Partial Least Squares (PLS-PM PM and PLS-R) and Generalized Procrustes Rotations for Multiple Table Analysis

Partial Least Squares (PLS

Partial Least Squares (PLS - - PM and PM and PLS PLS - - R) and Generalized R) and Generalized Procrustes Procrustes Rotations for Multiple Table Analysis Rotations for Multiple Table Analysis

Michel Tenenhaus Michel Tenenhaus

HEC School of Management HEC School of Management

GREGHEC, France GREGHEC, France Vincenzo Esposito Vinzi Vincenzo Esposito Vinzi

Department of Mathematics and Statistics Department of Mathematics and Statistics

University of Naples, Italy

University of Naples, Italy

Wine data (Asselin, Morlat & Pagès)

X ₁ = Smell at rest, X ₂ = View, X ₃ = Smell after shaking, X ₄ = Tasting

X

₁

X

₂

X

₃

2el (Saumur),1 1cha (Saumur),1 1fon (Bourgueil),1 1vau (Chinon),3 … t1 (Saumur),4 t2 (Saumur),4

Smell intensity at rest 3.07 2.96 2.86 2.81 … 3.70 3.71

Aromatic quality at rest 3.00 2.82 2.93 2.59 … 3.19 2.93

Fruity note at rest 2.71 2.38 2.56 2.42 … 2.83 2.52

Floral note at rest 2.28 2.28 1.96 1.91 … 1.83 2.04

Spicy note at rest 1.96 1.68 2.08 2.16 … 2.38 2.67

Visual intensity 4.32 3.22 3.54 2.89 … 4.32 4.32

Shading (orange to purple) 4.00 3.00 3.39 2.79 … 4.00 4.11

Surface impression 3.27 2.81 3.00 2.54 … 3.33 3.26

Smell intensity after shaking 3.41 3.37 3.25 3.16 … 3.74 3.73

Smell quality after shaking 3.31 3.00 2.93 2.88 … 3.08 2.88

Fruity note after shaking 2.88 2.56 2.77 2.39 … 2.83 2.60

Floral note after shaking 2.32 2.44 2.19 2.08 … 1.77 2.08

Spicy note after shaking 1.84 1.74 2.25 2.17 … 2.44 2.61

Vegetable note after shaking 2.00 2.00 1.75 2.30 … 2.29 2.17

Phenolic note after shaking 1.65 1.38 1.25 1.48 … 1.57 1.65

Aromatic intensity in mouth 3.26 2.96 3.08 2.54 … 3.44 3.10

Aromatic persisitence in mouth 3.26 2.96 3.08 2.54 … 3.44 3.10

Aromatic quality in mouth 3.26 2.96 3.08 2.54 … 3.44 3.10

Intensity of attack 2.96 3.04 3.22 2.70 … 2.96 3.33

Acidity 2.11 2.11 2.18 3.18 … 2.41 2.57

Astringency 2.43 2.18 2.25 2.18 … 2.64 2.67

Alcohol 2.50 2.65 2.64 2.50 … 2.96 2.70

Balance (Acid., Astr., Alco.) 3.25 2.93 3.32 2.33 … 2.57 2.77

Mellow ness 2.73 2.50 2.68 1.68 … 2.07 2.31

Bitterness 1.93 1.93 2.00 1.96 … 2.22 2.67

Ending intensity in mouth 2.86 2.89 3.07 2.46 … 3.04 3.33

Harmony 3.14 2.96 3.14 2.04 … 2.74 3.00

Global quality 3.39 3.21 3.54 2.46 … 2.64 2.85

X

₄

3 Appellations 4 Soils

Illustrative variable

SMELL AT REST

Component 1

1.0 .5

0.0 -.5

-1.0

Component 2

1.0

.5

0.0

-.5

-1.0

spicy note at rest

floral note at rest fruity note at rest aromatic quality at rest

smell intensity at rest

Component 1

1.0 .5

0.0 -.5

-1.0

Component 2

1.0

.5

0.0

-.5

-1.0

spicy note at rest

floral note at rest fruity note at rest aromatic quality at rest

smell intensity at rest

VIEW

Component 1

1.0 .5

0.0 -.5

-1.0

Component 2

1.0

.5

0.0

-.5

-1.0

surface impression

shading visual intensity

Component 1

1.0 .5

0.0 -.5

-1.0

Component 2

1.0

.5

0.0

-.5

-1.0

surface impression

shading visual intensity

SMELL AFTER SHAKING

Component 1

1.0 .5

0.0 -.5

-1.0

Component 2

1.0

.5

0.0

-.5

-1.0

aromatic quality in mouth

Aromatic persistence aromatic intensity in mouth phelonic note vegetable note

spicy note

floral note

fruity note smell quality smell intensity

Component 1

1.0 .5

0.0 -.5

-1.0

Component 2

1.0

.5

0.0

-.5

-1.0

aromatic quality in mouth

Aromatic persistence aromatic intensity in mouth phelonic note vegetable note

