CHAPTER 2

REVIEW OF RELATED WORKS 2.1 INTRODUCTION

Time series plays a vital role in the branches of Stock prices,

Global surface temperature, Economy Population growth and

agricultural production. Time series is the sequence of observations

ordered in time. Mostly these observations are collected at equally

spaced and discrete time interval. The basic assumption of time series

analysis is some aspects of the past pattern which will continue to

remain in the future. In recent years, many researchers have used

fuzzy time series to handle forecasting problems.

In this thesis, Markov modeling is a major statistical tool used

to predict the fuzzy time series data. The advantage of using Markov

modeling gives better forecasting accuracy for predicting time series

data. Many works are related to fuzzy time series in the applications of

stock price prediction, agriculture commodities prediction, and

Economy and university enrollment data. The works related to Markov

modeling fuzzy time series can be discussed.

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2.2 MARKOV MODEL FUZZY TIME SERIES (MM-FTS)

Fuzzy Time Series (FTS) is introduced by Song and

Chissom [1993a]. Several fuzzy time series model is developed in the

literature during the last two decades. Zadeh [1965] discussed fuzzy

sets and it is characterized by a membership function which assigns

the value to each object grade membership ranging from zero to one.

Song and Chissom [1993b] have initiated the study of time

invariant and time variant forecasting models using fuzzy time series

for enrollment data of University of Alabama. Also compare the

predicted values of fuzzy time series with those of non linear

regression models.

Song et al., [1993a] have used time-invariant fuzzy time series

model, while Song and Chissom [1994] used time-variant model for

the same problem. In their example, the crisp data is fuzzified into

linguistic values to illustrate the fuzzy time series method using fuzzy

set theory and its modeling by applying fuzzy relations equations and

approximate reasoning.

Sullivan et al., [1994] reviewed the first-order time-variant fuzzy

time series model and the first-order time-invariant fuzzy time series

model presented by Song and Chissom [1993b], where their models

are compared with each other and with a time-variant Markov model

using linguistic labels with probability distributions. The results of

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this model enrollment data are compared with three traditional time

series models such as a first order auto regressive model and second

order auto regressive models. However, the Markov model has an

advantage that any repeated transitions in the data are taken into the

account of estimating the model parameters, whereas the fuzzy time

series model based on the usual max-min operations cannot perform

the same.

Song et al., [1994] proposed an approach for developing the

time-variant fuzzy time series models with an example which

presented the process of forecasting the enrollment for the University

of Alabama. To defuzzify the output of the model, a 3-layer back

propagation neural network was trained and used as the defuzzifier.

Among the three different defuzzification methods, neural network

method yielded the best result.

Song et al., [1995] have constructed fuzzy time-series model by

means of defining some new operations on fuzzy numbers.

Chen [1996] developed a simplified method for time series forecasting

using the arithmetic operations rather than using complicated

maximum – minimum composition operations. The proposed method

not only gives good forecast for the University enrollments, but also

gives robust forecast even when the historical data are not accurate.

26

Ranaweera et al., [1996] proposed a fuzzy logic model for short

term load forecasting which was applied to historical weather and load

data. Since Conventional fuzzy systems cannot operate with random

phenomena, the Control processes in real-life plants were used.

They have deal the stochastic processes as Markov modeling

approach.

Song et al., [1997] have discussed fuzzy stochastic fuzzy time

series and three different models. Ping-Teng Chang [1997] has

developed a fuzzy technique for trends and seasonality forecasting

through fuzzy regression and fuzzy trend of time-series model. Fuzzy

forecasting and the analysis of the fuzziness of the forecasts are the

features of this method which differs from the traditional forecasting

techniques for seasonality.

Jeng- Ren Hwang et al., [1998] have proposed new method for

handling forecasting problems based on time variant fuzzy time series.

In the proposed method where the variation of enrollment of the

current year is related to the trend of the enrollments of past years.

Song [1999] has developed a seasonal forecasting model for fuzzy time

series with minor modifications of fuzzy logical relationship and the

membership functions. The model can be used directly for seasonality

forecast.

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Sullivan and Woodall [1999] have discussed three methods for

estimating Markov transition matrices when observed state

probabilities are not all either zeros or ones and a simulation-based

comparison of the performance of the estimators.

Fang-Mei Tseng et al., [1999] have proposed the method of fuzzy

seasonal time series for forecasting the production value of the

mechanical industry in Taiwan. This method provided the decision

makers with insight regarding the possible future situation. Chen et

al., [2000] have discussed fuzzy time series method and adapted to

temperature data in Taiwan.

