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AbstractIn this work an algorithm to adjust parameters

using a Bayesian method for cumulative rainfall time series

forecasting implemented by an ANN-filter is presented. The

criterion of adjustment comprises to generate a posterior

probability distribution of time series values from forecasted time

series, where the structure is changed by considering a Bayesian

inference. These are approximated by the ANN based predictor in

which a new input is taken in order for changing the structure

and parameters of the filter. The proposed technique is based on

the prior distribution assumptions. Predictions are obtained by

weighting up all possible models and parameter values accordingto their posterior distribution. Furthermore, if the time series is

smooth or rough, the fitting algorithm can be changed to suit, in

function of the long or short term stochastic dependence of the

time series, an on-line heuristic law to set the training process,

modify the NN topology, change the number of patterns and

iterations in addition to the Bayesian inference in accordance with

Hurst parameter H taking into account that the series forecasted

has the same H as the real time series.

The performance of the approach is tested over a time series

obtained from samples of the Mackey-Glass delay differential

equations and cumulative rainfall time series from some

geographical points of Cordoba, Argentina.

I ndex Terms Bayesian approach, Neural networks, timeseries forecast, Hursts parameter.

I. INTRODUCTION

he ANNs are mostly used as predictor filter with an

unknown number of parameters performed by a lot of

author, recently, such as in [2][13][16]. In turn, in this

Manuscript received April, 18, 2012. This work was supported in part by

Universidad Nacional de Cordoba, FONCYT-PDFT PRH N3 (UNC Program

RRHH03), SECYT UNC, National University of Catamarca, National Agency

for Scientific and Technological Promotion (ANPCyT), Departments ofElectrotechnics FCEFyN of Universidad Nacional de Cordoba.

C. Rodriguez Rivero is with Universidad Nacional de Cordoba, Department

of electrical and electronic engineering, Faculty of Exacts, Physical and Natural

Sciences, Cordoba, Argentina (phone 0054-351-4334147; e-mail:

[email protected]).

J. A. Pucheta is with Universidad Nacional de Cordoba, Department of

electrical and electronic engineering, Faculty of Exacts, Physical and Natural

Sciences, Cordoba, Argentina (phone 0054-351-4334147; e-mail:

[email protected]).

M. R. Herrera is with National University of Catamarca, Department of

electronic engineering, Faculty of Technology and Applied Sciences, (e-mail:

[email protected]).

work the main purpose is to estimated water availability

useful for control problems in agricultural activities such

as seedling growth and decision-making. The number of

filter parameters is set in function of the roughness of the

time series. These are considered as random variables

whose distribution is inferred by posterior probability

from the data, in which is included as an additional

parameter, the number of hidden neurons and modeling

uncertainty[1]. The model selectionproblem is, therefore,unavoidable; researchers must decide which model is best

at summarizing the data for each task of interest.The Bayesian approach permits propagation of uncertainty

in quantities which are unknown to other assumptions in the

model, which may be more generally valid or easier to guess in

the problem. For neural networks, the Bayesian approach was

pioneered in [2]-[3], and reviewed [5], [6]and [7]. The main

difficulty in model building is controlling the complexity of the

model. It is well known that the optimal number of degrees of

freedom in the model depends on the number of training

samples, amount of noise in the samples and the complexity of

the underlying function being estimated.

The procedure of determining the prior density and

likelihood functions associated with rainfall time seriesuncertainty is very complicated and there is a requirement to

assume a linear and normal distribution within the framework

of the proposed parameters. The problem of model selection is

often divided into discover an organization of a models

parameters that is well-matched such as the network topology,

e.g. number of patterns, layers, hidden units per layer, that

results in the best generalization performance. A common

result is with too many free parameters tend to overfit the

training data and, thus, show poor generalization performance.

