Statistics for high-dimensional data: methods, theory and applications
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2011
|
| Schriftenreihe: | Springer series in statistics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XVII, 556 S. graph. Darst. |
| ISBN: | 9783642201912 9783642268571 3642201911 9783642201929 |
Internformat
MARC
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| 100 | 1 | |a Bühlmann, Peter |d 1965- |e Verfasser |0 (DE-588)171745132 |4 aut | |
| 245 | 1 | 0 | |a Statistics for high-dimensional data |b methods, theory and applications |c Peter Bühlmann ; Sara van de Geer |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2011 | |
| 300 | |a XVII, 556 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
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IMAGE 1
CONTENTS
1 INTRODUCTION 1
1.1 THE FRAMEWORK 1
1.2 THE POSSIBILITIES AND CHALLENGES 2
1.3 ABOUT THE BOOK 3
1.3.1 ORGANIZATION OF THE BOOK 3
1.4 SOME EXAMPLES 4
1.4.1 PREDICTION AND BIOMARKER DISCOVERY IN GENOMICS 5
2 LASSO FOR LINEAR MODELS 7
2.1 ORGANIZATION OF THE CHAPTER 7
2.2 INTRODUCTION AND PRELIMINARIES 8
2.2.1 THE LASSO ESTIMATOR 9
2.3 ORTHONORMAL DESIGN 10
2.4 PREDICTION 11
2.4.1 PRACTICAL ASPECTS ABOUT THE LASSO FOR PREDICTION 12 2.4.2 SOME
RESULTS FROM ASYMPTOTIC THEORY 13
2.5 VARIABLE SCREENING AND \\SS - J3|| 9 -NORMS 14
2.5.1 TUNING PARAMETER SELECTION FOR VARIABLE SCREENING 17 2.5.2 MOTIF
REGRESSION FOR DNA BINDING SITES 18
2.6 VARIABLE SELECTION 19
2.6.1 NEIGHBORHOOD STABILITY AND IRREPRESENTABLE CONDITION 22 2.7 KEY
PROPERTIES AND CORRESPONDING ASSUMPTIONS: A SUMMARY 23 2.8 THE ADAPTIVE
LASSO: A TWO-STAGE PROCEDURE 25
2.8.1 AN ILLUSTRATION: SIMULATED DATA AND MOTIF REGRESSION 25 2.8.2
ORTHONORMAL DESIGN 27
2.8.3 THE ADAPTIVE LASSO: VARIABLE SELECTION UNDER WEAK CONDITIONS 28
2.8.4 COMPUTATION 29
2.8.5 MULTI-STEP ADAPTIVE LASSO 30
2.8.6 NON-CONVEX PENALTY FUNCTIONS 32
2.9 THRESHOLDING THE LASSO 33
2.10 THE RELAXED LASSO 34
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1010389874
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
2.11 DEGREES OF FREEDOM OF THE LASSO 34
2.12 PATH-FOLLOWING ALGORITHMS 36
2.12.1 COORDINATEWISE OPTIMIZATION AND SHOOTING ALGORITHMS 38 2.13
ELASTIC NET: AN EXTENSION 41
PROBLEMS 42
GENERALIZED LINEAR MODELS AND THE LASSO 45
3.1 ORGANIZATION OF THE CHAPTER 45
3.2 INTRODUCTION AND PRELIMINARIES 45
3.2.1 THE LASSO ESTIMATOR: PENALIZING THE NEGATIVE LOG-LIKELIHOOD. 46
3.3 IMPORTANT EXAMPLES OF GENERALIZED LINEAR MODELS 47
3.3.1 BINARY RESPONSE VARIABLE AND LOGISTIC REGRESSION 47 3.3.2 POISSON
REGRESSION 49
3.3.3 MULTI-CATEGORY RESPONSE VARIABLE AND MULTINOMIAL DISTRIBUTION 50
PROBLEMS 53
THE GROUP LASSO 55
4. 1 ORGANIZATION OF THE CHAPTER 55
4.2 INTRODUCTION AND PRELIMINARIES 56
4.2.1 THE GROUP LASSO PENALTY 56
