Fuzzy Integral-Based Kernel Regression Ensemble and Its Application

Fuzzy Integral-Based Kernel Regression Ensemble and Its Application

Yulin He (Shenzhen University, China), James N. K. Liu (The Hong Kong Polytechnic University, Hong Kong), Yanxing Hu (The Hong Kong Polytechnic University, Hong Kong) and Xizhao Wang (Shenzhen University, China)
DOI: 10.4018/978-1-4666-7258-1.ch012
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Abstract

Similar to ensemble learning for classification, regression ensemble also tries to improve the prediction accuracy through combining several “weak” estimators that are usually high-variance and thus unstable. In this chapter, the authors propose a new scheme of fusing the weak Priestley-Chao Kernel Estimators (PCKEs) based on Choquet fuzzy integral, which differs from all the existing models of regressor fusion. The new scheme uses Choquet fuzzy integral to fuse several target outputs from different PCKEs, in which the optimal bandwidths are obtained with cross-validation criteria. The key of applying fuzzy integral to PCKE fusion is the determination of fuzzy measure. Considering the advantage of Particle Swarm Optimization (PSO) algorithm on convergence rate, the authors use three different PSO algorithms (i.e., Standard PSO [SPSO], Gaussian PSO [GPSO], and GPSO with Gaussian Jump [GPSOGJ]) to determine the general and ? fuzzy measures. The experimental results on the standard testing functions and practical Fourier Transform Infrared Spectroscopy (FTIR) datasets show that the new paradigm for regression ensemble based on fuzzy integral is more accurate and stable in comparison with any individual PCKE and the Basic Ensemble Method (BEM). This demonstrates the feasibility and effectiveness of the proposed regression ensemble model.
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Introduction

Ensemble learning (Zhou, 2012; Zhang & Ma, 2012) is a fusion strategy which tries to make the final decision by integrating the multiple feedbacks from different base-learners so as to reduce the decision maker's variance and improve its robustness and accuracy. That is to say a strong learner will be produced by organizing some weak ones in a proper way. Commonly, these weak learners are integrated through the majority voting for classification and a linear combination for regression (Brown et al., 2005). In recent years, the ensemble learning for classification has been well studied. There are a number of classical works which introduce the ensemble strategies for different classifiers, e.g., boosting or bagging based ensembles for decision trees (Banfield et al., 2007), neural networks (Zhou et al., 2002) and support vector machines (Kim et al., 2003), et c. However, just as said in (Moreiraa et al., 2012), the successful ensemble learning approaches for classification techniques are often not directly applicable to regression. Thus, unlike the sophisticated ensemble methods for classification, the regression ensemble often uses the weighted or ordered weighted average of base-learners to conduct the prediction, where several different methods are developed to determine the weights (Moreiraa et al., 2012).

The weighted average and ordered weighted average operators are good choices to deal with the different importance of individual base-regressor, but these two methods are under an assumption that interaction does not exist among the individual regressors. However, this assumption may not be true in many real problems. If the interaction is considered, fuzzy integrals (Grabisch, 1995(b); Cho & Kim, 1995) may be a better choice. The fuzzy integral as a fusion tool, in which the non-additive measure can clearly express the interaction among regressors and the importance of each individual regressor, has its particular advantages. Motivated by the definition of fuzzy integral which can be considered as a mechanism of maximizing the potential efficiency of base-regressor, we construct a new approach for regression ensemble based on fuzzy integral in this chapter.

One difficulty for applying fuzzy integrals in regressor fusion is how to determine the fuzzy measures. There are some methods to determine fuzzy measures such as Gradient Descent (GD) (Grabisch, 1995(a)), Genetic Algorithm (GA) (Yang et al., 2008; Ganesan et al., 2011(b)), Neural Network (NN) (Wang & Wang, 1997; Ganesan et al., 2011(a)), etc. Although using GD, GA and NN to determine the fuzzy measures is successful to some extent, there exist many limitations in the application process. For example, GD and NN frequently fall into the local minimum, and GA is much slower. It is necessary to find some new computational techniques for determining fuzzy measures. In 2011, Wang et al. (Wang et al., 2011) proposed Particle Swarm Optimization (PSO) based fuzzy measure determination, where PSO is a kind of swarm intelligence optimization algorithm (Ganesan et al., 2012; Ganesan et al., 2013(a); Ganesan et al., 2013(a)). The theoretical analysis and experimental comparison demonstrated the superior performance of PSO based methods. Thus, we use PSO to determine the fuzzy measure in fuzzy integral based regression ensemble in this chapter. The main contributions of this chapter can be summarized as 1) using fuzzy integral to construct the regression ensemble and 2) applying three different PSOs (i.e., Standard PSO-SPSO (Kennedy & Eberhart, 1995), Gaussian PSO-GPSO (Krohling, 2004) and GPSO with Gaussian Jump-GPSOGJ (Krohling, 2005)) to determine the general and λ fuzzy measures. In our study, a kind of kernel regressor, i.e., Priestley-Chao Kernel Estimator-PCKE (Priestley & Chao, 1972), is selected as the base-regressor. We call the Kernel Regression Ensemble model based on Fuzzy Integrals with the general and λ fuzzy measures KREFIg and KREFIλ respectively.

Key Terms in this Chapter

Fuzzy Integral: When the classical measure is generalized to the fuzzy measure, the classical integral with respect to classical measure should be generalized. The generalized integrals with respect to fuzzy measures are called fuzzy integrals.

Particle Swarm Optimization (PSO): It is one of swarm intelligence optimization algorithms, which is originally designed to simulate the simple social systems and then to explain the complex social behavior. PSO exploits a population of potential solutions (called particles) to probe the search space. Each particle in PSO changes its positions by learning the experiences from itself and its neighbors.

Fuzzy Measure: The intuitive interpretation of measure on a given set is a set function to assign a number to each suitable subset of this set, where this number is usually used to measure the size of subset. Fuzzy measure is an improved set function which uses the monotonicity instead of additivity.

Bandwidth: It is an important parameter for kernel density estimation. Gaussian function is one of mostly-used kernels. In this case, the kernel density estimation can be regarded as the superposition of different normal density functions, where the bandwidth is the standard deviation of normal density function.

Parzen Window: It is a typical probability density function (p.d.f.) estimation method, which uses the superposition of different kernels to fit the true p.d.f., which includes two important parameters, (i.e., kernel and bandwidth).

Kernel Regression: It uses the kernel density estimation method to solve the conditional expectation of a random variable. The objective of kernel regression is to find a non-linear relation between a pair of random variables X and Y .

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