A model attempting to estimate the value of a random

variable may have potential access to a wide range of

measurements regarding the state of the environment. Some of

these quantities may provide the model with useful informationregarding the random variable, whereas others may not. In the

context of neural networks, only the useful quantities should be

used as inputs to a network. A network that receives both

useful inputs and nuisance inputs will contain too many free

parameters and, thus, be prone to overfitting the training data

leading to poor generalization.

Time series forecasting using Bayesian method:

application to cumulative rainfall

C. Rodriguez Rivero, J. Pucheta, M. Herrera, V. Sauchelli and S. Laboret,Member, IEEE

T

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II. BAYESIANAPPROACH

One of the most key principles of Bayesian technique is to

construct the posterior probability distributions for all the

unknown entities in a model, given the data sample. To use the

model, marginal distributions are constructed for all those

entities that we are interested in, i.e. the outputs of the ANNs.

These can be the parameters in parametric models, or the

predictions in (non-parametric) regression or classificationtasks. Use of the posterior probabilities requires explicit

definition of the prior probabilities for the parameters. The

posterior probability for the parameters u in a modelMgiven

the dataDis, according to the Bayes rule,

,)/(

)/(),/(),/(

MDP

MPMDPMDP

(1)

where P(/D,M)is the likelihood of the parameters , P(/M)

is the prior probability of , and P(D/M) is a normalizing

constant, called evidence of the model M. The termMdenotes

all the hypotheses and assumptions that are made in defining

the model. All the results are conditioned on these

assumptions, and to make this clear we prefer to have the term

Mexplicitly in the equations. In this notation the normalization

term P(D/M) is directly understandable as the marginal

probability of the data, conditional on M, integrated over

everything the chosen assumption M and prior P(/M)

comprise

dpxxpxxp nene )(),/()/( (2)

when having several models, P(D/M) is the likelihood of the

model i, which can be used in comparing the probabilities of

the models, hence the term evidence of the model. A widely

used Bayesian model choice method between two models is

based on Bayes factors,P(D/M1)/P(D/M2).The more common

notation of Bayes formula, with M dropped, more easily causes

misinterpreting the denominator P(0) as some kind of

probability of obtaining data D in the studied problem (or prior

probability of data before the modeling).

The prior distribution expresses our initial beliefs about

parameter values before any data is observed. After new data

D = {(x(1)

, xe(1)

), , (x(n)

, xe(n)

)} are observed, the prior

distribution will be updated to posterior distribution using

Bayes rule where L(/|D) is likelihood function of unknown

model parameters to the observed data. In case of independent

and exchangeable data points, the likelihood function is

),/()/(1

)()(

n

i

ii

e xxPDP (3)

where nis the number of data points.

To predict the new output xe(n+1) for the new input

x(n+1), predictive distribution is obtained by integrating the

predictions of the model with respect to the posterior

distribution of the model parameters

,)/(),/(

,/()/(

)1()1(

)1()1(

dDPxxD

xxPDP

nn

e

nn

e

(4)

where is the space of all possible parameters. Note that

predictive distribution for xe(n+1) is implicitly conditioned on

hypotheses that hold throughout and to be more explicit as the

following [51]

,),/(),,/(),/( )1()1()1(

dMDPDxxMDxP nn

e

n

e

(5)

whereM refers to the set of hypotheses or assumptions used to

define the model. Practically, the posterior distribution for

parameters in Eq. (4) is very complex and with many modes.

As a result, evaluating the above integral is a difficult task.

Neal [7] introduced Markov Chain Monte Carlo (MCMC)

method to perform this kind of difficult integrations. Then the

MCMC methods have been utilized by other authors.

III. ARQUITECTUREOFTHEANN

In Fig. 1 the block diagram of the nonlinear predictionscheme based on a ANN filter is shown. Here, a prediction

device [15]-[13] is designed such that starting from a given

sequence {xn} at time n corresponding to a time series it can be

obtained the best prediction {xe} for the following sequence of

18 values. Hence, it is proposed a predictor filter with an input

vector lx, which is obtained by applying the delay operator, Z-1

,

to the sequence {xn}. Then, the filter output will generatexeas

the next value, that will be equal to the present value xn. So,

the prediction error at time k can be evaluated as:

(6)

which is used for the learning rule to adjust the NN weights .