Nguyen et al.,[2002] has developed the mathematical modeling

of domains of linguistic variables, which gives a unified algebraic

approach to the natural structure of domains of linguistic variables.

Tapio Frantti et al., [2001] have obtained an anticipatory Fuzzy Logic

Advisory Tool (FLAT) for predicting the demand for signal transmission

device. The design principle for a general predictor is also explained.

The algorithm is implemented as a practical computer program and it

is applied to real manufacturing data.

Fang-Mei Tsenga et al., [2001] have developed Fuzzy ARIMA

(FARIMA) model and applied it for forecasting the exchange rate of NT

dollars to US dollars. With the help of this model it is possible for

decision makers to forecast the best and worst possible situations

based on fewer observations than the ARIMA model.

28

Fang-Mei Tsenga et al., [2002] have proposed fuzzy seasonal ARIMA

(FSARIMA) forecasting model. It is used to forecast two seasonal time

series data of the total production value of the Taiwan machinery

industry and the soft drink time series data.

Song [2003] has proposed the sample autocorrelation functions

of fuzzy time series and used in model selection. The main idea is to

select a number of different data sets from each fuzzy set and

calculate the sample auto correlation function for each data set. Three

different auto correlation functions are proposed and examples are

also presented. Javier Contreras et al., [2003] have proposed a method

to predict next-day electricity prices based on ARIMA methodology.

Yimin Xiong et al., [2004] have proposed a model-based method for

clustering univariate and simple multivariate ARMA time series.

Hsuan-Shih Lee et al., [2004] have presented an improved

method to forecast university enrollments based on the fuzzy time

series. The method proposed not only defines the supports of the fuzzy

numbers that represent the linguistic values of the linguistic variable

more appropriately, but also makes the RMSE smaller.

Melike Sah et al., [2005] have proposed a method to attain better

forecasting accuracy using time-invariant fuzzy time series. It should

be emphasized that it uses only historical data in the numerical form

29

without any addition pieces of knowledge for forecasting. In addition

the method is tested on different number of fuzzy sets for the purpose

of examining the forecasting accuracy.

Ruey-Chyn Tsaur et al., [2005] have proposed fuzzy relation

matrix affecting the forecasting performance and proposed an

arithmetic procedure for deriving fuzzy relation matrix method using

Fuzzy relation analysis in fuzzy time series. Fuzzy relation is a crucial

connector in presenting fuzzy time series model. Also the concept of

entropy is applied to measure the degrees of fuzziness when a time

invariant matrix is derived.

Hui-Kuang Yu [2005] has proposed weighted models to tackle

two issues in fuzzy time series forecasting, viz., recurrence and

weighting. It is argued that recurrent fuzzy relationships, which are

simply ignored in previous studies, should be considered in

forecasting. It also recommended that different weights shall be

assigned to various fuzzy relationships. The weighted model is

compared with local regression models in which weight function plays

an important role in forecasting.

Ping-Feng Pai et al., [2005] have proposed autoregressive

integrated moving average (ARIMA) model which has become one of

the most widely used linear models in time series forecasting.

Nai-Yi-Wang et al., [2009] have presented a new method to predict the

30

temperature and the Taiwan Futures Exchange (TAIFEX), based on

automatic clustering techniques and two-factor high-order fuzzy time

series. Tasha Inniss [2006] has applied this technique to weather and

aviation data to determine probabilistic distributions of arrival

capacity scenarios, which can be used for seasonal forecasting and

planning.

Alonsoa et al., [2006] have proposed time series method based

on the probability density of the forecasts using nonparametric kernel

estimator. Hao-Tien Liu [2007] has proposed improved time-variant

fuzzy time series method. The proposed method takes into

consideration of Window base, length of interval, degrees of

membership values, and existence of outliers. The improved method

provides decision makers with more precise forecasted values.

Sheng-Tun Li [2007] has proposed a deterministic forecasting

model to manage the crucial issues. In addition, an important

parameter, the maximum length of subsequence in a fuzzy time series

resulting in a certain state, is deterministically quantified.

Hao-Tien Liu [2009] has proposed a method to design an improved

fuzzy time series forecasting method in which the forecasted value

would be a trapezoidal fuzzy number instead of a single point value.

Also, the decision analyst can gather information about the possible

forecasted ranges under different degree of confidence.