spicy note

floral note

fruity note smell quality smell intensity

TASTING

Component 1

1.0 .5

0.0 -.5

-1.0

Component 2

1.0

.5

0.0

-.5

-1.0

harmony ending intensiry in mouth

bitterness

mellowness balance alcohol astringency acidity

intensity of attack

Component 1

1.0 .5

0.0 -.5

-1.0

Component 2

1.0

.5

0.0

-.5

-1.0

harmony ending intensiry in mouth

bitterness

mellowness balance alcohol astringency acidity

intensity of attack

PCA of each block

2 dimensions

2 dimensions 2 dimensions 1 dimension

1

.725 1

.833 .927 1

.714 .882 .940 1

PC1-Smell at rest PC1-View

PC1-Smell after shaking PC1-Tasting

PC1-Smell

at rest PC1-View

PC1-Smell

after shaking PC1-Tasting

1

-.376 1

.760 -.430 1

.741 -.029 .686 1

PC2-Smell at rest PC2-View

PC2-Smell after shaking PC2-Tasting

PC2-Smell

at rest PC2-View

PC2-Smell

after shaking PC2-Tasting

Some methods for multi-block

analysis optimizing a criterion

1. SUMCOR (Horst, 1961)

, ( _j, _k) or ( _j, _k)

j k j k

Max

∑

Cor F F Max

∑

Cor F

∑

F 2. MAXVAR (Horst, 1961) or

GCCA (Carroll, 1968):

[ ( , )] or 2( , )

first j k j j

Maxλ Cor F F Max

∑

Cor F F 3. SsqCor (Kettenring, 1971): ²

, ( _j, _k) Max

∑

j kCor F F 4. GenVar (Kettenring, 1971): Min det[Cor F F( _j, _k)]

5. MINVAR (Kettenring, 1971): Minλ_last[Cor F F( _j, _k)]

6. Lafosse (1989): ²( _j, _k)

j k

Max

∑

Cor F

∑

F 7. Mathes (1993), Hanafi (2004):

, | ( _j, _k) | Max

∑

j k Cor F F 8. ACOM (Hanafi & Chessel,

1996):

2 All 1, standardized

( , )

j

j j

w F j

Maximize Cov X w F

=

∑

9. MAXBET (Van de Geer,

1984 & Ten Berge, 1988): ^{All 1}

[ ( ) ( , )]

j

j j j j k k

j j k

w

Max Var X w _≠ Cov X w X w

=

∑

+

∑

10. MAXDIFF (Van de Geer,

1984 & Ten Berge, 1988): ^{All 1}

[ ( , )]

j

j j k k

w j k

Max _≠ Cov X w X w

=

∑

11. Generalized PCA (Casin, 2001):

2 2 ' 1 '

( , _j) ( _jh, _j( _{j j}) _j )

j h

Max

∑

R F X

∑

Cor x X X X ⁻ X F 12. Split-PCA (Lohmöller, 1989),

SUM-PCA (Smilde et al., 2003):

' 2

, _j j j j

MinF P

∑

X −FP 13. Multiple Factor Analysis

(Escofier & Pagès, 1994):

2 ' , j

'

1

[ ( , )]

j

j j

F P first jh jh

Min X FP

Cor x x −

∑

λ 14. Oblique Maximum Variance

method (Horst, 1965):

2

' 1/ 2 '

, j

(1 )

j

j j j j

MinF P X X X FP n

− −

∑

Generalized

Canonical Correlation Analysis

- Criteria searching for first order components - Auxiliary variables or super-block (not always) - Next (higher order) components by deflation on:

Super-block (usual) Each block

Both

- Variable nb of comp. in each block overcomes the dimension limitation due to each block

Generalized PLS Regresion

Split-PCA

' ' '

1 1 ... ...

j j

j j j jh j h j m j m j

X = F p + + F p + + F p + E

MBA is a factor analysis of tables :

(1) Factors (LV, Scores, Components)

uncorrelated, and are well explaining their own block.

1 ,...,

j jm

j

F F are standardized,

(2) Same order factors F _{1 h} ,..., F _Jh are well ( positively ) correlated subject to constraints :

1 1

' ' '

1 11 11 ... 1 _h 1 _h ... 1 _m 1 _m 1

X = F p + + F p + + F p + E

#

#

' ' '

1 1 ... ...

J J

J J J Jh J h J m J m J

X = F p + + F p + + F p + E

( to improve interpretation ).

Maximize Variance Maximize

Correlation

Step 1 :

Searching for « common » spaces ( ₁ ,..., )

j jm

j

L F F

1) For each block X _j the super-block X _-j is defined by :

2) PLS regression of X _-j on X _j provides m _j PLS components ( m _j is obtained by cross-validation).

