Hierarchical Fuzzy Rule Interpolation and its Application for Hotels Location Selection

1. Introduction

The “Curse of dimensionality” is a serious problem in fuzzy modelling of high-dimensional systems. It was discovered that the number of rules in a standard fuzzy system increases exponentially with the number of variables involved. Suppose that a fuzzy model contains K variables and each variable is partitioned into M fuzzy values. The order of the number of rules in the rule base is therefore O(M^K). This is usually referred to as the rule-explosion problem or curse of dimensionality. This problem of rules explosion reduces the transparency and interpretability, which is the important advantages of fuzzy systems. To address this problem, several methods have been proposed for optimizing the size of the rule-base. One of the most powerful ways to deal with this problem is through the use of hierarchical fuzzy systems (Raju and Zhou (1993); Wang (1998); Zeng and Keane (2005)). HFS can improve transparency and interpretability in many high dimensional situations. A hierarchical fuzzy system consists of a number of hierarchically connected low-dimensional fuzzy systems. The number of rules in the hierarchical fuzzy system increases linearly with an increasing number of input variables. In conventional (non-hierarchical) fuzzy systems, suppose that there are K input variables and M membership functions for each variable, then M^K rules are needed in order to construct a system that fully covers the underlying domain. An K-input hierarchical fuzzy system comprises K–1 low-dimensional fuzzy systems, if each sub-system has two inputs. In this case, given M fuzzy sets for each variable, the total number of rules is (K–1)M² which is a linear function of the number of input variables. A general example of hierarchical fuzzy systems is shown in Figure 1.

Figure 1.

General structure for hierarchical fuzzy systems

On the other hand, reducing the number of fuzzy terms K of each variable may result in sparse fuzzy rule base. In real world applications, it is hard to obtain a sufficient data set with many input variables that cover the whole input space. In most cases there usually are limited samples to train the hierarchical fuzzy systems, this is another reason which results in sparse fuzzy rule base. When a given observation has no overlap with antecedent values, no rule can be invoked in classical fuzzy inference, and no consequence can be derived. Fuzzy rule interpolation (FRI) was originally proposed in (K’oczy and Hirota (1993)). It is of particular significance for reasoning in the presence of insufficient knowledge or sparse rule bases. The techniques of FRI may not only support inference in such situations, but also help in reducing the complexity of fuzzy models.

In this paper, a novel technique termed hierarchical fuzzy interpolation (HFI) is introduced. The goal of HFI is to combine hierarchical fuzzy systems and fuzzy rule interpolation in order to overcome the “curse of dimensionality” problem and the sparse rule base problem simultaneously. The method of backward fuzzy rule interpolation (BFRI) is also used to calculate the crucial missing variables and intermediate variables. In so doing, the overall HFI model and its reasoning becomes more transparent and interpretable.

The rest of this paper is organised as follows. Section 2 introduces the general concepts of HFS, along with detailed descriptions of generating method of hierarchical fuzzy rule bases. Section 3 presents the scale and move transformation based fuzzy rule interpolation and backward fuzzy interpolation. The algorithm of hierarchical fuzzy interpolation approach is given in Section 4. An illustrative example is presented in Section 5. Section 6 presents the proposed framework for a hotel location selection process, and the results of the experiments demonstrate the efficacy of the proposed approach. Section 7 concludes the paper.