4.3 FACTOR VARIABLES AS COVARIATES 58
4.3.1 PREDICTION OF SPLICE SITES IN DNA SEQUENCES 59
4.4 PROPERTIES OF THE GROUP LASSO FOR GENERALIZED LINEAR MODELS 61 4.5
THE GENERALIZED GROUP LASSO PENALTY 64
4.5.1 GROUPWISE PREDICTION PENALTY AND PARAMETRIZATION INVARIANCE 65 4.6
THE ADAPTIVE GROUP LASSO 66
4.7 ALGORITHMS FOR THE GROUP LASSO 67
4.7.1 BLOCK COORDINATE DESCENT 68
4.7.2 BLOCK COORDINATE GRADIENT DESCENT 72
PROBLEMS 75
ADDITIVE MODELS AND MANY SMOOTH UNIVARIATE FUNCTIONS 77 5.1 ORGANIZATION
OF THE CHAPTER 77
5.2 INTRODUCTION AND PRELIMINARIES 78
5.2.1 PENALIZED MAXIMUM LIKELIHOOD FOR ADDITIVE MODELS 78 5.3 THE
SPARSITY-SMOOTHNESS PENALTY 79
5.3.1 ORTHOGONAL BASIS AND DIAGONAL SMOOTHING MATRICES 80 5.3.2 NATURAL
CUBIC SPLINES AND SOBOLEV SPACES 81
5.3.3 COMPUTATION 82
5.4 A SPARSITY-SMOOTHNESS PENALTY OF GROUP LASSO TYPE 85
5.4.1 COMPUTATIONAL ALGORITHM 86
5.4.2 ALTERNATIVE APPROACHES 88
5.5 NUMERICAL EXAMPLES 89
5.5.1 SIMULATED EXAMPLE 89
IMAGE 3
CONTENTS
5.5.2 MOTIF REGRESSION 90
5.6 PREDICTION AND VARIABLE SELECTION 91
5.7 GENERALIZED ADDITIVE MODELS 92
5.8 LINEAR MODEL WITH VARYING COEFFICIENTS 93
5.8.1 PROPERTIES FOR PREDICTION 95
5.8.2 MULTIVARIATE LINEAR MODEL 95
5.9 MULTITASK LEARNING 95
PROBLEMS 97
THEORY FOR THE LASSO 99
6.1 ORGANIZATION OF THIS CHAPTER 99
6.2 LEAST SQUARES AND THE LASSO 101
6.2.1 INTRODUCTION 101
6.2.2 THE RESULT ASSUMING THE TRUTH IS LINEAR 102
6.2.3 LINEAR APPROXIMATION OF THE TRUTH 108
6.2.4 A FURTHER REFINEMENT: HANDLING SMALLISH COEFFICIENTS 112 6.3 THE
SETUP FOR GENERAL CONVEX LOSS 114
6.4 THE MARGIN CONDITION 119
6.5 GENERALIZED LINEAR MODEL WITHOUT PENALTY 122
6.6 CONSISTENCY OF THE LASSO FOR GENERAL LOSS 126
6.7 AN ORACLE INEQUALITY 128
6.8 THE 4-ERROR FOR 1 Q 2 135
6.8.1 APPLICATION TO LEAST SQUARES ASSUMING THE TRUTH IS LINEAR . . .
136 6.8.2 APPLICATION TO GENERAL LOSS AND A SPARSE APPROXIMATION OF THE
TRUTH 137
6.9 THE WEIGHTED LASSO 139
6.10 THE ADAPTIVELY WEIGHTED LASSO 141
6.11 CONCAVE PENALTIES 144
6.11.1 SPARSITY ORACLE INEQUALITIES FOR LEAST SQUARES WITH ^-PENALTY 146
6.11.2 PROOFS FOR THIS SECTION (SECTION 6.11) 147
6.12 COMPATIBILITY AND (RANDOM) MATRICES 150
6.13 ON THE COMPATIBILITY CONDITION 156
6.13.1 DIRECT BOUNDS FOR THE COMPATIBILITY CONSTANT 158 6.13.2 BOUNDS
USING ||FT||? S\\SS S \\\ 161
6.13.3 SETS JV CONTAINING 5 167
6.13.4 RESTRICTED ISOMETRY 169
6.13.5 SPARSE EIGENVALUES 170
6.13.6 FURTHER COHERENCE NOTIONS 172
6.13.7 AN OVERVIEW OF THE VARIOUS EIGENVALUE FLAVORED CONSTANTS. 174
PROBLEMS 178
VARIABLE SELECTION WITH THE LASSO 183
7.1 INTRODUCTION 183
7.2 SOME RESULTS FROM LITERATURE 184
7.3 ORGANIZATION OF THIS CHAPTER 185
IMAGE 4
CONTENTS
7.4 THE BETA-MIN CONDITION 187
7.5 THE IRREPRESENTABLE CONDITION IN THE NOISELESS CASE 189
7.5.1 DEFINITION OF THE IRREPRESENTABLE CONDITION 190
