+

119902

sum

119894=1

120579119894120576119896minus119894

(17)

for 119896 = 1 2 119899 Based on the joint PDF relationshipdepicted by (15) the conditional log likelihood is then

ln 119871 (120579) = minus 119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576

minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896minus

119899

sum

119896=1

ln120590 (b 119905119896)

(18)

6 Mathematical Problems in Engineering

where 120582 equals the maximum one of 119901 and 119902 that is 120582 =max(119901 119902) Model parameters can be determined by solvingthe following equations

120597 ln 119871 (120579)120597119886119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119886119897

= 0 119897 = 0 1 119903 (19)

120597 ln 119871 (120579)120597119887119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119887119897

minus

119899

sum

119896=1

1

120590 (b 119905119896)

120597120590 (b 119905119896)

120597119887119897

= 0 119897 = 0 1 119904

(20)

120597 ln 119871 (120579)120597120593119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120593119897

= 0 119897 = 0 1 119901 (21)

120597 ln 119871 (120579)120597120579119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120579119897

= 0 119897 = 0 1 119902 (22)

120597 ln 119871 (120579)120597120590120576

= minus119899 minus 120582

120590120576

+1

1205903120576

119899

sum

119896=120582+1

1205762

119896= 0 (23)

Based on the initial values for 120576rsquos given above we canconclude that 120597120576

119896120597119886119897 120597120576119896120597119887119897 120597120576119896120597120593119897 and 120597120576

119896120597120579119897equal zero

for 119896 = 0 minus1 minus119902 + 1 Then these values for 119896 = 1 2 119899can be calculated by iterating on

120597120576119896

120597119886119897

= minus1

119868 (b 119905119896)

120597120583 (a 119905119896)

120597119886119897

+

119901

sum

119894=1

120593119894

119868 (b 119905119896minus119894)

120597120583 (a 119905119896minus119894)

120597119886119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119886119897

= 0

120597120576119896

120597119887119897

= minus119909119896minus 120597120583 (a 119905

119896)

1198682 (b 119905119896)

119868 (b 119905119896)

120597119886119897

+

119901

sum

119894=1

120578119894[119909119896minus119894minus 120583 (a 119905

119896minus119894)]

1198682 (b 119905119896minus119894)

120597119868 (b 119905119896minus119894)

120597119887119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119887119897

= 0

120597120576119896

120597120578119897

= minus119909119896minus119897minus 120597120583 (a 119905

119896minus119897)

119868 (b 119905119896minus119897)

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120578119897

= 0

120597120576119896

120597120579119897

= 120576119896minus119897+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120579119897

= 0

(24)

Then (23) can be rewritten as

1205902

120576=1

119899 minus 120582

119899

sum

119894=120582+1

1205762

119894= 0 (25)

Parameters 1198860 1198861 119886

119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 and

1205791 1205792 120579

119902can be obtained by solving (19) to (22) and

parameter 120590120576can be calculated through (25)

42 CCARMA Model Testing Here we focus on the modeltesting method which can demonstrate whether the pro-posed model is appropriate to describe the observation dataFor the nonstationary series 119909

119905 119905 = 1 2 119899 based on

the parameter estimation process given above we can obtainits mean function 120583(119905) and standard deviation function 120590(119905)Then we can implement the correlation coefficient stationarytest of series119909

119905through the covariance stationary test of series

119910119905 where 119910

119905= [119909

119905minus 120583(119905)]120590(119905) Several tests have been

proposed and applied to examine the covariance stationarityin literature In this paper we take the postsample predictiontesting method presented by Pagan and Schwert [35] asan example to illustrate the testing procedure Postsampleprediction test for covariance stationarity is a nonparametricmethod and it is facilitative to implement and familiar byscholars

First obtain the sequence 119910119905 119905 = 1 2 119899 through the

transformation119910119905= [119909119905minus120583(119905)]120590(119905) based on the determined

mean function 120583(119905) and standard deviation function 120590(119905)Second split series 119910

119905 119905 = 1 2 119899 averagely into

two parts and calculate the sample variance_1205902

(1)= 119864((2

119899)sum1198992

119894=11199102

119894) and

_1205902

(2)= 119864((2119899)sum

119899

119894=1+11989921199102

119894) for eachThen the

test statistic_120591=

_1205902

(2)minus

_1205902

(1)follows

radic2

119899

_120591sim 119873[0 2(119877

0+ 2

infin

sum

119894=1

119877119894)] (26)

If 1199102119905is a covariance stationary process with autocovariances

119877119894 and let ] = 119877

0+ 2suminfin

119894=1119877119894 then it can be estimated by

_]=

_1198770 + 2

8

sum

119894=1

_119877119894 (1 minus

119894

9) (27)

where_119877119894 is the estimated serial correlation coefficients of 1199102

119905

calculated over the whole sampleFinally define null hypothesis that119867

0 119910119905 119905 = 1 2 119899

is a covariance stationarity series versus alternative hypothe-sis that119867

119886 119910119905 119905 = 1 2 119899 is not a covariance stationarity

series Construct test statistic 119861 and the rejection region canbe expressed by

119861 =

1003816100381610038161003816100381610038161003816100381610038161003816

radic119899

2

_120591

radic2]

1003816100381610038161003816100381610038161003816100381610038161003816

gt 1199061minus1205722 (28)

where 119906119901is the 100119906th percentile of the standard normal

distribution and 120572 is the selected significance level indicatingthe probability of type I error

5 Simulation Experiment

To assess the computational performance of the proposedmethod and determine whether the approach seems to givereasonable results we study the effectiveness and the per-formance of presented methods in Sections 3 and 4 from aMonte Carlo simulated example

Mathematical Problems in Engineering 7

The following zero mean CCAR(1)model

119909119905

120590 (119905)= 120593

119909119905minus1

120590 (119905 minus 1)+ 120576119905 120576119905sim NID [0 1205902

120576] (29)

is considered with four simulation function forms for thestandard deviation including linear function quadraticfunction periodic function and the combination functionof periodic function and linear function The parametersassumed in each experiment are summarized in Table 1 forconvenience The sample size is 119899 = 200 The point numberof averaging and moving time interval for each experiment isalso listed in Table 1

Series |119909119905+1minus 119909119905| and |119909

119905| can be taken to deter-

mine the standard deviation function 120590(119905) by (3) (Theoremmethod) presented in Section 3 since the mean of series119909119905equals zero that is 120583(119905) = 0 Simulation results are

analyzed by the index of average percent relative errorerr = (1119899)sum119899

119905=1|120590(119905)minus

_120590(119905)| times 100120590(119905) which is presented

in Table 2 The results are based on 20000 Monte Carlosimulations with innovations drawn from an IID Gaussiandistribution

Nonzero mean CCAR(119901) and CCARMA(119901 119902) modelwith higher order and different parameter specifications arealso considered to study the authentication of the theoremand the results were very similar to those reported in thispaper

6 Empirical Results

In this section we focus on the practical performance ofthe proposed approach Experiments are presented for fourdifferent economic data sets presented in Section 61 InSection 62 the mean and standard deviation functions aredetermined Then in Section 63 we apply the statistical testmethod discussed in Section 4 Finally Section 64 is devotedto evaluating the forecasting performance of the correlationcoefficient stationary method

61 The Data Sets

Daily CIRThe daily returns to composite index for Shanghaifrom January 5 1999 through September 30 2003 (1131observations)

Monthly MPL The monthly maximum power load forGuangxi from January 1990 through December 1999 (120observations)

Monthly M2 The monthly money supply for China fromJanuary 2000 through December 2009 (112 observations)

Daily FX Rate EURUSD Euro to the United States dollarparity from January 1 2005 throughDecember 30 2005 (260observations)

62 Determination of Mean and Variance In order to test thestability of the three data sets we apply the determinationmethod presented in Sections 3 and 4 on the four data setsAll parameter results are summarized in Tables 3 and 4 Note

that the standard deviation functions 120590(119905) are obtained by (3)for Daily CIR Monthly MPL and Daily FX rate data sets andby (4) forMonthlyM2 because the theorem condition can besatisfied for the former three sequences

63 Testing for Correlation Coefficient Stationarity As asecond step we apply the correlation coefficient stationaritytest presented in Section 4 on the four data sets All resultsare summarized in Table 5 We first derive sequence 119910

119905by

transformation 119910119905= [119909119905minus 120583(119905)]120590(119905) with the results listed

in Table 3 Let significance level 120572 = 005 and the rejectionregion is 119861 gt 119906

1minus1205722= 196

From the results we can conclude that the four datasets are correlation coefficient stationary time series and theapplication of the proposed method is reasonable

64 Prediction Based on the CCAR models constructedabove the forecast of original sequence 119909

119905can be obtained by

(1) In this section we will consider autoregressive integratedmoving average (ARIMA) model variable differential (VD)model generalized autoregressive conditional heteroscedas-ticity (GARCH)model Grey predictionmodel GM(1 1) andmodified GM(1 1) model which are considered as standardremarks to study its forecasting accuracy The future 5daily Shanghai composite index data the future 24 monthlymaximum power load data for Guangxi from January 2000through December 2001 the future 8 monthly money supplydata for China fromMay 2009 through December 2009 andthe future 10 daily FX rate data EURUSD are forecasted bythe established models The prediction results are summa-rized from Tables 6 7 8 and 9 and they are analyzed bythe index of percent relative error (Per err) 120576

119894= |119909119899+119894minus

_119909119899+119894| times 100119909

119899+119894 119894 = 1 2 119897 and mean percent relative

error (Mper err) 120576 = (1119897) sum119897119894=1120576119894 where 119897 is the prediction

step aheadFrom the results we can conclude that the application

of the proposed method for correlation coefficient stationarytime series is reasonable and effective Furthermore thepresented methodology can be considered good and showsa promise for future applications in nonstationary time seriesanalysis and forecasting

7 Discussion

In this paper we discussed the category of correlationcoefficient stationary series a nonstationary time series withtime-varying mean and variance We proposed a movingdetermination method for its time-varying mean functionand standard deviation function We also discussed thecorrelation coefficient stationary test method

The determination principle of function form and orderof mean 120583(119905) and standard deviation 120590(119905) cannot be sep-arated from primary sequence analysis It is worth notingin prediction problem that the function models of 120583(119905) and120590(119905) should also trade off between sequence volatility andaccuracy requirements

For mean function determination the proposed movingmethod can be used to establish the mean function of allkinds of nonstationary time series and can help improve the

8 Mathematical Problems in Engineering

Table 1 Model specifications by experiments

Experiment [Δ120590(119905)120590(119905)]max 119898 Δ 120593 1198881

1198882

1198883

(1) 120590(119905) = 1198881+ 1198882119905 00040 40 20

04 5 002

06 5 002

08 5 002

(2) 120590(119905) = 1198881+ 1198882119905 + 11988831199053 00087 40 20

04 4 minus001 0000306 4 minus001 0000308 4 minus001 00003

(3) 120590(119905) = 1198881+ 1198882sin (003120587119905) 00241

04 13 2

06 13 2

08 13 2

(4) 120590(119905) = 1198881+ 1198882119905 + 1198883sin (003120587119905) 00154

04 13 002 206 13 002 208 13 002 2

Note119898 and Δ are point number of each averaging and moving time interval 120593 is the autocorrelation coefficient of simulation model 1198881 1198882 and 119888

3are model

parameters in standard deviation function [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) Symbol ldquordquo indicates that the parameter does not exist

Table 2 Results of parameter and the average percent relative error for each experiment

120593 04 06 08 04 06 08

Method Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Experiment 1 Experiment 21198881

