Sébastien Lê Agrocampus Rennes Multiple Factor Analysis, MFA, main features of the method The...

Sébastien Lê

Agrocampus Rennes

Multiple Factor Analysis, MFA, main features of the method

The wines of the Loire valley

Multiple Factor AnalysisThe wines of the Loire valley

2

Plan of the presentation

• Data table ; notation ; examples

• Weighting variables

• MFA is based on a weighted factor analysis applied to all active sets of variables

• Superimposed representation of the J clouds of individuals

• Looking for common factors

• Global representation of sets of variables

• Relationship between MFA and separate analysis of each set

• Case of categorical variables

• Conclusion

• References


3

Data: general framework

« Essai terroir » (soil trial) ; INRA Angers ; France (C. Asselin ; R. Morlat)

3 appellations of the Loire valley : Saumur ; Chinon ; Bourgueil


4

Data: general framework

20 parcels (plots of vineyards)

4 types of soil

Same : type of vine, viticulture, wine-making

21 wines: 1 wine proposed twice to the tasters

36 wine tasters (= judges)

29 questions (= descriptors = attributes)

Aim: to study the sensory variability of the wines in relation with the type of soil and the appellation

« Essai terroir » (soil trial) ; INRA Angers ; France (C. Asselin ; R. Morlat)

3 appellations of the Loire valley : Saumur ; Chinon ; Bourgueil


5

The questionnaire

Olfaction at rest 1 2 3 4 5

Olfactory intensity nil very high

Olfactory quality “

Fruity “

Floral “

Spicy “

Vision

Color intensity light dark

Shade orange/purple orange purple

Surface (size of tears ) nil very high

Olfaction after agitation

Olfactory intensity nil very high

Olfactory quality “

Fruity “

Floral “

Spicy “

Vegetal “

Phenolic “

Mouth aromas intensity “

Mouth aromas persistence “

Mouth aromas quality nil excellent


6

The questionnaire

Gustation

First mouth intensity nil very high

Sour “

Astringency “

Alcohol “

Balance sour -astr.-alc. “

Velvety “

Bitter “

End mouth intensity “

Harmony “

Overall judgment

Overall quality nil excellent

Typicity nil very high


7

Data setvariables

1 k 29 31

1

wines i xik

21

1k29 xik : average of the scores given by the 36 judges

to the wine i considering the variable k

k=30,31 xik : category of the wine i for the variable k.

Origin of the wines

Variable 30 (appellation) : Saumur, Chinon, Bourgueil

Variable 31 (type of soil) : reference soil, soil 2, soil 3, soil 4


8

Objectivevariables

1 k 29 31

1

wines i xik

21

Aim Highlight the main dimensions of sensory variability among the wines Relate these dimensions to the type of soil and the appellation

Usual methodologyApply PCA to the whole table sensory variables being active origin variables being supplementary


9

Examplesets 1 2 3 4 5 6

variables 1 31

1

wines

216 sets of variables

1 Olfaction at rest (5 variables)

2 Vision (3 variables)

3 Olfaction after agitation (10 variables)

4 Gustation (9 variables)

5 Overall judgement (2 variables)

6 Origin (2 variables ; 7 categories)

First problem: balancing the influence of the different sets in a global analysis


10

Weighting the variables

Each variable of the set j is weighted by 1/1j

1j : 1st eigenvalue of PCA applied to set j.

Balancing the sets of variables in MFA

Reference example

RI

set 1 : 2 var.

set 2 : 3 var.