7.5.2 THE KKT CONDITIONS 190
7.5.3 NECESSITY AND SUFFICIENCY FOR VARIABLE SELECTION 191 7.5.4 THE
IRREPRESENTABLE CONDITION IMPLIES THE COMPATIBILITY CONDITION 195
7.5.5 THE IRREPRESENTABLE CONDITION AND RESTRICTED REGRESSION 197 7.5.6
SELECTING A SUPERSET OF THE TRUE ACTIVE SET 199
7.5.7 THE WEIGHTED IRREPRESENTABLE CONDITION 200
7.5.8 THE WEIGHTED IRREPRESENTABLE CONDITION AND RESTRICTED REGRESSION
201
7.5.9 THE WEIGHTED LASSO WITH "IDEAL" WEIGHTS 203
7.6 DEFINITION OF THE ADAPTIVE AND THRESHOLDED LASSO 204
7.6.1 DEFINITION OF ADAPTIVE LASSO 204
7.6.2 DEFINITION OF THE THRESHOLDED LASSO 205
7.6.3 ORDER SYMBOLS 206
7.7 A RECOLLECTION OF THE RESULTS OBTAINED IN CHAPTER 6 206
7.8 THE ADAPTIVE LASSO AND THRESHOLDING: INVOKING SPARSE EIGENVALUES.
210 7.8.1 THE CONDITIONS ON THE TUNING PARAMETERS 210
7.8.2 THE RESULTS 211
7.8.3 COMPARISON WITH THE LASSO 213
7.8.4 COMPARISON BETWEEN ADAPTIVE AND THRESHOLDED LASSO 214 7.8.5 BOUNDS
FOR THE NUMBER OF FALSE NEGATIVES 215
7.8.6 IMPOSING BETA-MIN CONDITIONS 216
7.9 THE ADAPTIVE LASSO WITHOUT INVOKING SPARSE EIGENVALUES 218 7.9.1 THE
CONDITION ON THE TUNING PARAMETER 219
7.9.2 THE RESULTS 219
7.10 SOME CONCLUDING REMARKS 221
7.11 TECHNICAL COMPLEMENTS FOR THE NOISELESS CASE WITHOUT SPARSE
EIGENVALUES 222
7.11.1 PREDICTION ERROR FOR THE NOISELESS (WEIGHTED) LASSO 222 7.11.2
THE NUMBER OF FALSE POSITIVES OF THE NOISELESS (WEIGHTED) LASSO 224
7.11.3 THRESHOLDING THE NOISELESS INITIAL ESTIMATOR 225
7.11.4 THE NOISELESS ADAPTIVE LASSO 227
7.12 TECHNICAL COMPLEMENTS FOR THE NOISY CASE WITHOUT SPARSE EIGENVALUES
232
7.13 SELECTION WITH CONCAVE PENALTIES 237
PROBLEMS 241
THEORY FOR ^1/^2-PENALTY PROCEDURES 249
8.1 INTRODUCTION 249
8.2 ORGANIZATION AND NOTATION OF THIS CHAPTER 250
8.3 REGRESSION WITH GROUP STRUCTURE 252
8.3.1 THE LOSS FUNCTION AND PENALTY 253
IMAGE 5
CONTENTS
8.3.2 THE EMPIRICAL PROCESS 254
8.3.3 THE GROUP LASSO COMPATIBILITY CONDITION 255
8.3.4 A GROUP LASSO SPARSITY ORACLE INEQUALITY 256
8.3.5 EXTENSIONS 258
8.4 HIGH-DIMENSIONAL ADDITIVE MODEL 258
8.4.1 THE LOSS FUNCTION AND PENALTY 258
8.4.2 THE EMPIRICAL PROCESS 260
8.4.3 THE SMOOTHED LASSO COMPATIBILITY CONDITION 264 8.4.4 A SMOOTHED
GROUP LASSO SPARSITY ORACLE INEQUALITY 265 8.4.5 ON THE CHOICE OF THE
PENALTY 270
8.5 LINEAR MODEL WITH TIME-VARYING COEFFICIENTS 275
8.5.1 THE LOSS FUNCTION AND PENALTY 275
8.5.2 THE EMPIRICAL PROCESS 277
8.5.3 THE COMPATIBILITY CONDITION FOR THE TIME-VARYING COEFFICIENTS
MODEL 278
8.5.4 A SPARSITY ORACLE INEQUALITY FOR THE TIME-VARYING COEFFICIENTS
MODEL 279
8.6 MULTIVARIATE LINEAR MODEL AND MULTITASK LEARNING 281
8.6.1 THE LOSS FUNCTION AND PENALTY 281
8.6.2 THE EMPIRICAL PROCESS 282
8.6.3 THE MULTITASK COMPATIBILITY CONDITION 283
8.6.4 A MULTITASK SPARSITY ORACLE INEQUALITY 284
