Synopsis
1) Introduction 7
2) Forward Selection Component Analysis 11
A) Introduction to Forward Selection Component Analysis 12
B) The Mathematics and Code Examples 16
Maximizing the Explained Variance 18
Code for the Variance Maximization Criterion 20
Backward Refinement 24
Multi-Threading Backward Refinement 28
Orthogonalizing Ordered Components 36
C) Putting It All Together 39
Components From a Forward-Only Subset 44
Components From a Backward Refined Subset 46
D) An Example With Contrived Variables 48
3) Local Feature Selection 53
A) Intuitive Overview of the Algorithm 54
What This Algorithm Reports 60
B) A Brief Detour: the Simplex Algorithm 62
The Linear Programming Problem 63
Interfacing to the Simplex Class 64
A Little More Detail 67
C) A More Rigorous Approach to LFS 69
Intra-Class and Inter-Class Separation 73
Computing the Weights 77
Maximizing Inter-Class Separation 81
Minimizing Intra-Class Separation 86
Testing a Trial Beta 88
A Quick Note on Threads 93
D) CUDA Computation of Weights 94
Integrating the CUDA Code Into the Algorithm 95
Initializing the CUDA Hardware 97
Computing Differences from the Current Case 100
Computing the Distance Matrix 102
Computing the Minimum Distances 104
Computing the Terms for the Weight Equation 112
Transposing the Term Matrix 113
Summing the Terms For the Weights 114
Moving the Weights to the Host 116
E) An Example of Local Feature Selection 117
F) A Note on Run Time 118
4) Memory in Time Series Features 119
A) A Gentle Mathematical Overview 122
The Forward Algorithm 123
The Backward Algorithm 128
Correct Alpha and Beta, For Those Who Care 131
B) Some Mundane Computations 136
Means and Covariances 136
Densities 138
The Multivariate Normal Density Function 139
C) Starting Parameters 141
Outline of the Initialization Algorithm 141
Perturbing Means 142
Perturbing Covariances 143
Perturbing Transition Probabilities 144
A Note on Random Number Generators 145
D) The Complete Optimization Algorithm 146
Computing State Probabilities 147
Updating the Means and Covariances 151
Updating Initial and Transition Probabilities 153
E) Assessing HMM Memory in a Time Series 159
F) Linking Features to a Target 164
Linking HMM States to the Target 173
A Contrived and Inappropriate Example 183
A Sensible and Practical Example 186
5) Stepwise Selection on Steroids 189
A) The Feature Evaluation Model 192
Code For the Foundation Model 193
B) The Cross-Validated Performance Measure 198
C) The Stepwise Algorithm 201
Finding the First Variable 207
Adding a Variable to an Existing Model 210
D) Demonstrating the Algorithm Three Ways 214
6) Nominal-to-Ordinal Conversion 217
A) Implementation Overview 221
B) Testing For a Legitimate Relationship 222
C) An Example From Equity Price Changes 223
D) Code for Nominal-to-Ordinal Conversion 227
The Constructor 228
Printing the Table of Counts 232
Computing the Mapping Function 234
Monte-Carlo Permutation Tests 237
7) Index 353
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