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Gerhard Bohm, Günter ZechIntroduction to Statistics and DataAnalysis for PhysicistsVerlag Deutsches Elektronen-Synchrotron
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Copyright:Gerhard Bohm, Günter Zech
ISBN978-3-935702-41-6
DOI10.3204/DESY-BOOK/statistics (e-book)
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February 2010,
Gerhard Bohm, Günter Zech
Contents1 Introduction: Probability and Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Purpose of Statistics . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 1
1.2 Event, Observation and Measurement . . . . . . . . . . . . . . . . .. . . . . . . . . . . 2
1.3 How to Define Probability? . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3
1.4 Assignment of Probabilities to Events . . . . . . . . . . . . . . .. . . . . . . . . . . . . 4
1.5 Outline of this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 6
2 Basic Probability Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Random Events and Variables. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 9
2.2 Probability Axioms and Theorems . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 10
2.2.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 10
2.2.2 Conditional Probability, Independence, and Bayes" Theorem . . 11
3 Probability Distributions and their Properties. . . . . . . . . . . . . . . . . . . 15
3.1 Definition of Probability Distributions . . . . . . . . . . . . .. . . . . . . . . . . . . . 16
3.1.1 Discrete Distributions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 16
3.1.2 Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 16
3.1.3 Empirical Distributions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 20
3.2 Expected Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 20
3.2.1 Definition and Properties of the Expected Value. . . . . .. . . . . . . 21
3.2.2 Mean Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 22
3.2.3 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 23
3.2.4 Skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 26
3.2.5 Kurtosis (Excess). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 26
3.2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 27
3.2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 28
II Contents
3.3 Moments and Characteristic Functions . . . . . . . . . . . . . . .. . . . . . . . . . . . 32
3.3.1 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 32
3.3.2 Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 33
3.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 36
3.4 Transformation of Variables. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 38
3.4.1 Calculation of the Transformed Density . . . . . . . . . . . .. . . . . . . . 39
3.4.2 Determination of the Transformation Relating two Distributions 41
3.5 Multivariate Probability Densities . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 42
3.5.1 Probability Density of two Variables . . . . . . . . . . . . . .. . . . . . . . . 43
3.5.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 44
3.5.3 Transformation of Variables . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 46
3.5.4 Reduction of the Number of Variables. . . . . . . . . . . . . . .. . . . . . . 47
3.5.5 Determination of the Transformation between two Distributions 50
3.5.6 Distributions of more than two Variables . . . . . . . . . . .. . . . . . . . 51
3.5.7 Independent, Identically Distributed Variables . . .. . . . . . . . . . . 52
3.5.8 Angular Distributions . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 53
3.6 Some Important Distributions. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 55
3.6.1 The Binomial Distribution. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 55
3.6.2 The Multinomial Distribution . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 58
3.6.3 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 58
3.6.4 The Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 65
3.6.5 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 65
3.6.6 The Exponential Distribution . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 69
3.6.7 Theχ2Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6.8 The Gamma Distribution. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 72
3.6.9 The Lorentz and the Cauchy Distributions. . . . . . . . . . .. . . . . . . 74
3.6.10 The Log-normal Distribution . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 75
3.6.11 Student"stDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.6.12 The Extreme Value Distributions . . . . . . . . . . . . . . . . .. . . . . . . . . 77
4 Measurement errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1 General Considerations.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 81
4.1.1 Importance of Error Assignments. . . . . . . . . . . . . . . . . .. . . . . . . . 81
4.1.2 Verification of Assigned Errors . . . . . . . . . . . . . . . . . . .. . . . . . . . . 82
4.1.3 The Declaration of Errors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 82
Contents III
4.1.4 Definition of Measurement and its Error. . . . . . . . . . . . .. . . . . . . 83
4.2 Different Types of Measurement Uncertainty . . . . . . . . . . .. . . . . . . . . . . 84
4.2.1 Statistical Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 84
4.2.2 Systematic Errors (G. Bohm) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 88
4.2.3 Systematic Errors (G. Zech) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 90
4.2.4 Controversial Examples . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 94
4.3 Linear Propagation of Errors. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 94
4.3.1 Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 94