8.7 THE APPROXIMATION CONDITION FOR THE SMOOTHED GROUP LASSO 286 8.7.1
SOBOLEV SMOOTHNESS 286
8.7.2 DIAGONALIZED SMOOTHNESS 287
PROBLEMS 288
NON-CONVEX LOSS FUNCTIONS AND LI-REGULARIZATION 293
9.1 ORGANIZATION OF THE CHAPTER 293
9.2 FINITE MIXTURE OF REGRESSIONS MODEL 294
9.2.1 FINITE MIXTURE OF GAUSSIAN REGRESSIONS MODEL 294 9.2.2 \
-PENALIZED MAXIMUM LIKELIHOOD ESTIMATOR 295 9.2.3 PROPERTIES OF THE L\
-PENALIZED MAXIMUM LIKELIHOOD ESTIMATOR 299
9.2.4 SELECTION OF THE TUNING PARAMETERS 300
9.2.5 ADAPTIVE T\ -PENALIZATION 301
9.2.6 RIBOFLAVIN PRODUCTION WITH BACILLUS SUBTILIS 301
9.2.7 SIMULATED EXAMPLE 303
9.2.8 NUMERICAL OPTIMIZATION 304
9.2.9 GEM ALGORITHM FOR OPTIMIZATION 304
9.2.10 PROOF OF PROPOSITION 9.2 308
9.3 LINEAR MIXED EFFECTS MODELS 310
9.3.1 THE MODEL AND ^-PENALIZED ESTIMATION 311
9.3.2 THE LASSO IN LINEAR MIXED EFFECTS MODELS 312
9.3.3 ESTIMATION OF THE RANDOM EFFECTS COEFFICIENTS 312 9.3.4 SELECTION
OF THE REGULARIZATION PARAMETER 313
IMAGE 6
XIV CONTENTS
9.3.5 PROPERTIES OF THE LASSO IN LINEAR MIXED EFFECTS MODELS 313 9.3.6
ADAPTIVE ^-PENALIZED MAXIMUM LIKELIHOOD ESTIMATOR 314 9.3.7
COMPUTATIONAL ALGORITHM 314
9.3.8 NUMERICAL RESULTS 317
9.4 THEORY FOR I\ -PENALIZATION WITH NON-CONVEX NEGATIVE LOG-LIKELIHOOD
320 9.4.1 THE SETTING AND NOTATION 320
9.4.2 ORACLE INEQUALITY FOR THE LASSO FOR NON-CONVEX LOSS FUNCTIONS 323
9.4.3 THEORY FOR FINITE MIXTURE OF REGRESSIONS MODELS 326 9.4.4 THEORY
FOR LINEAR MIXED EFFECTS MODELS 329
9.5 PROOFS FOR SECTION 9.4 332
9.5.1 PROOF OF LEMMA 9.1 332
9.5.2 PROOF OF LEMMA 9.2 333
9.5.3 PROOF OF THEOREM 9.1 335
9.5.4 PROOF OF LEMMA 9.3 337
PROBLEMS 337
10 STABLE SOLUTIONS 339
10.1 ORGANIZATION OF THE CHAPTER 339
10.2 INTRODUCTION, STABILITY AND SUBSAMPLING 340
10.2.1 STABILITY PATHS FOR LINEAR MODELS 341
10.3 STABILITY SELECTION 346
10.3.1 CHOICE OF REGULARIZATION AND ERROR CONTROL 346
10.4 NUMERICAL RESULTS 351
10.5 EXTENSIONS 352
10.5.1 RANDOMIZED LASSO 352
10.6 IMPROVEMENTS FROM A THEORETICAL PERSPECTIVE 354
10.7 PROOFS 355
10.7.1 SAMPLE SPLITTING 355
10.7.2 PROOF OF THEOREM 10.1 356
PROBLEMS 358
11 P-VALUES FOR LINEAR MODELS AND BEYOND 359
11.1 ORGANIZATION OF THE CHAPTER 359
11.2 INTRODUCTION, SAMPLE SPLITTING AND HIGH-DIMENSIONAL VARIABLE
SELECTION 360
11.3 MULTI SAMPLE SPLITTING AND FAMILYWISE ERROR CONTROL 363 11.3.1
AGGREGATION OVER MULTIPLE P-VALUES 364
11.3.2 CONTROL OF FAMILYWISE ERROR 365
11.4 MULTI SAMPLE SPLITTING AND FALSE DISCOVERY RATE 367
11.4.1 CONTROL OF FALSE DISCOVERY RATE 368
11.5 NUMERICAL RESULTS 369
11.5.1 SIMULATIONS AND FAMILYWISE ERROR CONTROL 369
11.5.2 FAMILYWISE ERROR CONTROL FOR MOTIF REGRESSION IN COMPUTATIONAL
BIOLOGY 372
11.5.3 SIMULATIONS AND FALSE DISCOVERY RATE CONTROL 372
IMAGE 7
CONTENTS XV
11.6 CONSISTENT VARIABLE SELECTION 374
11.6.1 SINGLE SAMPLE SPLIT METHOD 374
11.6.2 MULTI SAMPLE SPLIT METHOD 377
