Wine Chemistry and Biochemistry

706 P.J. Mart ́ın-Alvarez ́

–40 –30 –20 –10 0 10 20 30 40 50

Canonical variable 1

–10

–8

–6

–4

–2

0

2

4

6

8

10

12

14

Canonical variable 2

Malvar

Airén

Trepat

Monastrell

Fig. 13.6Plot of the wines on the plane defined by the first two canonical variables

13.3.4 Multivariate Statistical Techniques to Study Dependence

The principal objective of these techniques is to study the dependence between two

groups of variables, (X 1 ,X 2 , ...,Xp)and(Y 1 , ...,Yq), fromnobservations in

thesep+qvariables:

X 1 X 2 ...Xp Y 1 Y 2 ...Yq

Observ.

1

2

...

n

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

x 1 , 1 x 1 , 2 ...x 1 ,p

x 2 , 1 x 2 , 2 ...x 2 ,p

... ... ... ...

xn, 1 xn, 2 ...xn,p

y 1 , 1 y 1 , 2 ...y 1 ,q

y 2 , 1 y 2 , 2 ...y 2 ,q

... ... ... ...

yn, 1 yn, 2 ...yn,q

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

Besides the descriptive values, it is also interesting to know the correlations

between the two groups of variables (rXi,Yj). The multivariate statistical methods

for this data matrix areCanonical Correlation Analysis(CCA) to investigate the

relationship between both sets of variables, andMultivariate Regressionwith a view

to predicting the values of the response variables in the Y-block in function of the

variables in the X-block, using a mathematical model.