4.3.2 Error of a Function of Several Measured Quantities . . .. . . . . . . 95
4.3.3 Averaging Uncorrelated Measurements . . . . . . . . . . . . .. . . . . . . . 98
4.3.4 Averaging Correlated Measurements . . . . . . . . . . . . . . .. . . . . . . . 98
4.3.5 Several Functions of Several Measured Quantities. . .. . . . . . . . . 100
4.3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 101
4.4 Biased Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 103
4.5 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 104
5 Monte Carlo Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 107
5.2 Generation of Statistical Distributions . . . . . . . . . . . .. . . . . . . . . . . . . . . 109
5.2.1 Computer Generated Pseudo Random Numbers . . . . . . . . . .. . . 109
5.2.2 Generation of Distributions by Variable Transformation . . . . . . 110
5.2.3 Simple Rejection Sampling . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 115
5.2.4 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 116
5.2.5 Treatment of Additive Probability Densities . . . . . . .. . . . . . . . . 119
5.2.6 Weighting Events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 120
5.2.7 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 120
5.3 Solution of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 123
5.3.1 Simple Random Selection Method . . . . . . . . . . . . . . . . . . .. . . . . . 123
5.3.2 Improved Selection Method . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 126
5.3.3 Weighting Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 127
5.3.4 Reduction to Expected Values . . . . . . . . . . . . . . . . . . . . .. . . . . . . 128
5.3.5 Stratified Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 129
5.4 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 129
6 Parameter Inference I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 131
IV Contents
6.2 Inference with Given Prior. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 133
6.2.1 Discrete Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 133
6.2.2 Continuous Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 135
6.3 Definition and Visualization of the Likelihood . . . . . . . .. . . . . . . . . . . . . 137
6.4 The Likelihood Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 140
6.5 The Maximum Likelihood Method for Parameter Inference .. . . . . . . . 142
6.5.1 The Recipe for a Single Parameter. . . . . . . . . . . . . . . . . .. . . . . . . 143
6.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 144
6.5.3 Likelihood Inference for Several Parameters . . . . . . .. . . . . . . . . . 148
6.5.4 Combining Measurements . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 151
6.5.5 Normally Distributed Variates andχ2. . . . . . . . . . . . . . . . . . . . . . 151
6.5.6 Likelihood of Histograms . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 152
6.5.7 Extended Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 154
6.5.8 Complicated Likelihood Functions . . . . . . . . . . . . . . . .. . . . . . . . . 155
6.5.9 Comparison of Observations with a Monte Carlo Simulation . . 155
6.5.10 Parameter Estimate of a Signal Contaminated by Background 160
6.6 Inclusion of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 163
6.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 163
6.6.2 Eliminating Redundant Parameters . . . . . . . . . . . . . . . .. . . . . . . . 164
6.6.3 Gaussian Approximation of Constraints . . . . . . . . . . . .. . . . . . . . 166
6.6.4 The Method of Lagrange Multipliers . . . . . . . . . . . . . . . .. . . . . . . 167
6.6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 168
6.7 Reduction of the Number of Variates. . . . . . . . . . . . . . . . . .. . . . . . . . . . . 168
6.7.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 168
6.7.2 Two Variables and a Single Linear Parameter . . . . . . . . .. . . . . . 169
6.7.3 Generalization to Several Variables and Parameters .. . . . . . . . . 169
6.7.4 Non-linear Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 171
6.8 Method of Approximated Likelihood Estimator. . . . . . . . .. . . . . . . . . . . 171
6.9 Nuisance Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 174
6.9.1 Nuisance Parameters with Given Prior . . . . . . . . . . . . . .. . . . . . . 175
6.9.2 Factorizing the Likelihood Function. . . . . . . . . . . . . .. . . . . . . . . . 176
6.9.3 Parameter Transformation, Restructuring . . . . . . . . .. . . . . . . . . 177
6.9.4 Profile Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 179
6.9.5 Integrating out the Nuisance Parameter . . . . . . . . . . . .. . . . . . . . 181
6.9.6 Explicit Declaration of the Parameter Dependence. . .. . . . . . . . 181
Contents V
6.9.7 Advice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 181
7 Parameter Inference II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.1 Likelihood and Information . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 183
7.1.1 Sufficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 183
7.1.2 The Conditionality Principle . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 185
7.1.3 The Likelihood Principle . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 186
7.1.4 Bias of Maximum Likelihood Results. . . . . . . . . . . . . . . .. . . . . . . 187
7.1.5 Stopping Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 190
7.2 Further Methods of Parameter Inference. . . . . . . . . . . . . .. . . . . . . . . . . . 191
7.2.1 The Moments Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 191
7.2.2 The Least Square Method . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 195
7.2.3 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 198
7.3 Comparison of Estimation Methods . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 199