11.7 EXTENSIONS 377
11.7.1 OTHER MODELS 378
11.7.2 CONTROL OF EXPECTED FALSE POSITIVE SELECTIONS 378 11.8 PROOFS 379
11.8.1 PROOF OF PROPOSITION 11.1 379
11.8.2 PROOF OF THEOREM 11.1 380
11.8.3 PROOF OF THEOREM 11.2 382
11.8.4 PROOF OF PROPOSITION 11.2 384
11.8.5 PROOF OF LEMMA 11.3 384
PROBLEMS 386
12 BOOSTING AND GREEDY ALGORITHMS 387
12.1 ORGANIZATION OF THE CHAPTER 387
12.2 INTRODUCTION AND PRELIMINARIES 388
12.2.1 ENSEMBLE METHODS: MULTIPLE PREDICTION AND AGGREGATION 388 12.2.2
ADABOOST 389
12.3 GRADIENT BOOSTING: A FUNCTIONAL GRADIENT DESCENT ALGORITHM 389
12.3.1 THE GENERIC FGD ALGORITHM 390
12.4 SOME LOSS FUNCTIONS AND BOOSTING ALGORITHMS 392
12.4.1 REGRESSION 392
12.4.2 BINARY CLASSIFICATION 393
12.4.3 POISSON REGRESSION 396
12.4.4 TWO IMPORTANT BOOSTING ALGORITHMS 396
12.4.5 OTHER DATA STRUCTURES AND MODELS 398
12.5 CHOOSING THE BASE PROCEDURE 398
12.5.1 COMPONENTWISE LINEAR LEAST SQUARES FOR GENERALIZED LINEAR MODELS
399
12.5.2 COMPONENTWISE SMOOTHING SPLINE FOR ADDITIVE MODELS 400 12.5.3
TREES 403
12.5.4 THE LOW-VARIANCE PRINCIPLE 404
12.5.5 INITIALIZATION OF BOOSTING 404
12.6 L 2 BOOSTING 405
12.6.1 NONPARAMETRIC CURVE ESTIMATION: SOME BASIC INSIGHTS ABOUT
BOOSTING 405
12.6.2 L 2 BOOSTING FOR HIGH-DIMENSIONAL LINEAR MODELS 409 12.7 FORWARD
SELECTION AND ORTHOGONAL MATCHING PURSUIT 413 12.7.1 LINEAR MODELS AND
SQUARED ERROR LOSS 414
12.8 PROOFS 418
12.8.1 PROOF OF THEOREM 12.1 418
12.8.2 PROOF OF THEOREM 12.2 420
12.8.3 PROOF OF THEOREM 12.3 426
PROBLEMS 430
IMAGE 8
XVI CONTENTS
13 GRAPHICAL MODELING 433
13.1 ORGANIZATION OF THE CHAPTER 433
13.2 PRELIMINARIES ABOUT GRAPHICAL MODELS 434
13.3 UNDIRECTED GRAPHICAL MODELS 434
13.3.1 MARKOV PROPERTIES FOR UNDIRECTED GRAPHS 434
13.4 GAUSSIAN GRAPHICAL MODELS 435
13.4.1 PENALIZED ESTIMATION FOR COVARIANCE MATRIX AND EDGE S E T . . .
436 13.4.2 NODEWISE REGRESSION 440
13.4.3 COVARIANCE ESTIMATION BASED ON UNDIRECTED GRAPH 442 13.5 ISING
MODEL FOR BINARY RANDOM VARIABLES 444
13.6 FAITHFULNESS ASSUMPTION 445
13.6.1 FAILURE OF FAITHFULNESS 446
13.6.2 FAITHFULNESS AND GAUSSIAN GRAPHICAL MODELS 448 13.7 THE
PC-ALGORITHM: AN ITERATIVE ESTIMATION METHOD 449
13.7.1 POPULATION VERSION OF THE PC-ALGORITHM 449
13.7.2 SAMPLE VERSION FOR THE PC-ALGORITHM 451
13.8 CONSISTENCY FOR HIGH-DIMENSIONAL DATA 453
13.8.1 AN ILLUSTRATION 455
13.8.2 THEORETICAL ANALYSIS OF THE PC-ALGORITHM 456
13.9 BACK TO LINEAR MODELS 462
13.9.1 PARTIAL FAITHFULNESS 463
13.9.2 THE PC-SIMPLE ALGORITHM 465
13.9.3 NUMERICAL RESULTS 468
13.9.4 ASYMPTOTIC RESULTS IN HIGH DIMENSIONS 471
13.9.5 CORRELATION SCREENING (SURE INDEPENDENCE SCREENING) 474 13.9.6
PROOFS 475
PROBLEMS 480
14 PROBABILITY AND MOMENT INEQUALITIES 481
14.1 ORGANIZATION OF THIS CHAPTER 481
14.2 SOME SIMPLE RESULTS FOR A SINGLE RANDOM VARIABLE 482
