1. Introduction

In 1965, Zadeh [1] proposed the concept of fuzzy sets as a tool to test the unknown degree of membership. Many fuzzy studies then attempted to use this method as a theoretical framework, which is widely used in the research fields of natural sciences and social sciences, obtaining good study achievements [2–22]. The fuzzy time series is also an analysis method derived from the concept of fuzzy sets. In 1993, Song and Chissom [18–21] successfully combined the concept of fuzzy sets with the time series model and began studies on fuzzy time series. Chen [3] proposed the simplified and easy-to-calculate method for Song and Chissom's model [18–21], so that the computation complexity of fuzzy time series is dramatically reduced. Lee and Chou [14] also proposed that rational settings of the lower and upper boundary in intervals of the universal set for fuzzy time series have improved their accuracy and reliability. Liaw [17] proposed a simple test method for whether a fuzzy time series has a fuzzy trend, in which the method is used to determine whether the data for analysis is in a steady state. Chou [12] added to Chen and Hsieh's defuzzification method [2] in the fuzzy time series, so that the long-term level for the series can be obtained, and the model originally used for single-point prediction can be applied to long-term prediction and interval prediction. This article mainly uses the algorithmic method of Chou's [12] research process for illustrating the fuzzy time series, taking the Taiwan Shipping and Transportation Index (STI) [23] as an example.

The remainder of this chapter is organized as follows. Section 2 presents the definition of fuzzy time series and Section 3 defines the long-term predictive

significance level process. A numerical example of STI is shown in Section 4, and concluding remarks are mentioned in conclusion.

1. y ¼ exp d tð Þ ⇔ ln y ¼ d tð Þ and

Fuzzy Forecast Based on Fuzzy Time Series DOI: http://dx.doi.org/10.5772/intechopen.82843

DL <sup>¼</sup> <sup>D</sup>min � stαð Þ <sup>n</sup> <sup>=</sup> ffiffiffi

distribution, then P Zð Þ¼ ≥ z<sup>α</sup> α.

8 >><

>>:

the critical region <sup>C</sup><sup>∗</sup> <sup>¼</sup> C C<sup>k</sup>

significance level α is 0.2.

satisfying:

satisfying:

1. min

2. min

27

DL <sup>¼</sup> <sup>D</sup>min � <sup>σ</sup>Zα<sup>=</sup> ffiffiffi

2. exp lnð Þ¼ d tð Þ d tð Þ, ln exp ð Þ¼ x d tð Þ.

Definition 10 [14]. The universe of discourse U ¼ ½ � DL; DU is defined such that

Definition 11 [14]. Assuming that there are m linguistic values under consider-

i Dð Þ <sup>U</sup> � DL

i Dð Þ <sup>U</sup> � DL m

n p when n ≤ 30 or

<sup>m</sup> , <sup>1</sup><sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>m</sup> � <sup>1</sup>

, i ¼ m:

! for all <sup>x</sup><sup>∈</sup> d tð Þ and

for all x∈ d tð Þ and.

<sup>n</sup> <sup>p</sup> when <sup>n</sup> . 30, where <sup>t</sup>αð Þ <sup>n</sup> is the

th linguistic value of the

, the initial value of the

! ≤ b.

 ≤ b.

, max ! ) is called

� � are the GMIR

<sup>n</sup> <sup>p</sup> and DU <sup>¼</sup> <sup>D</sup>max <sup>þ</sup> stαð Þ <sup>n</sup> <sup>=</sup> ffiffiffi

100 1ð Þ � α percentile of the t distribution with n degrees of freedom. z<sup>α</sup> is the 100 1ð Þ � α percentile of the standard normal distribution. Briefly, if Z is an N(0, 1)

linguistic variable, where 1≤ i≤ m. The support of Ai is defined as follows:

, DL þ

, DL þ

<sup>2</sup> . <sup>C</sup><sup>λ</sup> <sup>¼</sup> <sup>C</sup><sup>n</sup> <sup>2</sup> � ð Þ <sup>1</sup> � <sup>λ</sup> � � � �

Definition 13 [8]. Let d tð Þ be a set of real numbers d tð Þ⊆ R. An upper interval for d tð Þ is a number b such that x≤ b for all x∈ d tð Þ. The set d tð Þ is said to be an interval higher if d tð Þ has an upper interval. A number, max, is the maximum of d tð Þ

Definition 14 [8]. Let d tð Þ<sup>⊆</sup> <sup>R</sup>. The least upper interval of d tð Þ is a number max !

2. max ! is the least upper interval for d tð Þ, that is, <sup>x</sup><sup>≤</sup> <sup>b</sup> for all <sup>x</sup><sup>∈</sup> d tðÞ) max

Definition 15 [8]. Let d tð Þ be a set of real numbers d tð Þ⊆ R. A lower interval for d tð Þ is a number b such that x ≥ b for all x∈ d tð Þ. The set d tð Þ is said to be an interval below if d tð Þ has a lower interval. A number, min, is the minimum of d tð Þ if

Definition 16 [8]. Let d tð Þ<sup>⊆</sup> <sup>R</sup>. The least lower interval of d tð Þ is a number min

is the least lower interval for d tð Þ, that is, x ≥ b for all x∈ d tðÞ) min

Definition 18 [2]. Let Ai ¼ αi; βi; γ<sup>i</sup> ð Þ, i ¼ 1, 2, …, n, be n triangular fuzzy numbers. By using the graded mean integration representation (GMIR) method, the

Definition 12 [17]. For a test H<sup>0</sup> : nonfuzzy trend against H<sup>1</sup> : fuzzy trend, where

<sup>n</sup> <sup>p</sup> and DU <sup>¼</sup> <sup>D</sup>max <sup>þ</sup> <sup>σ</sup>Zα<sup>=</sup> ffiffiffi

DU � DL m

DU � DL m

<sup>2</sup> <sup>þ</sup> <sup>C</sup><sup>n</sup>�<sup>k</sup>

ation, let Ai be the fuzzy number that represents the i

DL þ ð Þ i � 1

DL þ ð Þ i � 1

if max is an upper interval for d tð Þ and max∈ d tð Þ.

min is a lower interval for d tð Þ and min∈ d tð Þ.

the static long-term predictive value interval.

1. max ! is an upper interval for d tð Þ such that <sup>x</sup><sup>≤</sup> max

is a lower interval for d tð Þ such that x ≥ min

Definition 17 [8]. The long-term predictive value interval (min

GMIR value P Að Þ<sup>i</sup> of Ai is P Að Þ¼<sup>i</sup> α<sup>i</sup> þ 4β<sup>i</sup> þ γ<sup>i</sup> ð Þ=6. P Að Þ<sup>i</sup> and P Aj

values of the triangular fuzzy numbers Ai and Aj, respectively.
