Principal component analysis in meteorology and oceanography
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| Format: | Buch |
| Sprache: | English |
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Amsterdam u.a.
Elsevier
1988
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| Schriftenreihe: | Developments in atmospheric science
17. |
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| 264 | 1 | |a Amsterdam u.a. |b Elsevier |c 1988 | |
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Datensatz im Suchindex
| _version_ | 1819624106827448320 |
|---|---|
| adam_text | Titel: Principal component analysis in meteorology and oceanography
Autor: Preisendorfer, Rudolph W.
Jahr: 1988
VII
Contents
Paj
List of Figures.................................................... xvi
List of Tables....................................................xvni
1. Introduction.....................................................l
a. An Overview of Principal Component Analysis (PCA)..................1
b. Outline of the Book..............................................2
c. A Brief History of PCA..........................................5
d. Acknowledgments...............................................9
2. Algebraic Foundations of PCA..................................11
a. Introductory Example: Bivariate Data Sets..........................11
Monterey, California air temperatures..............................11
Centering and rotating the data set................................12
Variances in the rotated frame....................................14
Principal angles...............................................15
Principal variances.............................................17
Principal covariance............................................17
Principal directions............................................18
Principal components; principal directions as basis vectors.............18
Matrix representation...........................................19
The PCA property..............................................20
Invariance of the total variance under rotation.......................21
Principal variances for standardized data sets.......................22
PCA and estimates of the statistical parameters of normal populations .... 22
PCA and the construction of Monte Carlo experiments.................23
Eigenvalues and eigenvectors of the covariance and scatter matrices......24
b. Principal Component Analysis: Real-valued Scalar Fields..............25
t-centering the data set..........................................26
The scatter probe and the scatter matrix............................26
The eigenstructures of PCA......................................27
The basic data set representations; analysis and synthesis formulas.......30
The PCA property..............................................32
Second-order properties of PCA; the total scatter.....................33
The singular value decomposition (SVD) of a data set..................36
Second-order properties of PCA; correlations........................37
PCA characterized by the PCA property............................38
The asymptotic PCA property and dynamical systems..................39
PCA of spatial composites of data sets..............................40
PCA of temporal composites of data sets............................43
c. Principal Component Analysis: Complex-valued Scalar Fields,
and Beyond...............................................44
PCA of complex-valued data sets (C-PCA)..........................44
?? CONTENTS
Complex algebra conventions.....................................45
The scatter probe and scatter matrix for C-PCA......................46
Derivation of the eigenstructures of C-PCA..........................47
The fundamental formulas of C-PCA...............................48
Generalization of PCA to quaternion-valued data sets (Q-PCA)..........49
Matrix representations of complex and quaternion numbers.............52
PCA of matrix-valued data sets (M-PCA)............................54
Reduction of M-PCA to C-PCA form...............................59
d. Bibliographic Notes and Miscellaneous Topics.......................61
Alternate interpretation of the scatter probe.........................62
Numerical calculations of eigenstructures of a scatter matrix............63
Some elementary properties of eigenstructures of a scatter matrix........63
Sample space vs. state space: choosing the dual computation...........64
PCA for continuous domains.....................................67
PCA for continuous domains: the viewpoint of empirical
orthogonal functions........................................75
The sixteen possible domain pairs for PCA: abstract PCA..............81
3. Dynamical Origins of PCA......................................89
a. One-dimensional Harmonic Motion................................89
A spring-linked-mass model; general form...........................89
A spring-linked-mass model; special form...........................90
A numerical example of the asymptotic PCA property..................91
Further investigations of the asymptotic PCA property and of EOF s......93
b. Two-dimensional Wave Motion..................................Ill
Solution of a two-dimensional damped-wave model...................Ill
Demonstration of the asymptotic PCA property (forcing and
friction absent)...........................................113
Demonstration of the asymptotic PCA property (forcing and
friction present)...........................................114
Physical basis for eigenframe rotations............................117
c. Dynamical Origins of Linear Regression (LR).......................117
From continuous to discrete solutions to the regression model..........118
The linear regression procedure..................................118
Comparison ofLRA and PCA....................................119
d. Random Processes and Karhunen-Loève Analysis....................120
Origins of random processes in linear settings......................120
Karhunen-Loève representation of random data sets and
comparison with PCA......................................121
e. Stationary Processes and PCA...................................123
Derivation of the PCA representation of a one-dimensional
stationary process via a simple wave model.....................123
