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Study for High-Order Models of Fuzzy Time Series Chao-Chih Tsai

Department of Mathematics Education

Abstract

This paper proposes two theorems and a new scheme for fuzzy time series. In view of the work by Song and ^{Chissom [4],} we find the model equation used is not theoretically clarified. ^Hence, in this study, two theorems are proposed and proved to set as a basis. ^Then, a new scheme based on the theorems is proposed. This new scheme can be applied to models of the fuzzy time series and makes the models high-order.

To illustrate the effect of this proposed new ^scheme, the forecasting enrollments are carried out. It is found that the root mean square error of the results can be improved from

327.38 for the Chen method to 59.75 for the current ^model, the model due to the injection of the new scheme into Chen model.

^Furthermore, forecasting population for Taiwan area is also implemented to reassure.

The results are compared with those of the other approaches. It is found that the root mean square error of the forecasts can be improved from 0.119 for the Song-Chissom method to

13. for the proposed model.

Key Words: Fuzzy time ^series; high-order ^{model; forecasts;} fuzzy relationship; linguistic value

1. Introduction

Since the creation in 1965 of fuzzy set theory by Zadeh [1], its applications have been steadily increasing. The increasingly widespread use of fuzzy set theory can be attributed to its ability to cope with the vagueness of human language. When a researcher desires to analysis historical data ^with linguistic ^variables, based on

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traditional time series methodologies may fail to work. Models of fuzzy time series have been developed to deal specially with research problems of this sort. These fuzzy time series models were pioneered by Zadeh [2-3], and were further developed by Song and Chissom [4], who employed fuzzy relational equations, defined various fuzzy time ^series, explored some important properties of fuzzy time ^series, and presented a step-by- step procedure for the implementation of a fuzzy time series ^with linguistic variables. Song and Chissom ^[5] subsequently published an application of their fuzzy time series ^model, a 1993 study which forecast enrollments for the University of ^Alabama, in this study, Song and Chissom developed a first-order model from historical data for enrollment. From which they proposed a step-by-step forecasting procedure. Song and Chissom were able to demonstrate from a comparison with other similar studies in the ^literature, that the average forecasting error of the fuzzy time series model is smaller than that of other approaches. In another study using this fuzzy time series ^model, Wu[ ^17] analyzed trends in quantities such as numbers of ^teachers, government expenditure, and exchange ^rates; he also found that the ability of fuzzy-based methods to predict future trends was superior to that of traditional forecasting methods. His results confirm that forecasting methods based on a fuzzy time series are an effective tool for a variety of problems. Although many applications of forecasting method based on fuzzy time series can now be found in the ^literature, the operations used by most are complicated. For this ^reason, Chen[9] presented a new fuzzy time series ^method, using fuzzy arithmetic operations. Chen employed fuzzy logical relationships to develop a step-by- step procedure for the implementation of a fuzzy time series with the ability to analysis a dynamic process with linguistic variables In order to test the robustness of his ^method, Chen applied his forecasting model to the same data used by Song and Chissom [5] to frecast enrollment at the University of Alabama. In comparison to other ^methods, Chen’s model is more efficient and simpler than most other approaches. ^However, the accuracy of his forecast method is quite limited. ^Also, the relational equations developed by Song and Chissom are not clear. In the present study, two theorems are proposed to clarify the theoretical foundation of Song-Chissom’s method and are developed as the basis for a new model of fuzzy time series.

The main advantage of fuzzy approach is that human experience and knowledge can be applied throughout the forecast procedure. ^However, most applications of fuzzy time series technology in the literature are based on a first-order fuzzy time ^series, that ^is, one which considers only two consecutive times. In order to enhance the accuracy of forecasting ^results, the present study introduces a new scheme to increase the order of the model. In order to demonstrate that a higher-order model is indeed more ^accurate, This study compare forecasts of enrollments and population using models of different orders. The new method of fuzzy forecasting has already been successfully applied by the author to a wide variety of problems, including nuclear energy supply, nuclear plant ^accidents, exchange ^rates, numbers of vocational high school ^teachers, and vegetable

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prices in Taiwan ^[10-16].

Section II reviews fuzzy set theory and fuzzy time series. In Section ^III, the new scheme for modeling the fuzzy time series is presented, and procedures for making forecasts are established. ^Then, in Section ^IV, forecasts of enrollments and popLilation are carried out using the new model. ^And, are compared with forecasts made with other approaches. The finally section presents conclusions.