47635 43543 44568 49790 41984 66076 47635 43543 44568 49790 41984 660761198882

00190 00174 00178 00200 00168 00268 00190 00174 00178 00200 00168 00268Per err 771 1348 1157 885 1615 3371 771 1348 1157 885 1615 3371

Experiment 3 Experiment 41198881

123941 113212 116007 129740 109307 172514 124157 113302 115889 129288 109200 1716421198882

19023 17321 17246 19175 16054 25234 00188 00173 00178 00203 00169 002761198883

18880 17320 17573 19839 16753 26349Per err 1022 1449 1121 832 1594 3330 862 1382 1189 1029 1617 3458

Note Experiments 1 to 4 are defined in Table 1 Symbol ldquordquo indicates that the parameter does not exist

Table 3 Determination results of mean and variance functions

Trend function 120583(119905)Daily CIR 0Monthly MPL 105762 + 20498119905 minus 10297 sin(120587119905

6) + 15926 cos (120587119905

6)

Monthly M2 11986292 + 22406119905 + 652801199052minus 07726119905

3+ 4077 times 10

minus31199054

Daily FX rate 12489 + 51943 times 10minus3119905 minus 11693 times 10

minus41199052+ 94416 times 10

minus71199053minus 33508 times 10

minus91199054+ 43846 times 10

minus121199055

+001006 cos(312058711990565) + 001587 cos(3120587119905

130)

Standard deviation function 120590(119905)Daily CIR 05456 + 31940 times 10

minus2119905 minus 19532 times 10

minus41199052+ 43798 times 10

minus71199053minus 41174 times 10

minus101199054+ 13751 times 10

minus131199055

Monthly MPL 11607 minus 79024 sin(1205871199056) minus 40838 cos (120587119905

6)

Monthly M2 257950 minus 35024119905 + 176221199052minus 02823119905

3+ 14410 times 10

minus31199054

Daily FX rate 54449 times 10minus3

Mathematical Problems in Engineering 9

Table 4 Model parameter results

Data set Model 119910119905= sum119901

119894=1120593119894119910119905minus119894+ 120576119905

119901 1205931

1205932

Daily CIR 1 002861

Monthly MPL 1 06353

Monthly M2 2 11572 minus01401Daily FX rate 1 08616

accuracy especially in situations of small sample size Forstandard deviation function fixing the presented theoremis only suitable for the correlation coefficient stationarysequence

In addition the mean and standard deviation functionsdepicted by (14) are general expressions Take the MonthlyMPL data set in Section 6 for example we can assume that1206010(119905) = 120595

0(119905) = 1 120601

1(119905) = 119905 120601

2(119905) = 120595

1(119905) = sin(1205871199056)

1206013(119905) = 120595

2(119905) = cos(1205871199056) and a = (119886

0 1198861 1198862 1198863)1015840 and b =

(1198870 1198871 1198872)1015840

Appendices

A Proof of Theorem 1

Suppose that 119909119905 119905 = 1 2 119899 is a correlation coefficient

stationary time series119864(119909119905) = 120583(119905) is a deterministic function

representing time-varying mean and radicVar(119909119905) = 120590(119905)

is a positive deterministic function denoting time-varyingstandard deviation function According to the concept ofcorrelation coefficient stationary time series we can express119909119905as

119909119905= 120583 (119905) + 120590 (119905) 120576

119905 120576119905sim NID [0 1] (A1)

We can derive the mean of sequence |119909119905+1minus120583(119905+1)minus119909

119905+

120583(119905)| 119905 = 2 3 119899 by

1198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816

= 1198641003816100381610038161003816120590 (119905 + 1) 120576119905+1 minus 120590 (119905) 120576119905

1003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816

120590 (119905) + Δ120590 (119905)

120590 (119905)120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

(A2)

Define a new randomvariable 120585(119905) = [1+Δ120590(119905)120590(119905)]120576119905+1minus

120576119905 We can conclude that 120585

119905is still a Gaussian random variable

because it is a linear combination of two Gaussian random

variables 120576119905+1

and 120576119905 And its mean and variance can be

derived by

119864 (120585119905) = 119864[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905 = 0

Var (120585119905) = 119864[1 +

Δ120590(119905)

120590(119905)] 120576119905+1minus 120576119905

2

= (1 +Δ120590 (119905)

120590 (119905))

2

+ 1 minus 21205881(1 +

Δ120590 (119905)

120590 (119905))

(A3)

Therefore Var(120585119905) is a constant when Δ120590(119905)120590(119905) is a

constant or [Δ120590(119905)120590(119905)]max is a negligible small amountcompared with one And

11986410038161003816100381610038161205851199051003816100381610038161003816 = 119864

1003816100381610038161003816[1 + Δ120590 (119905) 120590 (119905)] 120576119905+1 minus 1205761199051003816100381610038161003816 =radic2

120587Var (120585

119905)

(A4)

Let constant 119888 = 1radic(2120587)Var(120585119905) then the standard

deviation of original series 119909119905can be expressed as 120590(119905) =

119888119864|119909119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| Theorem 1 in Section 3 has

been proved

B Proof of PDF Relationship

Let us derive the relationship between joint PDF119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) and 119891

119884119899119884119899minus11198841

(119910119899 119910119899minus1

1199101) Cumulative distribution function (CDF) can be

expressed by

119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

= 119875119883119899minus 120583 (119905

119899)

120590 (119905119899)le 119910119899119883119899minus1minus 120583 (119905

119899minus1)

120590 (119905119899minus1)

le 119910119899minus1

1198831minus 120583 (119905

1)

120590 (1199051)le 1199101

= 119875 119883119899le 120583 (119905

119899) + 120590 (119905

119899) 119910119899 119883119899minus1le 120583 (119905

119899minus1)

+ 120590 (119905119899minus1) 119910119899minus1 119883

1le 120583 (119905

1) + 120590 (119905

1) 1199101

= 119875 119883119899le 119909119899 119883119899minus1le 119909119899minus1 119883

1le 1199091

= 119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

(B1)

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Then we implement moving 119898-point average on thewhole sample data length 119899 based on themoving time intervalΔ and obtain a group of mean value (119905

119894 119909119894) by

119905119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119905 119894 = 1 2 119897 (10)

119909119894=1

119898

Δ119894+119898minus2

sum

119905=Δ119894minus1

119909119905 119894 = 1 2 119897 (11)

where 119897 = ent((119899 minus 119898)Δ) + 1 indicates the total averagingtimes

Consequently fit the obtained group of mean value(119905119894 119909119894) 119894 = 1 2 119897 to get the regression function 119891(119905)

which is the nonperiodic trend function part of series 119909119905

119905 = 1 2 119899(3) Redetermine the periodic part of trend itemBased on the nonperiodic trend function 119891(119905) obtained

above the following series can be calculated

119909119905minus 119891 (119905) 119905 = 1 2 119899 (12)

Process the obtained series from (12) and then the periodicpart function expressed by (8) can be redetermined with theperiodogram method

(4) Repeat step (2) and step (3) until each parameterresults in periodic part function sum

119895120572119895cos[120596

119895(119905 minus 1)] +

120573119895sin[120596119895(119905minus1)] and nonperiodic part function119891(119905) becomes

numerical stabilized Then we can obtain the final trend itemexpression as

_120583(119905) = 119891 (119905) +sum

119895

120572119895cos [120596

119895(119905 minus 1)]

+ 120573119895sin [120596

119895(119905 minus 1)]

(13)

The mean function 120583(119905) can be directly determined byimplementing the above steps from (1) to (4) on series 119909

119905

119905 = 1 2 119899 Based on the theorem given in Section 3the standard deviation function 120590(119905) can be obtained byconducting the same steps from (1) to (4) on series |119909

119905+1minus

120583(119905 + 1) minus 119909119905+ 120583(119905)| = |119909

119905+1minus

_120583(119905 + 1) minus 119909

119905+

_120583(119905)|

119905 = 1 2 119899 minus 1 if the theorem condition can be satisfiedOtherwise the standard deviation function 120590(119905) can beobtained by conducting the same steps from (1) to (4) onseries on sequence |119909

119905minus 120583(119905)| = |119909

119905minus

_120583(119905)| 119905 = 1 2 119899

In a word the function forms of mean 120583(119905) and standarddeviation 120590(119905) can be determined In order to facilitate theparameter estimation process we depict the time-varyingfunctions of mean and standard deviation by

120583 (a 119905) =119903

sum

119894=0

119886119894120601119894(119905)

120590 (b 119905) =119904

sum

119895=0

119887119895120595119895(119905)

(14)

where 1206010(119905) = 120595

0(119905) = 1 120601

119894(119905) 119894 = 1 2 119903 and 120595

119895(119905)119895 =

1 2 119904 are functions that can be known through the above

determination process and a = (1198860 1198861 119886

119903)119879 and b = (119887

0

1198871 119887

119904)119879 are general sets of unknown parameters to be

calculated

4 Model Construction and Testing

41 CCARMA Model Parameter Estimation Let 119909119905be a

CCARMA series with mean 120583(a 119905) and standard deviation120590(b 119905) and then transformed sequence 119910

119905= [119909119905minus 120583(119905)]120590(119905)

is an ARMA series according to concept 2 in Section 21The relationship between 119909

119905and 119910

119905can be rewritten as

119909119905= 120583(119905) + 120590(119905)119910

119905 Joint probability density functions

(PDF) 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) can be derived by PDF

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) through

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus1sdotsdotsdot 1198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(15)

See Appendices for proofAccording to time series analysis theory [33] a common

approximation of the likelihood function for ARMA processconditions on initial values of both 119910rsquos and 120576rsquos Based on therecommendation given by Box and Jenkins [34] we set 120576rsquos tozero for 119896 = 0 minus1 minus119902 + 1 and 119910rsquos to their actual valuesfor 119896 = 0 minus1 minus119901 + 1 Then the sequence 120576

1 1205762 120576

119899

can be calculated from 1199101 1199102 119910

119899 and the conditional

log likelihood is then

ln 119871 = minus119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896 (16)

Considering the Gaussian CCARMA(119901 119902) process 119909119905

depicted by (1) suppose that we have a sample of 119899 obser-vations 119909

119905 119905 = 1 2 119899 Maximum likelihood estimation

with conditional likelihood function is utilized to estimatethe vector of population parameters 120579 = (119886

0 1198861

119886119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 1205791 1205792 120579

119902 120590120576)119879 Accord-

ingly a common approximation of the likelihood functionfor CCARMA process conditions on initial values of both119909rsquos and 120576rsquos We set 120576rsquos to zero for 119896 = 0 minus1 minus119902 + 1and 119909rsquos to their actual values for 119896 = 0 minus1 minus119901 + 1Then the sequence 120576

1 1205762 sdot sdot sdot 120576

119899 can be calculated from

1199091 1199092 119909

119899 by iterating on

120576119896=119909119896minus 120583 (a 119905

119896)

120590 (b 119905119896)minus

119901

sum

119894=1

120593119894

119909119896minus119894minus 120583 (a 119905

119896minus119894)

120590 (b 119905119896minus119894)+

119902

sum

119894=1

120579119894120576119896minus119894

(17)

for 119896 = 1 2 119899 Based on the joint PDF relationshipdepicted by (15) the conditional log likelihood is then

ln 119871 (120579) = minus 119899 minus 1205822

ln (2120587) minus (119899 minus 120582) ln120590120576

minus1

21205902120576

119899

sum

119896=120582+1

1205762

119896minus

119899

sum

119896=1

ln120590 (b 119905119896)

(18)

