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研究生： 郝沛毅
Hao, Pei-Yi
論文名稱： 使用支持向量學習的模糊決策模式-基於核心函式的方法論
Fuzzy Decision Model Using Support Vector Learning – A Kernel Function Based Approach
指導教授： 蔣榮先
Chiang, Jung-Hsien
學位類別： 博士
Doctor
系所名稱： 電機資訊學院 - 資訊工程學系
Department of Computer Science and Information Engineering
論文出版年： 2003
畢業學年度： 91
語文別： 英文
論文頁數： 179
中文關鍵詞： 模糊推論 、模糊聚類 、支持向量機 、類神經網路 、模糊
外文關鍵詞： SVM, neural networks, fuzzy, fuzzy inference, fuzzy clustering
相關次數： 點閱：156  下載：4 
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    • 支持向量機(Support Vector Machine; SVM) 是最近被提出來的一種機器學習的技術，它以Vapnik的統計學習理論為基礎，SVM不像傳統的機器學習技術以最小化經驗風險(Empirical Risk)為目標 — 即使得訓練資料的分類誤差最小，SVM以最小化結構風險(Structural Risk)為目標 — 即使得未知資料(測試資料)的分類誤差在一個機率上界以下。這種新的機器學習技術等同於最小化推理誤差的上界。
    當建構的模式充滿了不確定及複雜的現象，模糊模式是必須被考慮的，若以Zadeh所提出的模糊系統來表示，模糊決策在概念上是更接近於人類的決策方法。基本上，模糊理論提供了有效的方法擷取『近似於』、『不精確』等現實世界中的獨特特性。在這篇論文中，我們使用各種方法結合模糊理論與支持向量機的概念，嘗試去同時保留支持向量機的優點—即以統計學習理論保證的優良推理能力，與模糊理論的優點—即概念上更接近人類思考，以及能更有效的表達複雜與不確定的系統。具體而言我們提出了下面三種方法:
    方法1: 我們使用之支持向量學習機制，提供了一個架構能夠由訓練資料中萃取支持向量(support vector)，並經由適當的設計支持向量機中的核心函式(kernel function)，將每一個支持向量經轉化後解讀成為一條模糊『如果-則就』的規則。這樣的做法的好處為—我們不用再事先決定模糊『如果-則就』規則的數目，因為支持向量完全是由支持向量機學習機制自動決定的，以及最後得到的決策機制不再像傳統支持向量機是一個黑箱，而是可以由一條一條的模糊『如果-則就』規則來解讀。最後我們在經由實驗模擬證明其有效性。
    方法二: 我們將傳統的支持向量聚類機做了一個延伸，我們使用了模糊運算子中的『排序加權整合運算子(OWA operator)』，來建立一個適應性個體成長的支持向量聚類機，這樣處理的好處為我們可以得到每一個點對個別叢集的歸屬程度以及各個叢集的中心點。
    方法三: 在這裡我們結合了模糊集合理論與支持向量迴歸機的概念，在此時支持向量迴歸機中要被估計參數如權重與偏差量，不再是一個實數而是一個模糊數(fuzzy number)，而且迴歸時的期望輸出也是一個模糊數。所以我們迴歸最後得到的函數也是一個模糊函數，而迴歸的目標是將模糊函數輸出的模糊集合與期望輸出的模糊集合間的相配程度越密切越好。如今我們將一個充滿了曖昧、不確定的的模式使用了Zadeh所提出的模糊系統來表示。其概念更接近人類的思考與更符合現實世界的不確定性。

    Support Vector Machines (SVMs) have been recently introduced as a new technique for solving pattern recognition problems. According to the theory of SVMs, while traditional techniques for pattern recognition are based on the minimization of the empirical risk, that is, on the attempt to optimize the performance on the training set, SVMs minimize the structural risk, that is, the probability of misclassifying yet-to-been-seen patterns for a fixed but unknown probability distribution of the data.
    Fuzziness must be considered in systems where human estimation is influential. In this thesis, we incorporate the concept of fuzzy set theory into the support vector machine decision model in several approaches. We attempt to preserve the advantages of support vector machine (i.e. well generalization ability) and fuzzy set theory (i.e. closer to human thinking).