1 ,...,

j jm

j

t t

1 1

( , ..., ) ( , ..., )

j j

j jm j jm

L F F = L t t

1 ,..., 1 , 1 ,...,

j j j J

X ₋ = ⎣ ⎡ X X ₋ X ₊ X ⎤ ⎦

1 ,..., stand for the part of which explains and the other blocks.

j jm

j

j j

F F X X

“Smell at rest” block X ₁

PLS regression of [X ₂ , X ₃ , X ₄ ] on X ₁

Î Two components for the “Smell at rest” block X ₁

R2X = 0.750, R2Y = 0.503

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Comp[1] Comp[2]

Comp No.

SIMCA-P 10.5 - 10/08/2004 09:57:42

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

pc(corr)[Comp. 2]

pc(corr)[Comp. 1]

SMELL INTENSITY

AROMATIC QUALITY FRUITY NOTE

FLORAL NOTE SPICY NOTE

VISION1 VISION2

VISION3 AGITA1

AGITA2 AGITA3

AGITA4 AGITA5

AGITA6

AGITA7

AGITA8 AGITA9

AGITA10 GUSTA1 GUSTA2

GUSTA3

GUSTA4

GUSTA5 GUSTA6 GUSTA7

GUSTA8

GUSTA9

SIMCA-P 10.5 - 10/08/2004 10:07:59

T ₁

The “Smell at rest” block is summarized by T ₁ ^*

(*) = Standardized variables

t[1] t[2]

2EL 0.338937 -0.6825

1CHA -1.2798 -1.33349

1FON -1.37899 0.191776

1VAU -3.00289 0.756837

1DAM 3.31696 0.321494

- - -

1ROC -0.501942 -0.23561

2ING -2.37583 -0.785852

T1 1.0626 2.69374

T2 -0.0209994 3.486

“View” block X ₂

PLS regression of [X ₁ , X ₃ , X ₄ ] on X ₂

Î Two components for the “View” block X ₂

R2X = 0.995, R2Y = 0.454

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Comp[1] Comp[2]

Comp No.

SIMCA-P 10.5 - 10/08/2004 10:45:36

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60

T ₂

The “View” block is summarized by T ₂ ^*

0.80 1.00

pc(corr)[Comp. 2]

pc(corr)[Comp. 1]

REPOS1 REPOS2 REPOS3

REPOS4

REPOS5

AGITA1 AGITA2 AGITA3

AGITA4

AGITA5 AGITA6

AGITA7

AGITA8

AGITA9 AGITA10

GUSTA1 GUSTA2

GUSTA3 GUSTA4 GUSTA5

GUSTA6 GUSTA7 GUSTA8GUSTA9

SIMCA-P 10.5 - 10/08/2004 11:36:25

SHADING SURFACE IMPRESSION

VISUAL INTENSITY

t[1] t[2]

2EL 0.858738 -0.200721

1CHA -2.34146 0.00945552

1FON -1.182 0.0270623

1VAU -3.49799 -0.394901

1DAM 1.21566 0.0442866

- - -

1ROC -0.618729 0.498721

2ING -4.27007 -0.295052

T1 0.991179 -0.024048

T2 0.952648 -0.326391

“Smell after shaking” block X ₃

PLS regression of [X ₁ , X ₂ , X ₄ ] on X ₃

Î Two components for the “Smell after shaking” block X ₃

R2X = 0.715, R2Y = 0.626

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Comp[1] Comp[2]

Comp No.

SIMCA-P 10.5 - 10/08/2004 12:52:04

T ₃

The “Smell after shaking” block is summarized by T ₃ ^*

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

pc(corr)[Comp. 2]

pc(corr)[Comp. 1]

Smell intensity

Smell quality Fruity note

Floral note Spicy note

Vegetable note

Phelonic note

Aromatic intensity Aromatic persistence

Aromatic quality REPOS1

REPOS2 REPOS3

REPOS4 REPOS5

VISION1 VISION2

VISION3 GUSTA1 GUSTA2

GUSTA3

GUSTA4

GUSTA5 GUSTA6

GUSTA7

GUSTA8

GUSTA9

t[1] t[2]

2EL 0.37056 -1.17115

1CHA -2.09278 -1.82405

1FON -1.27842 -1.07693

1VAU -4.87052 0.537523

1DAM 3.08777 -0.349279

- - -

1ROC -0.144715 -1.16469

2ING -5.33619 -0.929866

T1 -0.57741 3.84257

T2 -0.972137 3.64734

“Tasting” block X ₄

PLS regression of [X ₁ , X ₂ , X ₃ ] on X ₄

Î Two components for the “Tasting” block X ₄

R2X = 0.822, R2Y = 0.555

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Comp[1] Comp[2]

Comp No.