6 Mathematical Problems in Engineering

where 120582 equals the maximum one of 119901 and 119902 that is 120582 =max(119901 119902) Model parameters can be determined by solvingthe following equations

120597 ln 119871 (120579)120597119886119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119886119897

= 0 119897 = 0 1 119903 (19)

120597 ln 119871 (120579)120597119887119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119887119897

minus

119899

sum

119896=1

1

120590 (b 119905119896)

120597120590 (b 119905119896)

120597119887119897

= 0 119897 = 0 1 119904

(20)

120597 ln 119871 (120579)120597120593119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120593119897

= 0 119897 = 0 1 119901 (21)

120597 ln 119871 (120579)120597120579119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120579119897

= 0 119897 = 0 1 119902 (22)

120597 ln 119871 (120579)120597120590120576

= minus119899 minus 120582

120590120576

+1

1205903120576

119899

sum

119896=120582+1

1205762

119896= 0 (23)

Based on the initial values for 120576rsquos given above we canconclude that 120597120576

119896120597119886119897 120597120576119896120597119887119897 120597120576119896120597120593119897 and 120597120576

119896120597120579119897equal zero

for 119896 = 0 minus1 minus119902 + 1 Then these values for 119896 = 1 2 119899can be calculated by iterating on

120597120576119896

120597119886119897

= minus1

119868 (b 119905119896)

120597120583 (a 119905119896)

120597119886119897

+

119901

sum

119894=1

120593119894

119868 (b 119905119896minus119894)

120597120583 (a 119905119896minus119894)

120597119886119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119886119897

= 0

120597120576119896

120597119887119897

= minus119909119896minus 120597120583 (a 119905

119896)

1198682 (b 119905119896)

119868 (b 119905119896)

120597119886119897

+

119901

sum

119894=1

120578119894[119909119896minus119894minus 120583 (a 119905

119896minus119894)]

1198682 (b 119905119896minus119894)

120597119868 (b 119905119896minus119894)

120597119887119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119887119897

= 0

120597120576119896

120597120578119897

= minus119909119896minus119897minus 120597120583 (a 119905

119896minus119897)

119868 (b 119905119896minus119897)

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120578119897

= 0

120597120576119896

120597120579119897

= 120576119896minus119897+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120579119897

= 0

(24)

Then (23) can be rewritten as

1205902

120576=1

119899 minus 120582

119899

sum

119894=120582+1

1205762

119894= 0 (25)

Parameters 1198860 1198861 119886

119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 and

1205791 1205792 120579

119902can be obtained by solving (19) to (22) and

parameter 120590120576can be calculated through (25)

42 CCARMA Model Testing Here we focus on the modeltesting method which can demonstrate whether the pro-posed model is appropriate to describe the observation dataFor the nonstationary series 119909

119905 119905 = 1 2 119899 based on

the parameter estimation process given above we can obtainits mean function 120583(119905) and standard deviation function 120590(119905)Then we can implement the correlation coefficient stationarytest of series119909

119905through the covariance stationary test of series

119910119905 where 119910

119905= [119909

119905minus 120583(119905)]120590(119905) Several tests have been

proposed and applied to examine the covariance stationarityin literature In this paper we take the postsample predictiontesting method presented by Pagan and Schwert [35] asan example to illustrate the testing procedure Postsampleprediction test for covariance stationarity is a nonparametricmethod and it is facilitative to implement and familiar byscholars

First obtain the sequence 119910119905 119905 = 1 2 119899 through the

transformation119910119905= [119909119905minus120583(119905)]120590(119905) based on the determined

mean function 120583(119905) and standard deviation function 120590(119905)Second split series 119910

119905 119905 = 1 2 119899 averagely into

two parts and calculate the sample variance_1205902

(1)= 119864((2

119899)sum1198992

119894=11199102

119894) and

_1205902

(2)= 119864((2119899)sum

119899

119894=1+11989921199102

119894) for eachThen the

test statistic_120591=

_1205902

(2)minus

_1205902

(1)follows

radic2

119899

_120591sim 119873[0 2(119877

0+ 2

infin

sum

119894=1

119877119894)] (26)

If 1199102119905is a covariance stationary process with autocovariances

119877119894 and let ] = 119877

0+ 2suminfin

119894=1119877119894 then it can be estimated by

_]=

_1198770 + 2

8

sum

119894=1

_119877119894 (1 minus

119894

9) (27)

where_119877119894 is the estimated serial correlation coefficients of 1199102

119905

calculated over the whole sampleFinally define null hypothesis that119867

0 119910119905 119905 = 1 2 119899

is a covariance stationarity series versus alternative hypothe-sis that119867

119886 119910119905 119905 = 1 2 119899 is not a covariance stationarity

series Construct test statistic 119861 and the rejection region canbe expressed by

119861 =

1003816100381610038161003816100381610038161003816100381610038161003816

radic119899

2

_120591

radic2]

1003816100381610038161003816100381610038161003816100381610038161003816

gt 1199061minus1205722 (28)

where 119906119901is the 100119906th percentile of the standard normal

distribution and 120572 is the selected significance level indicatingthe probability of type I error

5 Simulation Experiment

To assess the computational performance of the proposedmethod and determine whether the approach seems to givereasonable results we study the effectiveness and the per-formance of presented methods in Sections 3 and 4 from aMonte Carlo simulated example

Mathematical Problems in Engineering 7

The following zero mean CCAR(1)model

119909119905

120590 (119905)= 120593

119909119905minus1

120590 (119905 minus 1)+ 120576119905 120576119905sim NID [0 1205902

120576] (29)

is considered with four simulation function forms for thestandard deviation including linear function quadraticfunction periodic function and the combination functionof periodic function and linear function The parametersassumed in each experiment are summarized in Table 1 forconvenience The sample size is 119899 = 200 The point numberof averaging and moving time interval for each experiment isalso listed in Table 1

Series |119909119905+1minus 119909119905| and |119909

119905| can be taken to deter-

mine the standard deviation function 120590(119905) by (3) (Theoremmethod) presented in Section 3 since the mean of series119909119905equals zero that is 120583(119905) = 0 Simulation results are

analyzed by the index of average percent relative errorerr = (1119899)sum119899

119905=1|120590(119905)minus

_120590(119905)| times 100120590(119905) which is presented

in Table 2 The results are based on 20000 Monte Carlosimulations with innovations drawn from an IID Gaussiandistribution

Nonzero mean CCAR(119901) and CCARMA(119901 119902) modelwith higher order and different parameter specifications arealso considered to study the authentication of the theoremand the results were very similar to those reported in thispaper

6 Empirical Results

In this section we focus on the practical performance ofthe proposed approach Experiments are presented for fourdifferent economic data sets presented in Section 61 InSection 62 the mean and standard deviation functions aredetermined Then in Section 63 we apply the statistical testmethod discussed in Section 4 Finally Section 64 is devotedto evaluating the forecasting performance of the correlationcoefficient stationary method

61 The Data Sets

Daily CIRThe daily returns to composite index for Shanghaifrom January 5 1999 through September 30 2003 (1131observations)

Monthly MPL The monthly maximum power load forGuangxi from January 1990 through December 1999 (120observations)

Monthly M2 The monthly money supply for China fromJanuary 2000 through December 2009 (112 observations)

Daily FX Rate EURUSD Euro to the United States dollarparity from January 1 2005 throughDecember 30 2005 (260observations)

62 Determination of Mean and Variance In order to test thestability of the three data sets we apply the determinationmethod presented in Sections 3 and 4 on the four data setsAll parameter results are summarized in Tables 3 and 4 Note

that the standard deviation functions 120590(119905) are obtained by (3)for Daily CIR Monthly MPL and Daily FX rate data sets andby (4) forMonthlyM2 because the theorem condition can besatisfied for the former three sequences

63 Testing for Correlation Coefficient Stationarity As asecond step we apply the correlation coefficient stationaritytest presented in Section 4 on the four data sets All resultsare summarized in Table 5 We first derive sequence 119910

119905by

transformation 119910119905= [119909119905minus 120583(119905)]120590(119905) with the results listed

in Table 3 Let significance level 120572 = 005 and the rejectionregion is 119861 gt 119906

1minus1205722= 196

From the results we can conclude that the four datasets are correlation coefficient stationary time series and theapplication of the proposed method is reasonable

64 Prediction Based on the CCAR models constructedabove the forecast of original sequence 119909

119905can be obtained by

(1) In this section we will consider autoregressive integratedmoving average (ARIMA) model variable differential (VD)model generalized autoregressive conditional heteroscedas-ticity (GARCH)model Grey predictionmodel GM(1 1) andmodified GM(1 1) model which are considered as standardremarks to study its forecasting accuracy The future 5daily Shanghai composite index data the future 24 monthlymaximum power load data for Guangxi from January 2000through December 2001 the future 8 monthly money supplydata for China fromMay 2009 through December 2009 andthe future 10 daily FX rate data EURUSD are forecasted bythe established models The prediction results are summa-rized from Tables 6 7 8 and 9 and they are analyzed bythe index of percent relative error (Per err) 120576

119894= |119909119899+119894minus

_119909119899+119894| times 100119909

119899+119894 119894 = 1 2 119897 and mean percent relative

error (Mper err) 120576 = (1119897) sum119897119894=1120576119894 where 119897 is the prediction

step aheadFrom the results we can conclude that the application

of the proposed method for correlation coefficient stationarytime series is reasonable and effective Furthermore thepresented methodology can be considered good and showsa promise for future applications in nonstationary time seriesanalysis and forecasting

7 Discussion

In this paper we discussed the category of correlationcoefficient stationary series a nonstationary time series withtime-varying mean and variance We proposed a movingdetermination method for its time-varying mean functionand standard deviation function We also discussed thecorrelation coefficient stationary test method

The determination principle of function form and orderof mean 120583(119905) and standard deviation 120590(119905) cannot be sep-arated from primary sequence analysis It is worth notingin prediction problem that the function models of 120583(119905) and120590(119905) should also trade off between sequence volatility andaccuracy requirements

For mean function determination the proposed movingmethod can be used to establish the mean function of allkinds of nonstationary time series and can help improve the

8 Mathematical Problems in Engineering

Table 1 Model specifications by experiments

Experiment [Δ120590(119905)120590(119905)]max 119898 Δ 120593 1198881

1198882

1198883

(1) 120590(119905) = 1198881+ 1198882119905 00040 40 20

04 5 002

06 5 002

08 5 002

(2) 120590(119905) = 1198881+ 1198882119905 + 11988831199053 00087 40 20

04 4 minus001 0000306 4 minus001 0000308 4 minus001 00003

(3) 120590(119905) = 1198881+ 1198882sin (003120587119905) 00241

04 13 2

06 13 2

08 13 2

(4) 120590(119905) = 1198881+ 1198882119905 + 1198883sin (003120587119905) 00154

04 13 002 206 13 002 208 13 002 2

Note119898 and Δ are point number of each averaging and moving time interval 120593 is the autocorrelation coefficient of simulation model 1198881 1198882 and 119888

3are model

parameters in standard deviation function [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) Symbol ldquordquo indicates that the parameter does not exist

Table 2 Results of parameter and the average percent relative error for each experiment

120593 04 06 08 04 06 08

Method Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Experiment 1 Experiment 21198881

47635 43543 44568 49790 41984 66076 47635 43543 44568 49790 41984 660761198882

00190 00174 00178 00200 00168 00268 00190 00174 00178 00200 00168 00268Per err 771 1348 1157 885 1615 3371 771 1348 1157 885 1615 3371

