A New Efficient and Effective Fuzzy Modeling Method for Binary Classification

Introduction

Binary classification refers to the task of classifying an unknown object into one of the two known classes. Many real world problems can be formulated as binary classification problems. The examples are numerous, which include determining whether a patient has certain disease or not, determining whether an artifact is defective or not, determining whether an email is a spam or not, and so on.

Before classification can be carried out, a classification system or a classifier must be built first. A classification system or classifier built for performing binary classification is called a binary classifier. Many machine learning methods such as decision trees, neural networks, support vector machines, and fuzzy models are suitable for building binary classifiers. Of interest in this paper are fuzzy models.

A fuzzy model is comprised of a set of fuzzy if-then rules. Various forms of fuzzy rules are possible. Most well-known among them are Mamdani linguistic rules and TSK rules, as described below for an n-input and single-output system.

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  Mamdani linguistic rules are of the form of R_i: If X₁ is A_1i and X₂ is A_2i, …, and X_n is A_ni, then Y is B_i. In each Mamdani rule, X_n denotes the nth input variable, A_ni the fuzzy set associated with A_n, and B_i the fuzzy set associated with output variable Y of rule R_i. B_i can be represented by its modal value or as a fuzzy singleton for fuzzy control and classification applications.

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  TSK rules are of the form of R_i: If X₁ is A_1i and X₂ is A_2i, …, and X_n is A_ni, then Y = b_0i + b_1iX₁ + b_2iX₂ + … + b_niX_n. In each TSK rule, X_n denotes the nth input variable, A_ni the fuzzy set associated with A_n, and b_i is a parameter vector and b_0i is a scalar offset associated with rule R_i. If all parameters in the vector b_i are zeros, then TSK rules are identical to Mamdani rules with B_i being fuzzy singletons.

A fuzzy model built for classification consists of a set of fuzzy classification rules of the following form R_i: If X₁ is A_1i and X₂ is A_2i, …, and X_n is A_ni, then Y is B_i. In each classification rule, X_n denotes the nth input variable, A_ni the fuzzy set associated with A_n, and B_i ∈ {B₁, B₂, …, B_m} representing the class label of rule R_i. For binary classification problems, there are only two values, i.e., B_i ∈ {B₁, B₂}. Fuzzy classification rules are identical to Mamdani rules with B_i being fuzzy singletons or TSK rules with b_i being zeros. Each rule could be assigned a rule weight. If not explicitly assigned, each rule is considered having rule weight of one.

Fuzzy models can be built manually or interactively using a tool designed to do that, e.g. the FIS Editor in the Matlab fuzzy toolbox. These approaches, however, require a priori knowledge about the model. The alternative is to build fuzzy models from data for which numerous methods have been proposed. These data-driven methods can be grouped into the following categories (Liao, 2006):