SIMCA-P 10.5 - 10/08/2004 14:23:34

-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00

-1.00 -0.80 -0.60 -0.40 -0.20 0.00

The “Tasting” block is summarized by T ₄ ^*

T ₄

0.20 0.40 0.60 0.80 1.00

pc(corr)[Comp. 2]

pc(corr)[Comp. 1]

Attack intensity Acidity

Astringency

Alcohol

Balance Mellowness Bitterness

Ending intensity

Harmony REPOS1

REPOS2 REPOS3

REPOS4 REPOS5

VISION1 VISION2

VISION3 AGITA1

AGITA2 AGITA3

AGITA4 AGITA5 AGITA6

AGITA7

AGITA8 AGITA9

AGITA10

t[1] t[2]

2EL -0.933524 -1.35483

1CHA -1.64902 -1.09668

1FON -0.355978 -1.26974

1VAU -5.37923 1.67518

1DAM 2.23994 -0.215513

- - -

1ROC -0.345487 0.0565316

2ING -6.8104 -1.57181

T1 -0.965254 2.50417

T2 0.293074 3.47162

Step 2 : Searching for factors ₁ ,...,

j jm

j

F F

I. Searching for the first factors :

F ₁₁ = T ₁ ^* a ₁₁ , F ₂₁ = T ₂ ^* a ₂₁ , F ₃₁ = T ₃ ^* a ₃₁ , F ₄₁ = T ₄ ^* a ₄₁

Horst’s Generalized Canonical Correlation Analysis is used to obtain positively correlated factors:

* *

1 ' '1

'

( _{j j} , _{j j} )

j j

Maximize cor T a T a

∑ ≠

or else

*

1 1

( _{j j} , )

j

Maximize ∑ cor T a F where 1 _j ^* _j 1

j

F = ∑ T a

Causal model associated with Horst’s GCCA ( Hierarchical PLS model )

PLS-Graph output

Super-block or auxiliary variable

The general PLS Path Modeling algorithm

Mode A: w _h = (1/n)T _h ´z _h

Mode B: w _h = (T _h ´T _h ) ^-1 T _h ´z _h Choice of weights e

_hh’

:

- Centroid: correlation signs - Factorial: correlations

- Path weighting scheme: multiple regression coefficients or correlations

w _h

Initial step

v _h ∝±T ^* _h w _h

Outer estimation

v _h2 v _h1

v _hm

z _h

#

e _h1 e _h2

e _hm

Inner estimation

Update weights W

Reiterate till convergence

After convergence: OLS multiple regression or PLS-R for path coefficients

Causal model associated with Horst’s GCCA ( Hierarchical PLS model )

PLS approach :

Mode B, Centroid scheme ^{1 1}

( _j , )

j

Maximize ∑ cor F F

<=>

PLS-Graph output

Causal model associated with Mathes-Hanafi criterion

PLS approach :

Mode B, Centroid scheme

* *

1 ' '1 '

| ( _{j j} , _{j j} ) |

j j

Maximize cor T a T a

∑ ≠

<=>

PLS (Mode B + Centroid) is solution to the problem :

* * * * * *

1 1 1 1 1 1

| (

_{SR SR}

,

_{T T}

) | | (

_{V V}

,

_{T T}

) | | (

_{SS SS}

,

_{T T}

) | Maximize ⎡ ⎣ cor T a T a + cor T a T a + cor T a T a ⎤ ⎦

One result for a simple causal model

(Mathes, 1993, Hanafi, 2004)

Another result (Mathes, 1993, Hanafi, 2004)

PLS (Mode B + Factorial) is solution to the problem :

2 * * 2 * * 2 * *

1 1 1 1 1 1

(

_{SR SR}

,

_{T T}

) (

_{V V}

,

_{T T}

) (

_{SS SS}

,

_{T T}

)

Maximize ⎡ ⎣ cor T a T a + cor T a T a + cor T a T a ⎤ ⎦

Another result (Tenenhaus, 2004)

PLS (Mode A + Centroid) solution is obtained by maximizing :

subject to : ||a _SR1 || = ||a _V1 || = ||a _SS1 || = ||a _T1 || = 1

* * * * * *

1 1 1 1 1 1

* * *

1 1 1

* * * * * * *

1 1 1 1 1 1 1

( ) | ( , ) | ( ) | ( , ) |

( ) | ( , ) |

( ) (| ( , ) | | ( , ) | | ( , ) |)

SR SR SR SR T T V V V V T T

SS SS SS SS T T

T T T T SR SR T T V V T T SS SS

Var T a Cor T a T a Var T a Cor T a T a Var T a Cor T a T a

Var T a Cor T a T a Cor T a T a Cor T a T a +

+

+ + +

Some modified multi-block methods for SEM

c _jk = 1 if blocks are linked, 0 otherwise

SUMCOR (Horst, 1961)