31

Guoqi Qian et al., [2007] have proposed a feasible computing

method based on the Gibbs sampler. By this method, model selection

is performed through a random sample generation algorithm and also

a model of fixed dimension in which the parameter estimation is done

through the maximum likelihood method.

Hye-young Jung et al., [2008] have proposed rearranged interval

method to reflect the fluctuation of historical data and to improve the

forecasting accuracy of fuzzy time series. This is discussed and the

forecasting accuracy is evaluated. Empirical analysis shows that this

method in forecasting accuracy is superior to existing methods and it

fully reflects the fluctuation of historical data.

Ashraf Abd-Elaal et al.,[2010] have proposed a fuzzy time series

technique based on fuzzy c- mean clustering. This method is adapted

to university enrollment data and it is improved the forecasting

accuracy compared with existing models.

Srivastava [2011] has presented a new method which gives

short term forecast of agricultural production based on fuzzy time

series. The study used fuzzy set theory and applied in fuzzy time

series model and artificial neural network to forecast the production of

food grains. Pushparani Suri et al., [2011] have presented a simple,

systematic and iterative methodology for forecasting gold price. This

method is realized on fuzzy clustering and weighted least square.

32

Hemant Kumar [2011] has proposed relation matrix method and

it is used to reduce its calculation. This method is most

remarkable one which gives the most plausible range of forecasting

and it is in the form of interval rather than a single value.

Seyed Morteza Hosseini et al., [2011] have demonstrated the results of

test data from the currency market shows that this combined

approach have to be more careful from existing fuzzy time series

model. The practical results obtained show that this model have a

good application.

2.3 HIDDEN MARKOV MODEL FUZZY TIME SERIES (HMM – FTS)

Rabiner et al., [1986] have proposed the theory and

implementation of Markov modeling technique and applied it to

speech recognition problems. Rabiner [1989] has developed theory of

hidden Markov models from the simplest concepts to the most

sophisticated models and focused on physical explanations of the

basic mathematics. Rabiner also illustrated the applications of the

theory of HMM’s to simple problems in speech recognition and pointed

out how the techniques could be applied to more advanced speech

recognition problems.

Juang et al., [1991] have reviewed the statistical method of

HMM's. It reveals that the strengths of the method lie in the consistent

statistical framework that is flexible and versatile, particularly for

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speech applications, and the ease of implementation that makes the

method practically attractive. It is pointed out some areas of the

general HMM method that deserves more attention with the hope that

increased understanding will lead to performance improvements for

many applications.

Chen et al., [2000] have proposed two factor time variant

algorithms based on fuzzy time series model and applied it to forecast

the daily temperature. Both the algorithms have the advantage of

obtaining good forecasting results.

Rafiul Hassan et al., [2006] have proposed a single Hidden

Markov Model based on clustering method, which identifies a suitable

number of clusters in a given dataset without using prior knowledge

about the number of clusters. Initially, the dataset is partitioned into

windows of fixed size based on the HMM log likelihood values. This

model provides a framework for identifying the most appropriate

number of clusters (windows of varying sizes). After determining the

number of clusters, the data values are then labeled and allocated to

clusters.

Ching-Hsue Cheng et al., [2007] have discussed the fuzzy time

series method based on rough set theory and this method is applied to

stock price index forecasting problem. Sheng - Tun Li [2008] has

studied fuzzy time series which has increasingly

34

attracted much attention due to its salient capabilities of tackling

uncertainty and vagueness inherent in the data collected.

Ching-Hsue Cheng et al., [2008] have introduced a new multiple-

attribute fuzzy time series method based on fuzzy clustering

technique. The methods of fuzzy clustering are integrated in the

processes of fuzzy time series to partition datasets objectively and

enable processing of multiple attributes.

Sheng-Tun Li et al., [2009] have proposed a new forecasting

model based on Hidden Markov Model for fuzzy time series to realize

the probabilistic state transition and also conducted experiments of

forecasting a real-world temperature application to validate the better

accuracy of the proposed model achieved over traditional fuzzy time

series models.

Rafiul Hassan [2009] has presented a novel combination of

hidden Markov model and fuzzy models for forecasting stock market

data. The model is tested by preparing forecasts for the financial time

series data of six stock prices.

Ming Dong et al., [2009] have proposed the states of hidden

semi Markov models are used to represent the PM2.5 concentration

levels. The model parameters are estimated through modified forward

backward training algorithm. It can be used to predict PM2.5

concentration levels. Jens Runi Poulson [2009] has been developed

35

new fuzzy time series method which combines aggregation and

particle swarm optimization techniques. By combining these

techniques, forecast rule can be individually tuned to match the data.