Experiment 3 Experiment 41198881

123941 113212 116007 129740 109307 172514 124157 113302 115889 129288 109200 1716421198882

19023 17321 17246 19175 16054 25234 00188 00173 00178 00203 00169 002761198883

18880 17320 17573 19839 16753 26349Per err 1022 1449 1121 832 1594 3330 862 1382 1189 1029 1617 3458

Note Experiments 1 to 4 are defined in Table 1 Symbol ldquordquo indicates that the parameter does not exist

Table 3 Determination results of mean and variance functions

Trend function 120583(119905)Daily CIR 0Monthly MPL 105762 + 20498119905 minus 10297 sin(120587119905

6) + 15926 cos (120587119905

6)

Monthly M2 11986292 + 22406119905 + 652801199052minus 07726119905

3+ 4077 times 10

minus31199054

Daily FX rate 12489 + 51943 times 10minus3119905 minus 11693 times 10

minus41199052+ 94416 times 10

minus71199053minus 33508 times 10

minus91199054+ 43846 times 10

minus121199055

+001006 cos(312058711990565) + 001587 cos(3120587119905

130)

Standard deviation function 120590(119905)Daily CIR 05456 + 31940 times 10

minus2119905 minus 19532 times 10

minus41199052+ 43798 times 10

minus71199053minus 41174 times 10

minus101199054+ 13751 times 10

minus131199055

Monthly MPL 11607 minus 79024 sin(1205871199056) minus 40838 cos (120587119905

6)

Monthly M2 257950 minus 35024119905 + 176221199052minus 02823119905

3+ 14410 times 10

minus31199054

Daily FX rate 54449 times 10minus3

Mathematical Problems in Engineering 9

Table 4 Model parameter results

Data set Model 119910119905= sum119901

119894=1120593119894119910119905minus119894+ 120576119905

119901 1205931

1205932

Daily CIR 1 002861

Monthly MPL 1 06353

Monthly M2 2 11572 minus01401Daily FX rate 1 08616

accuracy especially in situations of small sample size Forstandard deviation function fixing the presented theoremis only suitable for the correlation coefficient stationarysequence

In addition the mean and standard deviation functionsdepicted by (14) are general expressions Take the MonthlyMPL data set in Section 6 for example we can assume that1206010(119905) = 120595

0(119905) = 1 120601

1(119905) = 119905 120601

2(119905) = 120595

1(119905) = sin(1205871199056)

1206013(119905) = 120595

2(119905) = cos(1205871199056) and a = (119886

0 1198861 1198862 1198863)1015840 and b =

(1198870 1198871 1198872)1015840

Appendices

A Proof of Theorem 1

Suppose that 119909119905 119905 = 1 2 119899 is a correlation coefficient

stationary time series119864(119909119905) = 120583(119905) is a deterministic function

representing time-varying mean and radicVar(119909119905) = 120590(119905)

is a positive deterministic function denoting time-varyingstandard deviation function According to the concept ofcorrelation coefficient stationary time series we can express119909119905as

119909119905= 120583 (119905) + 120590 (119905) 120576

119905 120576119905sim NID [0 1] (A1)

We can derive the mean of sequence |119909119905+1minus120583(119905+1)minus119909

119905+

120583(119905)| 119905 = 2 3 119899 by

1198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816

= 1198641003816100381610038161003816120590 (119905 + 1) 120576119905+1 minus 120590 (119905) 120576119905

1003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816

120590 (119905) + Δ120590 (119905)

120590 (119905)120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

(A2)

Define a new randomvariable 120585(119905) = [1+Δ120590(119905)120590(119905)]120576119905+1minus

120576119905 We can conclude that 120585

119905is still a Gaussian random variable

because it is a linear combination of two Gaussian random

variables 120576119905+1

and 120576119905 And its mean and variance can be

derived by

119864 (120585119905) = 119864[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905 = 0

Var (120585119905) = 119864[1 +

Δ120590(119905)

120590(119905)] 120576119905+1minus 120576119905

2

= (1 +Δ120590 (119905)

120590 (119905))

2

+ 1 minus 21205881(1 +

Δ120590 (119905)

120590 (119905))

(A3)

Therefore Var(120585119905) is a constant when Δ120590(119905)120590(119905) is a

constant or [Δ120590(119905)120590(119905)]max is a negligible small amountcompared with one And

11986410038161003816100381610038161205851199051003816100381610038161003816 = 119864

1003816100381610038161003816[1 + Δ120590 (119905) 120590 (119905)] 120576119905+1 minus 1205761199051003816100381610038161003816 =radic2

120587Var (120585

119905)

(A4)

Let constant 119888 = 1radic(2120587)Var(120585119905) then the standard

deviation of original series 119909119905can be expressed as 120590(119905) =

119888119864|119909119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| Theorem 1 in Section 3 has

been proved

B Proof of PDF Relationship

Let us derive the relationship between joint PDF119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) and 119891

119884119899119884119899minus11198841

(119910119899 119910119899minus1

1199101) Cumulative distribution function (CDF) can be

expressed by

119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

= 119875119883119899minus 120583 (119905

119899)

120590 (119905119899)le 119910119899119883119899minus1minus 120583 (119905

119899minus1)

120590 (119905119899minus1)

le 119910119899minus1

1198831minus 120583 (119905

1)

120590 (1199051)le 1199101

= 119875 119883119899le 120583 (119905

119899) + 120590 (119905

119899) 119910119899 119883119899minus1le 120583 (119905

119899minus1)

+ 120590 (119905119899minus1) 119910119899minus1 119883

1le 120583 (119905

1) + 120590 (119905

1) 1199101

= 119875 119883119899le 119909119899 119883119899minus1le 119909119899minus1 119883

1le 1199091

= 119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

(B1)

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

where 120582 equals the maximum one of 119901 and 119902 that is 120582 =max(119901 119902) Model parameters can be determined by solvingthe following equations

120597 ln 119871 (120579)120597119886119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119886119897

= 0 119897 = 0 1 119903 (19)

120597 ln 119871 (120579)120597119887119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597119887119897

minus

119899

sum

119896=1

1

120590 (b 119905119896)

120597120590 (b 119905119896)

120597119887119897

= 0 119897 = 0 1 119904

(20)

120597 ln 119871 (120579)120597120593119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120593119897

= 0 119897 = 0 1 119901 (21)

120597 ln 119871 (120579)120597120579119897

= minus1

1205902120576

119899

sum

119896=120582+1

120576119896

120597120576119896

120597120579119897

= 0 119897 = 0 1 119902 (22)

120597 ln 119871 (120579)120597120590120576

= minus119899 minus 120582

120590120576

+1

1205903120576

119899

sum

119896=120582+1

1205762

119896= 0 (23)

Based on the initial values for 120576rsquos given above we canconclude that 120597120576

119896120597119886119897 120597120576119896120597119887119897 120597120576119896120597120593119897 and 120597120576

119896120597120579119897equal zero

for 119896 = 0 minus1 minus119902 + 1 Then these values for 119896 = 1 2 119899can be calculated by iterating on

120597120576119896

120597119886119897

= minus1

119868 (b 119905119896)

120597120583 (a 119905119896)

120597119886119897

+

119901

sum

119894=1

120593119894

119868 (b 119905119896minus119894)

120597120583 (a 119905119896minus119894)

120597119886119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119886119897

= 0

120597120576119896

120597119887119897

= minus119909119896minus 120597120583 (a 119905

119896)

1198682 (b 119905119896)

119868 (b 119905119896)

120597119886119897

+

119901

sum

119894=1

120578119894[119909119896minus119894minus 120583 (a 119905

119896minus119894)]

1198682 (b 119905119896minus119894)

120597119868 (b 119905119896minus119894)

120597119887119897

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597119887119897

= 0

120597120576119896

120597120578119897

= minus119909119896minus119897minus 120597120583 (a 119905

119896minus119897)

119868 (b 119905119896minus119897)

+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120578119897

= 0

120597120576119896

120597120579119897

= 120576119896minus119897+

119902

sum

119894=1

120579119894

120597120576119896minus119894

120597120579119897

= 0

(24)

Then (23) can be rewritten as

1205902

120576=1

119899 minus 120582

119899

sum

119894=120582+1

1205762

119894= 0 (25)

Parameters 1198860 1198861 119886

119903 1198870 1198871 119887

119904 1205931 1205932 120593

119901 and

1205791 1205792 120579

119902can be obtained by solving (19) to (22) and

parameter 120590120576can be calculated through (25)

42 CCARMA Model Testing Here we focus on the modeltesting method which can demonstrate whether the pro-posed model is appropriate to describe the observation dataFor the nonstationary series 119909

119905 119905 = 1 2 119899 based on

the parameter estimation process given above we can obtainits mean function 120583(119905) and standard deviation function 120590(119905)Then we can implement the correlation coefficient stationarytest of series119909

119905through the covariance stationary test of series

119910119905 where 119910

119905= [119909

119905minus 120583(119905)]120590(119905) Several tests have been

proposed and applied to examine the covariance stationarityin literature In this paper we take the postsample predictiontesting method presented by Pagan and Schwert [35] asan example to illustrate the testing procedure Postsampleprediction test for covariance stationarity is a nonparametricmethod and it is facilitative to implement and familiar byscholars

First obtain the sequence 119910119905 119905 = 1 2 119899 through the

transformation119910119905= [119909119905minus120583(119905)]120590(119905) based on the determined

mean function 120583(119905) and standard deviation function 120590(119905)Second split series 119910

119905 119905 = 1 2 119899 averagely into

two parts and calculate the sample variance_1205902

(1)= 119864((2

119899)sum1198992

119894=11199102

119894) and

_1205902

(2)= 119864((2119899)sum

119899

119894=1+11989921199102

119894) for eachThen the

test statistic_120591=

_1205902

(2)minus

_1205902

(1)follows

radic2

119899

_120591sim 119873[0 2(119877

0+ 2

infin

sum

119894=1

119877119894)] (26)

If 1199102119905is a covariance stationary process with autocovariances

119877119894 and let ] = 119877

0+ 2suminfin

119894=1119877119894 then it can be estimated by

_]=

_1198770 + 2

8

sum

119894=1

_119877119894 (1 minus

119894

9) (27)

where_119877119894 is the estimated serial correlation coefficients of 1199102

119905

calculated over the whole sampleFinally define null hypothesis that119867

0 119910119905 119905 = 1 2 119899

is a covariance stationarity series versus alternative hypothe-sis that119867

119886 119910119905 119905 = 1 2 119899 is not a covariance stationarity

series Construct test statistic 119861 and the rejection region canbe expressed by

119861 =

1003816100381610038161003816100381610038161003816100381610038161003816

radic119899

2

_120591

radic2]

1003816100381610038161003816100381610038161003816100381610038161003816

gt 1199061minus1205722 (28)

where 119906119901is the 100119906th percentile of the standard normal

distribution and 120572 is the selected significance level indicatingthe probability of type I error

5 Simulation Experiment

To assess the computational performance of the proposedmethod and determine whether the approach seems to givereasonable results we study the effectiveness and the per-formance of presented methods in Sections 3 and 4 from aMonte Carlo simulated example

Mathematical Problems in Engineering 7

The following zero mean CCAR(1)model

119909119905

120590 (119905)= 120593

119909119905minus1

120590 (119905 minus 1)+ 120576119905 120576119905sim NID [0 1205902

120576] (29)

is considered with four simulation function forms for thestandard deviation including linear function quadraticfunction periodic function and the combination functionof periodic function and linear function The parametersassumed in each experiment are summarized in Table 1 forconvenience The sample size is 119899 = 200 The point numberof averaging and moving time interval for each experiment isalso listed in Table 1