    First, we propose a fuzzy modeling framework based on support vector machine, a rule-based framework that explicitly characterizes the representation in fuzzy inference procedure. The support vector learning mechanism provides an architecture to extract support vectors for generating fuzzy IF-THEN rules from the training data set, and a method to describe the fuzzy system in terms of kernel functions. Thus, it has the inherent advantage that the model does not have to determine the number of rules in advance. Moreover, the decision model is not a black box anymore.
    Second, we enlarge SVM clustering by using a generalized ordered weighted averaging (OWA) operator to make multi-shpere SV clustering is capable of adaptively growing cell when a new point (but doesn’t belong to any existed cluster) is presented. Each sphere in the feature space corresponds to a cluster in original space and, whereby, it is possible to obtain the grade of fuzzy memberships, as well as cluster prototypes (sphere center) in partition.
    Third, we incorporate the concept of fuzzy set theory into the SVM regression. The parameters to be identified in SVM regression, such as the components within the weight vector and the bias term, are fuzzy numbers. The desired outputs in training samples are also fuzzy numbers. The SVM’s theory characterizes properties of learning machines which enable them to generalize well the unseen data and the fuzzy set theory might be very useful for finding a fuzzy structure in an evaluation system.

    Abstract------------------------------------------------------------------------i Acknowledgments-----------------------------------------------------------------v Table of Contents--------------------------------------------------------------vi List of Tables-----------------------------------------------------------------ix List of Figures-----------------------------------------------------------------x Chapter 1 Introduction----------------------------------------------------------1 1.1 Support Vector Machines----------------------------------------------------1 1.2 Fuzzy Theory---------------------------------------------------------------3 1.3 Motivation-----------------------------------------------------------------5 1.3.1 Support Vector Learning Mechanism for Fuzzy Rule-Based Modeling---------5 1.3.2 Fuzzy SVM Clustering with OWA-based Adaptively Cell Growing-------------6 1.3.3 A Fuzzy Model of Support Vector Machine Regression----------------------7 1.3.4 Redundant Support Vectors Pruning Algorithm-----------------------------8 1.4 Organization of Dissertation ----------------------------------------------9 Chapter 2 An Overview of Statistical Learning Theory---------------------------11 2.1 The General Setting of Learning Problem-----------------------------------11 2.1.1 The Problem of Pattern Recognition-------------------------------------12 2.1.2 The Problem of Regression Estimation-----------------------------------12 2.1.3 The Problem of Density Estimation--------------------------------------13 2.1.4 Setting of the Learning Problem----------------------------------------13 2.1.5 Empirical Risk Minimization Inductive Principle------------------------14 2.2 Consistency and Uniform Convergence for Learning Processes--------------- 15 2.3 Entropy of the Set of Functions and the Related Theory to Uniform Convergence---------------------------------17 2.3.1 Entropy of the Set of Indicator Functions------------------------------17 2.3.2 Entropy of the Set of Real Functions-----------------------------------19 2.4 Three Milestones in Learning Theorem--------------------------------------21 2.5 The Relationship between Growth Function and VC dimension-----------------23 2.6 An Equivalent Definition of the VC Dimension------------------------------25 2.7 A Bound on the Generalization Performance-------------------------------- 28 2.8 Minimizing the Risk Bound by Minimizing the VC Dimension------------------29 2.9 Structural Risk Minimization----------------------------------------------30 Chapter 3 Support Vector Machine Frameworks------------------------------------33 3.1 Support Vector Machine for Classification---------------------------------33 3.1.1 An Optimal Margin Classifier ------------------------------------------32 3.1.2 The Connection Between Margin and VC Dimension-------------------------34 3.1.3 Training Support Vector Machine ---------------------------------------36 3.1.4 Nonlinear Decision Surfaces--------------------------------------------39 3.2 Support Vector Machine for Regression-------------------------------------43 3.3 Support Vector Machine for Clustering-------------------------------------46 Chapter 4 Support Vector Learning Mechanism for Fuzzy Rule-Based Modeling------52 4.1 Fuzzy Inference Systems---------------------------------------------------53 4.1.1 Three