Using Non-Additive Measure for Optimization-Based Nonlinear Classification

Over the past few decades, numerous optimization-based methods have been proposed for solving the classification problem in data mining. Classic optimization-based methods do not consider attribute interactions toward classification. Thus, a novel learning machine is needed to provide a better understanding on the nature of classification when the interaction among contributions from various attributes cannot be ignored. The interactions can be described by a non-additive measure while the Choquet integral can serve as the mathematical tool to aggregate the values of attributes and the corresponding values of a non-additive measure. As a main part of this research, a new nonlinear classification method with non-additive measures is proposed. Experimental results show that applying non-additive measures on the classic optimization-based models improves the classification robustness and accuracy compared with some popular classification methods. In addition, motivated by well-known Support Vector Machine approach, we transform the primal optimization-based nonlinear classification model with the signed non-additive measure into its dual form by applying Lagrangian optimization theory and Wolfes dual programming theory. As a result, 2 – 1 parameters of the signed non-additive measure can now be approximated with m (number of records) Lagrangian multipliers by applying necessary conditions of the primal classification problem to be optimal. This method of parameter approximation is a breakthrough for solving a non-additive measure practically when there are a relatively small number of training cases available (). Furthermore, the kernel-based learning method engages the nonlinear classifiers to achieve better classification accuracy. The research produces practically deliverable nonlinear models with the non-additive measure for classification problem in data mining when interactions among attributes are considered

Genetic Algorithm Optimization for Determining Fuzzy Measures from Fuzzy Data

Fuzzy measures and fuzzy integrals have been successfully used in many real applications. How to determine fuzzy measures is a very difficult problem in these applications. Though there have existed some methodologies for solving this problem, such as genetic algorithms, gradient descent algorithms, neural networks, and particle swarm algorithm, it is hard to say which one is more appropriate and more feasible. Each method has its advantages. Most of the existed works can only deal with the data consisting of classic numbers which may arise limitations in practical applications. It is not reasonable to assume that all data are real data before we elicit them from practical data. Sometimes, fuzzy data may exist, such as in pharmacological, financial and sociological applications. Thus, we make an attempt to determine a more generalized type of general fuzzy measures from fuzzy data by means of genetic algorithms and Choquet integrals. In this paper, we make the first effort to define the σ-λ rules. Furthermore we define and characterize the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on σ-λ rules. In addition, we design a special genetic algorithm to determine a type of general fuzzy measures from fuzzy data

Fuzzy Operator Trees for Modeling Utility Functions

In this thesis, we propose a method for modeling utility (rating) functions based on a novel concept called textbf{Fuzzy Operator Tree} (FOT for short). As the notion suggests, this method makes use of techniques from fuzzy set theory and implements a fuzzy rating function, that is, a utility function that maps to the unit interval, where $0$ corresponds to the lowest and $1$ to the highest evaluation. Even though the original motivation comes from quality control, FOTs are completely general and widely applicable. Our approach allows a human expert to specify a model in the form of an FOT in a quite convenient and intuitive way. To this end, he simply has to split evaluation criteria into sub-criteria in a recursive manner, and to determine in which way these sub-criteria ought to be combined: conjunctively, disjunctively, or by means of an averaging operator. The result of this process is the qualitative structure of the model. A second step, then, it is to parameterize the model. To support or even free the expert form this step, we develop a method for calibrating the model on the basis of exemplary ratings, that is, in a purely data-driven way. This method, which makes use of optimization techniques from the field of evolutionary algorithms, constitutes the second major contribution of the thesis. The third contribution of the thesis is a method for evaluating an FOT in a cost-efficient way. Roughly speaking, an FOT can be seen as an aggregation function that combines the evaluations of a number of basic criteria into an overall rating of an object. Essentially, the cost of computing this rating is hence given by sum of the evaluation costs of the basic criteria. In practice, however, the precise utility degree is often not needed. Instead, it is enough to know whether it lies above or below an important threshold value. In such cases, the evaluation process, understood as a sequential evaluation of basic criteria, can be stopped as soon as this question can be answered in a unique way. Of course, the (expected) number of basic criteria and, therefore, the (expected) evaluation cost will then strongly depend on the order of the evaluations, and this is what is optimized by the methods that we have developed