Sheng-Tun Li et al., [2010] have proposed a novel forecasting

model based on the Hidden Markov model by enhancing Sullivan and

Woodall’s work to allow handling of two-factor forecasting problems.

It is built upon an HMM in which the fuzzy relationships are

formulated as state transitions. So that it can handle two factor

forecasting problems. Moreover, in order to make a nature of

conjecture and randomness of forecasting more realistic, the Monte

Carlo method is used to estimate the outcome.

Edmundo de Souza e Silva et al., [2010] have investigated the

performance of a hidden Markov model in predicting future crude oil

price movements. Additionally, they developed forecasting

methodologies consists of three steps. First, they employed wavelet

analysis to remove high frequency price movements, and then the

hidden Markov model is used to forecast the probability distribution of

the price return accumulated over the next F days.

Suresh et al., [2011] have analyzed Global surface temperature

with CO2 data using hidden Markov model in fuzzy time series. This

method is used to forecast successive year’s of global surface

temperature data.

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2.4 HIGHER ORDER MARKOV MODEL FUZZY TIME SERIES

(HOMM – FTS)

Adrian Raftery et al., [1994] have introduced the mixture

transition distribution model for higher order Markov chains. Also

proposed a computational algorithm for maximum likelihood

estimation which is based on a way of reducing the large number of

constrains.

Chen [2002] has presented present a new method for handling

forecasting problem using a high-order fuzzy time series model, where

the historical enrollments of the University of Alabama are used to

illustrate the forecasting process.

Vasek Chavatal [1983] has discussed linear programming

problem formulated from optimization problem and solved by simplex

method. Adrian Raftery [1985] has introduced higher order Markov

chain model which combines realism and parsimony.

Ching et al., [2002] have proposed a multivariate Markov chain

model for modeling multiple categorical data sequences and also

developed efficient estimation methods for the model parameters.

Ching et al., [2003] have proposed a new higher-order Markov model

whose number of states is linear and also developed a new parameter

estimation method based on linear programming.

37

Tie Liu [2010] has adopted Markov chains model to analyze and

predict the time series. Some series can be expressed by a first-order

discrete-time Markov chain and others must be expressed by a higher-

order Markov chain model. Ching et al.,[2008] has proposed an nth-

order multivariate Markov chain model for modeling multiple

categorical data sequences such that the total number of parameters

are of O(ns2m2).

Yutong Li et al., [2008] have presented a stochastic simulation

approach to synthetic series of weather data to evaluate the

performance of open cycle solar desiccant air condition system of

Hong Kong. The results reveal that the open cycle desiccant system

can meet most of the latent load through out the cooling season if the

components are proper sized and energy savings.

Zhilong Wang et al., [2009] have presented a higher-order

multivariate Markov chain model combined with particle swarm

optimization algorithm, capable of searching for the optimal parameter

values η for level characteristics value to obtain a high accuracy model

for forecasting of multidimensional time series. Particle swarm

optimization algorithm is employed to optimize the coefficient of level

characteristics value.

Ersoy Oz [2011] has discussed the application of monthly

changes of the US Dollar selling rates and the monthly changes of the

38

Euro selling rates. The two changes are taken into consideration as

two categorical data sequences and it is revealed with multivariate

Markov chain model to what degree these sequences affect each other

in Turkey.

Chi-Chen Wang et al., [2011a] have compared the applications

of the forecasting methods Auto Regressive Integrated Moving Average

(ARIMA) time series model and fuzzy time series model by heuristic

models on the amount of export values in Taiwan.

Chi-Chen Wang [2011b] has attempted to use information of

export values in Taiwan as an example to test whether the fuzzy time

series is indeed practical in its forecast of macro economic variables.

By comparing fuzzy time series with ARIMA time series method, better

understanding of the appropriateness of forecasting model could be

obtained.

Hongxing Yang et al., [2011] have introduced a new method to

generate the annual weather data by using the first order multivariate

Markov chain model. The weather variables are described in a

stochastic way and multiple categorical sequences are generated by

similar source.

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2.5 HIGHER ORDER COMPUTATIONAL FUZZY TIME SERIES

Huarng [2001a] has proposed an effective length of intervals to

improve the forecasting accuracy and introduced two methods such as

average based and distribution based methods. Distribution based

length is the largest length which is smaller than at least half of the

first differences of data. Average based length is set to one half of the

average of the first differences of data. Both can be used as effective

lengths to improve forecasting in fuzzy time series.