Commonly Used Fuzzy Reasoning Mechasisms-------------------------54 4.1.2 Fuzzy Basis Function Inference System----------------------------------55 4.2 The Decision Procedure of Support Vector Machines-------------------------57 4.3 Modeling SVM-based Fuzzy Basis Function Inference System------------------59 4.4 Geometrical Analysis of the Fuzzy Rule Extraction-------------------------65 4.4.1 Comparing with Clustering Partition Based Fuzzy Rule Extracting--------65 4.4.2 Comparing with OLS based Fuzzy Rules Extracting------------------------67 4.5 Experimental Results------------------------------------------------------69 4.5.1 The Iris Pattern Classification Problem--------------------------------69 4.5.2 The Two Spirals Problem------------------------------------------------70 4.5.3 Prediction of the Chaotic Time-Series----------------------------------71 4.5.4 The Ball and Beam Problem----------------------------------------------74 4.6 Concluding Remark---------------------------------------------------------78 Chapter 5 A New Kernel-Based Fuzzy Clustering Approach: Support Vector Clustering with Cell Growing--------------79 5.1 Review of Clustering Methods----------------------------------------------80 5.1.1 Fuzzy Clustering-------------------------------------------------------80 5.1.2 Kernel-based Clustering -----------------------------------------------81 5.2 Proposed Multi-Sphere SV clustering---------------------------------------83 5.3 Fuzzy Memberships Computation and Heuristic Analysis----------------------90 5.3.1 Computation of cluster memberships-------------------------------------90 5.3.2 Estimation of winning cluster validity---------------------------------92 5.3.3 Two heuristic methods to reduce the computation time-------------------94 5.3.4 An incrementally learning algorithm for support vector clustering------96 5.3.4.1 The Kuhn-Trucker (KT) Conditions------------------------------------97 5.3.4.2 Adiabatic increments------------------------------------------------98 5.3.4.3 Bookkeeping: upper limit on increment------------------------------100 5.3.4.4 Recursive Matrix P updates-----------------------------------------101 5.3.4.5 Incremental procedure----------------------------------------------102 5.4 Experimental Results-----------------------------------------------------103 5.4.1 Synthesis Data Set----------------------------------------------------103 5.4.1.1 Unequal Populations Data Set---------------------------------------103 5.4.1.2 Random Noise Data Set----------------------------------------------104 5.4.1.3 The "XO" Data Set--------------------------------------------------105 5.4.2 Handwritten Digits Data Set-------------------------------------------105 5.5 Concluding Remark--------------------------------------------------------109 Chapter 6 A Fuzzy Model of Support Vector Machine Regression------------------110 6.1 Fuzzy Linear Function with Fuzzy Parameters------------------------------111 6.2 The Fitting Degree of two Fuzzy Number-----------------------------------113 6.3 Proposed Fuzzy SVM regression--------------------------------------------114 6.4 Experiment---------------------------------------------------------------119 6.5 Concluding Remark--------------------------------------------------------121 Chapter 7 Pruning Support Vectors in the SVM Framework------------------------123 7.1 Some notes on the SVM framework------------------------------------------124 7.2 An Example of Redundant Support Vectors----------------------------------126 7.3 Support Vector Pruning Algorithm-----------------------------------------128 7.3.1 Method I (Penalty Term Method)----------------------------------------129 7.3.2 Method II (Sensitivity Calculation Method)----------------------------133 7.4 Experiments--------------------------------------------------------------135 7.4.1 Synthesis Experiment--------------------------------------------------135 7.4.2 Face Detection Experiment---------------------------------------------137 7.5 Concluding Remark--------------------------------------------------------140 Chapter 8 Conclusions and Future Directions-----------------------------------141 References--------------------------------------------------------------------145 Appendix 1 SVM-based Fuzzy Inference Systems are Universal Approximators------164 Appendix 2 Derivation of Optimality Condition for Eq.(5.10)-------------------167 Appendix 3 Derivation of Incremental Step in Section 5.5.2--------------------171 Appendix 4 Derivation of Matrix Update Rules Eqs (5.25) and (5.26)------------173 Appendix 5 Proof of Proposition 6.1 - the Fuzzy Linear Function --------------176 List of Publications----------------------------------------------------------178 Vita -------------------------------------------------------------------------179

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