14.2.1 SUB-EXPONENTIAL RANDOM VARIABLES 482
14.2.2 SUB-GAUSSIAN RANDOM VARIABLES 483
14.2.3 JENSEN'S INEQUALITY FOR PARTLY CONCAVE FUNCTIONS 485 14.3
BERNSTEIN'S INEQUALITY 486
14.4 HOEFFDING'S INEQUALITY 487
14.5 THE MAXIMUM OF P AVERAGES 489
14.5.1 USING BERNSTEIN'S INEQUALITY 489
14.5.2 USING HOEFFDING'S INEQUALITY 491
14.5.3 HAVING SUB-GAUSSIAN RANDOM VARIABLES 493
14.6 CONCENTRATION INEQUALITIES 494
14.6.1 BOUSQUET'S INEQUALITY 494
14.6.2 MASSART'S INEQUALITY 496
14.6.3 SUB-GAUSSIAN RANDOM VARIABLES 496
14.7 SYMMETRIZATION AND CONTRACTION 497
IMAGE 9
CONTENTS XVII
14.8 CONCENTRATION INEQUALITIES FOR LIPSCHITZ LOSS FUNCTIONS 500 14.9
CONCENTRATION FOR SQUARED ERROR LOSS WITH RANDOM DESIGN 504 14.9.1 THE
INNER PRODUCT OF NOISE AND LINEAR FUNCTIONS 505 14.9.2 SQUARED LINEAR
FUNCTIONS 505
14.9.3 SQUARED ERROR LOSS 508
14.10 ASSUMING ONLY LOWER ORDER MOMENTS 508
14.10.1 NEMIROVSKI MOMENT INEQUALITY 509
14.10.2 A UNIFORM INEQUALITY FOR QUADRATIC FORMS 510
14.11 USING ENTROPY FOR CONCENTRATION IN THE SUB-GAUSSIAN CASE 511 14.12
SOME ENTROPY RESULTS 516
14.12.1 ENTROPY OF FINITE-DIMENSIONAL SPACES AND GENERAL CONVEX HULLS
518
14.12.2 SETS WITH RESTRICTIONS ON THE COEFFICIENTS 518
14.12.3 CONVEX HULLS OF SMALL SETS: ENTROPY WITH LOG-TERM 519 14.12.4
CONVEX HULLS OF SMALL SETS: ENTROPY WITHOUT LOG-TERM 520 14.12.5 FURTHER
REFINEMENTS 523
14.12.6 AN EXAMPLE: FUNCTIONS WITH (M- L)-TH DERIVATIVE OF BOUNDED
VARIATION 523
14.12.7 PROOFS FOR THIS SECTION (SECTION 14.12) 525
PROBLEMS 535
AUTHOR INDEX 539
INDEX 543
REFERENCES 547 |
| any_adam_object | 1 |
| author | Bühlmann, Peter 1965- Geer, Sara van de 1958- |
| author_GND | (DE-588)171745132 (DE-588)113851472 |
| author_facet | Bühlmann, Peter 1965- Geer, Sara van de 1958- |
| author_role | aut aut |
| author_sort | Bühlmann, Peter 1965- |
| author_variant | p b pb s v d g svd svdg |
| building | Verbundindex |
| bvnumber | BV037410789 |
| classification_rvk | QH 230 SK 830 |
| classification_tum | MAT 627f MAT 625f |
| ctrlnum | (OCoLC)712538943 (DE-599)DNB1010389874 |
| dewey-full | 519.54 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.54 |
| dewey-search | 519.54 |
| dewey-sort | 3519.54 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV037410789 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T11:08:37Z |
| institution | BVB |
| isbn | 9783642201912 9783642268571 3642201911 9783642201929 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022563252 |
| oclc_num | 712538943 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-703 DE-824 DE-521 DE-11 DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-384 DE-188 DE-83 |
| owner_facet | DE-19 DE-BY-UBM DE-703 DE-824 DE-521 DE-11 DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-384 DE-188 DE-83 |
