Modern Statistical Methods for Astronomy: With R Applications

Cover
......Page 1
Modern Statistical Methods for Astronomy......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Motivation and goals......Page 17
Outline and classroom use......Page 18
Acknowledgements......Page 19
1.1.1 Astronomy and astrophysics......Page 21
1.1.2 Probability and statistics......Page 23
1.1.3 Statistics and science......Page 24
1.2.1 Antiquity through the Renaissance......Page 26
1.2.2 Foundations of statistics in celestial mechanics......Page 27
1.2.3 Statistics in twentieth-century astronomy......Page 28
1.3 Recommended reading......Page 30
2.1 Uncertainty in observational science......Page 33
2.2 Outcome spaces and events......Page 34
2.3 Axioms of probability......Page 35
2.4 Conditional probabilities......Page 37
2.4.1 Bayes' theorem......Page 38
2.4.2 Independent events......Page 39
2.5 Random variables......Page 40
2.5.1 Density and distribution functions......Page 41
2.5.2 Independent and identically distributed random variables......Page 44
2.6 Quantile function......Page 45
2.7 Discrete distributions......Page 46
2.8 Continuous distributions......Page 47
2.9 Distributions that are neither discrete nor continuous......Page 49
2.11 Recommended reading......Page 50
2.12 R applications......Page 51
3.1 The astronomical context......Page 55
3.2 Concepts of statistical inference......Page 56
3.3 Principles of point estimation......Page 58
3.4 Techniques of point estimation......Page 60
3.4.1 Method of moments......Page 61
3.4.2 Method of least squares......Page 62
3.4.3 Maximum likelihood method......Page 63
3.4.4 Confidence intervals......Page 65
3.4.5 Calculating MLEs with the EM algorithm......Page 67
3.5 Hypothesis testing techniques......Page 68
3.6.1 Jackknife......Page 72
3.6.2 Bootstrap......Page 74
3.7 Model selection and goodness-of-fit......Page 77
3.7.1 Nonparametric methods for goodness-of-fit......Page 78
3.7.2 Likelihood-based methods for model selection......Page 80
3.7.3 Information criteria for model selection......Page 81
3.7.4 Comparing different model families......Page 82
3.8 Bayesian statistical inference......Page 83
3.8.1 Inference for the binomial proportion......Page 84
3.8.2 Prior distributions......Page 85
3.8.3 Inference for Gaussian distributions......Page 87
3.8.4 Hypotheses testing and the Bayes factor......Page 89
3.8.5 Model selection and averaging......Page 90
3.8.6 Bayesian computation......Page 91
3.9 Remarks......Page 92
3.10 Recommended reading......Page 93
3.11 R applications......Page 94
4.1 Binomial and multinomial......Page 96
4.1.1 Ratio of binomial random variables......Page 99
4.2.1 Astronomical context......Page 100
4.2.2 Mathematical properties......Page 101
4.2.3 Poisson processes......Page 103
4.3 Normal and lognormal......Page 105
4.4 Pareto (power-law)......Page 107
4.4.1 Least-squares estimation......Page 109
4.4.2 Maximum likelihood estimation......Page 110
4.4.3 Extensions of the power-law......Page 111
4.4.4 Multivariate Pareto......Page 112
4.4.5 Origins of power-laws......Page 113
4.5 Gamma......Page 114
4.7 R applications......Page 116
4.7.1 Comparing Pareto distribution estimators......Page 117
4.7.2 Fitting distributions to data......Page 121
4.7.3 Scope of distributions in R and CRAN......Page 123
5.1 The astronomical context......Page 125
5.2 Concepts of nonparametric inference......Page 126
5.3.1 Kolmogorov–Smirnov and other e.d.f. tests......Page 127
5.3.2 Robust statistics of location......Page 130
5.4 Hypothesis testing......Page 131
5.4.2 Two-sample and k-sample tests......Page 132
5.5 Contingency tables......Page 133
5.6 Bivariate and multivariate tests......Page 135
5.7 Remarks......Page 136
5.9.1 Exploratory plots and summary statistics......Page 137
5.9.2 Empirical distribution and quantile functions......Page 141
5.9.3 Two-sample tests......Page 144
5.9.4 Contingency tables......Page 145
5.9.5 Scope of nonparametrics in R and CRAN......Page 147
6.2 Concepts of density estimation......Page 148
6.3 Histograms......Page 149
6.4.1 Basic properties......Page 151
6.4.2 Choosing bandwidths by cross-validation......Page 152
6.4.3 Multivariate kernel density estimation......Page 153
6.5.1 Adaptive kernel estimators......Page 154
6.5.2 Nearest-neighbor estimators......Page 155
6.6.1 Nadaraya–Watson estimator......Page 156
6.6.2 Local regression......Page 157
6.8 Recommended reading......Page 158