Huarng [2001b] has demonstrated how the heuristic model

outperforms of Chen’s and other models using the enrollment

forecasting at the University of Alabama. This study assumes that

there is heuristic knowledge showing the increase or decrease in

enrollment for the next year. With this heuristic, the heuristic model

forecasts the enrollment better than the other models. In the heuristic

models, the heuristic knowledge is used to guide the search for

suitable fuzzy sets for index forecasting.

Chen [2002] has presented a forecasting method based on high-

order fuzzy time series. From the proposed model he developed an

algorithm to forecast the enrollments of the University of Alabama,

where the historical enrollment data at the University of Alabama are

used to illustrate the forecasting process.

40

Chen et al., [2004] has proposed fuzzy time series belonging to

first order and time–variant methods. It is possible to get higher

forecasting accuracy rate for forecasting enrollments than the existing

methods.

Sheng –Tun -Li et al., [2004] have presented a new approach to

handling the issue by applying the natural partitioning technique,

which can recursively partitioned the universe of discourse level by

level in a natural way. Experimental results found that the enrollment

data of the University of Alabama results can forecast the data

effectively and efficiently and outperforms the existing models.

Chung-Ming Own et al., [2005] have proposed an enhanced

fuzzy time series model, called heuristic high-order fuzzy time series

model, to deal forecasting problems. The proposed method eliminates

ambiguities at forecasting and requires a vast memory for data

storage. The empirical analysis reveals that the proposed method yield

more accurate forecasts. Moreover, the forecasting model can be

restricted in the acceptable-order fuzzy time series to reduce the

memory needed for the data storage.

Huarng et al., [2006] have suggested that ratios, instead of

equal lengths of intervals, including distribution based length and

average based length can more properly represent the intervals among

41

observations. The method also suggests that ratio-based lengths of

interval can be applied to improve fuzzy time series forecasting.

Ching-Hsue Cheng et al., [2006] have proposed two approaches

to improve the persuasiveness in determining the universe of

discourse, length of intervals and membership functions of fuzzy time

series. The first approach is using Minimize entropy principle

approach (MEPA) to partition the universe of discourse and build

membership functions, and the second is using Trapezoid fuzzification

approach (TFA).

Li-Wei Lee et al., [2006] have proposed a new method for

forecasting temperature and TAIEX, based on two factor high order

fuzzy time series. The two factor high-order fuzzy logical relationship

is used to increase the forecasting accuracy rate of prediction.

Shiva Raj Singh [2007] proposed a simple time variant method

for fuzzy time series forecasting. This method overcomes the deficiency

of ambiguity in trends of data and also does not need the heuristic

function. It provides simple computational algorithms for complexity

in linear order. It minimizes the time of generating relation equations

by using min-max composition operations and the time consumed by

the various defuzzification processes to be applied for getting crisp

forecast. The proposed algorithm is implemented for forecasting

42

enrollments of university of Alabama and the results are compared

with the existing methods to show its superiority.

Tahseen Ahmed Jilani et al., [2008d] have presented new fuzzy

metrics for high order multivariate fuzzy time series forecasting for car

road accident casualties in Belgium. Sun Xihao et al. [2008] have

proposed Average – based fuzzy time series model, which can be used

to adjust the lengths of intervals determined during the early stages of

forecasting and that method is applied to daily stock index in

shanghai.

Tahseen Ahmed Jilani et al., [2008b] have proposed a new

method for time series forecasting, having simple computational

algorithm of complexity of linear order. The method first predicts the

trend of the future value and then use the quantile based fuzzy

forecasting approach.

Tahseen Ahmed Jilani et al., [2008a] have presented two new

multivariate fuzzy time series forecasting methods. These methods

assumed m-factors with one main factor of interest. Stochastic fuzzy

dependence of order k is assumed to define general methods of

multivariate fuzzy time series forecasting and control. This method

provided a general work for forecasting, that can be increased by

increasing the stochastic fuzzy dependence and the simplicity of

computation used triangular membership function.

43

Tahseen Ahmed Jilani et al., [2008c] have proposed other

method based on frequency density partitioning applied for historical

enrollments of Alabama. The proposed method uses heuristic

approach to define frequency-density –based partitions of the universe

of discourse and also uses a trend predictor to calculate the forecasted

value. The trend predictor is used to adjust the weights of the

proposed fuzzy metric for forecasting. The method is found to be

robust and can handle the problem of inaccuracy in the data set.

