Preface |
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ix | |
Author |
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xi | |
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Chapter 1 Probability Basics: A Retrospective |
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1 | (54) |
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1.1 What Is "Probability"? |
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1 | (2) |
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2 | (1) |
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3 | (2) |
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1.3 Conditional Probability and Independence |
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5 | (4) |
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Summary: Important Laws of Probability |
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8 | (1) |
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1.4 Permutations and Combinations |
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9 | (1) |
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1.5 Continuous Random Variables |
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10 | (6) |
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Summary: Important Facts about Continuous Random Variables |
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15 | (1) |
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1.6 Countability and Measure Theory |
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16 | (2) |
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18 | (3) |
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Summary: Important Facts about Expected Value and Moments |
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21 | (1) |
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1.8 Derived Distributions |
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21 | (3) |
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Summary: Important Facts about Change of Variable |
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24 | (1) |
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1.9 The Normal or Gaussian Distribution |
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24 | (4) |
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Summary: Important Equations Involving the Normal (Gaussian) Distribution |
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28 | (1) |
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1.10 Multivariate Statistics |
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28 | (2) |
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1.11 The Bivariate Probability Density Functions |
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30 | (5) |
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34 | (1) |
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Summary: Important Equations for Bivariate Random Variables |
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35 | (1) |
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1.12 The Bivariate Gaussian Distribution |
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35 | (4) |
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38 | (1) |
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Summary of Important Equations for the Bivariate Gaussian |
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39 | (1) |
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1.13 Sums of Random Variables |
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39 | (5) |
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43 | (1) |
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Summary of Important Equations for Sums of Random Variables |
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44 | (1) |
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1.14 The Multivariate Gaussian |
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44 | (2) |
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1.15 The Importance of the Normal Distribution |
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46 | (9) |
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47 | (8) |
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Chapter 2 Random Processes |
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55 | (20) |
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2.1 Examples of Random Processes |
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55 | (6) |
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2.2 The Mathematical Characterization of Random Processes |
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61 | (6) |
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Summary: The First and Second Moments of Random Processes |
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64 | (3) |
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2.3 Prediction: The Statistician's Task |
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67 | (8) |
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69 | (6) |
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Chapter 3 Analysis of Raw Data |
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75 | (36) |
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3.1 Stationarity and Ergodicity |
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75 | (2) |
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3.2 The Limit Concept in Random Processes |
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77 | (2) |
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3.3 Spectral Methods for Obtaining Autocorrelations |
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79 | (3) |
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3.4 Interpretation of the Discrete Time Fourier Transform |
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82 | (1) |
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3.5 The Power Spectral Density |
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83 | (6) |
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3.6 Interpretation of the Power Spectral Density |
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89 | (2) |
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3.7 Engineering the Power Spectral Density |
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91 | (4) |
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3.8 Back to Estimating the Autocorrelation |
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95 | (4) |
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99 | (1) |
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3.9 Optional Reading the Secret of Bartlett's Method |
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99 | (5) |
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3.10 Spectral Analysis for Continuous Random Processes |
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104 | (7) |
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Summary: Spectral Properties of Discrete and Continuous Random Processes |
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105 | (1) |
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105 | (6) |
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Chapter 4 Models for Random Processes |
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111 | (40) |
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4.1 Differential Equations Background |
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111 | (1) |
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112 | (3) |
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115 | (1) |
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4.4 The Yule--Walker Equations |
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116 | (2) |
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118 | (1) |
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4.5 Construction of ARMA Models |
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118 | (1) |
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4.6 Higher-Order ARMA Processes |
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119 | (3) |
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122 | (3) |
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124 | (1) |
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4.8 The Bernoulli and Binomial Processes |
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125 | (3) |
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Summary: Bernoulli Process |
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125 | (1) |
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126 | (2) |
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Summary: Binomial Process |
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128 | (1) |
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4.9 Shot Noise and the Poisson Process |
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128 | (8) |
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Online Sources and Demonstrations |
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136 | (1) |
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4.10 Random Walks and the Wiener Process |
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136 | (3) |
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138 | (1) |
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139 | (12) |
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144 | (1) |
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Summary: Common Random Process Models |
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144 | (2) |
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146 | (5) |
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Chapter 5 Least Mean-Square Error Predictors |
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151 | (18) |
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5.1 The Optimal Constant Predictor |
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151 | (1) |
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5.2 The Optimal Constant-Multiple Predictor |
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152 | (1) |
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5.3 Digression: Orthogonality |
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152 | (2) |
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5.4 Multivariate LMSE Prediction: The Normal Equations |
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154 | (2) |
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156 | (1) |
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157 | (1) |
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5.6 The Best Straight-Line Predictor |
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157 | (2) |
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5.7 Prediction for a Random Process |
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159 | (1) |
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5.8 Interpolation, Smoothing, Extrapolation, and Back-Prediction |
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160 | (1) |
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161 | (8) |
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166 | (1) |
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166 | (3) |
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Chapter 6 The Kalman Filter |
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169 | (24) |
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6.1 The Basic Kalman Filter |
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169 | (2) |
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6.2 Kalman Filter with Transition: Model and Examples |
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171 | (5) |
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Digression: Examples of the Kalman Model |
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172 | (1) |
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173 | (3) |
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6.3 Scalar Kalman Filter with Noiseless Transition |
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176 | (1) |
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6.4 Scalar Kalman Filter with Noisy Transition |
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177 | (2) |
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6.5 Iteration of the Scalar Kalman Filter |
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179 | (3) |
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6.6 Matrix Formulation for the Kalman Filter |
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182 | (11) |
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188 | (1) |
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189 | (4) |
Index |
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193 | |