, _jk

(

_j

,

_k

) Max ∑

j k

c Cor F F Mathes (1993), Hanafi (2004):

²

, _jk

(

_j

,

_k

) Max ∑

j k

c Cor F F Mathes (1993), Hanafi (2004)

, _jk

| (

_j

,

_k

) | Max ∑

j k

c Cor F F Tenenhaus (2004):

All 1

( ) | ( , ) |

j

j j jk j j k k

j k j

w

Max Var X w

_≠

c Cor X w X w

=

∑ ∑

MAXBET (Van de Geer, 1984 & Ten Berge, 1988):

All 1

[ ( ) ( , )]

j

j j jk j j k k

j j k

w

Max Var X w

_≠

c Cov X w X w

=

∑ + ∑

MAXDIFF (Van de Geer, 1984 & Ten Berge, 1988):

All 1

[ ( , )]

j

jk j j k k

w j k

Max

_≠

c Cov X w X w

=

∑

Mathes-Hanafi 1 Mode B and Factorial scheme

Mathes-Hanafi 2 Mode B and Centroid scheme

Tenenhaus Mode A and Centroid scheme

Results for the wine data example

1) The causal model for Horst’s GCCA is obtained by relating each block to the super-block yielded by concatenating all the blocks (Hierarchical model).

2) The solution to Horst’s GCCA is obtained by PLS estimation (options « Mode B and Centroid scheme ») of the hierarchical model.

3) Factors :

^{* *}

11 11 12

* *

21 21 22

* *

31 31 32

* *

41 41 42

.999 .021 .956 .296 .986 .165 .994 .108

F t t

F t t

F t t

F t t

= × + ×

= × + ×

= × + ×

= × + ×

Latent variables

===============================================================

smell view smell tasting global

at rest after sh.

(F

₁₁

) (F

₂₁

) (F

₃₁

) (F

₄₁

) (F

₁

)

--- 2EL 0.222 0.345 0.047 -0.515 0.026 1CHA -0.904 -1.355 -1.178 -0.802 -1.122 1FON -0.947 -0.664 -0.718 -0.264 -0.686 1VAU -2.053 -2.343 -2.227 -2.162 -2.324 1DAM 2.291 0.742 1.412 0.941 1.425

. . .

1ROC -0.348 0.029 -0.192 -0.145 -0.174 2ING -1.652 -2.722 -2.607 -3.054 -2.655 T1 0.776 0.561 0.144 -0.201 0.339 T2 0.046 0.296 -0.060 0.423 0.187

====================================================================

F

₁

∝ F

₁₁

+F

₂₁

+F

₃₁

+F

₄₁

Results

Results

Correlations of latent variables

=====================================================

smell view smell tasting

at rest after shaking

---

smell at rest 1.000

view 0.781 1.000

smell after shaking 0.878 0.914 1.000

tasting 0.741 0.916 0.915 1.000 global 0.900 0.955 0.981 0.945

=======================================================

High positive correlations

between the factors

II. Searching for the second order factors

F ₁₂ = T ₁ ^* a ₁₂ , F ₂₂ = T ₂ ^* a ₂₂ , F ₃₂ = T ₃ ^* a ₃₂ , F ₄₂ = T ₄ ^* a ₄₂

subject to the constraints :

Horst’s Generalized Canonical Correlation Analysis:

2 '2

'

(

_j

,

_j

)

j j

Maximize cor F F

∑

≠

or else

2 2

(

_j

, )

j

Maximize ∑ cor F F ^where

_{2 j2}

j

F = ∑ F

Cor(F _j1 , F _j2 ) = 0, j = 1 to 4

Application to wine data

Cor(F

_j1

,F

_j2

) = 0, j = 1 to 4

F₁₂ Smell at rest

T₁^*

F₁₁

E₁

view T₂^*

E₂

F₂₁

Smell after shaking T₃^* F₃₂

F₃₁

E₃

Global

components :

F ₁ ∝ F ₁₁ +F ₂₁ +F ₃₁ +F ₄₁ F ₂ ∝ F ₁₂ +F ₂₂ +F ₃₂ +F ₄₂

F₂₂

Tasting T₄^* F₄₂

F₄₁

E₄

Advantage of orthonormal factors

defined on blocks of orthonormal variables

* *

1 ,..., 1 ,...,

j j

j jm j jm j

F F t t A

⎡ ⎤ ⎡ = ⎤

⎣ ⎦ ⎣ ⎦

Factors F _jh are deduced from standardized PLS components by « rotation ».

