Prediction‐based adaptive compositional model for seasonal time series analysis

• DOI: http://doi.org/10.1002/for.2474
• Author: Rong Chen,Kun Chang,Thomas B. Fomby
• Date: 01 November 2017
• Published date: 01 November 2017

Received: 1 July 2016 Revised: 29 January 2017 Accepted: 21 March 2017

DOI: 10.1002/for.2474

RESEARCH ARTICLE

Prediction-based adaptive compositional model for seasonal time

series analysis

Kun Chang1Rong Chen1Thomas B. Fomby2

1Department of Statistics and Biostatistics,

Rutgers, the State University of New Jersey,

Piscataway,NJ, USA

2Department of Economics, Southern

Methodist University,Dallas, TX, USA

Correspondence

Rong Chen, Department of Statistics and

Biostatistics, Rutgers, the State University of

New Jersey,Piscat away, NJ 08854, USA.

Email: rongchen@stat.rutgers.edu

Funding information

NSF, Grant/Award Number:DMS-1540863,

DMS-1209085; Department of Homeland

Security, Grant/Award Number:

2009-ST-061-CCI002-05 and

2009-ST-061-CCI002-06

Abstract

In this paper we propose a new class of seasonal time series models, based on a stable

seasonal composition assumption. With the objective of forecasting the sum of the

next 𝓁observations, the concept of rolling season is adopted and a structure of rolling

conditional distributions is formulated. The probabilistic properties, estimation and

prediction procedures, and the forecasting performance of the model are studied and

demonstrated with simulations and real examples.

KEYWORDS

adaptive forecasting, prediction, stable seasonal pattern

1INTRODUCTION

Seasonal time series are encountered in a wide range of

applications. Traditionally, there are three general classes of

seasonal time series models, namely the seasonal autore-

gressive integrated moving average (ARIMA) models (Box

& Jenkins, 1994), the trend-and-seasonal models (Franzini

& Harvey, 1983), and the stable seasonal pattern models

(Chen & Fomby, 1999; Oliver, 1987). All these models pro-

vide different perspectives in dealing with seasonality. In

particular, standard seasonal ARIMA models are in a mul-

tiplicative form, whereas trend-and-seasonal models are in

an additive form. There is a vast literature on seasonal time

series analysis and seasonal adjustment (e.g., Bell & Hillmer,

1984; Box & Jenkins, 1994; Cleveland & Tiao,1976; Findley,

Monsell, Bell, Otto, & Chen, 1998; Ghysels & Osborn, 2001;

Zellner, 1978).

On the other hand, compositional models of Aitchison

(1986) concentrate on the proportion of each component

relative to the whole. Compositional models, by modeling

ratios of proportions, successfully release the unit sum con-

straint, which makes it possible to apply standard statistical

methodologies. This type of model has been used in many

applications. For example, statistical analysis of percentages

by weight of major oxides in rock specimens can be used to

identify new types of rock specimens, as shown in a series

of research topics by Thomas and Aitchison (2006). Another

example is the study of budget patterns of a household

reflected by the proportions of total expenditures allocated to

several commodity groups. Aitchison (1986) analyzed such

an example on five commodity groups.

Seasonality in a time series can often be viewed as a cer-

tain type of regular composition of seasons over time. For

example, for a monthly time series with an annual season-

ality, the 12 months can be seen as 12 components of the

year (the composition), and the seasonality can be seen as a

certain systematic distributive pattern of the measurements

among 12 months with respect to the total measurement of

the year. In the sales industry, the percentage of sales amount

in each quarter out of the year is often stable across dif-

ferent years, whereas the yearly total may vary. Chen and

Fomby (1999) touched upon this observation and introduced

a stable seasonal pattern model, by assuming that the propor-

tion (composition) of each part in a period remains the same

(probability-wise) across seasons.

842 Copyright © 2017 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/for Journal of Forecasting.2017;36:842–853.