Hervé Abdi - Cedric-Cnam

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Hervé Abdi

The University of Texas at Dallas

Paris CNAM November 2011

A^NALYSE MÛLTI-TÂBLEAUX: LÂ FÂMILLE STATIS

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CNAM: 7 N^OVEMBRE 2011 Part two: La Famille STATIS

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CNAM: 7 N^OVEMBRE 2011 La Famille STATIS

Biblio. Allez voir à www.utdallas.edu/~herve A87: STATIS & DISTATIS

A71, A59: DISTATIS C71: Rv Coefficient

C40: Multiple Factor Analysis C33: STATIS

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S^TART ^WITH K ^TABLES The STATIS Family

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S^TART ^WITH K ^TABLES The STATIS Family

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B^EFORE STATIS

•  Center and Normalize

(or not) columns (almost always)

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B^EFORE STATIS

•  Normalize (or not) rows (rarely) exception: CA

è sum of x = 1

Escofier/Volle/Rao/Hellinger (etc.) è sum of x² = 1

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B^EFORE STATIS

•  What about the tables?

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N^ORMALIZING T^ABLES: W^HY?

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H^OW ^TO ^NORMALIZE T^ABLE K.

•  Divide all elements of X_kby J_k

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•  Divide all elements of X_kby J_kor better by J_k^½

•  Plain multi-block èTucker 1 & consensus PCA

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•  Divide all elements of X_kby (sum X_k) ^½

•  Plain multi-block è SUM-PCA

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•  Divide all elements of X_kby (sum X_kX_k^T) ^½

•  Plain multi-block è R_V-PCA

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•  Divide all elements of X_kby first singular value

•  Plain multi-block è Multiple Factor Analysis

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TÂBLE NORMALIZATION: MÔST ÎMPORTANT ^STEP!

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W^HEN ^THE ^TABLES ^ARE ^NORMALIZED: W^HAT ^TO ^DO?

•  STATIS: 1,2, 3 …

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S^TART ^WITH ^A ^MULTI ^TABLE ^MATRIX

1

i I

……

1 ^… j ^… J₁ 1 ^… j ^… J_k 1^… j^…J_K

X

C

< S_k,S_k’ >

K

K 1

2

PCA of C

GPCA weighted by a

Compromise Tables in the compromise

… …

…

J variables in K studies

I Observations

Inner Product matrix

Inner Product map

a

… …

X₁ X_k X_K

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C^OMPUTE ^THE ^BETWEEN ^TABLE ^SIMILARITY

1

i I

……

1 ^… j ^… J₁ 1 ^… j ^… J_k 1^… j^…J_K

X

C

< S_k,S_k’ >

K

K 1

2

PCA of C

GPCA weighted by a

… …

…

I Observations

Inner Product map

a

… …

X₁ X_k X_K

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GÊT Â PCA ÔF ^THE ^BETWEEN ^TABLE ^SIMILARITY

1

i I

……

1 ^… j ^… J₁ 1 ^… j ^… J_k 1^… j^…J_K

X

C

< S_k,S_k’ >

K

K 1

2

PCA of C

GPCA weighted by a

… …

…

I Observations

Inner Product map

a

… …

X₁ X_k X_K

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PCA ÔF C ^GIVES ÔPTIMAL ÂLPHA ^WEIGHTS

1

i I

……

1 ^… j ^… J₁ 1 ^… j ^… J_k 1^… j^…J_K

X

C

< S_k,S_k’ >

K

K 1

2

PCA of C

GPCA weighted by a

… …

…

I Observations

Inner Product map

a

… …

X₁ X_k X_K

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A^LPHA ^WEIGHTS ÂRE ÛSED ^FOR GPCA ÔF X

1

i I

……

1 ^… j ^… J₁ 1 ^… j ^… J_k 1^… j^…J_K

X

C

< S_k,S_k’ >

K

K 1

2

PCA of C

GPCA weighted by a

… …

…

I Observations

Inner Product map

a

… …

X₁ X_k X_K

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S^TART ^WITH ^A ^MULTI ^TABLE ^MATRIX

1

i I

……

1 ^… j ^… J₁ 1 ^… j ^… J_k 1^… j^…J_K

X

C

< S_k,S_k’ >

K

K 1

2

PCA of C

GPCA weighted by a

… …

…

I Observations

Inner Product map

a

… …

X₁ X_k X_K

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GPCA ^OF X è ^FACTOR ^SCORES F (^COMPROMISE)

1

i I

……

1 ^… j ^… J₁ 1 ^… j ^… J_k 1^… j^…J_K

X

C

< S_k,S_k’ >

K

K 1

2

PCA of C

GPCA weighted by a

… …

…

I Observations

Inner Product map

a

… …

X₁ X_k X_K

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P^ROJECT ^THE XK ON COMPROMISE è FK

1

i I

……

1 … j … J₁ 1 ^… j ^… J_k 1^…j ^…J_K

X

C

< S_k,S_k’ >

K

K 1

2

PCA of C

GPCA weighted by a

… …

…

I Observations

Inner Product map

a

… …

X₁ X_k X_K

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A^N E^XAMPLE

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E^XAMPLE: 10 PARTICIPANTS TASTE 3*4 =12 ^WINES

•  Sauvignon Blanc Wines

•  From New-Zealand, France, and Canada

•  Chemical/Physical measurements

•  Specific scales + Four commons scales:

cat-pee, passion, green pepper, mineral

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R^EMEMBER: T^HE ^STEPS ^OF ^STATIS

•  1. Between Table Structure

•  2. Derive Optimal Weights

•  3. Compute Compromise from Weights

•  4. Eigen-decompose Compromise

•  5. Project Original Tables (factor scores)