Shiva Raj Singh [2008] has proposed a method which minimizes

the complicated computations of fuzzy relational matrices and a

suitable defuzzification process and also provided the forecasted

values of better accuracy. The developed method is a generalized

method of forecast thereby proving the forecast of better accuracy

than the existing models.

Satyendra Nath-Mandal [2008] has proposed the method to

fuzzify the original data based on Gaussian function, triangular

function, s-function, Trapezoidal and L –function. After fuzzifying, all

fuzzified data were defuzzified to get normal form. The error analysis

indicates that the membership function was appropriate for

fuzzification of data and it was used to predict the short length at

maturity.

44

Hui -Li Hsu et al., [2008] have presented evaluation techniques

for interval forecasting which can provide a more objective decision

space in interval forecasting to policy makers.

Adesh Kumar Pandey et al., [2008] have proposed a comparative study

of neural network and fuzzy time series forecasting techniques. It is

successfully implemented for forecasting wheat production at pant

Nagar farm.

Chen et al., [2008] has proposed a comprehensive fuzzy time-

series, which factors linear relationships between recent periods of

stock prices and fuzzy logical relationships (nonlinear relationships)

mined from time-series into forecasting processes.

Dug Hun Hong [2005] has considered the expanded results to the non

homogeneous fuzzy time series and the general fuzzy time series by

using the weakest t-norm based algebraic fuzzy operations and solved

the open problem.

Taylor [2008] has used minute-by-minute British electricity

demand observations to evaluate methods for prediction between 10

and 30 minutes ahead.

Muhammad Hisyam Lee et al., [2009] have proposed the

adoption for the weighted and the difference between actual data

toward midpoint interval based on fuzzy time series. The weights are

determined according to chronological number of fuzzy logical

45

relations in fuzzy logic group and modification is also done in reversal

of weight elements of transpose matrix for forecasting rule.

Arkov et al., [2009] provided the method with high

computational speed because it utilizes only operations of move and

comparison. However, this approach allows the simulation of a limited

number of systems with states depending on state quantization.

Specific application is system state prediction.

Nai-Yi-Wang et al., [2009] have presented a new method to

predict the temperature and the Taiwan Futures Exchange (TAIFEX),

based on automatic clustering techniques and two-factor high-order

fuzzy time series. First, they had applied an automatic clustering

algorithm to cluster the historical data into intervals of different

lengths. Then, they applied the same based on two-factor high-order

fuzzy time series.

Kuo et al., [2009] have used the particle swarm optimization to

find the proper content of the main factors. A new hybrid forecasting

model which combined particle swarm optimization with fuzzy time

series is proposed to improve the forecasted accuracy. The

experimental results of forecasting enrollments of students of the

University of Alabama found that the new model is better than any

existing models, and it can get better quality solutions based on the

first-order and the high-order fuzzy time series, respectively.

46

Chen et al., [2009] have presented a new method to forecast

enrollments based on automatic clustering techniques and fuzzy

logical relationships. First, he presented an automatic clustering

algorithm for clustering the historical enrollments of the University of

Alabama into intervals of different lengths. Then, each obtained

interval is divided into p sub-intervals.

Shiva Raj Singh [2009] has proposed a computational method of

forecasting based on higher order fuzzy time series. The developed

methods avoid the computations of complicated fuzzy logical relational

matrices and search for suitable defuzzification method.

Suresh et al., [2009] have presented a modified method of

forecasting in fuzzy time series using transition probability

membership function. In addition, the modified method is identified

outlier in time series data using cook’s distance method.

Okan Duru et al., [2010] have investigated the predictive

performance of fuzzy time series analysis method for dry bulk freight

method. The advantages of this method indicated the lack of several

diagnostic tests like as normality and stationarity.

47

Narendra Kumar [2010] has developed time variant fuzzy time

series models and its implementation testing for forecasting of wheat

crop production. Also the results are compared with other known

existing methods.

Ismail et al., [2011] have discussed a modified weight for fuzzy

time series and it shows a significant reduction of mean square error

and average forecasting error when compared with known existing

method.

Lazim Abdullah et al., [2010] have presented a combination of

fourth-order fuzzy time series with the multi-period adaptation model.

It is adopted to forecast the KLCI stock index. The results found that

the forecasting model that combines with the multi-period adaptation

model produce lower RMSE compared existing approaches. It is

proves that the multi-period adaptation models can effectively improve

the forecasting performance.