Connections between PCA and stationary processes: the case of
one dimension............................................129
CONTENTS À÷
Connections between PC A and stationary processes: extension to
two dimensions...........................................146
f. Bibliographic Notes...........................................156
4. Extensions of PC A to Multivariate Fields.......................159
a. Categories of Data and Modes of Analysis..........................159
Examples....................................................159
Generalized notation: the concepts of individual and variable
in PCA..................................................161
b. Local PCA of a General Vector Field..............................162
The PCA formalism............................................162
Squared correlations...........................................165
Variational origin of the scatter matrix............................166
Examples....................................................166
c. Global PCA of a General Vector Field: Time-Modulation Form........167
The PCA formalism............................................167
Squared correlations...........................................171
Degeneracy of global PCA to local PCA...........................172
Variational origin of the scatter matrix............................172
d. Global PCA of a General Vector Field: Space-Modulation Form........173
The PCA formalism............................................173
Squared correlations...........................................176
Variational origin of the scatter matrix............................176
e. PCA of Spectral Components of a General Vector Field...............178
Fourier analysis of the vector field components......................178
The scatter matrix in the spectral setting...........................179
Example of spectral PCA of a windfield............................181
f. Bibliographic Notes and Miscellaneous Topics......................182
The eight modes of analysis and Cotteli s classifications...............182
Time-modulation PCA as a special case of matrix-valued PCA.........182
Applications to the PCA of wind fields.............................183
Distinction between time-modulation PCA and complex PCA...........184
Applications to the PCA of storm tracks............................187
5. Selection Rules for PCA.......................................192
a. Random Reference Data Sets....................................193
b. Dynamical Origins of the Dominant-Variance Selection Rules..........195
A dynamical model............................................195
Rationale for selection rules.....................................196
÷ CONTENTS
c. Rule A4.....................................................197
Statistical basis and discussion...................................197
ChoiceofX0..................................................199
d. Rule N......................................................199
Statistical basis and discussion...................................199
Adjustments for correlated data: effective sample size................202
Asymptotic eigenvalues for large data sets..........................204
e. Rule M.....................................................205
f. Comments on Dominant-Variance Rules...........................207
g. Dynamical Origins of the Time-History Selection Rules...............207
h. Rule KS2....................................................208
The white spectrum and the cumulative periodogram.................209
Statement of Rule KS2..........................................209
i. Rules ÁÌÑë.................................................211
Fisher s test..................................................211
Siegel stest..................................................212
Statement of Rules ÁÌÑë.......................................214
j. Rule Q......................................................214
k. Selection Rules for Vector-Valued Fields..........................215
Local PCA rules..............................................215
Global PCA (time-modulated) rules...............................217
Global PCA (space-modulated) rules..............................218
1. A Space-map Selection Rule.....................................219
Canonic direction angles.......................................220
Differential relations between unit vectors and canonic direction angles .. 223
An r-tile metric for comparing canonic direction angles...............224
Statistical aspects: critical values for class errors...................226
Statement of the selection rule...................................233
m. Bibliographic Notes and Miscellaneous Topics......................234
Puzzles and problems underlying RuleN; the logarithmic eigenvalue
curve...................................................234
Numerical intractability of the classical formulas for the eigenvalues
of a random matrix........................................237
Monte Carlo approaches to the eigenvalue distribution problem........240
Comparison of Monte Carlo methods and asymptotic formulas for
eigenvalue distributions....................................241
The problem of closely spaced eigenvalues; tests for equal eigenvalues. .. 247
The generalized basis for dominant variance selection rules............248
Parallel work in atomic physics..................................251
CONTENTS XI
6. Factor Analysis (FA) and PCA..................................253
a. Comparison of PCA, LRA, and FA...............................253
Similarities between PCA, LRA, andFA............................253
Dissimilarities between PCA, LRA, andFA.........................254
The usual algebraic form of FA; its PC andLR interpretations.........255
b. The Central Problems of FA.....................................257
The matrix formulation of FA....................................257
The detailed sub-problems of FA.................................259
c. Bibliographic Notes...........................................260
The selection rule problem in FA.................................261
The parameter estimation problem in FA...........................263
7. Diagnostic Procedures via PCA and FA........................265
a. Dual Interpretations of a Data Set: State Space and Sample Space.......265
b. Interpreting Å-frames in PCA State Space..........................267