Series |119909119905+1minus 119909119905| and |119909

119905| can be taken to deter-

mine the standard deviation function 120590(119905) by (3) (Theoremmethod) presented in Section 3 since the mean of series119909119905equals zero that is 120583(119905) = 0 Simulation results are

analyzed by the index of average percent relative errorerr = (1119899)sum119899

119905=1|120590(119905)minus

_120590(119905)| times 100120590(119905) which is presented

in Table 2 The results are based on 20000 Monte Carlosimulations with innovations drawn from an IID Gaussiandistribution

Nonzero mean CCAR(119901) and CCARMA(119901 119902) modelwith higher order and different parameter specifications arealso considered to study the authentication of the theoremand the results were very similar to those reported in thispaper

6 Empirical Results

In this section we focus on the practical performance ofthe proposed approach Experiments are presented for fourdifferent economic data sets presented in Section 61 InSection 62 the mean and standard deviation functions aredetermined Then in Section 63 we apply the statistical testmethod discussed in Section 4 Finally Section 64 is devotedto evaluating the forecasting performance of the correlationcoefficient stationary method

61 The Data Sets

Daily CIRThe daily returns to composite index for Shanghaifrom January 5 1999 through September 30 2003 (1131observations)

Monthly MPL The monthly maximum power load forGuangxi from January 1990 through December 1999 (120observations)

Monthly M2 The monthly money supply for China fromJanuary 2000 through December 2009 (112 observations)

Daily FX Rate EURUSD Euro to the United States dollarparity from January 1 2005 throughDecember 30 2005 (260observations)

62 Determination of Mean and Variance In order to test thestability of the three data sets we apply the determinationmethod presented in Sections 3 and 4 on the four data setsAll parameter results are summarized in Tables 3 and 4 Note

that the standard deviation functions 120590(119905) are obtained by (3)for Daily CIR Monthly MPL and Daily FX rate data sets andby (4) forMonthlyM2 because the theorem condition can besatisfied for the former three sequences

63 Testing for Correlation Coefficient Stationarity As asecond step we apply the correlation coefficient stationaritytest presented in Section 4 on the four data sets All resultsare summarized in Table 5 We first derive sequence 119910

119905by

transformation 119910119905= [119909119905minus 120583(119905)]120590(119905) with the results listed

in Table 3 Let significance level 120572 = 005 and the rejectionregion is 119861 gt 119906

1minus1205722= 196

From the results we can conclude that the four datasets are correlation coefficient stationary time series and theapplication of the proposed method is reasonable

64 Prediction Based on the CCAR models constructedabove the forecast of original sequence 119909

119905can be obtained by

(1) In this section we will consider autoregressive integratedmoving average (ARIMA) model variable differential (VD)model generalized autoregressive conditional heteroscedas-ticity (GARCH)model Grey predictionmodel GM(1 1) andmodified GM(1 1) model which are considered as standardremarks to study its forecasting accuracy The future 5daily Shanghai composite index data the future 24 monthlymaximum power load data for Guangxi from January 2000through December 2001 the future 8 monthly money supplydata for China fromMay 2009 through December 2009 andthe future 10 daily FX rate data EURUSD are forecasted bythe established models The prediction results are summa-rized from Tables 6 7 8 and 9 and they are analyzed bythe index of percent relative error (Per err) 120576

119894= |119909119899+119894minus

_119909119899+119894| times 100119909

119899+119894 119894 = 1 2 119897 and mean percent relative

error (Mper err) 120576 = (1119897) sum119897119894=1120576119894 where 119897 is the prediction

step aheadFrom the results we can conclude that the application

of the proposed method for correlation coefficient stationarytime series is reasonable and effective Furthermore thepresented methodology can be considered good and showsa promise for future applications in nonstationary time seriesanalysis and forecasting

7 Discussion

In this paper we discussed the category of correlationcoefficient stationary series a nonstationary time series withtime-varying mean and variance We proposed a movingdetermination method for its time-varying mean functionand standard deviation function We also discussed thecorrelation coefficient stationary test method

The determination principle of function form and orderof mean 120583(119905) and standard deviation 120590(119905) cannot be sep-arated from primary sequence analysis It is worth notingin prediction problem that the function models of 120583(119905) and120590(119905) should also trade off between sequence volatility andaccuracy requirements

For mean function determination the proposed movingmethod can be used to establish the mean function of allkinds of nonstationary time series and can help improve the

8 Mathematical Problems in Engineering

Table 1 Model specifications by experiments

Experiment [Δ120590(119905)120590(119905)]max 119898 Δ 120593 1198881

1198882

1198883

(1) 120590(119905) = 1198881+ 1198882119905 00040 40 20

04 5 002

06 5 002

08 5 002

(2) 120590(119905) = 1198881+ 1198882119905 + 11988831199053 00087 40 20

04 4 minus001 0000306 4 minus001 0000308 4 minus001 00003

(3) 120590(119905) = 1198881+ 1198882sin (003120587119905) 00241

04 13 2

06 13 2

08 13 2

(4) 120590(119905) = 1198881+ 1198882119905 + 1198883sin (003120587119905) 00154

04 13 002 206 13 002 208 13 002 2

Note119898 and Δ are point number of each averaging and moving time interval 120593 is the autocorrelation coefficient of simulation model 1198881 1198882 and 119888

3are model

parameters in standard deviation function [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) Symbol ldquordquo indicates that the parameter does not exist

Table 2 Results of parameter and the average percent relative error for each experiment

120593 04 06 08 04 06 08

Method Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Experiment 1 Experiment 21198881

47635 43543 44568 49790 41984 66076 47635 43543 44568 49790 41984 660761198882

00190 00174 00178 00200 00168 00268 00190 00174 00178 00200 00168 00268Per err 771 1348 1157 885 1615 3371 771 1348 1157 885 1615 3371

Experiment 3 Experiment 41198881

123941 113212 116007 129740 109307 172514 124157 113302 115889 129288 109200 1716421198882

19023 17321 17246 19175 16054 25234 00188 00173 00178 00203 00169 002761198883

18880 17320 17573 19839 16753 26349Per err 1022 1449 1121 832 1594 3330 862 1382 1189 1029 1617 3458

Note Experiments 1 to 4 are defined in Table 1 Symbol ldquordquo indicates that the parameter does not exist

Table 3 Determination results of mean and variance functions

Trend function 120583(119905)Daily CIR 0Monthly MPL 105762 + 20498119905 minus 10297 sin(120587119905

6) + 15926 cos (120587119905

6)

Monthly M2 11986292 + 22406119905 + 652801199052minus 07726119905

3+ 4077 times 10

minus31199054

Daily FX rate 12489 + 51943 times 10minus3119905 minus 11693 times 10

minus41199052+ 94416 times 10

minus71199053minus 33508 times 10

minus91199054+ 43846 times 10

minus121199055

+001006 cos(312058711990565) + 001587 cos(3120587119905

130)

Standard deviation function 120590(119905)Daily CIR 05456 + 31940 times 10

minus2119905 minus 19532 times 10

minus41199052+ 43798 times 10

minus71199053minus 41174 times 10

minus101199054+ 13751 times 10

minus131199055

Monthly MPL 11607 minus 79024 sin(1205871199056) minus 40838 cos (120587119905

6)

Monthly M2 257950 minus 35024119905 + 176221199052minus 02823119905

3+ 14410 times 10

minus31199054

Daily FX rate 54449 times 10minus3

Mathematical Problems in Engineering 9

Table 4 Model parameter results

Data set Model 119910119905= sum119901

119894=1120593119894119910119905minus119894+ 120576119905

119901 1205931

1205932

Daily CIR 1 002861

Monthly MPL 1 06353

Monthly M2 2 11572 minus01401Daily FX rate 1 08616

accuracy especially in situations of small sample size Forstandard deviation function fixing the presented theoremis only suitable for the correlation coefficient stationarysequence

In addition the mean and standard deviation functionsdepicted by (14) are general expressions Take the MonthlyMPL data set in Section 6 for example we can assume that1206010(119905) = 120595

0(119905) = 1 120601

1(119905) = 119905 120601

2(119905) = 120595

1(119905) = sin(1205871199056)

1206013(119905) = 120595

2(119905) = cos(1205871199056) and a = (119886

0 1198861 1198862 1198863)1015840 and b =

(1198870 1198871 1198872)1015840

Appendices

A Proof of Theorem 1

Suppose that 119909119905 119905 = 1 2 119899 is a correlation coefficient

stationary time series119864(119909119905) = 120583(119905) is a deterministic function

representing time-varying mean and radicVar(119909119905) = 120590(119905)

is a positive deterministic function denoting time-varyingstandard deviation function According to the concept ofcorrelation coefficient stationary time series we can express119909119905as

119909119905= 120583 (119905) + 120590 (119905) 120576

119905 120576119905sim NID [0 1] (A1)

We can derive the mean of sequence |119909119905+1minus120583(119905+1)minus119909

119905+

120583(119905)| 119905 = 2 3 119899 by

1198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816

= 1198641003816100381610038161003816120590 (119905 + 1) 120576119905+1 minus 120590 (119905) 120576119905

1003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816

120590 (119905) + Δ120590 (119905)

120590 (119905)120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

(A2)

Define a new randomvariable 120585(119905) = [1+Δ120590(119905)120590(119905)]120576119905+1minus

120576119905 We can conclude that 120585

119905is still a Gaussian random variable

because it is a linear combination of two Gaussian random

variables 120576119905+1

and 120576119905 And its mean and variance can be

derived by

119864 (120585119905) = 119864[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905 = 0

Var (120585119905) = 119864[1 +

Δ120590(119905)

120590(119905)] 120576119905+1minus 120576119905

2

= (1 +Δ120590 (119905)

120590 (119905))

2

+ 1 minus 21205881(1 +

Δ120590 (119905)

120590 (119905))

(A3)

Therefore Var(120585119905) is a constant when Δ120590(119905)120590(119905) is a

constant or [Δ120590(119905)120590(119905)]max is a negligible small amountcompared with one And

11986410038161003816100381610038161205851199051003816100381610038161003816 = 119864

1003816100381610038161003816[1 + Δ120590 (119905) 120590 (119905)] 120576119905+1 minus 1205761199051003816100381610038161003816 =radic2

120587Var (120585

119905)

(A4)

Let constant 119888 = 1radic(2120587)Var(120585119905) then the standard

deviation of original series 119909119905can be expressed as 120590(119905) =

119888119864|119909119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| Theorem 1 in Section 3 has

been proved

B Proof of PDF Relationship

Let us derive the relationship between joint PDF119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) and 119891

119884119899119884119899minus11198841

(119910119899 119910119899minus1

1199101) Cumulative distribution function (CDF) can be

expressed by

119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

= 119875119883119899minus 120583 (119905

119899)

120590 (119905119899)le 119910119899119883119899minus1minus 120583 (119905

119899minus1)

120590 (119905119899minus1)

le 119910119899minus1

1198831minus 120583 (119905

1)

120590 (1199051)le 1199101

= 119875 119883119899le 120583 (119905

119899) + 120590 (119905

119899) 119910119899 119883119899minus1le 120583 (119905

119899minus1)

+ 120590 (119905119899minus1) 119910119899minus1 119883

1le 120583 (119905

1) + 120590 (119905

1) 1199101

= 119875 119883119899le 119909119899 119883119899minus1le 119909119899minus1 119883

1le 1199091

= 119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

(B1)