Fig. 1. Block diagram of the nonlinear prediction.

The coefficients of the nonlinear filter are adjusted on-linein the learning process, by considering an online heuristic

criterion that modifies at each pass of the time series the

number of patterns, the number of iterations and the length of

the tapped-delay line, in function of the Hursts value H

calculated from the time series taking into account the

Bayesian inference of the output values[9].

According to the stochastic behavior of the series, H can

be greater or smaller than 0.5, which means that the series

tends to present long or short term dependence [14],

respectively.

Estimation of

prediction error

Z-1 I Error-correctionsignal

One-step

prediction

NN-Based

Nonlinear Filter

Input signal

kxkxke en

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139 3

Furthermore, the NNs weights are tuned by means of the

Levenberg-Marquardt rule, which considers the long or short

term stochastic dependence of the time series measured by the

Hursts parameter H. The learning rule consists of changing the

number of patterns, the filters length and the number of

iterations for each corresponding time series using the Bayesian

inference in accordance with Hurst parameter H taking into

account that the series forecasted has the same H as the real

time series.

A. Application to ANN predictor

When a short or long series is being analyzed, it is

important to make use of the simplest possible models.

Specifically, the number of unknown parameters must be kept

at a minimum.

The gamma distributions have been considered in the

literature for this purpose. When a Bayesian analysis is

conducted, inferences about the unknown parameters are

derived from the posterior distribution. This is a probability

model which describes the knowledge gained after observing aset of data. The application of the regression problem involving

the correspond neural network functiony(x,w) and the data set

consisting of N pairs, input vector lx and targets tn(n=1,.,N)

Assuming Gaussian noise on the target, the likelihood

function takes the form:

,);(2

exp2

),/(1

22/

N

n

nn

N

twxyMwDP

(7)

where is a hyper-parameter representing the inverse of the

noise variance. We consider in this work a single hidden layer

of tanh units and a linearoutputs units.

To complete the Bayesian approach for this work, prior

information for the network is required. It is proposed to use,

analogous to penalties terms, the following equation

,2

exp2)(2

2

2/2

w

wwwP

N

(8)

assuming that the expected scale of the weights is given by w

set by hand. This was carried out considering that the network

function f(xn+1,w) is approximately linear with respect to win

the vicinity of this mode, in fact, the predictive distribution for

yn+1will be another multivariate Gaussian.

B. Performance measure of the ANN predictor filter

In order to test the proposed design procedure of the ANN

predictor, an experiment with time series obtained from the

MG solution and cumulative rainfall time series was performed.

The performance of the filter is evaluated using the Symmetric

Mean Absolute Percent Error (SMAPE) proposed in the most

of metric evaluation, defined by

1002

1

1

n

t tt

tt

S

FX

FX

nSMAPE (9)

where tis the observation time, nis the size of the test set,sis

each time series, Xt and Ftare the actual and the forecasted

time series values at time t respectively. The SMAPE of each

series s calculates the symmetric absolute error in percent

between the actualXt and its corresponding forecast value Ft,

across all observations tof the test set of size nfor each time

series s.

IV.

MAINRESULTS

A. MG and Santa Francisa Rainfall generations

The time series are obtained from the solution of the MG

equation[10]. The MG equation is explained by the time delay

differential equation defined as:

),()(1

)()( ty

ty

tyty

c

(10)

and rainfall time series obtained from Santa Francisca, San

Bartolom and La Sevillana, from Cordoba, Argentina, with

parameters shown in Table 1. This collection of coefficients

was chosen to generate time series MG1 and MG2 whose H

parameters vary between 0 and 1. The chosen one was selected

in accordance to its roughness.

Table 1. Parameters to generate the times series.

