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

2. Multiple Factor Analysis The 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. Multiple Factor Analysis The wines of the Loire valley 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. Multiple Factor Analysis The wines of the Loire valley 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. Multiple Factor Analysis The wines of the Loire valley 5 The questionnaire Mouth aromas persistence “ Mouth aromas quality nil  excellent

6. Multiple Factor Analysis The wines of the Loire valley 6 The questionnaire

7. Multiple Factor Analysis The wines of the Loire valley 7 Data set 1  k  29x ik : average of the scores given by the 36 judges to the wine i considering the variable k k=30,31x ik : 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. Multiple Factor Analysis The wines of the Loire valley 8 Objective Aim Highlight the main dimensions of sensory variability among the wines Relate these dimensions to the type of soil and the appellation Usual methodology Apply PCA to the whole table sensory variables being active origin variables being supplementary

9. Multiple Factor Analysis The wines of the Loire valley 9 Example 6 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. Multiple Factor Analysis The wines of the Loire valley 10 Weighting the variables Each variable of the set j is weighted by 1/ 1 j 1 j : 1 st eigenvalue of PCA applied to set j. Balancing the sets of variables in MFA Reference example R I set 1 : 2 var. set 2 : 3 var. 0.5 1 1 1

11. Multiple Factor Analysis The wines of the Loire valley 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. Multiple Factor Analysis The wines of the Loire valley 12 MFA : representation of the wines -5.0-2.502.5 -2 0 2 4 6 F2 - 19.49 % 2EL 1CHA 1FON 1VAU 1DAM 2BOU 1BOI 3EL DOM1 1TUR 4EL PER1 2DAM 1POY 1ING 1BEN 2BEA 1ROC 2ING T1 T2 F1 - 49.38 %

13. Multiple Factor Analysis The wines of the Loire valley 13 MFA : representation of the quantitative variables F1 - 49.38 % Intens. olfac. rest Quality arom. rest Fruity (rest) Floral (rest) Spicy (rest) Color intensity Shade orange/purple Surface Olfactory intensity Olfactory quality Fruity Floral Spicy Vegetal Phenolic Mouth aromas intens. Mouth aromas persist. Mouth aromas qual. First mouth intens. Sour Astringency Alcohol Balance so-ast-alc Velvety Bitter End mouth intensity Harmony F2 - 19.49 %

14. Multiple Factor Analysis The wines of the Loire valley 14 Superimposed representation of the J clouds of individuals Extract (6 wines) -5.0-2.502.5 -2 0 2 4 6 1 F1 - 49.38 % 1VAU 1DAM 1POY 2ING T1 T2 1 2 3 4 1 2 3 4 1 2 3 4 3 2 4 1 2 2 3 4 1 4 3 F2 - 19.49 %

15. Multiple Factor Analysis The wines of the Loire valley 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 Some values for the three variables of the set 1 the most correlated to the first factor

16. Multiple Factor Analysis The wines of the Loire valley 16 Looking for factors common to several sets of variables For each s and each j, correlation between the common factor F s (projection of the mean cloud N I ) the canonical variable F s j (projection of the partial cloud N I j ) How to highlight common factors ?

17. Multiple Factor Analysis The wines of the Loire valley 17 Global representation of sets of variables F1 F2 olfaction at rest (.617) olfaction after agitation (.784) taste (.772) vision (.729) 1 1 0 G1 G2 G3 G4 overall appreciation (.444) G6 G5 origin (.189)

18. Multiple Factor Analysis The wines of the Loire valley 18 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. Multiple Factor Analysis The wines of the Loire valley 19 Case of categorical variables Illustrative : as in PCA For each category : labeled individuals co-ordinate of the centre of gravity

20. Multiple Factor Analysis The wines of the Loire valley 20 Case of categorical variables

21. Multiple Factor Analysis The wines of the Loire valley 21 Case of categorical variables -5.0-2.502.5 -2 0 2 4 6 F1 - 49.38 % Saumur Bourgueil Chinon Reference soil Soil 2 Soil 3 Soil 4 1 2 3 4 1 2 3 4 F2 - 19.49 %