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

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