Elsevier

Applied Soft Computing

Volume 9, Issue 2, March 2009, Pages 647-651
Applied Soft Computing

A new approach for determining the length of intervals for fuzzy time series

https://doi.org/10.1016/j.asoc.2008.09.002Get rights and content

Abstract

In the implementations of fuzzy time series forecasting, the identification of interval lengths has an important impact on the performance of the procedure. However, the interval length has been chosen arbitrarily in many papers. Huarng developed a new approach which is called ratio-based lengths of intervals in order to identify the length of intervals. In our paper, we propose a new approach which uses a single-variable constrained optimization to determine the ratio for the length of intervals. The proposed approach is applied to the two well-known time series, which are enrollment data at The University of Alabama and inventory demand data. The obtained results are compared to those of other methods. The proposed method produces more accurate predictions for the future values of used time series.

Introduction

Fuzzy set theory was introduced by Zadeh [13]. Since then, fuzzy set has been adopted in many applications such as regression, time series forecasting and ext. Fuzzy time series was firstly introduced by Song and Chissom [9], [10], [11]. All of these studies have been inspired by knowledge presented in the paper [14], [15]. Fuzzy time series has been widely studied for recent years for the aim of forecasting. The traditional time series approaches require having the linearity assumption and at least 50 observations. In fuzzy time series approaches, there is not only a limitation for the number of observations but also there is no need for the linearity assumption. The fuzzy time series approaches consist of three steps. The first step is the fuzzification of observations. In the second step, fuzzy relationships are established and the defuzzification is done in the third step. In recent years, researchers have been doing many studies to improve and explore all of these three steps. Chen’s method consists of considerably simple calculations. The method proposed by Chen [1] is easier than the one Song and Chissom suggested. The two of other contributions in establishing fuzzy relationships were done by [6], [7].

In Hurang’s paper [2], it has been investigated that the identification of the length of intervals in the fuzzification stage affects the performance of the fuzzy time series approach in forecasting. However, the length of intervals has been chosen arbitrarily in many research papers [1], [3], [8], [10], [11]. In order to approach the problem computationally, Huarng proposed two novel approaches which are based on the average and the distribution [2]. Some other studies in the step of the fuzzification were conducted by [5], [12]. Huarng [4] suggested a different method which is called ratio-based lengths of intervals. The method compared to the others of which the length is chosen arbitrarily has generated more accurate forecasts for enrollment, inventory demand and TAIEX stock price data. In the ratio-based approach of Hurang [4], intensive calculations such as relative difference, cumulative distributions of relative differences are done for the certain numbers of the ratio values. The ratios were identified only for the considered sample percentiles.

We proposed a novel approach to determine the length of interval in order to obtain more accurate forecasts in fuzzy time series. The proposed approach is based on a single variable constrained optimization. We improved Huarng’s method [4] with our approach. The advantages of the proposed approach can be summarized as follows:

  • There is no need to do some calculations such as relative difference, cumulative distributions of relative differences.

  • This approach is more comprehensive than Huarng’s method because proposed method implementing optimization determines the ratio, which leads to the best forecasts. However, Huarng’s method employs just a few numbers of selected ratios.

  • It has been observed that the forecasting accuracies for the two well-known data sets were significantly improved when the proposed method is employed.

The proposed method was applied to the enrollment data at The University of Alabama and the inventory demand used in [4] time series and the results were compared to those obtained by Chen [1] and Huarng’s methods [4]. Huarng [4] compared his approach, which is ratio-based lengths of intervals, with other proposed method to show how accurate his method produces forecasting results. Huarng showed that his method outperformed the other methods. Therefore, we compare our proposed method with Huarng’s method in order to show the efficiency of the proposed method. It has been investigated that our proposed method considerably improves the forecasting performance.

In Section 2, the general knowledge about fuzzy time series is given. In Sections 3 Related research, 4 A new method to obtain more efficient forecasting Chen’s algorithm and Huarng’s method are presented, respectively. Section 5 introduces the proposed method. In Section 6, the results obtained by applying our proposed method to the enrollment data and the inventory demand data are presented. In final section, the results are discussed.

Section snippets

Fuzzy time series

The definition of fuzzy time series was firstly introduced by [9], [10]. In contrast to conventional time series procedures, various theoretical assumptions do not need to be checked in fuzzy time series approach. The most important advantage of fuzzy time series approach is to be able to work with a very small set of data and not to require the linearity assumption. General definitions of fuzzy time series are given as follows:

Let U be the universe of discourse, where U = {u1, u2, …, ub}. A fuzzy

Related research

Many researches on improving procedures for forecasting of fuzzy time series have been recently made. One of those was done by [1]. The procedure proposed by Chen [1] is simpler than the method proposed by Song and Chissom [9], [10]. In Chen’s method, matrix calculations are not necessary. The other considerable study was presented by Huarng [2]. Huarng pointed out that it is a very critical to decide on what the interval length will be for the forecasting accuracy. In his latter study, Huarng

A new method to obtain more efficient forecasting

There will be no fluctuations in the fuzzy time series when an effective length of intervals is too large. On the other hand, the meaning of fuzzy time series will be diminished when the length is too small [2]. Therefore, the effective lengths of intervals are an important issue for improving fuzzy time series forecasting. In the ratio-based length approach, a ratio is determined with respect to the sample percentile α which is defined by the cumulative distribution of relative differences.

Application

In order to show what we have achieved from the proposed approach, we applied it to the enrollment and the inventory demand data and then have compared the obtained results with those obtained from Chen [1] and Huarng’s method [4]. The first application was on yearly data on enrollments at the University of Alabama. Years and the corresponding enrollment observations are listed in the first and the second column of Table 1, respectively. The enrollment observations from 1971 to 1988 were used

Conclusion

The approaches of the fuzzy time series are recently getting quiet popular. Traditional time series analyses assume that the number of observations should be greater than 50 and the structure of time series should be linear. Fuzzy time series approaches do not require such assumptions. Although this makes fuzzy approaches very attractive, there are still problems that need to be solved. One of these problems is to determine the lengths of intervals. The decision on what the lengths will be is

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