•  6. … Is the Earth round? …

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T^HE ^STEPS ^OF ^STATIS

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T^HE D^ATA: 10 A^SSESSORS ^BY 3*4 = 12 ^WINES

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SUPPLEMENTARY TABLE: C^HEMISTRY

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A ^MATRIX: A^SSESSOR 1

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P^RE-P^ROCESSED (^CENTER, S^UM ^OF ^X² = 1)

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A C^ROSS-P^RODUCT M^ATRIX: X₁X₁^T

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CÔSINE (ÔR R_V) MÂTRIX

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E^IGEN DECOMPOSITION OF C

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EÎGEN ÔF ^C: T^HE ÂSSESSORS’ ^MAP

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F^ACTOR ^SCORES ^FROM C

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R^ESCALE ^FACTOR ^SCORES ^DIMENSION 1 ^TO ^SUM ^OF 1

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W^EIGHTS (^EQUAL ^WEIGHTS = .10)

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W^EIGHTS (^EQUAL ^WEIGHTS = .10)

•  From α get diagonal matrix Α

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•  3. Compute Compromise from α weights

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MÂSSES ÂRE ^FOR ^THE ^ROWS (ÊQUAL ^MASSES = .08)

•  Get diagonal matrix of masses for rows M

•  M = ¹/_II (equal masses)

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G^ET C^OMPROMISE. 1 GENERALIZED SVD OF X

•  X = PΔQ^Twith Q^TAQ = P^TMP = I

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GÊT CÔMPROMISE. 2 FÂCTOR ^SCORES

•  F = PΔ = XAQ

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C^OMPROMISE: P^LOT ^OF ^FACTOR ^SCORES

1 2

2 2

4 3

1 3

4

2 1 3

4 1 2

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P^ARTIAL ^FACTOR ^SCORES

•  F = PΔ = XAQ

•  F_k = X_kQ_k

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P^ARTIAL ^FACTOR ^SCORES: B^ARYCENTRIC ^PROPERTY

•  F = PΔ = XAQ

•  F_k = X_kQ_k

•  F =

Σ

α_kF_k =

Σ

α_kX_kQ_k

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C^OMPROMISE ^WITH “T^ABLES”

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T^HE T^ABLES ^AS “^BIPLOTS”

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W^HAT ^ARE ^THE ^IMPORTANT ^TABLES?

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CONTRIBUTIONS TO I^NERTIA

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P^ARTIAL I^NERTIA

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H^OW ^TO ^HELP: P^ROJECTING ^NEW ^TABLES

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P^HYSICO ^AS ^SUP (^FACTOR ^SCORES + ^LOADINGS)

1 2

1 1

1

2 2

3 2

3

4 3 4

4

Acidity pH

Alcohol Sugar

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P^HYSICO. R^V ^AS ^SUP

1

2 3 4 5

6 7

8 9

101 2

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•  6… ? Is the Earth …

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I^S ^THE ^EARTH ^ROUND P < .05?

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I^S ^THE ^EARTH ^ROUND P < .05?

•  Bootstrap again: The assessors are random

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C^OMPUTE ^CONFIDENCE ^INTERVALS

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B^OOTSTRAP 95% CI

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BÔOTSTRAP RÂTIOS (^WHAT ÎS ^THAT?)

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E^XTENSIONS ^OF STATIS

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P^ARTIAL T^RIADIC A^NALYSIS

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P^ARTIAL T^RIADIC ^ANALYSIS

•  Same variables all over:

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use X_kin lieu of S_k

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use X_kin lieu of S_k

Possible problem: negative cosine

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PTA F^ACTOR S^CORES

PeeCat

Passion Fruit

Green Pepper

Mineral

1

2

4 3

1

3 4

2 1 4 3

2 1

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D^ISTATIS

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D^ISTATIS: S^TART ^WITH ^DISTANCE ^MATRICES

•  K (squared Euclidean) distance matrices D_k

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T^RANSFORMS ^THE DÎSTANCES ÎNTO CÔVARIANCE

I

S

I

D

Double Centering

Distance

With a formula:

With matrices:

s_i,j = d_i,j – (d_i,+ – d_+,+) – (d_+,j – d_+,+)

S = –.5ΞDΞ^T with Ξ = I – 1m^T and m^T1 = 1

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A^ND ^BACK ^TO ^STANDARD STATIS

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N ^GROUPS: C^ANONICAL S^TATIS: ^CANOSTATIS.

•  Here 3 groups:

France, Canada, New Zealand

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N ^GROUPS

•  Compute Mahalanobis distance per Table

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N ^GROUPS

•  Compute Mahalanobis distance per Table

•  And back to DISTATIS

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W^INE E^XAMPLE

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CANOSTATIS: T^HE ^ASSESSORS

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C^ANONICAL S^TATIS: 3 G^ROUPS

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CANOSTATIS ^WITH ^CONFIDENCE

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O^NE ^MORE ^TABLE: (K+1) ^STATIS

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•  Use S_k* = HX_kinstead of S_k

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•  Use S_k* = HX_kinstead of S_k

•  and back to STATIS

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(K+1) STATIS

1) C^OMPROMISE & 2) P^HYSICO

A B

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A^NISO-^STATIS

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A^NISO-^STATIS

•  One weight per column

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A^NISOSTATIS

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DÔUBLE ^STATIS: ^DO-^STATIS ÔR ^DO-ÂCT

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DÔUBLE ^STATIS: ^DO-^STATIS ÔR ^DO-ÂCT

•  Two sets of matrices

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R^ELATED ^TECHNIQUES

•  Generalized canonical correlation

•  Multiple factor analysis & SUM-PCA

•  INDSCAL