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

The following zero mean CCAR(1)model

119909119905

120590 (119905)= 120593

119909119905minus1

120590 (119905 minus 1)+ 120576119905 120576119905sim NID [0 1205902

120576] (29)

is considered with four simulation function forms for thestandard deviation including linear function quadraticfunction periodic function and the combination functionof periodic function and linear function The parametersassumed in each experiment are summarized in Table 1 forconvenience The sample size is 119899 = 200 The point numberof averaging and moving time interval for each experiment isalso listed in Table 1

Series |119909119905+1minus 119909119905| and |119909

119905| can be taken to deter-

mine the standard deviation function 120590(119905) by (3) (Theoremmethod) presented in Section 3 since the mean of series119909119905equals zero that is 120583(119905) = 0 Simulation results are

analyzed by the index of average percent relative errorerr = (1119899)sum119899

119905=1|120590(119905)minus

_120590(119905)| times 100120590(119905) which is presented

in Table 2 The results are based on 20000 Monte Carlosimulations with innovations drawn from an IID Gaussiandistribution

Nonzero mean CCAR(119901) and CCARMA(119901 119902) modelwith higher order and different parameter specifications arealso considered to study the authentication of the theoremand the results were very similar to those reported in thispaper

6 Empirical Results

In this section we focus on the practical performance ofthe proposed approach Experiments are presented for fourdifferent economic data sets presented in Section 61 InSection 62 the mean and standard deviation functions aredetermined Then in Section 63 we apply the statistical testmethod discussed in Section 4 Finally Section 64 is devotedto evaluating the forecasting performance of the correlationcoefficient stationary method

61 The Data Sets

Daily CIRThe daily returns to composite index for Shanghaifrom January 5 1999 through September 30 2003 (1131observations)

Monthly MPL The monthly maximum power load forGuangxi from January 1990 through December 1999 (120observations)

Monthly M2 The monthly money supply for China fromJanuary 2000 through December 2009 (112 observations)

Daily FX Rate EURUSD Euro to the United States dollarparity from January 1 2005 throughDecember 30 2005 (260observations)

62 Determination of Mean and Variance In order to test thestability of the three data sets we apply the determinationmethod presented in Sections 3 and 4 on the four data setsAll parameter results are summarized in Tables 3 and 4 Note

that the standard deviation functions 120590(119905) are obtained by (3)for Daily CIR Monthly MPL and Daily FX rate data sets andby (4) forMonthlyM2 because the theorem condition can besatisfied for the former three sequences

63 Testing for Correlation Coefficient Stationarity As asecond step we apply the correlation coefficient stationaritytest presented in Section 4 on the four data sets All resultsare summarized in Table 5 We first derive sequence 119910

119905by

transformation 119910119905= [119909119905minus 120583(119905)]120590(119905) with the results listed

in Table 3 Let significance level 120572 = 005 and the rejectionregion is 119861 gt 119906

1minus1205722= 196

From the results we can conclude that the four datasets are correlation coefficient stationary time series and theapplication of the proposed method is reasonable

64 Prediction Based on the CCAR models constructedabove the forecast of original sequence 119909

119905can be obtained by

(1) In this section we will consider autoregressive integratedmoving average (ARIMA) model variable differential (VD)model generalized autoregressive conditional heteroscedas-ticity (GARCH)model Grey predictionmodel GM(1 1) andmodified GM(1 1) model which are considered as standardremarks to study its forecasting accuracy The future 5daily Shanghai composite index data the future 24 monthlymaximum power load data for Guangxi from January 2000through December 2001 the future 8 monthly money supplydata for China fromMay 2009 through December 2009 andthe future 10 daily FX rate data EURUSD are forecasted bythe established models The prediction results are summa-rized from Tables 6 7 8 and 9 and they are analyzed bythe index of percent relative error (Per err) 120576

119894= |119909119899+119894minus

_119909119899+119894| times 100119909

119899+119894 119894 = 1 2 119897 and mean percent relative

error (Mper err) 120576 = (1119897) sum119897119894=1120576119894 where 119897 is the prediction

step aheadFrom the results we can conclude that the application

of the proposed method for correlation coefficient stationarytime series is reasonable and effective Furthermore thepresented methodology can be considered good and showsa promise for future applications in nonstationary time seriesanalysis and forecasting

7 Discussion

In this paper we discussed the category of correlationcoefficient stationary series a nonstationary time series withtime-varying mean and variance We proposed a movingdetermination method for its time-varying mean functionand standard deviation function We also discussed thecorrelation coefficient stationary test method

The determination principle of function form and orderof mean 120583(119905) and standard deviation 120590(119905) cannot be sep-arated from primary sequence analysis It is worth notingin prediction problem that the function models of 120583(119905) and120590(119905) should also trade off between sequence volatility andaccuracy requirements

For mean function determination the proposed movingmethod can be used to establish the mean function of allkinds of nonstationary time series and can help improve the

8 Mathematical Problems in Engineering

Table 1 Model specifications by experiments

Experiment [Δ120590(119905)120590(119905)]max 119898 Δ 120593 1198881

1198882

1198883

(1) 120590(119905) = 1198881+ 1198882119905 00040 40 20

04 5 002

06 5 002

08 5 002

(2) 120590(119905) = 1198881+ 1198882119905 + 11988831199053 00087 40 20

04 4 minus001 0000306 4 minus001 0000308 4 minus001 00003

(3) 120590(119905) = 1198881+ 1198882sin (003120587119905) 00241

04 13 2

06 13 2

08 13 2

(4) 120590(119905) = 1198881+ 1198882119905 + 1198883sin (003120587119905) 00154

04 13 002 206 13 002 208 13 002 2

Note119898 and Δ are point number of each averaging and moving time interval 120593 is the autocorrelation coefficient of simulation model 1198881 1198882 and 119888

3are model

parameters in standard deviation function [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) Symbol ldquordquo indicates that the parameter does not exist

Table 2 Results of parameter and the average percent relative error for each experiment

120593 04 06 08 04 06 08

Method Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Experiment 1 Experiment 21198881

47635 43543 44568 49790 41984 66076 47635 43543 44568 49790 41984 660761198882

00190 00174 00178 00200 00168 00268 00190 00174 00178 00200 00168 00268Per err 771 1348 1157 885 1615 3371 771 1348 1157 885 1615 3371

Experiment 3 Experiment 41198881

123941 113212 116007 129740 109307 172514 124157 113302 115889 129288 109200 1716421198882

19023 17321 17246 19175 16054 25234 00188 00173 00178 00203 00169 002761198883

18880 17320 17573 19839 16753 26349Per err 1022 1449 1121 832 1594 3330 862 1382 1189 1029 1617 3458

Note Experiments 1 to 4 are defined in Table 1 Symbol ldquordquo indicates that the parameter does not exist

Table 3 Determination results of mean and variance functions

Trend function 120583(119905)Daily CIR 0Monthly MPL 105762 + 20498119905 minus 10297 sin(120587119905

6) + 15926 cos (120587119905

6)

Monthly M2 11986292 + 22406119905 + 652801199052minus 07726119905

3+ 4077 times 10

minus31199054

Daily FX rate 12489 + 51943 times 10minus3119905 minus 11693 times 10

minus41199052+ 94416 times 10

minus71199053minus 33508 times 10

minus91199054+ 43846 times 10

minus121199055

+001006 cos(312058711990565) + 001587 cos(3120587119905

130)

Standard deviation function 120590(119905)Daily CIR 05456 + 31940 times 10

minus2119905 minus 19532 times 10

minus41199052+ 43798 times 10

minus71199053minus 41174 times 10

minus101199054+ 13751 times 10

minus131199055

Monthly MPL 11607 minus 79024 sin(1205871199056) minus 40838 cos (120587119905

6)

Monthly M2 257950 minus 35024119905 + 176221199052minus 02823119905

3+ 14410 times 10

minus31199054

Daily FX rate 54449 times 10minus3

Mathematical Problems in Engineering 9

Table 4 Model parameter results

Data set Model 119910119905= sum119901

119894=1120593119894119910119905minus119894+ 120576119905

119901 1205931

1205932

Daily CIR 1 002861

Monthly MPL 1 06353

Monthly M2 2 11572 minus01401Daily FX rate 1 08616

accuracy especially in situations of small sample size Forstandard deviation function fixing the presented theoremis only suitable for the correlation coefficient stationarysequence

In addition the mean and standard deviation functionsdepicted by (14) are general expressions Take the MonthlyMPL data set in Section 6 for example we can assume that1206010(119905) = 120595

0(119905) = 1 120601

1(119905) = 119905 120601

2(119905) = 120595

1(119905) = sin(1205871199056)

1206013(119905) = 120595

2(119905) = cos(1205871199056) and a = (119886

0 1198861 1198862 1198863)1015840 and b =

(1198870 1198871 1198872)1015840

Appendices

A Proof of Theorem 1

Suppose that 119909119905 119905 = 1 2 119899 is a correlation coefficient

stationary time series119864(119909119905) = 120583(119905) is a deterministic function

representing time-varying mean and radicVar(119909119905) = 120590(119905)

is a positive deterministic function denoting time-varyingstandard deviation function According to the concept ofcorrelation coefficient stationary time series we can express119909119905as

119909119905= 120583 (119905) + 120590 (119905) 120576

119905 120576119905sim NID [0 1] (A1)

We can derive the mean of sequence |119909119905+1minus120583(119905+1)minus119909

119905+

120583(119905)| 119905 = 2 3 119899 by

1198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816

= 1198641003816100381610038161003816120590 (119905 + 1) 120576119905+1 minus 120590 (119905) 120576119905

1003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816

120590 (119905) + Δ120590 (119905)

120590 (119905)120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

(A2)

Define a new randomvariable 120585(119905) = [1+Δ120590(119905)120590(119905)]120576119905+1minus

120576119905 We can conclude that 120585

119905is still a Gaussian random variable

because it is a linear combination of two Gaussian random

variables 120576119905+1

and 120576119905 And its mean and variance can be

derived by

119864 (120585119905) = 119864[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905 = 0

Var (120585119905) = 119864[1 +

Δ120590(119905)

120590(119905)] 120576119905+1minus 120576119905

2

= (1 +Δ120590 (119905)

120590 (119905))

2

+ 1 minus 21205881(1 +

Δ120590 (119905)

120590 (119905))

(A3)

Therefore Var(120585119905) is a constant when Δ120590(119905)120590(119905) is a

constant or [Δ120590(119905)120590(119905)]max is a negligible small amountcompared with one And

11986410038161003816100381610038161205851199051003816100381610038161003816 = 119864

1003816100381610038161003816[1 + Δ120590 (119905) 120590 (119905)] 120576119905+1 minus 1205761199051003816100381610038161003816 =radic2

120587Var (120585

119905)

(A4)

Let constant 119888 = 1radic(2120587)Var(120585119905) then the standard

deviation of original series 119909119905can be expressed as 120590(119905) =

119888119864|119909119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| Theorem 1 in Section 3 has

been proved

B Proof of PDF Relationship

Let us derive the relationship between joint PDF119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) and 119891

119884119899119884119899minus11198841

(119910119899 119910119899minus1

1199101) Cumulative distribution function (CDF) can be

expressed by

119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

= 119875119883119899minus 120583 (119905

119899)

120590 (119905119899)le 119910119899119883119899minus1minus 120583 (119905

119899minus1)

120590 (119905119899minus1)

le 119910119899minus1

1198831minus 120583 (119905

1)

120590 (1199051)le 1199101

= 119875 119883119899le 120583 (119905

119899) + 120590 (119905

119899) 119910119899 119883119899minus1le 120583 (119905

119899minus1)