Time Series Type Parameters H

1 MG1 =1.9 0.22

2 MG2 =2.1 0.13

3 Cumulative Rainfall Santa Francisca 0.022

4 Cumulative Rainfall San Bartolom 0.0132

5 Cumulative Rainfall La Sevillana 0.259

B. Set-up of the ANN algorithm

The initial conditions for the filter and learning algorithm

are shown in Table 2. Note that the first number of hidden

neurons and iteration are set in function of the input number.

These initiatory conditions of the learning algorithm were used

to forecast the primitive of the time series, whose length is 102

values.

TABLE 2. INITIAL CONDITIONS OF THE PARAMETERS

Variable Initial Conditions

lx 12

Ho 7

it 100

H 0.5

C. Time Series Prediction Results

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Each time series is composed of the MG solutions and

cumulative rainfall time series. However, there are three classes

of data sets: one is the original time series used for the

algorithm in order to give the forecast, which comprises 102 or

79 as in La Sevillana rainfall series values. The other one is the

primitive obtained by integrating the values of original time

series and the last one is used to compare if the forecast is

acceptable or not where the 18 last values can be used to

validate the performance of the prediction system, which 102or 61 values form the data set, and 120 or 79 values constitute

the Forecasted and the Real ones. A comparison is made

between this work and earlier presented in [18] ANN H

dependent predictor filter.

The Monte Carlo method was used to forecast the next 18

values from MG time series and rainfall time series. Such

outcomes are shown from Fig. 2 to Fig. 6.

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [samples].Real Media = 0.10534. Forescated Mean = 0.1261.

H = 0.2263. He= 0.17241. l

x= 20. SMAPE = 1.1693.

H dependent algorithm.Neural Network based predictor

Mackey Glass parameters: =1.9,=30, c=10, =100.

Mean Forescasted

Data

Real

Fig. 2. Results obtained from MG1 time series.

0 20 40 60 80 100 120 1400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time [samples].Real Media = 0.11226. Forescated Mean = 0.097659.H = 0.13835. H

e= 0.1472. l

x= 20. SMAPE = 2.2398.


Mackey Glass parameters: =2.1,=40, c=10, =100.

Mean Forescasted

Data

Real

Fig. 3. Results obtained from MG2 time series.

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [samples].Real Media = 0.20763. Forescated Mean = 0.25411.

H = 0.022508. He= 0.17071. l

x= 20. SMAPE = 2.5966.


Lluvia Mensual Acumulado Santa Francisca

Mean Forescasted

Data

Real

Fig. 4. Results obtained from cumulative rainfall time series

Santa Francisca, Crdoba, Argentina.

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [samples].Real Media = 0.14141. Forescated Mean = 0.20693 .H = 0.12252. H

e= 0.23774. l

x= 20. SMAPE = 8.7109.


Lluvia Mensual Acumulado San Bartolom

Mean Forescasted

Data

Real


San Bartolom, Crdoba, Argentina.

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [samples].Real Media = 0.25847. Forescated Mean = 0.21356 .

H = 0.25994. He= 0.027721. l

x= 20. SMAPE = 4.8701.


Lluvia Mensual Acumulado La Sevillana

Mean Forescasted

Data

Real


San Bartolom, Crdoba, Argentina.

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D. Comparative Results

TABLE 3. FIGURES OBTAINED BY THE PROPOSED APPROACH

Series No. H He

Real

Mean

Mean

Forecasted SMAPE

MG1 0.22 0.17 0.105 0.126 1.16

MG2 0.13 0.14 0.11 0.097 2.23

Santa

Francisca 0.022 -0.17 0.207 0.254 2.59

San

Bartolom 0.12 0.23 0.141 0.206 8.71

La Sevillana 0.25 0.027 0.258 0.217 4.87

The assessments of the obtained results by comparing the

performance of the proposed filter with the earlier work [18]

[19]and[20], both are based on ANN.

