Simulation of time series using periodic gamma autoregressive models

• DOI: http://doi.org/10.1111/itor.12593
• Author: Pedro Guilherme Costa Ferreira,Victor Eduardo Leite de Almeida Duca,Fernando Luiz Cyrino Oliveira,Reinaldo Castro Souza
• Date: 01 July 2019
• Published date: 01 July 2019

Intl. Trans. in Op. Res. 26 (2019) 1315–1338

DOI: 10.1111/itor.12593

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Simulation of time series using periodic

gamma autoregressive models

Victor Eduardo Leite de Almeida Ducaa, Reinaldo Castro Souzac,

Pedro Guilherme Costa Ferreiraband Fernando Luiz Cyrino Oliveirac

aDepartment of Electrical Engineering, PUC-Rio, Rio de Janeiro – RJ 22451-000, Brazil

bBrazilian Institute of Economics, FGVIBRE, Rio de Janeiro, Brazil

cDepartment of Industrial Engineering, PUC-Rio, Rio de Janeiro – RJ 22451-000, Brazil

E-mail: victorduca08@gmail.com [Leite de Almeida Duca]; reinaldo@puc-rio.br[Castro Souza];

pedro.guilherme@fgv.br[Costa Ferreira]; cyrino@puc-rio.br [Cyrino Oliveira]

Received 27 October 2017; receivedin revised form 9 August 2018; accepted 24 August 2018

Abstract

Periodic autoregressive models are frequently used to model hydrologic series. In the literature, annual

streamflowseries are approximated bynor mal distribution.However,for short periods (daily,weekly, monthly)

this is no longer the case, particularlydue to the data skewness. A new class of first-order model was,therefore,

studied in an attempt to overcome this problem. The model has an autoregressive structure and can be

additive, multiplicative, or hybrid, but with gamma marginal distribution. Furthermore, the classical model

assumes that the method of moments is effective for parameters estimation. For the first time, this paper

undertakes a complete analysis of the hybrid model for this context and the novelty lies in the parameter

estimation via maximization of the likelihood function. For an application in the Brazilian case, the additive

model proved to be more effective than previously reported and the method allows high orders and skewed

distributions.

Keywords:PAGAR(1); PMGAR(1); PGAR(1)

1. Introduction

Synthetic streamflow and natural hydropower (NH) series are frequently used in hydroelectric

system planning and operation studies. Many autoregressive models have been developed to model

such series under the assumption of normality and are described in the literature.

According to Fernandez and Salas (1986), annual streamflow series can be approximated to

normal distributions. However, for short periods (e.g., daily, weekly, or monthly series) this is

no longer the case, particularly because of the problem of skewness. To overcome this difficulty,

various methods are discussed in the literature in an attempt to achieve an approximation to a

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1316 V.E. Leite de Almeida Duca et al. / Intl. Trans. in Op. Res. 26 (2019) 1315–1338

normal distribution, but none of them has met with success. Many of these models have constant

or periodic first-order autoregressive structures for all the data in the series and assume a normal

distribution for the residuals.

Fernandez and Salas (1986) proposeda model that addressed the abovelimitations through the use

of a gamma distribution and an autoregressive structure. In addition to additive and multiplicative

formulations, they built a hybrid model that incorporated properties of both these formulations

to take into account the seasonal behavior of the marginal distribution and the autocorrelation

function. Basically, the modeling procedure involves estimating the periodic parameters with the

method of moments and generating the residuals. It is important to emphasize that the authors

did not explore larger orders because of the complexity the model would become, needing the

build of a multivariate gamma distribution. Fernandez (1984), in his thesis, elucidates the build of

only the periodic additive gamma autoregressive (PAGAR) model for order 2 through the same

ideas and mathematical tools used in order model 1. In addition to the mathematical complexity

that the model becomes, the author also emphasizes that PAGAR(2) cannot easily be reduced

to the particular case when the asymmetry coefficient is constant for every period, in addition

to problems in assigning exponential marginal. These constraints make order 2 model with little

practical use. The author also points out that in the literature, there is no specific treatment for

gamma models or moving average structures with larger orders, except for the simplest case of

exponential.

The model, which can be linear or nonlinear and assumes a gamma distribution, is restric-

tive, but complementary in the way it treats the periodic skewness coefficient. The particu-

lar form of the model used here has an autoregressive structure and is periodic and nonlin-

ear. The advantage of the model is that it can be applied directly to skewed periodic series,

thus avoiding the need for a transformation to achieve normality, as is the case with energy

series.

The model in question was used by Fernandez and Salas (1986) and other authors on various

occasions. Fernandezand Salas (1990) proposed a method for correcting the bias in estimates of the

parameters of a first-order gamma autoregressive model (GAR(1)) based on computer simulation

studies, while in Chebaane et al. (1995) the periodic gamma autoregressive (PGAR) model was

used with the periodic discrete autoregressive (PDAR) model to model hydrological series in dry

regions. This combined model wasknown as PGAR–PDAR. S¸arlak and S¸ orman (2007) studied the

GAR(1) model with hydrological series and built estimates using a modified maximum likelihood

(ML) method.

A number of recent studies have used the PGAR model in the Brazilian electricity sector (BES).

In Ferreira (2013), the hybrid model is discussed as an alternative to the periodic autoregressive

(PAR(p)) model for generating synthetic series in the context of the BES. However, the author

identifies some restrictions associated with this type of modeling. Bragaand Calmon (2017) compare

the results of PGAR with the periodic normal and lognormal models adopted in the sector. They

found that the PGAR model yielded good results compared with the periodic normal model, and

it was a good alternative to the lognormal model.

The aim in generating synthetic series is to approximate the stochastic behavior of an autore-

gressive model to a historical series and synthetically build new series that are different from the

original series but statistically possible. This method is frequently used in the BES. In Oliveira

(2010), this approach is used to propose methodological improvements to the PAR(p) model.

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2018 The Authors.

International Transactionsin Operational Research C

2018 International Federation ofOperational Research Societies