Matrix tricks for linear statistical models: our personal top twenty
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2011
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 486 S. Ill., graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783642104725 |
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MARC
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| 024 | 3 | |a 9783642104725 | |
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| 100 | 1 | |a Puntanen, Simo |e Verfasser |0 (DE-588)136877214 |4 aut | |
| 245 | 1 | 0 | |a Matrix tricks for linear statistical models |b our personal top twenty |c Simo Puntanen ; George P. H. Styan ; Jarkko Isotalo |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2011 | |
| 300 | |a XVII, 486 S. |b Ill., graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 689 | 0 | 1 | |a Lineare Algebra |0 (DE-588)4035811-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Styan, George P. H. |e Verfasser |4 aut | |
| 700 | 1 | |a Isotalo, Jarkko |e Verfasser |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-642-10473-2 |
| 856 | 4 | 2 | |m Digitalisierung UB Passau |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024495093&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-024495093 | ||
Record in the Search Index
| _version_ | 1804148502093103104 |
|---|---|
| adam_text | Contents
Introduction
.................................................. 1
Opening Figures
............................................ 1
Preliminaries to Linear Algebra and Statistics
.................. 3
Preliminaries to Random Vectors and Data Matrices
............ 16
Preliminaries to Linear Models
............................... 28
Exercises
.................................................. 46
Opening Figures: Continued
—
see page
2....................... 56
1
Easy Column Space Tricks
................................ 57
1.1
Equality f(X VX)
=
^(X V) when V Is
Nonnegative
Definite
.............................................. 59
1.2
Estimability in a Simple ANOVA
........................ 60
1.3
Column Space of the Covariance Matrix
.................. 62
1.4
Exercises
............................................. 66
2
Easy Projector Tricks
..................................... 71
2.1
Minimal Angle
........................................ 77
2.2
Minimizing vai d(y
—
Xob)
............................. 79
2.3
Minimizing (Y
-
XB) (Y
-
XB)
........................ 80
2.4
Orthogonal Projector with Respect to V
................. 81
2.5
Generalized Orthogonal Projector
....................... 85
2.6
Exercises
............................................. 88
3
Easy Correlation Tricks
................................... 91
3.1
Orthogonality and Uncorrelatedness
..................... 92
3.2
On the Role of the Vector
1............................. 95
3.3
Partial Correlation Geometrically
........................ 101
3.4
Exercises
............................................. 102
4
Generalized Inverses in a Nutshell
........................ 105
4.1
Generalized Inverse
&
the Singular Value Decomposition
. . . 109
xiv Contents
4.2
Generalized Inverse
&
Oblique Projector
..................112
4.3
Generalized Inverse
&
Solutions to a Linear Equation
......114
4.4
A Simple Example
.....................................118
4.5
Exercises
.............................................119
5
Rank of the Partitioned Matrix and the Matrix Product
. 121
5.1
Decomposition
<ář(X
:
V)
=
^(X
:
VX1)
.................123
5.2
Consistency Condition
у Є ^(Х
:
V)
.....................125
5.3
Rank of a Partitioned
Nonnegative
Definite Matrix
........126
5.4
Equality of ^(AF) and
« (AG)
.........................126
5.5
Rank of the Model Matrix
..............................127
5.6
The Rank of the Extended Model Matrix
.................130
5.7
Matrix Volume and the Sample Generalized Variance
......131
5.8
Properties of X VX
...................................132
5.9
Number of Unit Eigenvalues of
РдРв
....................133
5.10
Properties of X VX
-
X VZ(Z VZ)-Z VX
..............134
5.11
Intersection ci{K)
Π %?(Β)..............................
137
5.12
Solution (in Y) to Y(A
:
B)
= (0 :
B)
...................139
5.13
Some Properties of
<^(X)^„1
.............................140
5.14
Exercises
.............................................143
6
Rank Cancellation Rule
...................................145
6.1
Simple Properties of the Hat Matrix
Η
...................146
6.2
Properties of
Χ(Χ ν+Χ)-χ ν+γ
.......................147
7
Sum of Orthogonal Projectors
............................151
8
A Decomposition of the Orthogonal Projector
............155
8.1
Mahalanobis Distance
&
the Hat Matrix
Η
...............157
8.2
Subvector of
β
........................................158
8.3
A Single Regression Coefficient
..........................161
8.4
The Reduced Model
&
the Frisch-Waugh-Lovell Theorem
. . 163
8.5
The Subvectors
ßo
and /3X
..............................166
8.6
The Decomposition SST
=
SSR + SSE
....................168
8.7
Representations of R2
..................................171
8.8
R
as a Maximal Correlation
.............................172
8.9
Testing a Linear Hypothesis
............................175
8.10
A Property of the Partial Correlation
....................178
8.11
Another Property of the Partial Correlation
...............179
8.12
Deleting an Observation: cov(y)
=
σ2Ι
...................180
8.13
Geometrical Illustrations
...............................183
8.14
Exercises
.............................................184
9
Minimizing cov(y
—
Fx)
...................................191
9.1
Best Linear Predictor
..................................194
9.2
What Is Regression?