Although the difference between filters resides only in the

adjustment algorithm, the coefficients that each filter has, each

ones performs different behaviors, in which it can be noted that

rainfall time series have more roughness than MG solutions, so

the Bayesian approach applied to the parameter of the ANN

demonstrate a level improvement, in which an adequate prior

distribution model was chosen in order for tuning the

parameters and outputs of the predictor filter.

V. CONCLUSIONS

In this work an algorithm to adjust parameters using a

Bayesian method for cumulative rainfall time series forecasting

implemented by an ANN-filter was presented. The contributionof this work resides in modeling the number of NN in the filter

that has an uncertainty number of parameters, which is adjusted

by Bayesian approach. Thereby, these are considered as

random variables whose posterior probability distribution is

inferred from the data.

The learning rule proposed to adjust the NNs weights is

based on the Levenberg-Marquardt method. Furthermore, in

function of the long or short term stochastic dependence of the

time series, the proposed approach changes the number of

patterns and iterations using the Bayesian inference in

accordance with parameter H evaluated of each time series. An

on-line heuristic adaptive law was proposed to update the

ANN topology at each time-stage taking into account that theseries forecasted has the same H as the real time series. The

major result shows that the predictor system has an optimal

performance from several MG time series and cumulative

rainfall against the classical predictor filter presented in[20], in

particular to time series whose H parameter has a high

roughness of signal, which is evaluated by H, respectively.

These results show a better performance compared with [19].

This fact encourages us to apply the proposed approach to

meteorological time series when the observations are taken

from a single point.

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[15] Pucheta, J., Patino, D. and Kuchen, B. A Statistically Dependent

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[18] J. Pucheta, M., C. Rodrguez Rivero, M. Herrera, C. Salas, D. Patio

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[19] C. Rodrguez Rivero, J. Pucheta, J. Baumgartner, M. Herrera, D. Patio

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[20] C. Rivero Rodrguez, J. Pucheta, J. Baumgartner, H.D. Patio and B.

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Stochastic Processes Through Time-Lagged feed-forward neural

network. The 2010 World Congress in Computer Science, Computer

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C. Rodriguez Riveroreceived the Electrical Engineering degree from Faculty

of Exact, Physical and Natural Sciences at National University of Cordoba,

Argentina, in 2007, and he is currently pursuing the Ph.D. degree in electrical

engineering at University of Cordoba, Argentina, under supervision of Dr.

Pucheta and Grant PRH National Agency for Scientific and Technological

Promotion (ANPCyT), in the field of automatic control for slow dynamics

processes, stochastic control, optimization and time series forecast. He joined

the Mathematics Research Laboratory applied to Control in 2009. IEEE

member of CSS and CIS.

J. Pucheta received the Electrical Engineering degree from NationalTechnological University - Cordoba Regional Faculty, Argentina, the M.S. and

Ph.D. degrees from National University of San Juan, Argentina, in 1999, 2002

and 2006, respectively. He is currently Professor at the Laboratory of research

in mathematics applied to control, National University of Cordoba, Argentina.

His research interests include stochastic and optimal control, time series

forecast and machine learning. IEEE member.

M. Herrera received the Electrical Engineering degree in 2007 from Facultyof Exact, Physical and Natural Sciences at National University of Cordoba,

Argentina. He is Associated Professor and researcher from Faculty of AppliedSciences and Technology at National University of Catamarca. His research

interests include control systems and automation.

V. Sauchelli received the Electrical Engineering degree and Ph.D. degreesfrom National University of Cordoba, Major in University Education from

National Technological University, Crdoba Regional Faculty (UTN, FRC,)

Argentina, in 1973 and 1997, respectively. He is currently Principal at

Telecommunication Postgraduate Program, National University of Cordoba,

and Professor of Control Systems and Signal Processing in electrical and

electronic engineering. He is also Director of Director of Mathematics Research

Laboratory applied to Control (LIMAC)and Research Projects at Secretary of

Sciences and Technology (SECyT), National University of Cordoba. His

interests include automatic control, neural networks, fractional-order controllers

and signal processing.
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