+ 120590 (119905119899minus1) 119910119899minus1 119883

1le 120583 (119905

1) + 120590 (119905

1) 1199101

= 119875 119883119899le 119909119899 119883119899minus1le 119909119899minus1 119883

1le 1199091

= 119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

(B1)

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 1 Model specifications by experiments

Experiment [Δ120590(119905)120590(119905)]max 119898 Δ 120593 1198881

1198882

1198883

(1) 120590(119905) = 1198881+ 1198882119905 00040 40 20

04 5 002

06 5 002

08 5 002

(2) 120590(119905) = 1198881+ 1198882119905 + 11988831199053 00087 40 20

04 4 minus001 0000306 4 minus001 0000308 4 minus001 00003

(3) 120590(119905) = 1198881+ 1198882sin (003120587119905) 00241

04 13 2

06 13 2

08 13 2

(4) 120590(119905) = 1198881+ 1198882119905 + 1198883sin (003120587119905) 00154

04 13 002 206 13 002 208 13 002 2

Note119898 and Δ are point number of each averaging and moving time interval 120593 is the autocorrelation coefficient of simulation model 1198881 1198882 and 119888

3are model

parameters in standard deviation function [Δ120590(119905)120590(119905)]max is the maximum of Δ120590(119905)120590(119905) Symbol ldquordquo indicates that the parameter does not exist

Table 2 Results of parameter and the average percent relative error for each experiment

120593 04 06 08 04 06 08

Method Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Equation(3)

Equation(4)

Experiment 1 Experiment 21198881

47635 43543 44568 49790 41984 66076 47635 43543 44568 49790 41984 660761198882

00190 00174 00178 00200 00168 00268 00190 00174 00178 00200 00168 00268Per err 771 1348 1157 885 1615 3371 771 1348 1157 885 1615 3371

Experiment 3 Experiment 41198881

123941 113212 116007 129740 109307 172514 124157 113302 115889 129288 109200 1716421198882

19023 17321 17246 19175 16054 25234 00188 00173 00178 00203 00169 002761198883

18880 17320 17573 19839 16753 26349Per err 1022 1449 1121 832 1594 3330 862 1382 1189 1029 1617 3458

Note Experiments 1 to 4 are defined in Table 1 Symbol ldquordquo indicates that the parameter does not exist

Table 3 Determination results of mean and variance functions

Trend function 120583(119905)Daily CIR 0Monthly MPL 105762 + 20498119905 minus 10297 sin(120587119905

6) + 15926 cos (120587119905

6)

Monthly M2 11986292 + 22406119905 + 652801199052minus 07726119905

3+ 4077 times 10

minus31199054

Daily FX rate 12489 + 51943 times 10minus3119905 minus 11693 times 10

minus41199052+ 94416 times 10

minus71199053minus 33508 times 10

minus91199054+ 43846 times 10

minus121199055

+001006 cos(312058711990565) + 001587 cos(3120587119905

130)

Standard deviation function 120590(119905)Daily CIR 05456 + 31940 times 10

minus2119905 minus 19532 times 10

minus41199052+ 43798 times 10

minus71199053minus 41174 times 10

minus101199054+ 13751 times 10

minus131199055

Monthly MPL 11607 minus 79024 sin(1205871199056) minus 40838 cos (120587119905

6)

Monthly M2 257950 minus 35024119905 + 176221199052minus 02823119905

3+ 14410 times 10

minus31199054

Daily FX rate 54449 times 10minus3

Mathematical Problems in Engineering 9

Table 4 Model parameter results

Data set Model 119910119905= sum119901

119894=1120593119894119910119905minus119894+ 120576119905

119901 1205931

1205932

Daily CIR 1 002861

Monthly MPL 1 06353

Monthly M2 2 11572 minus01401Daily FX rate 1 08616

accuracy especially in situations of small sample size Forstandard deviation function fixing the presented theoremis only suitable for the correlation coefficient stationarysequence

In addition the mean and standard deviation functionsdepicted by (14) are general expressions Take the MonthlyMPL data set in Section 6 for example we can assume that1206010(119905) = 120595

0(119905) = 1 120601

1(119905) = 119905 120601

2(119905) = 120595

1(119905) = sin(1205871199056)

1206013(119905) = 120595

2(119905) = cos(1205871199056) and a = (119886

0 1198861 1198862 1198863)1015840 and b =

(1198870 1198871 1198872)1015840

Appendices

A Proof of Theorem 1

Suppose that 119909119905 119905 = 1 2 119899 is a correlation coefficient

stationary time series119864(119909119905) = 120583(119905) is a deterministic function

representing time-varying mean and radicVar(119909119905) = 120590(119905)

is a positive deterministic function denoting time-varyingstandard deviation function According to the concept ofcorrelation coefficient stationary time series we can express119909119905as

119909119905= 120583 (119905) + 120590 (119905) 120576

119905 120576119905sim NID [0 1] (A1)

We can derive the mean of sequence |119909119905+1minus120583(119905+1)minus119909

119905+

120583(119905)| 119905 = 2 3 119899 by

1198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816

= 1198641003816100381610038161003816120590 (119905 + 1) 120576119905+1 minus 120590 (119905) 120576119905

1003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816

120590 (119905) + Δ120590 (119905)

120590 (119905)120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

(A2)

Define a new randomvariable 120585(119905) = [1+Δ120590(119905)120590(119905)]120576119905+1minus

120576119905 We can conclude that 120585

119905is still a Gaussian random variable

because it is a linear combination of two Gaussian random

variables 120576119905+1

and 120576119905 And its mean and variance can be

derived by

119864 (120585119905) = 119864[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905 = 0

Var (120585119905) = 119864[1 +

Δ120590(119905)

120590(119905)] 120576119905+1minus 120576119905

2

= (1 +Δ120590 (119905)

120590 (119905))

2

+ 1 minus 21205881(1 +

Δ120590 (119905)

120590 (119905))

(A3)

Therefore Var(120585119905) is a constant when Δ120590(119905)120590(119905) is a

constant or [Δ120590(119905)120590(119905)]max is a negligible small amountcompared with one And

11986410038161003816100381610038161205851199051003816100381610038161003816 = 119864

1003816100381610038161003816[1 + Δ120590 (119905) 120590 (119905)] 120576119905+1 minus 1205761199051003816100381610038161003816 =radic2

120587Var (120585

119905)

(A4)

Let constant 119888 = 1radic(2120587)Var(120585119905) then the standard

deviation of original series 119909119905can be expressed as 120590(119905) =

119888119864|119909119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| Theorem 1 in Section 3 has

been proved

B Proof of PDF Relationship

Let us derive the relationship between joint PDF119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) and 119891

119884119899119884119899minus11198841

(119910119899 119910119899minus1

1199101) Cumulative distribution function (CDF) can be

expressed by

119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

= 119875119883119899minus 120583 (119905

119899)

120590 (119905119899)le 119910119899119883119899minus1minus 120583 (119905

119899minus1)

120590 (119905119899minus1)

le 119910119899minus1

1198831minus 120583 (119905

1)

120590 (1199051)le 1199101

= 119875 119883119899le 120583 (119905

119899) + 120590 (119905

119899) 119910119899 119883119899minus1le 120583 (119905

119899minus1)

+ 120590 (119905119899minus1) 119910119899minus1 119883

1le 120583 (119905

1) + 120590 (119905

1) 1199101

= 119875 119883119899le 119909119899 119883119899minus1le 119909119899minus1 119883

1le 1199091

= 119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

(B1)

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 4 Model parameter results

Data set Model 119910119905= sum119901

119894=1120593119894119910119905minus119894+ 120576119905

119901 1205931

1205932

Daily CIR 1 002861

Monthly MPL 1 06353

Monthly M2 2 11572 minus01401Daily FX rate 1 08616

accuracy especially in situations of small sample size Forstandard deviation function fixing the presented theoremis only suitable for the correlation coefficient stationarysequence

In addition the mean and standard deviation functionsdepicted by (14) are general expressions Take the MonthlyMPL data set in Section 6 for example we can assume that1206010(119905) = 120595

0(119905) = 1 120601

1(119905) = 119905 120601

2(119905) = 120595

1(119905) = sin(1205871199056)

1206013(119905) = 120595

2(119905) = cos(1205871199056) and a = (119886

0 1198861 1198862 1198863)1015840 and b =

(1198870 1198871 1198872)1015840

Appendices

A Proof of Theorem 1

Suppose that 119909119905 119905 = 1 2 119899 is a correlation coefficient

stationary time series119864(119909119905) = 120583(119905) is a deterministic function

representing time-varying mean and radicVar(119909119905) = 120590(119905)

is a positive deterministic function denoting time-varyingstandard deviation function According to the concept ofcorrelation coefficient stationary time series we can express119909119905as

119909119905= 120583 (119905) + 120590 (119905) 120576

119905 120576119905sim NID [0 1] (A1)

We can derive the mean of sequence |119909119905+1minus120583(119905+1)minus119909

119905+

120583(119905)| 119905 = 2 3 119899 by

1198641003816100381610038161003816119909119905+1 minus 120583 (119905 + 1) minus 119909119905 + 120583 (119905)

1003816100381610038161003816

= 1198641003816100381610038161003816120590 (119905 + 1) 120576119905+1 minus 120590 (119905) 120576119905

1003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816

120590 (119905) + Δ120590 (119905)

120590 (119905)120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

= 120590 (119905) 119864

10038161003816100381610038161003816100381610038161003816[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905

10038161003816100381610038161003816100381610038161003816

(A2)

Define a new randomvariable 120585(119905) = [1+Δ120590(119905)120590(119905)]120576119905+1minus

120576119905 We can conclude that 120585

119905is still a Gaussian random variable

because it is a linear combination of two Gaussian random

variables 120576119905+1

and 120576119905 And its mean and variance can be

derived by

119864 (120585119905) = 119864[1 +

Δ120590 (119905)

120590 (119905)] 120576119905+1minus 120576119905 = 0

Var (120585119905) = 119864[1 +

Δ120590(119905)

120590(119905)] 120576119905+1minus 120576119905

2

= (1 +Δ120590 (119905)

120590 (119905))

2

+ 1 minus 21205881(1 +

Δ120590 (119905)

120590 (119905))

(A3)

Therefore Var(120585119905) is a constant when Δ120590(119905)120590(119905) is a

constant or [Δ120590(119905)120590(119905)]max is a negligible small amountcompared with one And

11986410038161003816100381610038161205851199051003816100381610038161003816 = 119864

1003816100381610038161003816[1 + Δ120590 (119905) 120590 (119905)] 120576119905+1 minus 1205761199051003816100381610038161003816 =radic2

120587Var (120585

119905)

(A4)

Let constant 119888 = 1radic(2120587)Var(120585119905) then the standard

deviation of original series 119909119905can be expressed as 120590(119905) =

119888119864|119909119905+1minus 120583(119905 + 1) minus 119909

119905+ 120583(119905)| Theorem 1 in Section 3 has

been proved

B Proof of PDF Relationship

Let us derive the relationship between joint PDF119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) and 119891

119884119899119884119899minus11198841

(119910119899 119910119899minus1

1199101) Cumulative distribution function (CDF) can be

expressed by

119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

= 119875119883119899minus 120583 (119905

119899)

120590 (119905119899)le 119910119899119883119899minus1minus 120583 (119905

119899minus1)

120590 (119905119899minus1)

le 119910119899minus1

1198831minus 120583 (119905

1)

120590 (1199051)le 1199101

= 119875 119883119899le 120583 (119905

119899) + 120590 (119905

119899) 119910119899 119883119899minus1le 120583 (119905

119899minus1)