...................................198
9.3
Throwing Two Dice
....................................201
9.4
Predictive Approach to Principal Component Analysis
.....203
9.5
A Property of the Prediction Errors
......................207
9.6
Prediction in the Autocorrelation Structure
...............208
9.7
Minimizing cov(Hy
-
FMy)
............................210
9.8
Exercises
.............................................211
10
BLUE
....................................................215
10.1
BLUE
&
Pandora s Box when X
=
ln
...................222
10.2
OLSE vs BLUE when X
=
ln
...........................224
10.3
OLSE vs BLUE under the Intraclass Correlation Structure.
. 225
10.4
General Expressions for the BLUE
.......................228
10.5
Multivariate Linear Model
..............................229
10.6
Random Sample without Replacement
...................234
10.7
Efficiency of the OLSE
.................................234
10.8
Efficiency and Canonical Correlations
....................241
10.9
Best Linear Unbiased Predictor
.........................245
10.10
Examples on the BLUPs
...............................250
10.11
Mixed Linear Models
..................................253
10.12
Linear Sufficiency
......................................257
10.13
Admissibility
..........................................259
10.14
Exercises
.............................................260
11
General Solution to AYB
=
С
............................267
11.1
Equality of the BLUEs of
Xß
under Two Models
..........269
11.2
Model Jii vs Model Jiv
...............................272
11.3
Stochastic Restrictions and the BLUPs
...................272
11.4
Two Mixed Models
....................................276
11.5
Two Models with New Unobserved Future Observations
.... 278
11.6
Exercises
.............................................280
12
Invariance
with Respect to the Choice of Generalized
Inverse
...................................................283
12.1
Invariance
of X(X X)-X
..............................284
12.2
Estimability and
Invariance
.............................284
12.3
Properties of X W~X
.................................286
12.4
Exercises
.............................................288
13
Block-Diagonalization and the
Schur
Complement
........291
13.1
Inverting a Partitioned nnd Matrix
—Schur
Complement
. . 294
13.2
Inverting X X
........................................295
13.3
Determinants, Ranks and
Schur
Complements
.............297
13.4
Inverting a Sum of Matrices
.............................300
13.5
Generating Observations with a Given Covariance Matrix.
.. 301
13.6
Exercises
.............................................303
14
Nonnegative
Definiteness of a Partitioned Matrix
.........305
14.1
When Is a Correlation Matrix a Correlation Matrix?
.....307
14.2
A <L
Β φ=φ·
^(A)
с
■žf(B)
and ch!(AB+)
< 1 ..........311
14.3
BLUE s Covariance Matrix as a Shorted Matrix
...........312
14.4
Exercises
.............................................314
15
The Matrix
M
............................................317
15.1
Representations for the BLUE
...........................324
15.2
The BLUE s Covariance Matrix
.........................325
15.3
Partitioned Linear Model
...............................327
15.4
The Frisch-Waugh-Lovell Theorem: General Case
.........328
15.5
The Watson Efficiency of a Subvector
....................331
15.6
Equality of the BLUEs of
Хі/Зі
under Two Models
........333
15.7
Adding a Variable: a Recursive Formula for the BLUE
.....335
15.8
Deleting an Observation: cov(y)
=
σ2ν
..................337
15.9
Weighted Sum of Squares of Errors
......................338
15.10
Exercises
.............................................340
16
Disjointness of Column Spaces
............................343
16.1
Estimability of X2/32
...................................345
16.2
When is (V
+
XUX ) a G-inverse of V?
................346
16.3
Usual Constraints
.....................................347
16.4
Exercises
.............................................348
17
Full Rank Decomposition
.................................349
17.1
Some Properties of an Idempotent Matrix
................350
17.2
Rank Additivity
.......................................351
17.3
Cochran s Theorem: a Simple Version
....................352
17.4
Proof of the RCR Using the Full Rank Decomposition
......354
17.5
Exercises
.............................................354
18
Eigenvalue Decomposition
................................357
18.1
The Maximal Value of x Ax
............................360
18.2
Principal Components
..................................362
18.3
Eigenvalues of
Σ2χ2
...................................364
18.4
Eigenvalues of the Intraclass Correlation Matrix
...........366
18.5
Proper Eigenvalues:
В
Positive Definite
..................367
18.6
Statistical Distance
....................................372
18.7
Proper Eigenvalues;
В
Nonnegative
Definite
..............374
18.8
Eigenvalues of the BLUE s Covariance Matrix
.............378
18.9
Canonical Correlations and Proper Eigenvalues
............379
18.10
More about Canonical Correlations and the Watson
Efficiency
.............................................382
18.11
Exercises
.............................................386
19 Singular
Value
Decomposition
............................391
19.1
Maximal Value
of x Ay
................................395
19.2
When is AyA
=
B B?