A _j is an orthogonal matrix.

orthonormal orthonormal

Application to the wine data

[

11 12

]

1^{* 1 1} 1^*

1 1

cos sin .999 .021

, sin cos .021 .999

F F T ⎡ θ θ ⎤ T ⎡ − ⎤

= ⎢ ⎣ − θ θ ⎥ ⎦ = ⎢ ⎣ ⎥ ⎦

[

21 22

]

2^{* 2 2} 2^*

2 2

cos sin .956 .296

, sin cos .296 .956

F F T ⎡ θ θ ⎤ T ⎡ ⎤

= ⎢ ⎣ θ − θ ⎥ ⎦ = ⎢ ⎣ − ⎥ ⎦

The factors F _jh are deduced from the standardized PLS components by “rotation”, checking for positive correlation.

[

31 32

]

3^{* 3 3} 3^*

3 3

cos sin .986 .165

, sin cos .165 .986

F F T ⎡ θ θ ⎤ T ⎡ − ⎤

= ⎢ ⎣ − θ θ ⎥ ⎦ = ⎢ ⎣ ⎥ ⎦

[

41 42

]

4^{* 4 4} 4^*

4 4

cos sin .994 .108

, sin cos .108 .994

F F T ⎡ θ θ ⎤ T ⎡ − ⎤

= ⎢ ⎣ − θ θ ⎥ ⎦ = ⎢ ⎣ ⎥ ⎦

Results for second order components

Smell at rest

View Smell after shaking

Tasting Global

Smell at rest 1

View .407 1

Smell after shaking .803 .398 1

Tasting .822 .145 .780 1

Global .928 .394 .950 .906 1

All correlations between the factors are positive.