6.9.1 Histogram, quantile function and measurement errors......Page 159
6.9.2 Kernel smoothers......Page 160
6.9.3 Nonparametric regressions......Page 164
6.9.4 Scope of smoothing in R and CRAN......Page 168
7.1 Astronomical context......Page 170
7.2 Concepts of regression......Page 171
7.3.1 Ordinary least squares......Page 174
7.3.2 Symmetric least-squares regression......Page 175
7.3.3 Bootstrap error analysis......Page 176
7.3.4 Robust regression......Page 178
7.3.5 Quantile regression......Page 180
7.3.6 Maximum likelihood estimation......Page 181
7.4 Weighted least squares......Page 182
7.5 Measurement error models......Page 184
7.5.1 Least-squares estimators......Page 186
7.5.2 SIMEX algorithm......Page 188
7.6 Nonlinear models......Page 189
7.6.1 Poisson regression......Page 190
7.6.2 Logistic regression......Page 191
7.7 Model validation, selection and misspecification......Page 192
7.7.1 Residual analysis......Page 193
7.7.2 Cross-validation and the bootstrap......Page 195
7.8 Remarks......Page 196
7.10 R applications......Page 197
7.10.1 Linear modeling......Page 199
7.10.2 Generalized linear modeling......Page 201
7.10.3 Robust regression......Page 202
7.10.4 Quantile regression......Page 203
7.10.5 Nonlinear regression of galaxy surface brightness profiles......Page 204
7.10.6 Scope of regression in R and CRAN......Page 209
8.1 The astronomical context......Page 210
8.2 Concepts of multivariate analysis......Page 211
8.2.1 Multivariate distances......Page 212
8.2.2 Multivariate normal distribution......Page 214
8.3 Hypothesis tests......Page 215
8.4.1 Multiple linear regression......Page 217
8.4.2 Principal components analysis......Page 219
8.4.3 Factor and canonical correlation analysis......Page 220
8.4.4 Outliers and robust methods......Page 221
8.4.5 Nonlinear methods......Page 222
8.5 Multivariate visualization......Page 223
8.6 Remarks......Page 224
8.7 Recommended reading......Page 225
8.8.1 Univariate tests of normality......Page 226
8.8.2 Preparing the dataset......Page 228
8.8.3 Bivariate relationships......Page 229
8.8.4 Principal components analysis......Page 232
8.8.5 Multiple regression and MARS......Page 234
8.8.6 Multivariate visualization......Page 236
8.8.7 Interactive graphical displays......Page 237
8.8.8 Scope of multivariate analysis R and CRAN......Page 240
9.1 The astronomical context......Page 242
9.2.1 Definitions and scopes......Page 244
9.2.2 Metrics, group centers and misclassifications......Page 245
9.3.1 Agglomerative hierarchical clustering......Page 246
9.3.2 k-means and related nonhierarchical partitioning......Page 248
9.4 Clusters with substructure or noise......Page 249
9.5 Mixture models......Page 251
9.6.1 Multivariate normal clusters......Page 252
9.6.2 Linear discriminant analysis and its generalizations......Page 253
9.6.3 Classification trees......Page 254
9.6.4 Nearest-neighbor classifiers......Page 256
9.6.5 Automated neural networks......Page 257
9.6.6 Classifier validation, improvement and fusion......Page 258
9.7 Remarks......Page 259
9.8 Recommended reading......Page 261
9.9.1 Unsupervised clustering of COMBO-17 galaxies......Page 262
9.9.2 Mixture models......Page 266
9.9.3 Supervised classification of SDSS point sources......Page 270
9.9.4 LDA, k-nn and ANN classification......Page 271
9.9.5 CART and SVM classification......Page 275
9.9.6 Scope of R and CRAN......Page 279
10.1 The astronomical context......Page 281
10.2 Concepts of survival analysis......Page 283
10.3.1 Parametric estimation......Page 286
10.3.2 Kaplan–Meier nonparametric estimator......Page 288
10.3.3 Two-sample tests......Page 289
10.4.1 Correlation coefficients......Page 291
10.4.2 Regression models......Page 292
10.5 Truncation......Page 294
10.5.2 Nonparametric Lynden-Bell–Woodroofe estimator......Page 295
10.6 Remarks......Page 297
10.7 Recommended reading......Page 298
10.8.1 Kaplan–Meier estimator......Page 299
10.8.2 Two-sample tests with censoring......Page 301
10.8.3 Bivariate and multivariate problems with censoring......Page 304
10.8.4 Lynden-Bell–Woodroofe estimator for truncation......Page 307
10.8.5 Scope of censoring and truncation in R and CRAN......Page 310
11.1 The astronomical context......Page 312
11.2 Concepts of time series analysis......Page 314
11.3.1 Smoothing......Page 316
11.3.2 Autocorrelation and cross-correlation......Page 317
11.3.3 Stochastic autoregressive models......Page 318
11.3.4 Regression for deterministic models......Page 321