8 Interval Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 201
8.2 Error Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 202
8.2.1 Parabolic Approximation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 203
8.2.2 General Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 204
8.3 Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 205
8.3.1 Averaging Measurements . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 205
8.3.2 Approximating the Likelihood Function . . . . . . . . . . . .. . . . . . . . 208
8.3.3 Incompatible Measurements . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 209
8.3.4 Error Propagation for a Scalar Function of a Single Parameter 210
8.3.5 Error Propagation for a Function of Several Parameters . . . . . . 210
8.4 One-sided Confidence Limits . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 214
8.4.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 214
8.4.2 Upper Poisson Limits, Simple Case . . . . . . . . . . . . . . . . .. . . . . . . 215
8.4.3 Poisson Limit for Data with Background . . . . . . . . . . . . .. . . . . . 216
8.4.4 Unphysical Parameter Values . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 219
8.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 219
9 Deconvolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 221
9.1.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 221
VI Contents
9.1.2 Deconvolution by Matrix Inversion . . . . . . . . . . . . . . . .. . . . . . . . 224
9.1.3 The Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 226
9.1.4 Regularization Methods . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 226
9.2 Deconvolution of Histograms . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 227
9.2.1 Fitting the Bin Content . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 227
9.2.2 Iterative Deconvolution . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 231
9.2.3 Regularization of the Transfer Matrix . . . . . . . . . . . . .. . . . . . . . . 232
9.3 Binning-free Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 234
9.3.1 Iterative Deconvolution . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 234
9.3.2 The Satellite Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 235
9.3.3 The Maximum Likelihood Method . . . . . . . . . . . . . . . . . . . .. . . . . 237
9.4 Comparison of the Methods. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 239
9.5 Error Estimation for the Deconvoluted Distribution . . .. . . . . . . . . . . . . 241
10 Hypothesis Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 245
10.2 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 246
10.2.1 Single and Composite Hypotheses . . . . . . . . . . . . . . . . .. . . . . . . . 246
10.2.2 Test Statistic, Critical Region and Significance Level . . . . . . . . . 246
10.2.3 Errors of the First and Second Kind, Power of a Test . . .. . . . . 247
10.2.4 P-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 248
10.2.5 Consistency and Bias of Tests . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 248
10.3 Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 250
10.3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 250
10.3.2 P-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 252
10.3.3 Theχ2Test in Generalized Form . . . . . . . . . . . . . . . . . . . . . . . . . . 254
10.3.4 The Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 261
10.3.5 The Kolmogorov-Smirnov Test. . . . . . . . . . . . . . . . . . . .. . . . . . . . 263
10.3.6 Tests of the Kolmogorov-Smirnov - and Cramer-von Mises
Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 265
10.3.7 Neyman"s Smooth Test. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 266
10.3.8 TheL2Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
10.3.9 Comparing a Data Sample to a Monte Carlo Sample and the
Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 269
10.3.10 The k-Nearest Neighbor Test . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 270
10.3.11 The Energy Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 270
Contents VII
10.3.12 Tests Designed for Specific Problems . . . . . . . . . . . . .. . . . . . . . . 272
10.3.13 Comparison of Tests . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 273
10.4 Two-Sample Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 275
10.4.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 275
10.4.2 Theχ2Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
10.4.3 The Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 276
10.4.4 The Kolmogorov-Smirnov Test. . . . . . . . . . . . . . . . . . . .. . . . . . . . 277
10.4.5 The Energy Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 277
10.4.6 The k-Nearest Neighbor Test . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 278
10.5 Significance of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 279
10.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 279
10.5.2 The Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 281
10.5.3 Tests Based on the Signal Strength . . . . . . . . . . . . . . . .. . . . . . . . 286
11 Statistical Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 289
11.2 Smoothing of Measurements and Approximation by Analytic
Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 291
11.2.1 Smoothing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 292
11.2.2 Approximation by Orthogonal Functions . . . . . . . . . . .. . . . . . . . 294
11.2.3 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 298
11.2.4 Spline Approximation . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 300
11.2.5 Approximation by a Combination of Simple Functions .. . . . . . 302
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