0.50.5

1

1

1


11

Weighted factor analysis

Quantitative variables : MFA is based on weighted PCA

standardized variables

unstandardized variables

mixed

Equivalence

When each set is composed by 1 quantitative variable : MFA=PCA

MFA is based on a factor analysis applied to all active sets of variables


12

MFA : representation of the wines

-5.0 -2.5 0 2.5

-2

0

2

4

6 F2 - 19.49 %

2EL

1CHA

1FON

1VAU

1DAM2BOU1BOI

3EL

DOM1

1TUR

4ELPER1

2DAM

1POY1ING

1BEN 2BEA

1ROC2ING

T1

T2

F1 - 49.38 %


13

MFA : representation of the quantitative variables

F1 - 49.38 %

Intens. olfac. rest

Quality arom. rest

Fruity (rest)

Floral (rest)

Spicy (rest)

Color intensityShade orange/purple

Surface

Olfactory intensity

Olfactory qualityFruity

Floral

Spicy

Vegetal

Phenolic

Mouth aromas intens.

Mouth aromas persist.

Mouth aromas qual.

First mouth intens.

SourAstringency

Alcohol

Balance so-ast-alcVelvety

Bitter

End mouth intensity

Harmony

F2 - 19.49 %


14

Superimposed representation of the J clouds of individuals

Extract (6 wines)

-5.0 -2.5 0 2.5

-2

0

2

4

6

1

F1 - 49.38 %

1VAU

1DAM1POY

2ING

T1

T2

1

2

3

4

12

3

4

1

2

3

4

3

2

4

12

2

3

4

1

4

3

F2 - 19.49 %


15

Superimposed representation of the J clouds of individuals

Some values for the four variables of the set 4 the most correlated to the first factor

velvety end mouthintensity

harmony first mouthintensity

maximum1POY1DAMmean

3.2863.2313.0362.674

3.6763.6673.6433.166

3.7863.7863.6433.148

3.5193.5193.4643.156

Some values for the three variables of the set 1 the most correlated to the first factor

Aromasquality (at rest)

Fruity (at rest) olfactive intensity(at rest)

maximum1DAM1POYmean

3.4293.4293.1073.046

3.1543.1542.7312.714

3.7083.6073.0713.111


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Looking for factors common to several sets of variables

axis 1 axis 2 axis 3 axis 4 axis 5 axis 6Set 1 (olfaction at rest) .89 .96 .89 .48 .42 .27Set 2 (vision) .93 .22 .16 .22 .17 .08Set 3 (olf. after agitation) .97 .89 .90 .57 .66 .49Set 4 (taste) .95 .87 .30 .25 .52 .57

For each s and each j, correlation between the common factor Fs (projection of the mean cloud NI )

the canonical variable Fsj (projection of the partial cloud NI

j )

How to highlight common factors ?


17

Global representation of sets of variables

F1

F2

olfaction at rest (.617)

olfaction after agitation (.784)

taste (.772)

vision (.729)

1

10

G1

G2

G3

G4

overall appreciation (.444)

G6

G5

origin (.189)


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Relationship between MFA and separate analysis of each set

G2F1 : first factor of PCA applied to the set 2

F1 - 49.38 %

G1F1

G1F2

G2F1

G2F2

G3F1

G3F2

G4F1

G4F2

G6F1

G6F2

F2 - 19.49 %


19

Case of categorical variables

-5.0 -2.5 0 2.5

-2

0

2

4

6

F1 - 49.38 %

RefRef

RefRef

Ref

Ref

Ref

F2 - 19.49 %

Illustrative : as in PCAFor each category :

labeled individualsco-ordinate of the centre of gravity


20


-5.0 -2.5 0 2.5

-2

0

2

4

6

1VAU

F1 - 49.38 %Saumur

Bourgueil

Chinon

Reference soilSoil 2

Soil 3

Soil 4

F2 - 19.49 %

2ING

1DAM

1POY

T1

T2


21


-5.0 -2.5 0 2.5

-2

0

2

4

6

F1 - 49.38 %

Saumur

Bourgueil

ChinonReference soilSoil 2

Soil 3

Soil 4

1

2

3

4

1

2

3

4

F2 - 19.49 %

Sébastien Lê Agrocampus Rennes Multiple Factor Analysis, MFA, main features of the method The...

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