Example: graphical display of eigenvectors........................267
Rationales for interpreting eigenmaps and time series................269
PCA as a means, rather than an end..............................270
c. Informative and Uninformative Å-frames in PCA State Space..........271
d. Rotating Å-frames in PCA State Space (varimax)....................273
A two-dimensional example of the varimax procedure.................273
The general varimax procedure..................................274
The loss of the PCA property for rotated E-frames...................277
e. Projections onto Å-frames in PCA State Space (procmstes)............278
Derivation of the procrustes technique.............................278
Some observations on the generality of the procrustes technique........281
f. Interpreting A-frames in PCA Sample Space........................282
g. Rotating A-frames in PCA Sample Space (varimax)..................282
h. Projections onto A-frames in PCA Sample Space (procrustes)..........284
i. Detecting Clusters of Points in PCA State or Sample Spaces...........285
Minimal spanning trees.........................................285
Defining cluster pairs, and tests for significance.....................286
j. The Analogous PCA Interpretations and Transformations in FA.........288
XII CONTENTS
k. Bibliographic Notes...........................................289
The factor transformation problem in FA...........................289
Chimerical selection rules and the principle of mimicry...............290
8. Canonical Correlation Analysis (CCA) and PCA...............293
a. The Singular Value Decomposition (S VD) of Two Data Sets...........294
b. The Correlation Probe..........................................295
Definitions of the spans of A andB ...............................295
Definition of the correlation probe y(r,s)...........................296
c. Maximizing the Correlation Function y(r,s).........................297
Variational origin of the canonical components......................297
Observations on rank and orthonormality..........................298
d. Canonical Correlations.........................................300
Correlations and orthonormality of the canonical component vectors .... 300
Useful relations for CCA.......................................301
Projector matrix form of canonical correlation theory................302
e. Canonical Component Representations of Data Sets..................304
State space and sample space representations; canonic maps...........304
Properties of the canonic vectors.................................308
f. Selection Rules for CCA........................................309
Geometric constraints on canonical correlations.....................310
Constructing random pairs of data sets............................311
Selection rule formulation.......................................313
g. Bibliographic Notes...........................................315
Hotelling s original formulation of CCA...........................316
Miscellaneous comments.......................................317
Prohaska s PCA of correlation matrices...........................318
9. Linear Regression Analysis (LRA) and PCA....................322
a. Basic Regression Equations.....................................322
Norm probe..................................................322
Variational origin of linear regression.............................323
Projection form of LRA.........................................324
b. Regression Using PCA Frames...................................325
Regressing A andB frames on each other.........................325
Using PCA to avoid singular data sets in LRA.......................326
Using PC selection rules to improve LR models.....................327
CONTENTS XIII
c. Regression Using CCA Frames..................................327
Regressing U and V frames on each other..........................327
Regressing Æ on Y, using canonical component vectors................328
Potential truncation representations..............................330
Prediction formulas...........................................331
d. Regression Hindcast Skill.......................................334
Definition of the classic hindcast skill SH...........................334
Analysis of S„ using PCA and CCA concepts........................334
Invariance of the total SH in A and B frame settings..................337
Analysis ofSy and TSH using CCA concepts........................338
Definition of the canonie hindcast skill QH..........................339
e. Regression Signal-to-Noise Ratio.................................340
Noise estimators..............................................340
The signal-to-noise ratio ñ......................................341
The distribution ofQH and confidence intervals for ñ.................342
Significant regression models....................................344
f. Significant Hindcast Skill.......................................344
Significance test for QH.........................................344
intuitive connection between significant models and significant skill.....345
Significance test for SH.........................................346
Effects of autocorrelation on significance tests......................346
Significance tests for total hindcast skill TSH........................347
TSH and the distance between PC frames...........................348
g. Bibliographic Notes...........................................350
10. Statistical-Dynamical Models and PCA.........................352
a. Example 1: A Linear Two-dimensional Damped-wave Model..........352
Continuously extended EOF s....................................353
Applying selection rules to the EOFs..............................354
The statistical-dynamical model..................................355
Integration of the model; initial and boundary conditions..............356
b. Example 2: Linearized Primitive Equations for the Atmosphere
and Oceans..............................................357
Eckart s equations............................................357
Continuously extended EOFs....................................359
Applying selection rules to the EOFs..............................359
Evaluating typical terms........................................360
The statistical-dynamical model..................................361