Map of the correlations with the “Global” components

Correlation with global component 1

1.0 .8

.6 .4

.2 -.0

-.2 -.4

-.6

Correlati on with global component 2

1.0

.8

.6

.4

.2

-.0

-.2

-.4

-.6

GLOBAL QUALITY Harmony

Ending intensity in mouth Bitterness

Mellowness Balance

Alcohol Astringency Acidity

Intensity of attack

Aromatic qualityin mouth Aromatic persistence

Aromatic intensityin mouth Phelonic note

Vegetable note Spicy note

Floral note

Fruity note Smell quality Smell intensity

Surface impression Shading

Visual intensity Spicy note at rest

Floral note at rest

Fruity note at rest

Aromatic quality at rest Smell intensity atrest

Correlation with global component 1

1.0 .8

.6 .4

.2 -.0

-.2 -.4

-.6

Correlation with global component 2

1.0

.8

.6

.4

.2

-.0

-.2

-.4

-.6

GLOBAL QUALITY Harmony

Ending intensity in mouth Bitterness

Mellowness Balance

Alcohol Astringency Acidity

Intensity of attack

Aromatic qualityin mouth Aromatic persistence

Aromatic intensityin mouth Phelonic note

Vegetable note Spicy note

Floral note

Fruity note Smell quality Smell intensity

Surface impression Shading

Visual intensity Spicy note at rest

Floral note at rest

Fruity note at rest

Aromatic quality at rest Smell intensity atrest

Wine visualization in the “Global” components space Wines marked by Appellation

Global component 1

2 1

0 -1

-2 -3

Glob al c o mpo n e n t 2

3.5 3.0 2.5 2.0 1.5 1.0 .5 0.0 -.5 -1.0 -1.5

Appellation

Saumur Chinon Bourgueil T2

T1

2ING 1ROC

2BEA 1BEN1ING 1POY

2DAM 4EL PER1

1TUR DOM1 3EL

2BOU 1BOI 1DAM 1VAU

1FON

1CHA 2EL

GOOD QUALITY

Wine visualization in the global space component space Wines marked by Soil

Global component 1

2 1

0 -1

-2 -3

Global component 2

3.5 3.0 2.5 2.0 1.5 1.0 .5 0.0 -.5 -1.0 -1.5

Soil

Reference Soil 4 Soil 2 Soil 1 T2

T1

2ING

1ROC

2BEA 1BEN

1ING

1POY 2DAM 4EL PER1

1TUR

DOM1 3EL

1BOI 2BOU

1DAM 1VAU

1FON

1CHA 2EL

Global component 1

2 1

0 -1

-2 -3

Global component 2

3.5 3.0 2.5 2.0 1.5 1.0 .5 0.0 -.5 -1.0 -1.5

Soil

Reference Soil 4 Soil 2 Soil 1 T2

T1

2ING

1ROC

2BEA 1BEN

1ING

1POY 2DAM 4EL PER1

1TUR

DOM1 3EL

1BOI 2BOU

1DAM 1VAU

1FON

1CHA 2EL

GOOD QUALITY

Smell at Rest - Score 1

1,2 1,0

,8 ,6

,4 ,2

0,0 -,2

Smell at Rest - Score 2

1,2 1,1 1,0 ,9 ,8 ,7 ,6 ,5 ,4 ,3 ,2 ,1 -,0 -,1 -,2 -,3 -,4 -,5

Global Quality 1 Spicy note

Floral note

Fruity note

Aromatic quality Smell intensity

Map of the correlations with the “Smell at rest” components

Smell at rest - Score 1

3 2

1 0

-1 -2

-3

Smell at rest - Score 2

3

2

1

0

-1

-2

Appellation

Saumur Chinon Bourgueil

T1

2ING

1ROC

2BEA 1BEN

1ING 1POY

2DAM 4ELPER1

1TUR DOM1

3EL

2BOU 1BOI

1DAM 1VAU

1FON

1CHA

2EL

Wine visualization in the “Smell at rest” space

T2

View - Component 1

1,2 1,0

,8 ,6

,4 ,2

0,0 -,2

-,4

View - Component 2

,5

,4

,3

,2

,1

0,0

-,1

-,2

Global Quality 2

Impression of surfac Shading

Visual intensity

Map of the correlations with the “View” components

View - Score 1

2 1

0 -1

-2 -3

View - Score 2

3

2

1

0

-1

-2

Appellation

Saumur Chinon Bourgueil

T2

T1 2ING

1ROC

2BEA 1BEN

1ING 1POY

2DAM PER1

4EL 1TUR

DOM1 3EL

1BOI

2BOU

1DAM 1VAU

1FON 1CHA

2EL

Wine visualization in the “View” space

Wine visualization in the “View” space

View - Score 1

2 1

0 -1

-2 -3

View - Score 2

3

2

1

0

-1

-2

Soil

Reference Milieu 4 Milieu 2 Milieu 1

T2

T1 2ING

1ROC

2BEA 1BEN

1ING 1POY

2DAM PER1

4EL 1TUR

DOM1 3EL

1BOI

2BOU

1DAM 1VAU

1FON 1CHA

2EL

Smell after shaking - Component 1

1,4 1,2

1,0 ,8

,6 ,4

,2 0,0 -,2

-,4 -,6

-,8

Smell after shaking - Component 2

1,0

,8

,6

,4

,2

0,0

-,2 -,4

-,6 -,8

Global Quality 3 Aromatic quality

Aromatic persistence Aromatic Intensity Phenolic note

Vegetable note Spicy note

Floral note

Fruity note Smell quality Smell intensity

Map of the correlations with the “Smell after shaking” components

Smell after shaking - Score 1

2 1

0 -1

-2 -3

Smell after shaking - Score 2

3

2

1

0

-1

-2

Appellation

Saumur Chinon Bourgueil

T2T1

2ING

1ROC

2BEA

1BEN

1ING 1POY

2DAM PER1

4EL 1TUR

DOM1 3EL

1BOI

2BOU

1DAM 1VAU

1FON 1CHA

2EL

Wine visualization in the “Smell after shaking” space

Tasting - Component 1

1,2 1,0

,8 ,6

,4 ,2

0,0 -,2

-,4

Tasting - Component 2

1,0

,8

,6

,4

,2

-,0

-,2

-,4 -,6

Global Quality 4 Harmony Ending intensity Bitterness

Mellowness Balance

Alcohol Astringency Acidity

Intensity of attack

Map of the correlations with the “Tasting” components

Tasting - Score 1

2 1

0 -1

-2 -3

-4

Tasting - Score 2

3

2

1

0

-1

-2

Appellation

Saumur Chinon Bourgueil

T2

T1

2ING

1ROC2BEA

1BEN 1ING

1POY 2DAM PER1

4EL

1TUR

DOM1 3EL

1BOI 2BOU

1DAM 1VAU

1FON 1CHA

2EL

Wine visualization in the “Tasting” space

2 1

0 -1

-2 -3

3

2

1

0

-1

-2

GLOBAL SCORE Tasting

Smell after shaking View

Smell at rest

2BEA T1

3EL 1POY

1CHA 1VAU

Visualization of wine variability among the blocks

Star-plots of scores for some wines

2,0 1,0

0,0 -1,0

-2,0 -3,0

3,0

2,0

1,0

0,0

-1,0

-2,0

GLOBAL SCORE Tasting

Smell after shaking View

Smell at rest

2DAM

Visualization of wine variability among the blocks Star-plot of the “best wine” – 2DAM SAUMUR

DAM =

Dampierre-sur-Loire

A soft, warm, blackberry nose. A good core of fruit on the palate with quite well worked tannin and acidity on the finish; Good length and a lot of potential.