11.4.1 Discrete correlation function......Page 322
11.5 Spectral analysis of evenly spaced data......Page 324
11.5.1 Fourier power spectrum......Page 325
11.5.2 Improving the periodogram......Page 327
11.6.1 Lomb–Scargle periodogram......Page 328
11.6.2 Non-Fourier periodograms......Page 330
11.6.3 Statistical significance of periodogram peaks......Page 332
11.6.4 Spectral analysis of event data......Page 333
11.6.5 Computational issues......Page 334
11.7 State-space modeling and the Kalman filter......Page 335
11.8 Nonstationary time series......Page 337
11.9 1/f noise or long-memory processes......Page 339
11.10 Multivariate time series......Page 342
11.11 Remarks......Page 343
11.12 Recommended reading......Page 344
11.13 R applications......Page 345
11.13.1 Exploratory time series analysis......Page 346
11.13.2 Spectral analysis......Page 349
11.13.3 Modeling as an autoregressive process......Page 350
11.13.4 Modeling as a long-memory process......Page 353
11.13.5 Wavelet analysis......Page 354
11.13.6 Scope of time series analysis in R and CRAN......Page 356
12.1 The astronomical context......Page 357
12.2 Concepts of spatial point processes......Page 358
12.3 Tests of uniformity......Page 360
12.4.1 Global measures of spatial autocorrelation......Page 361
12.4.2 Local measures of spatial autocorrelation......Page 363
12.5 Spatial interpolation......Page 364
12.6.1 Cumulative second-moment measures......Page 366
12.6.2 Two-point correlation function......Page 368
12.7.1 Models for galaxy clustering......Page 371
12.7.2 Models in geostatistics......Page 373
12.8 Graphical networks and tessellations......Page 374
12.9 Points on a circle or sphere......Page 375
12.10 Remarks......Page 377
12.11 Recommended reading......Page 378
12.12 R applications......Page 379
12.12.1 Characterization of autocorrelation......Page 381
12.12.2 Variogram analysis......Page 382
12.12.3 Characterization of clustering......Page 384
12.12.4 Tessellations......Page 388
12.12.5 Spatial interpolation......Page 390
12.12.6 Spatial regression and modeling......Page 393
12.12.7 Circular and spherical statistics......Page 394
12.12.8 Scope of spatial analysis in R and CRAN......Page 397
Notation......Page 399
Acronyms......Page 400
B.2 Session environment......Page 402
B.3 R object classes......Page 405
B.4 Basic operations on classes......Page 406
B.5 Input/output......Page 408
B.6 A sample R session......Page 409
B.8 Computational efficiency......Page 414
B.9 Learning more about R......Page 417
B.10 Recommended reading......Page 418
Appendix C: Astronomical datasets......Page 427
Statistical issues: univariate estimation, measurement errors, multimodality......Page 428
Statistical exercises......Page 429
Statistical issues: contingency tables, hypothesis tests......Page 430
Statistical issues: univariate two-sample tests, parametric estimation, truncation......Page 431
Statistical exercises......Page 432
Statistical issues: bivariate censoring, survival analysis, correlation......Page 433
Statistical issues: spatial point processes, multivariate clustering, Poisson processes, measurement errors......Page 434
Statistical exercises......Page 435
Statistical issues: multivariate clustering, mixture models, regression, spatial point processes......Page 436
Statistical exercises......Page 437
Statistical issues: multivariate analysis, regression......Page 438
Statistical issues: multivariate analysis, regression, censoring......Page 439
Statistical issues: multivariate classification, mixture models, regression......Page 441
Statistical issues: density estimation, multivariate clustering, mixture models......Page 447
Statistical issues: nonlinear regression, model selection......Page 448
Statistical issues: time series analysis, Fourier analysis, autoregressive models, long-memory processes......Page 449
Statistical issues: time series analysis, periodicities......Page 450
Statistical issues: time series analysis, periodicities, multivariate analysis......Page 451
Statistical issues: time series analysis, periodicities, autoregressive models......Page 453
Statistical issues: data mining, multivariate analysis, multivariate classification, spatial point processes, parametric modeling......Page 456
Statistical issues: time series analysis, heteroscedastic measurement errors, censoring......Page 458
Statistical exercises......Page 460
References......Page 462
Subject index......Page 490
R and CRAN commands......Page 498