c. Bibliographic Notes...........................................362
XIV CONTENTS
11. The Eigenvector-Partition Problem............................365
a. PCA on Partitioned Domains....................................365
Partitioning the given domain....................................365
Solving the two PCA subproblems................................366
Applying selection rules to the sub-eigenstructures...................366
Optimal combinations of the sub-eigenstructures....................367
Connections between the exact and approximate eigenstructures........368
b. Iterative Improvement of the Optimal Combinations..................369
c. Generalizations of the Eigenvector-Partition Problem.................371
d. Bibliographic Notes...........................................371
12. Complex Harmonic PCA (CH-PCA) of Random
Multivariate Fields........................................373
a. Elementary Moving-Pattern Analysis via Real Harmonic
Analysis (RHA)..........................................374
b. Single Data Set; RHA first, PCA second: Standing Waves.............375
c. Essentials of Complex Harmonic Analysis (CHA)...................376
d. Single Data Set; PCA first, CHA second: Standing Waves.............379
Analysis; standing waves..............v........................379
Verification of the eigenstructures of S and §.......................380
e. Single Data Set; CHA first, PCA second: Traveling Waves............380
Analysis.....................................................381
Moving waves................................................383
Kinematic basis for CH-PCA....................................384
f. CH-PCA of an Ensemble of Data Sets.............................385
Defining the ensemble of data sets................................385
The scatter probe and the cross power spectrum matrix...............386
Reduction to principal harmonic parts; the PCA property..............387
g. Traveling-Wave Analysis by CH-PCA of Random Multivariate Fields ... 388
The phase spectrum and the gain factor............................388
The power spectrum and the coherence............................389
h. Selection Rules for CH-PCA....................................390
Modifying Rule N.............................................391
Direct application of Rule KS2...................................392
i. Return to the Time Domain in CH-PCA............................393
Convolution and covariance theorems.............................393
CONTENTS ֒
CH-PCA on the time domain....................................394
Physical interpretation of time-domain CH-PCA formulas.............395
Connection between the cross power spectrum matrix and the average-
scatter matrix; the distinction between PC A and CH-PCA.........396
j. Bibliographic Notes...........................................397
Brillinger s development of CH-PCA on the time domain..............399
Alternate approaches to the traveling-wave problem..................399
References.........................................................402
Index..............................................................419
|
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| author | Preisendorfer, Rudolph W. |
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| discipline | Geologie / Paläontologie Geographie |
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| id | DE-604.BV001303219 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T10:12:28Z |
| institution | BVB |
| isbn | 0444430148 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000786471 |
| oclc_num | 299848240 |
| open_access_boolean | |
| owner | DE-12 DE-19 DE-BY-UBM DE-703 DE-20 DE-11 |
| owner_facet | DE-12 DE-19 DE-BY-UBM DE-703 DE-20 DE-11 |
| physical | XVIII, 425 S. graph. Darst. |
| publishDate | 1988 |
| publishDateSearch | 1988 |
| publishDateSort | 1988 |
| publisher | Elsevier |
| record_format | marc |
| series | Developments in atmospheric science |
| series2 | Developments in atmospheric science |
| spellingShingle | Preisendorfer, Rudolph W. Principal component analysis in meteorology and oceanography Developments in atmospheric science Analyse factorielle Méthodologie - Océanographie Météorologie - Modèles mathématiques ram Météorologie - Observations Océanographie - Modèles mathématiques ram Hauptkomponentenanalyse (DE-588)4129174-8 gnd Meteorologie (DE-588)4038953-4 gnd Komponentenanalyse (DE-588)4133251-9 gnd Faktorenanalyse (DE-588)4016338-6 gnd Meereskunde (DE-588)4074685-9 gnd |
| subject_GND | (DE-588)4129174-8 (DE-588)4038953-4 (DE-588)4133251-9 (DE-588)4016338-6 (DE-588)4074685-9 |
| title | Principal component analysis in meteorology and oceanography |
| title_auth | Principal component analysis in meteorology and oceanography |
| title_exact_search | Principal component analysis in meteorology and oceanography |
| title_full | Principal component analysis in meteorology and oceanography posthumously comp. and ed. by Curtis D. Mobley |
| title_fullStr | Principal component analysis in meteorology and oceanography posthumously comp. and ed. by Curtis D. Mobley |
| title_full_unstemmed | Principal component analysis in meteorology and oceanography posthumously comp. and ed. by Curtis D. Mobley |
| title_short | Principal component analysis in meteorology and oceanography |
| title_sort | principal component analysis in meteorology and oceanography |
| topic | Analyse factorielle Méthodologie - Océanographie Météorologie - Modèles mathématiques ram Météorologie - Observations Océanographie - Modèles mathématiques ram Hauptkomponentenanalyse (DE-588)4129174-8 gnd Meteorologie (DE-588)4038953-4 gnd Komponentenanalyse (DE-588)4133251-9 gnd Faktorenanalyse (DE-588)4016338-6 gnd Meereskunde (DE-588)4074685-9 gnd |
| topic_facet | Analyse factorielle Méthodologie - Océanographie Météorologie - Modèles mathématiques Météorologie - Observations Océanographie - Modèles mathématiques Hauptkomponentenanalyse Meteorologie Komponentenanalyse Faktorenanalyse Meereskunde |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000786471&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001891242 |
| work_keys_str_mv | AT preisendorferrudolphw principalcomponentanalysisinmeteorologyandoceanography AT mobleycurtisd principalcomponentanalysisinmeteorologyandoceanography |