Smoothing and Regression: Approaches, Computation, and Application

Smoothing and Regression: Approaches, Computation, and Application......Page 5
Contents......Page 11
Foreword......Page 17
Preface......Page 19
1.1 Introduction......Page 25
1.2 General Form of the Estimator......Page 29
1.3 The Linear Smoothing Spline......Page 31
1.4 Large-Sample Efficiency......Page 33
1.5 Bayesian Motivation......Page 35
1.6 Extensions and Implementations......Page 39
References......Page 40
2.1 Introduction and Some Definitions......Page 43
2.2 Interpretation of the Smoothing Parameter......Page 46
2.3 Quantifying the Complexity of a Smoothing Spline......Page 48
2.4 Estimation of σ2......Page 53
2.5 Determination of λ......Page 56
References......Page 62
3.1 Introduction......Page 67
3.2 The Nadaraya–Watson Kernel Regression Estimate......Page 68
3.3 Pointwise Bias Properties of the Nadaraya–Watson Estimate......Page 72
3.4 Pointwise Variance Properties of the Nadaraya–Watson Estimate......Page 75
3.5 Trade-off Between Bias and Variance: The Mean Squared Error......Page 77
3.6 Global Results: Mean Integrated Squared Error Properties......Page 80
3.7 L∞ Convergence Properties of the NadarayaW–atson Estimate......Page 82
3.8 Complementary Bibliography......Page 87
References......Page 91
4.1 Introduction......Page 95
4.2 Nonparametric Variance Estimators......Page 98
4.3 Bandwidth Choice for Kernel Regression Estimators......Page 112
References......Page 127
5.1 Introduction......Page 133
5.2 Description of the Main Methods......Page 134
5.3 A Comparative View......Page 143
5.4 Examples......Page 146
5.5 Software Hints......Page 152
References......Page 154
6.1 Introduction......Page 159
6.2 Approaches for a Known Autocorrelation Function......Page 162
6.3 Approaches for an Unknown Autocorrelation Function......Page 168
6.4 A Bayesian Approach to Smoothing Dependent Data......Page 172
6.5 Applications of Smoothing Dependent Data......Page 175
References......Page 180
7.1 Introduction......Page 183
7.2 Wavelet Expansions......Page 184
7.3 The Discrete Wavelet Transform in S......Page 187
7.4 Wavelet Shrinkage......Page 194
7.5 Estimators for Data With Correlated Noise......Page 201
7.6 Implementation of the Wavelet Transform......Page 205
7.7 How to Obtain and Install the Wavelet Software......Page 211
References......Page 213
8.1 Introduction......Page 217
8.2 Smoothing Contingency Tables......Page 218
8.3 Smoothing Approaches to Categorical Regression......Page 231
8.4 Conclusion......Page 248
References......Page 249
9.1 Introduction......Page 253
9.2 Properties of Local Polynomial Fitting......Page 261
9.3 Choice of Bandwidth......Page 271
9.4 Choice of the Degree......Page 281
9.5 Local Modeling......Page 283
9.6 Some More Applications......Page 292
References......Page 297
10.1 Introduction......Page 301
10.2 The Additive Model......Page 303
10.3 Generalized Additive Models......Page 321
10.4 Alternating Conditional Expectations Additivity, and Variance Stabilization......Page 339
10.5 Smoothing Parameter and Bandwidth Determination......Page 345
10.6 Model Diagnostics......Page 346
10.7 New Developments......Page 347
References......Page 348
11.1 Introduction......Page 353
11.2 Smoothing Splines as Bayes Estimates......Page 354
11.3 ANOVA Decomposition on Product Domains......Page 356
11.4 Tensor Product Splines......Page 357
11.5 Computation......Page 360
11.6 Bayesian Confidence Intervals......Page 362
11.7 Software......Page 364
11.8 Cosine Diagnostics......Page 368
11.9 Partial Splines......Page 372
11.10 Thin-Plate Splines......Page 374
11.11 Non-Gaussian Regression......Page 376
References......Page 378
12.2 Multidimensional Smoothing with Kernels......Page 381
12.3 Semiparametric Generalized Regression Models......Page 401
12.4 Practical Application and Software Hints......Page 410
References......Page 413
13.1 Introduction......Page 425
13.2 Thin-Plate Splines......Page 429
13.3 Spatial-Process Estimates......Page 433
13.4 Ridge-Regression Estimates and Shrinkage......Page 439
13.5 A Response-Surface Example......Page 443
13.6 Predicting Ambient Ozone......Page 451
13.7 Future Directions......Page 453
References......Page 455
14.1 Introduction......Page 457
14.2 The Idea of Bootstrap......Page 458
14.3 Bootstrap in Nonparametric Regression......Page 462
14.4 Bootstrap Confidence Intervals and Bands......Page 466
14.5 Bootstrap-Bandwidth Choice......Page 469
14.6 Bootstrap Tests in Nonparametrics......Page 471
14.7 Bootstrap Inference on the Shape of a Curve......Page 474
14.8 Extensions......Page 477
References......Page 478
15.1 Introduction......Page 483
15.2 Data-Point Visualization......Page 484
15.3 Functional Visualization in One and Two Variables......Page 488
15.4 Averaged Shifted Histograms......Page 490
15.5 Functional Visualization in Three Variables and Beyond......Page 494
15.6 Visualization of Regression Functions......Page 499
References......Page 500
16.1 Introduction......Page 503
16.2 The Basic PPR Algorithm......Page 505
16.3 Quality of Approximation......Page 506
16.4 Number of Terms to Choose......Page 507
16.5 Interpretable PPR......Page 508
16.6 Convergence Rates......Page 509
16.7 Modifications......Page 510
16.8 PPR and Neural Networks......Page 516
16.9 Optimization Methods for PPR and Neural Networks......Page 519
16.10 The Implementation of PPR in S-PLUS R, and Xplore......Page 520
16.11 An Example......Page 522
References......Page 525
17.1 Introduction......Page 529
17.2 The Idea......Page 531
17.3 Statistical Properties......Page 533
17.4 The Unknown Dimensionality......Page 534
17.5 Slicing Strategies......Page 535
17.6 Implementation......Page 536
17.7 Modifications......Page 539
17.8 An Example......Page 540
References......Page 543
18.1 Introduction......Page 545
18.2 Linear Dynamic Models and Optimal Smoothing for Time Series Data......Page 546
18.3 Non-Gaussian Observation Models......Page 552
18.4 Generalized Additive and Varying Coefficient Models......Page 568
18.5 Conclusions......Page 572
References......Page 574
19.1 Introduction......Page 577
19.2 Bayesian Model and Subset Selection......Page 579
19.3 Bivariate Surface Estimation......Page 586
19.4 Robust Surface Estimation......Page 600
19.5 Surface Estimation for Time Series Data......Page 607
19.6 Alternative Bases and Model Mixing......Page 608
References......Page 610
Index......Page 613