Introduction | p. ix |
Stochastic Processes | p. 1 |
Introduction | p. 1 |
Foundations of Probability | p. 1 |
Intuitive Definition of Probability | p. 3 |
Random Variables | p. 11 |
Stochastic Processes | p. 15 |
Probability Distribution and Probability Density | p. 19 |
Expectation and Conditional Mathematical Expectation | p. 23 |
Characteristic Function | p. 35 |
Orthogonal and Uncorrelated Sequences | p. 37 |
On Convergence | p. 41 |
Finite Markov Chains | p. 46 |
Renewal Processes | p. 62 |
Martingale, Supermartingale, Submartingale | p. 68 |
Examples of Martingales | p. 69 |
Martingale Convergence Theorems | p. 80 |
Some Definitions | p. 84 |
Conclusions | p. 88 |
Estimation of Probability Densities | p. 93 |
Introduction | p. 93 |
Main Probability Distributions | p. 93 |
Bernoulli Distribution | p. 94 |
Binomial Distribution | p. 94 |
Hypergeometric Distribution | p. 95 |
Poisson Distribution | p. 96 |
Gaussian Distribution | p. 99 |
Truncated Normal Distribution | p. 102 |
Lognormal Distribution | p. 102 |
Laplace Distribution | p. 104 |
Cauchy Distribution | p. 105 |
Exponential Distribution | p. 106 |
Geometric Distribution | p. 108 |
Rayleigh Distribution | p. 109 |
Gamma Distribution | p. 110 |
Weibull Distribution | p. 111 |
Skewness and Kurtosis Measures | p. 117 |
Classification of Probability Distributions | p. 121 |
Transformation of Random Variables | p. 123 |
Estimation of Probability Density Functions | p. 130 |
Method of Moments | p. 131 |
Series of Rectangular Pulses | p. 132 |
Modeling Using Polynomials | p. 133 |
Kernel Estimators | p. 134 |
Maximum Likelihood | p. 136 |
Expectation Maximization | p. 138 |
Neural Networks | p. 141 |
Numerical Examples | p. 146 |
EM algorithm | p. 146 |
Kurtosis-based EM-algorithm | p. 147 |
Maximum Likelihood Estimation Using SNN | p. 149 |
Model Validation | p. 153 |
The x[superscript 2] Approach | p. 153 |
Kolmogorov Criterion | p. 156 |
Stochastic Approximation | p. 157 |
Conclusion | p. 160 |
Optimization Techniques | p. 167 |
Introduction | p. 167 |
Stochastic Approximation Techniques | p. 168 |
Unconstrained Optimization Using Gradient Measurements | p. 168 |
Unconstrained Optimization Using Function Measurements | p. 170 |
Optimization under Constraints | p. 171 |
Learning Automata | p. 173 |
Learning Automaton | p. 173 |
Unconstrained Optimization | p. 177 |
Optimization under Constraints | p. 183 |
Applications of Learning Automata | p. 188 |
Simulated Annealing | p. 211 |
Genetic Algorithms | p. 214 |
Conclusions | p. 216 |
Analysis of Recursive Algorithms | p. 223 |
Introduction | p. 223 |
The Analysis of Recursive Algorithms | p. 224 |
Vector Form | p. 224 |
Lyapunov Approach | p. 226 |
Robbins-Monro Approach | p. 229 |
The Ordinary Differential Equation | p. 232 |
Summary | p. 233 |
Use of Some Inequalities, Lemmas and Theorems | p. 234 |
Case 1: Single Learning Automaton | p. 254 |
Learning Automata | p. 255 |
Projection Procedure | p. 256 |
Stochastic Learning Model | p. 257 |
Asymptotic Properties | p. 260 |
Numerical Example | p. 263 |
Conclusions | p. 269 |
Case 2: Team of Binary Learning Automata | p. 269 |
Stochastic Learning Automata | p. 271 |
Optimization Algorithm | p. 272 |
Asymptotic Properties | p. 279 |
Numerical Example | p. 285 |
Conclusions | p. 287 |
Convergence Rate | p. 288 |
Convergence with Probability 1 | p. 290 |
Normalized Deviation | p. 293 |
Rate of Mean Squares Convergence | p. 295 |
Asymptotic Normality and Rate of Convergence in Distribution Sense | p. 296 |
Rate of Almost Sure Convergence | p. 298 |
Optimization of the Convergence Rate on the Basis of the Gain Matrix Selection | p. 298 |
Feasible Realization of the Optimal Gain Matrix | p. 301 |
Summary | p. 305 |
Inequalities, Lemmas and Theorems | p. 315 |
Inequalities | p. 315 |
Lemmas | p. 320 |
Theorems | p. 323 |
Matlab Program | p. 327 |
Index | p. 329 |
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