Learning with Fractional Orthogonal Kernel Classifiers in Support Vector Machines. Theory, Algorithms and Applications

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Preface Contents Editors and Contributors Part I Basics of Support Vector Machines 1 Introduction to SVM 1.1 What Is Machine Learning? 1.1.1 Classification of Machine Learning Techniques 1.2 What Is the Pattern? 1.3 An Introduction to SVM with a Geometric Interpretation 1.4 History of SVMs 1.5 SVM Applications References 2 Basics of SVM Method and Least Squares SVM 2.1 Linear SVM Classifiers 2.1.1 Hard Margin SVM 2.1.2 Soft Margin SVM 2.2 Nonlinear SVM Classifiers 2.2.1 Kernel Trick and Mercer Condition 2.3 SVM Regressors 2.4 LS-SVM Classifiers 2.5 LS-SVM Regressors References Part II Special Kernel Classifiers 3 Fractional Chebyshev Kernel Functions: Theory and Application 3.1 Introduction 3.2 Preliminaries 3.2.1 Properties of Chebyshev Polynomials 3.2.2 Properties of Fractional Chebyshev Functions 3.3 Chebyshev Kernel Functions 3.3.1 Ordinary Chebyshev Kernel Function 3.3.2 Other Chebyshev Kernel Functions 3.3.3 Fractional Chebyshev Kernel 3.4 Application of Chebyshev Kernel Functions on Real Datasets 3.4.1 Spiral Dataset 3.4.2 Three Monks' Dataset 3.5 Conclusion References 4 Fractional Legendre Kernel Functions: Theory and Application 4.1 Introduction 4.2 Preliminaries 4.2.1 Properties of Legendre Polynomials 4.2.2 Properties of Fractional Legendre Functions 4.3 Legendre Kernel Functions 4.3.1 Ordinary Legendre Kernel Function 4.3.2 Other Legendre Kernel Functions 4.3.3 Fractional Legendre Kernel 4.4 Application of Legendre Kernel Functions on Real Datasets 4.4.1 Spiral Dataset 4.4.2 Three Monks' Dataset 4.5 Conclusion References 5 Fractional Gegenbauer Kernel Functions: Theory and Application 5.1 Introduction 5.2 Preliminaries 5.2.1 Properties of Gegenbauer Polynomials 5.2.2 Properties of Fractional Gegenbauer Polynomials 5.3 Gegenbauer Kernel Functions 5.3.1 Ordinary Gegenbauer Kernel Function 5.3.2 Validation of Gegenbauer Kernel Function 5.3.3 Other Gegenbauer Kernel Functions 5.3.4 Fractional Gegenbauer Kernel Function 5.4 Application of Gegenbauer Kernel Functions on Real Datasets 5.4.1 Spiral Dataset 5.4.2 Three Monks' Dataset 5.5 Conclusion References 6 Fractional Jacobi Kernel Functions: Theory and Application 6.1 Introduction 6.2 Preliminaries 6.2.1 Properties of Jacobi Polynomials 6.2.2 Properties of Fractional Jacobi Functions 6.3 Jacobi Kernel Functions 6.3.1 Ordinary Jacobi Kernel Function 6.3.2 Other Jacobi Kernel Functions 6.3.3 Fractional Jacobi Kernel 6.4 Application of Jacobi Kernel Functions on Real Datasets 6.4.1 Spiral Dataset 6.4.2 Three Monks' Dataset 6.5 Summary and Conclusion References Part III Applications of Orthogonal Kernels 7 Solving Ordinary Differential Equations by LS-SVM 7.1 Introduction 7.2 LS-SVM Formulation 7.2.1 Collocation Form of LS-SVM 7.3 Rational Legendre Kernels 7.4 Collocation Form of LS-SVM for Lane-Emden Type Equations 7.5 Numerical Examples 7.6 Conclusion References 8 Solving Partial Differential Equations by LS-SVM 8.1 Introduction 8.2 LS-SVM Method for Solving Second-Order Partial Differential Equations 8.2.1 Temporal Discretization 8.2.2 LS-SVM Collocation Method 8.3 Numerical Simulations 8.3.1 Fokker–Planck Equation 8.3.2 Generalized Fitzhugh–Nagumo Equation 8.4 Conclusion References 9 Solving Integral Equations by LS-SVR 9.1 Introduction 9.2 Integral Equations 9.2.1 Fredholm Integral Equations 9.2.2 Volterra Integral Equations 9.2.3 Volterra-Fredholm Integral Equations 9.2.4 Integro-Differential Equations 9.2.5 Multi-dimensional Integral Equations 9.2.6 System of Integral Equations 9.3 LS-SVR for Solving IEs 9.3.1 One-Dimensional Case 9.3.2 Multi-dimensional Case 9.3.3 System of Integral Equations 9.3.4 CLS-SVR Method 9.3.5 GLS-SVR Method 9.4 Numerical Simulations 9.5 Conclusion References 10 Solving Distributed-Order Fractional Equations by LS-SVR 10.1 Introduction 10.1.1 A Brief Review of Other Methods Existing in the Literature 10.2 Preliminaries 10.2.1 Fractional Derivative 10.2.2 Numerical Integration 10.3 LS-SVR Method for Solving Distributed-Order Fractional Differential Equations 10.4 Numerical Results and Discussion 10.4.1 Test Problem 1 10.4.2 Test Problem 2 10.4.3 Test Problem 3 10.4.4 Test Problem 4 10.4.5 Test Problem 5 10.5 Conclusion References Part IV Orthogonal Kernels in Action 11 GPU Acceleration of LS-SVM, Based on Fractional Orthogonal Functions 11.1 Parallel Processing 11.2 GPU Architecture 11.2.1 CUDA Programming with Python 11.3 Analyzing Codes and Functions 11.3.1 Analyzing the Training Function 11.3.2 Analyzing the Test Function 11.4 Hardware and Software Requirements 11.5 Accelerating the Chebyshev Kernel 11.6 More Optimizations 11.7 Accelerating the QP Solver 11.8 Conclusion References 12 Classification Using Orthogonal Kernel Functions: Tutorial on ORSVM Package 12.1 Introduction 12.1.1 ORSVM 12.2 How to Install 12.3 Model Class 12.4 SVM Class 12.4.1 Chebyshev Class 12.4.2 Legendre Class 12.4.3 Gegenbauer Class 12.4.4 Jacobi Class 12.5 Transformation Function 12.6 How to Use Appendix Appendix: Python Programming Prerequisite A.1 Introduction A.2 Basics of Python How to Use Python? Python Basics Basic Syntax Comments Variable Types Numbers and Casting Strings Lists Dictionary If ... Else While Loops For Loops Range, Break, and Continue Try, Expect Functions Libraries A.3 Pandas A.4 Numpy A.5 Matplotlib Pyplot Formatting the Style of Your Plot Plotting with Keyword Strings Plotting with Categorical Variables