+ 120590 (119905119899minus1) 119910119899minus1 119883

1le 120583 (119905

1) + 120590 (119905

1) 1199101

= 119875 119883119899le 119909119899 119883119899minus1le 119909119899minus1 119883

1le 1199091

= 119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

(B1)

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 5 Testing results of correlation coefficient stationarity

Data set Daily CIR Monthly MPL Monthly M2 Daily FX rateTesting statistic 119861 06420 03340 09236 09741

Table 6 Prediction results for future five daily Shanghai composite indexes

Time Real valueCCAR(1) model AR model GARCHmodel

Predictionvalue Per err Prediction

value Per err Predictionvalue Per err

Oct-8th-03 137169 136750 030 136617 040 136627 040Oct-9th-03 136917 137169 018 136583 024 136693 016Oct-10th-03 140401 136917 248 136484 278 136568 273Oct-13th-03 139966 140401 031 136486 248 136573 242Oct-14th-03 138817 139966 083 136497 167 136580 161

Mper Err 082 151 146

Table 7 Prediction results of the Guangxi monthly maximum power load ()

Time Per err Time Per err Time Per errCCAR(1) model VD model CCAR(1) model VD model CCAR(1) model VD model

Jan-00 334 148 Sep-00 943 510 May-00 134 029Feb-00 113 125 Oct-00 632 235 Jun-00 285 378Mar-00 211 550 Nov-00 320 114 Jul-00 318 318Apr-00 188 578 Dec-00 440 112 Aug-01 147 201May-00 164 366 Jan-01 173 151 Sep-01 268 202Jun-00 611 126 Feb-01 386 577 Oct-01 116 096Jul-00 760 465 Mar-01 025 076 Nov-01 502 551Aug-00 778 453 Apr-01 141 095 Dec-01 059 109

Mper Err of CCAR(1) Model Mper Err of VDMode335 273

Table 8 Prediction results of China money supply

Time Real value10minus1Billion RMB

CCAR(2) model ARIMA(620) modelPrediction value10minus1

Billion RMB Per err Prediction value10minus1Billion RMB Per err

May-09 54826351 55063876 043 54803339 004Jun-09 56891620 56110615 137 56510199 067Jul-09 57310285 57189307 021 57977354 116Aug-09 57669895 58300986 109 59550825 326Sep-09 58540534 59446716 155 61140940 444Oct-09 58664329 60627588 335 62591163 669Nov-09 59460472 61844718 401 63698009 713Dec-09 61022452 63099250 340 65130143 673

Mper Err 19274 37665

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 9 Prediction results of Daily FX rate

Time Real value CCAR(1) model GM(11) MGM(11)Prediction value per err Prediction value per err Prediction value per err

Dec-19th-05 12007 11986 01751 11701 25474 11517 40793Dec-20th-05 11864 12006 11993 11695 14220 11507 30054Dec-21th-05 11830 12026 16565 11689 11881 11498 28053Dec-22th-05 11879 12045 13948 11684 16448 11489 32809Dec-23th-05 11865 12062 16615 11678 15779 11481 32377Dec-26th-05 11856 12078 18730 11672 15524 11473 32312Dec-27th-05 11832 12092 21993 11666 14020 11465 30982Dec-28th-05 11830 12104 23197 11660 14346 11458 31410Dec-29th-05 11843 12115 22925 11654 15919 11452 33025Dec-30th-05 11832 12122 24541 11649 15496 11446 32638

Mper Err 17226 15911 32245

Then PDF 119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1) can be derived by

119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119910119899120597119910119899minus1sdot sdot sdot 1205971199101

=120597119899119865119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot120597119909119899

120597119910119899

sdot120597119909119899minus1

120597119910119899minus1

sdot sdot sdot1205971199091

1205971199101

=120597119899119865119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

120597119909119899120597119909119899minus1sdot sdot sdot 1205971199091

sdot

119899

prod

119896=1

120590 (119905119896)

= 119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1) sdot

119899

prod

119896=1

120590 (119905119896)

(B2)

That is

119891119883119899119883119899minus11198831

(119909119899 119909119899minus1 119909

1)

=119891119884119899119884119899minus11198841

(119910119899 119910119899minus1 119910

1)

120590 (119905119899) 120590 (119905119899minus1) sdot sdot sdot 120590 (119905

1)

(B3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper

Acknowledgments

The authors are grateful to the anonymous reviewers fortheir critical and constructive review of the paper This studywas cosupported by the National Natural Science Foundationof China (Grant nos 11202011 and 61203093) the NationalBasic Research Program of China (973 Program) (Grantno 2012CB720000) and the ldquoWeishirdquo Foundation for YoungScholars (Grant no YWF13HK11)

References

[1] Z Li S Zhou C Sievenpiper and S Choubey ldquoChange detec-tion in the Cox proportional hazardsmodels fromdifferent reli-ability datardquo Quality and Reliability Engineering Internationalvol 26 no 7 pp 677ndash689 2010

[2] G Zhang B E Patuwo and M Y Hu ldquoForecasting with arti-ficial neural networks the state of the artrdquo International Journalof Forecasting vol 14 no 1 pp 35ndash62 1998

[3] E Vanhatalo B Bergquist and K Vannman ldquoTowards im-proved analysis method for two-level factorial experimentswith time series responsesrdquo Quality and Reliability EngineeringInternational vol 29 no 5 pp 725ndash741 2012

[4] A BenvenisteMMetivier and P PriouretAdaptive Algorithmsand Stochastic Approximations vol 22 of Applications of Math-ematics Springer New York NY USA A V Balakrishnan IKaratzas M Yor Eds 1990

[5] AGelman and JHillDataAnalysis Using Regression andMulti-levelHierarchical Models Cambridge University Press NewYork NY USA 2007

[6] R Ghazali A J Hussain N M Nawi and B Mohamad ldquoNon-stationary and stationary prediction of financial time seriesusing dynamic ridge polynomial neural networkrdquo Neurocom-puting vol 72 no 10ndash12 pp 2359ndash2367 2009

[7] E Kayacan B Ulutas and O Kaynak ldquoGrey system theory-based models in time series predictionrdquo Expert Systems withApplications vol 37 no 2 pp 1784ndash1789 2010

[8] A Monin and G Salut ldquoARMA lattice identification a newhereditary algorithmrdquo IEEE Transactions on Signal Processingvol 44 no 2 pp 360ndash370 1996

[9] J A R Blais ldquoOptimal modeling and filtering of stochastic timeseries for geoscience applicationsrdquo Mathematical Problems inEngineering vol 2013 Article ID 895061 8 pages 2013

[10] R S Tsay Analysis of Financial Time Series Wiley Series inProbability and Statistics John Wiley amp Sons New York NYUSA 2nd edition 2005

[11] V Venkatasubramanian R Rengaswamy K Yin and S NKavuri ldquoA review of process fault detection and diagnosis partI quantitativemodel-basedmethodsrdquoComputers and ChemicalEngineering vol 27 no 3 pp 293ndash311 2003

[12] S A Yourstone and D C Montgomery ldquoTime-series approachto discrete real-time process quality controlrdquo Quality and Reli-ability Engineering International vol 5 no 4 pp 309ndash317 1989

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[13] D C Baillie and J Mathew ldquoA comparison of autoregressivemodeling techniques for fault diagnosis of rolling elementbearingsrdquoMechanical Systems and Signal Processing vol 10 no1 pp 1ndash17 1996

[14] R J Triolo and G D Moskowitz ldquoThe experimental demon-stration of a multichannel time-series myoprocessor systemtesting and evaluationrdquo IEEE Transactions on Biomedical Engi-neering vol 36 no 10 pp 1018ndash1027 1989

[15] C T Seppala T J Harris and D W Bacon ldquoTime series meth-ods for dynamic analysis of multiple controlled variablesrdquo Jour-nal of Process Control vol 12 no 2 pp 257ndash276 2002

[16] WWMelek Z Lu A Kapps andWD Fraser ldquoComparison oftrend detection algorithms in the analysis of physiological time-series datardquo IEEE Transactions on Biomedical Engineering vol52 no 4 pp 639ndash651 2005

[17] D Gujarati Essentials of Econometrics McGrawHill New YorkNY USA 1998

[18] Y Zhang H Zhou S J Qin and T Chai ldquoDecentralized faultdiagnosis of large-scale processes using multiblock kernel par-tial least squaresrdquo IEEE Transactions on Industrial Informaticsvol 6 no 1 pp 3ndash10 2010

[19] Y Zhang ldquoActuator fault-tolerant control for discrete systemswith strong uncertaintiesrdquo Computers and Chemical Engineer-ing vol 33 no 11 pp 1870ndash1878 2009

[20] J Yu ldquoA nonlinear kernel Gaussian mixture model based infer-ential monitoring approach for fault detection and diagnosis ofchemical processesrdquo Chemical Engineering Science vol 68 no1 pp 506ndash519 2012

[21] J Yu ldquoNonlinear bioprocess monitoring using multiway kernellocalized fisher discriminant analysisrdquo Industrial and Engineer-ing Chemistry Research vol 50 no 6 pp 3390ndash3402 2011

[22] H Albazzaz and X Z Wang ldquoStatistical process control chartsfor batch operations based on independent component ana-lysisrdquo Industrial and EngineeringChemistry Research vol 43 no21 pp 6731ndash6741 2004

[23] Y W Zhang J Y An and C Ma ldquoFault detection of non-Gaussian processes based on model migrationrdquo IEEE Transac-tions on Control Systems Technology vol 21 no 5 pp 1517ndash15252013

[24] R F Engle ldquoAutoregressive conditional heteroscedasticity withestimates of the variance of United Kingdom inflationrdquo Econo-metrica vol 50 no 4 pp 987ndash1007 1982

[25] A Ibrahim ldquoA complementary test for the KPSS test with anapplication to the US DollarEuro exchange raterdquo EconomicsBulletin vol 3 no 4 pp 1ndash5 2004

[26] M Loretan and P C B Phillips ldquoTesting the covariancestationarity of heavy-tailed time series an overview of thetheory with applications to several financial datasetsrdquo Journalof Empirical Finance vol 1 no 2 pp 211ndash248 1994

[27] M Versace R Bhatt O Hinds and M Shiffer ldquoPredicting theexchange traded fund DIA with a combination of genetic algo-rithms and neural networksrdquo Expert Systems with Applicationsvol 27 no 3 pp 417ndash425 2004

[28] MKhashei andMBijari ldquoAnew class of hybridmodels for timeseries forecastingrdquo Expert Systems with Applications vol 39 no4 pp 4344ndash4357 2012

[29] S Van Bellegem and R Von Sachs ldquoForecasting economic timeseries with unconditional time-varying variancerdquo InternationalJournal of Forecasting vol 20 no 4 pp 611ndash627 2004

[30] H-M Fu and C-R Liu ldquoAnalysis method of correlationcoefficient ARMA(pq) seriesrdquo Journal of Aerospace Power vol18 no 2 pp 161ndash166 2003

[31] P H Franses Time Series Models For Business and EconomicForecasting Cambridge University Press New York NY USA1998

[32] H YWangNonstationary Random Signal Analysis and Process-ing National Defence Industry Press Beijing China 1998

[33] J D HamiltonTime Series Analysis PrincetonUniversity PressPrinceton NJ USA 1994

[34] G E P Box and GM Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[35] A R Pagan and GW Schwert ldquoTesting for covariance station-arity in stock market datardquo Economics Letters vol 33 no 2 pp165ndash170 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of