DECANTER (mai 1997)

(DECANTER AWARD ***** : Outstanding quality, a virtually perfect example)

Cuvée Lisagathe 1995

F₁₂ Smell at rest

T₁^*

F₁₁

E₁

view T₂^*

E₂

F₂₁

Smell after shaking T₃^* F₃₂

F₃₁

E₃

F₂₂

Tasting T₄^* F₄₂

F₄₁

E₄

Generalized Procrustes Analysis : Global solution to Horst’s GCCA

*

1

,

2

, 1,...,4

j j j j

F F T A j

⎡ ⎤ = =

⎣ ⎦

2

2 '

1 '

|| ||

j

jh j h

A h j j

Minimize F F

= ≠

∑∑ − GPA

2

'

1 '

( , )

j

jh j h

A h j j

Maximize Cor F F

= ≠

∑∑

Global Horst’s GCCA

Use of GenStat GPA

Input : Significant PLS components for each block T _j ^*

Output : New common factors and rotation matrices F _j = T A _j ^* _j

GPA is available also in XLSTAT by Addinsoft

Coordinates for Configuration 1

1 2 1 0.0333 -0.0892 2 -0.1414 -0.1678 3 -0.1458 0.0293

...

20 0.1237 0.3440 21 0.0118 0.4498

* Rotation matrix * 1 2

1 1.000 -0.031 2 0.031 1.000

F

₁₁

F

₁₂

A

₁

Use of GenStat GPA

The “GPA” factors are normalized to : Square root(4/8).

They can be standardized.

Coordinates for Configuration 2

1 2 1 0.0520 0.1026 2 -0.2087 -0.0709 3 -0.1026 -0.0444

...

20 0.0858 0.0378 21 0.0448 0.1540

* Rotation matrix * 1 2

1 0.952 0.305 2 0.305 -0.952

F

₂₁

F

₂₂

A

₂

Coordinates for Configuration 4

1 2 1 -0.0777 -0.1597 2 -0.1221 -0.1236 3 -0.0385 -0.1530

...

20 -0.0346 0.3122 21 0.0601 0.4225

* Rotation matrix * 1 2

1 0.995 -0.095 2 0.095 0.995

F

₄₁

F

₄₂

A

₄

Coordinates for Configuration 3

1 2 1 0.0039 -0.1221 2 -0.1864 -0.1526 3 -0.1131 -0.0895

...

20 0.0332 0.3916 21 0.0010 0.3776

* Rotation matrix * 1 2

1 0.981 -0.192 2 0.192 0.981

F

₃₁

F

₃₂

A

₃

F₁₂ Smell at rest

T₁^*

F₁₁

E₁

view T₂^*

E₂

F₂₁

Smell after shaking T₃^* F₃₂

F₃₁

E₃

F₂₂

Tasting T₄^* F₄₂

F₄₁

E₄

Generalized Procrustes Analysis : Results

[

11 12

]

1^*

.999 .031

, .031 .999

F F T ⎡ − ⎤

= ⎢ ⎥

⎣ ⎦

[

41 42

]

4^*

.995 .095

, .095 .995

F F T ⎡ − ⎤

= ⎢ ⎥

⎣ ⎦

#

2

' 1 '

( , )

8.505 for iterative Horst

8.502 for global Horst (GPA)

jh j h

h j j

Cor F F

= <

=

=

∑∑

(!!)

J blocks : X ₁ ,…, X _J

Conclusion on Multi-block analysis

PLS regression of X _-j on X _j Î X _j replaced by T _j ^*

GPA of ( T ₁ ^* ,…,T _J ^* ) Horst’s GCCA of

( T ₁ ^* ,…,T _J ^* ) on m dimensions

=

Hierarchical PLS (Mode B + Centroid)

F _j =T _j ^* A _j

Confirmatory Factor Analysis for 1 ^st factors

X

₁

X

₂

X

_J

X

₁₁

X

₂₁

X

_J1

F ₁₁

F ₂₁

F _J1

#

Block Deflated

Block

(residual)

PLS Confirmatory Factor Analysis

PLS approach :

Mode B, Centroid scheme

| (

_{j j}

,

_{k k}

) |

j k

M axim ize cor X a X a

∑

≠

<=>

PLS Confirmatory Factor Analysis

PLS approach :

Mode B, Factorial scheme

2

(

_{j j}

,

_{k k}

)

j k

M axim ize C or X a X a

∑

≠

<=>

PLS Confirmatory Factor Analysis

Mode A

Centroid scheme

^{j1 1} ₁

⁽

¹

^{) (}

¹

^,

¹

⁾

J

j j j j k k

a j k j

M ax V ar X a C or X a X a

=

∑

=

∑

≠

Tends

to

Dimension 1

PLS Confirmatory Factor Analysis

Mode A

Centroid scheme

^{j2 1} ₁

⁽

^{1 2}

^{) (}

^{1 2}

^,

^{1 2}

⁾

J

j j j j k k

a j k j

M ax V ar X a C or X a X a

=

∑

=

∑

≠

Dimension 2

Expressed in term of original variables Tends

to

Deflated block