Forecasting in high order fuzzy times series by using neural networks to define fuzzy relations
Introduction
In recent years, fuzzy time series approach introduced by Song and Chissom, 1993a, Song and Chissom, 1993b has been used widely. Chen (1996) proposed a method which is simpler than the method proposed by Song and Chissom, 1993a, Song and Chissom, 1993b in forecasting fuzzy time series. The method proposed by Chen (1996) does not include complex matrix operations in defining fuzzy relation. Huarng and Hui-Kuang (2006) uses a simple feed forward neural network to define fuzzy relation. Due to the first order fuzzy times series approach implementation in Huarng and Hui-Kuang (2006), his proposed method includes a simple feed forward neural network model which has one input neuron, two hidden layers’ neurons, and one output neuron. Because of not losing the generalization ability of neural network model, Hurang used two neurons in hidden layer.
Hwang et al., 1998, Chen, 2002 used high order fuzzy time series model. Chen’s (2002) model consists of defining fuzzy relation based on previous observations. The implementation of Chen’s approach becomes more difficult when the order of fuzzy time series increases. However, neural networks can be used easily for high order fuzzy time series. In this study, feed forward neural networks are employed to define fuzzy relation by trying various architectures for high order fuzzy time series. The proposed approach based on neural networks is applied to well-known enrollment data for University of Alabama. Obtained results are compared with other methods and it is clearly seen that our proposed method has better forecasting accuracy when compared with other methods proposed in the literature.
Section 2 includes the definitions of first and high order time series. The brief information related to neural networks is given in Section 3. The new proposed method is introduced and the implementation results of enrollment data are given in Sections 4 The proposed method, 5 Application respectively. Final section is for conclusion.
Section snippets
Fuzzy time series
The definition of fuzzy time series was firstly introduced by Song and Chissom, 1993a, Song and Chissom, 1993b. In fuzzy time series approximation, you do not need various theoretical assumptions just as you need in conventional time series procedures. The most important advantage of fuzzy time series approximations is to be able to work with a very small set of data and not to require the linearity assumption. The some general definitions of fuzzy time series are given as follows:
Let U be the
Artificial neural networks
‘What is an artificial neural network?’ is the first question that should be answered. Picton (1994) answered this question by separating this question into two parts. The first part is why it is called an artificial neural network. It is called an artificial neural network because it is a network of interconnected elements. These elements were inspired from studies of biological nervous systems. In other words, artificial neural networks are an attempt at creating machines that work in a
The proposed method
In order to construct high order fuzzy time series model, various feed forward neural networks architectures are employed to define fuzzy relation. The stages of the proposed method based on neural networks are given below.
- Stage 1.
Define and partition the universe of discourse
The universe of discourse for observations, U = [starting, ending], is defined. After the length of intervals, l, is determined, the U can be partitioned into equal-length intervals u1, u2,…,ub, b = 1,… and their corresponding
Application
The proposed method is applied to the enrollment data of University of Alabama which is shown in Table 2. First, second, third, and fourth order fuzzy time series model are used in the application. In the first stage of the proposed method, as Huarng (2001) did, the length of intervals are chosen as 200, 300, 400, 500, 600, 700, 800, 900 and 1000. After following stages 2 and 3 in algorithm given in Section 4, the number of neurons of hidden layer is altered 1 through 4 not to lose
Results and discussion
A given observation in time series does not only depend on preceding one but also previous ones in general. Therefore, high order fuzzy time series approach might represent the fuzzy relation better than does the first order fuzzy time series approach. However, defining fuzzy relation in high order is more difficult than that in the first order. In this study, we proposed a method to define a fuzzy relation by using neural networks for high order fuzzy time series. The new proposed method is
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