Pattern recognition using type-II fuzzy sets

Abstract

Type II fuzzy sets are a generalization of the ordinary fuzzy sets in which the membership value for each member of the set is itself a fuzzy set in [0,1]. We introduce a similarity measure for measuring the similarity, or compatibility, between two type-II fuzzy sets. With this new similarity measure we show that type-II fuzzy sets provide us with a natural language for formulating classification problems in pattern recognition.

Introduction

Pattern recognition problems typically involve the classification of an unknown pattern $\text{Q}$ given a set of K prototypes $\text{P}{}_{\mathrm{k}}$, k∈{1,2,…,K} [1], [2], [3], [4], [5], [6]. Each prototype $\text{P}{}_{\mathrm{k}}$ belongs to a given class C_m, m∈{1,2,…,M}, which is specified by the indicator function $\text{A}{}_{\mathrm{k}}$:

$$\text{A}{}_{\mathrm{k}}\text{=}\text{C}{}_{\mathrm{m}}\text{if}\text{P}{}_{\mathrm{k}}\text{belongs}\text{to}\text{the}\text{m}\text{th}\text{class}\text{C}{}_{\mathrm{m}}\text{.}$$

Let $\text{S}\text{(}\text{Q}\text{,}\text{P}{}_{\mathrm{k}}\text{)}$ be a similarity measure which measures the degree of similarity, or compatibility, between the unknown pattern $\text{Q}$ and the kth prototype $\text{P}{}_{\mathrm{k}}$. Then, formally, we may write the process of classifying, or assigning, the unknown pattern $\text{Q}$ to the class , where

$$\text{k}{}^{\mathrm{\ast }}\text{=}\text{arg}\text{max}\text{k}\text{(}\text{S}\text{(}\text{Q}\text{,}\text{P}{}_{\mathrm{k}}\text{)).}$$

Very often we apply the techniques of pattern recognition to situations which are inherently vague and uncertain [7], [8]. Such situations arise when the information regarding the prototypes is “linguistic” and is based on the opinions and judgements of human experts. Examples of such situations are: handwritten character recognition, fingerprint recognition, human face recognition, classification of X-ray images, classification of remotely sensed data. One way of handling such situations is to make the information precise and to give it a mathematically well-defined form by using the concepts and techniques of fuzzy logic.

Let U denote the feature space. Then, mathematically, we represent the unknown pattern $\text{Q}$, and the prototypes, $\text{P}{}_{\mathrm{k}}$, k∈{1,2,…,K}, with (ordinary) fuzzy sets Q≡Q(u) and P_k≡P_k(u), where u∈U. The result is the following optimization problem:

$$\text{k}{}^{\mathrm{\ast }}\text{=}\text{arg}\text{max}\text{k}\text{(}\text{S}\text{(}\text{Q}\text{(u),}\text{P}{}_{\mathrm{k}}\text{(u))),}$$

where S denotes an appropriate similarity, or compatibility, measure [9].

Unfortunately, (3) is sometimes too precise in the sense that no uncertainty whatsoever is allowed in specifying the fuzzy sets Q(u) and P_k(u).

Dengfeng and Chuntian [10] suggested that one way to introduce a controlled amount of uncertainty into (3), is to replace the ordinary fuzzy sets Q(u) and P_k(u), with intuitionistic fuzzy sets [11], and to replace S with an appropriate intuitionistic similarity measure [10], [12], [13], [14]. This is correct if we model the uncertainty with a uniform distribution. However, in many pattern recognition problems, it is more accurate to model the uncertainty with a nonuniform distribution. In this case, we use type-II fuzzy sets $\text{Q}\text{(u)}$ and $\text{P}{}_{\mathrm{k}}\text{(u)}$ and (3) becomes

$$\text{k}{}^{\mathrm{\ast }}\text{=}\text{arg}\text{max}\text{k}\text{(}\text{S}\text{(}\text{Q}\text{(u),}\text{P}{}_{\mathrm{k}}\text{(u))),}$$

where $\text{S}$ denotes a type-II similarity measure.

The main advantages of using a type-II framework are twofold:

• By using type-II fuzzy sets we transform a vague pattern classification problem into a precise, well-defined, optimization problem.

• Type-II fuzzy sets, unlike ordinary fuzzy sets, retain a controlled degree of uncertainty.

The main disadvantage to using a type-II formulation is

• The relatively high computational complexity.

The article is organized as follows. In Section 2 we discuss the concept of type-II fuzzy sets. In Section 3 we describe a new type-II similarity measure $\text{S}$ which is specifically designed to measure the compatibility of two type-II fuzzy sets. In Section 4 we show how we may reformulate the problem of pattern classification using type-II fuzzy sets. In Section 5 we analyze a real-left pattern classification problem and show how we may solve it, and similar classification problems, using the new approach. Finally the article concludes with a brief summary in Section 6.