..................................396
19.3
Approximation by a Matrix of Lower Rank
...............397
19.4
Principal Component Analysis
..........................402
19.5
Generalized Inverses through the
SVD
...................407
19.6
Canonical Correlations and the
SVD
.....................408
19.7
Exercises
.............................................409
20
The Cauchy—
Schwarz
Inequality
..........................415
20.1
Specific Versions of the Cauchy-Schwarz Inequality
........ 416
20.2
Maximizing cor(y, b x)
................................. 417
20.3
The Kantorovich Inequality
............................. 418
20.4
The Wielandt Inequality
............................... 419
20.5
How Deviant Can You Be?
............................. 420
20.6
Two Inequalities Related to Cronbach s alpha
............. 422
20.7
Matrix Versions of the Cauchy-Schwarz and Kantorovich
Inequality
............................................ 423
20.8
Exercises
............................................. 425
Notation
......................................................427
List of Figures and Philatelic Items
...........................435
References
....................................................439
Author Index
.................................................469
Subject Index
................................................475
|
| any_adam_object | 1 |
| author | Puntanen, Simo Styan, George P. H. Isotalo, Jarkko |
| author_GND | (DE-588)136877214 |
| author_facet | Puntanen, Simo Styan, George P. H. Isotalo, Jarkko |
| author_role | aut aut aut |
| author_sort | Puntanen, Simo |
| author_variant | s p sp g p h s gph gphs j i ji |
| building | Verbundindex |
| bvnumber | BV039645266 |
| classification_rvk | SK 840 |
| ctrlnum | (OCoLC)755981546 (DE-599)DNB1000032892 |
| dewey-full | 519.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5 |
| dewey-search | 519.5 |
| dewey-sort | 3519.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV039645266 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:08:09Z |
| institution | BVB |
| isbn | 9783642104725 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024495093 |
| oclc_num | 755981546 |
| open_access_boolean | |
| owner | DE-384 DE-739 DE-824 DE-188 DE-83 DE-11 |
| owner_facet | DE-384 DE-739 DE-824 DE-188 DE-83 DE-11 |
| physical | XVII, 486 S. Ill., graph. Darst. 235 mm x 155 mm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| spelling | Puntanen, Simo Verfasser (DE-588)136877214 aut Matrix tricks for linear statistical models our personal top twenty Simo Puntanen ; George P. H. Styan ; Jarkko Isotalo Berlin [u.a.] Springer 2011 XVII, 486 S. Ill., graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lineares Modell (DE-588)4134827-8 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Lineares Modell (DE-588)4134827-8 s Lineare Algebra (DE-588)4035811-2 s DE-604 Styan, George P. H. Verfasser aut Isotalo, Jarkko Verfasser aut Erscheint auch als Online-Ausgabe 978-3-642-10473-2 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024495093&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Puntanen, Simo Styan, George P. H. Isotalo, Jarkko Matrix tricks for linear statistical models our personal top twenty Lineares Modell (DE-588)4134827-8 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4134827-8 (DE-588)4035811-2 |
| title | Matrix tricks for linear statistical models our personal top twenty |
| title_auth | Matrix tricks for linear statistical models our personal top twenty |
| title_exact_search | Matrix tricks for linear statistical models our personal top twenty |
| title_full | Matrix tricks for linear statistical models our personal top twenty Simo Puntanen ; George P. H. Styan ; Jarkko Isotalo |
| title_fullStr | Matrix tricks for linear statistical models our personal top twenty Simo Puntanen ; George P. H. Styan ; Jarkko Isotalo |
| title_full_unstemmed | Matrix tricks for linear statistical models our personal top twenty Simo Puntanen ; George P. H. Styan ; Jarkko Isotalo |
| title_short | Matrix tricks for linear statistical models |
| title_sort | matrix tricks for linear statistical models our personal top twenty |
| title_sub | our personal top twenty |
| topic | Lineares Modell (DE-588)4134827-8 gnd Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Lineares Modell Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024495093&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT puntanensimo matrixtricksforlinearstatisticalmodelsourpersonaltoptwenty AT styangeorgeph matrixtricksforlinearstatisticalmodelsourpersonaltoptwenty AT isotalojarkko matrixtricksforlinearstatisticalmodelsourpersonaltoptwenty |