| physical | XVII, 556 S. graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer series in statistics |
| spelling | Bühlmann, Peter 1965- Verfasser (DE-588)171745132 aut Statistics for high-dimensional data methods, theory and applications Peter Bühlmann ; Sara van de Geer Berlin [u.a.] Springer 2011 XVII, 556 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in statistics Hier auch später erschienene, unveränderte Nachdrucke Boosting (DE-588)4839853-6 gnd rswk-swf Dimensionsreduktion (DE-588)4224279-4 gnd rswk-swf Lasso-Methode (DE-588)7750854-3 gnd rswk-swf Inferenzstatistik (DE-588)4247120-5 gnd rswk-swf Inferenzstatistik (DE-588)4247120-5 s Dimensionsreduktion (DE-588)4224279-4 s Lasso-Methode (DE-588)7750854-3 s Boosting (DE-588)4839853-6 s DE-604 Geer, Sara van de 1958- Verfasser (DE-588)113851472 aut Erscheint auch als Online-Ausgabe 978-3-642-20192-9 (DE-604)BV047116031 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3681287&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022563252&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bühlmann, Peter 1965- Geer, Sara van de 1958- Statistics for high-dimensional data methods, theory and applications Boosting (DE-588)4839853-6 gnd Dimensionsreduktion (DE-588)4224279-4 gnd Lasso-Methode (DE-588)7750854-3 gnd Inferenzstatistik (DE-588)4247120-5 gnd |
| subject_GND | (DE-588)4839853-6 (DE-588)4224279-4 (DE-588)7750854-3 (DE-588)4247120-5 |
| title | Statistics for high-dimensional data methods, theory and applications |
| title_auth | Statistics for high-dimensional data methods, theory and applications |
| title_exact_search | Statistics for high-dimensional data methods, theory and applications |
| title_full | Statistics for high-dimensional data methods, theory and applications Peter Bühlmann ; Sara van de Geer |
| title_fullStr | Statistics for high-dimensional data methods, theory and applications Peter Bühlmann ; Sara van de Geer |
| title_full_unstemmed | Statistics for high-dimensional data methods, theory and applications Peter Bühlmann ; Sara van de Geer |
| title_short | Statistics for high-dimensional data |
| title_sort | statistics for high dimensional data methods theory and applications |
| title_sub | methods, theory and applications |
| topic | Boosting (DE-588)4839853-6 gnd Dimensionsreduktion (DE-588)4224279-4 gnd Lasso-Methode (DE-588)7750854-3 gnd Inferenzstatistik (DE-588)4247120-5 gnd |
| topic_facet | Boosting Dimensionsreduktion Lasso-Methode Inferenzstatistik |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3681287&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022563252&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT buhlmannpeter statisticsforhighdimensionaldatamethodstheoryandapplications AT geersaravande statisticsforhighdimensionaldatamethodstheoryandapplications |