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Canonical Correlation Analysis (CCA). CCA. This is it! The mother of all linear statistical analysis. When ? We want to find a structural relation between a set of independent variables and a set of dependent variables. CCA. When ? (part 2)

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Canonical Correlation Analysis (CCA)

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Canonical correlation analysis cca

Canonical Correlation Analysis (CCA)


Canonical correlation analysis cca

CCA

  • This is it!

    • The mother of all linear statistical analysis

  • When ?

    • We want to find a structural relation between a set of independent variables and a set of dependent variables.


Canonical correlation analysis cca

CCA

  • When ? (part 2)

    • To what extend can one set of two or more variables be predicted or “explained” by another set of two or more variables?

    • What contribution does a single variable make to the explanatory power to the set of variables to which the variable belongs?

    • What contribution does a single variable contribute to predicting or “explaining” the composite of the variables in the variable set to which the variable does not belong?

    • What different dynamics are involved in the ability of one variable set to “explain” in different ways different portions of other variable set?

    • What relative power do different canonical functions have to predict or explain relationships?

    • How stable are canonical results across samples or sample subgroups?

    • How closely do obtained canonical results conform to expected canonical results?


Canonical correlation analysis cca

CCA

  • Assumptions

    • Linearity: if not, nonlinear canonical correlation analysis.

    • Absence of multicollinearity: If not, Partial Least Squares (PLS) regression to reduce the space.

    • Homoscedasticity: If not, data transformation.

    • Normality: If not, re-sampling.

    • A lot of data: Max(p, q)20nb of pairs.

    • Absence of outliers.


Canonical correlation analysis cca

CCA

  • Toy example

IVs

DVs

=X


Canonical correlation analysis cca

CCA

  • Z score transformation

IV1

IV1

DV2

DV2

=Z


Canonical correlation analysis cca

CCA

  • Canonical Correlation Matrix


Canonical correlation analysis cca

CCA

  • Relations with other subspace methods


Canonical correlation analysis cca

CCA

  • Eigenvalues and eigenvectors decomposition

R =

PCA


Canonical correlation analysis cca

CCA

  • Eigenvalues and eigenvectors decomposition

  • The roots of the eigenvalues are the canonical correlation values


Canonical correlation analysis cca

CCA

  • Significance test for the canonical correlation

  • A significant output indicates that there is a variance share between IV and DV sets

  • Procedure:

    • We test for all the variables (m=1,…,min(p,q))

    • If significant, we removed the first variable (canonical correlate) and test for the remaining ones (m=2,…, min(p,q)

    • Repeat


Canonical correlation analysis cca

CCA

  • Significance test for the canonical correlation

Since all canonical variables are significant, we will keep them all.


Canonical correlation analysis cca

CCA

  • Canonical Coefficients

    • Analogous to regression coefficients

BY=

Eigenvectors

Correlation matrix of the dependant variables

Bx=


Canonical correlation analysis cca

CCA

  • Canonical Variates

    • Analogous to regression coefficients


Canonical correlation analysis cca

CCA

  • Loading matrices

    • Matrices of correlations between the variables and the canonical coefficients

Ax

Ay


Canonical correlation analysis cca

CCA

  • Loadings and canonical correlations for both canonical variate pairs

    • Only coefficient higher than |0.3| are interpreted.

Loading

Canonical correlation


Canonical correlation analysis cca

CCA

  • Proportion of variance extracted

    • How much variance does each of the canonical variates extract form the variables on its own side of the equation?

First

First

Second

Second


Canonical correlation analysis cca

CCA

  • Redundancy

    • How much variance the canonical variates form the IVs extract from the DVs, and vice versa.

rdyx

Eigenvalues


Canonical correlation analysis cca

CCA

  • Redundancy

    • How much variance the canonical variates form the IVs extract from the DVs, and vice versa.

Summary

The first canonical variate from IVs extract 40% of the variance in the y variable.

The second canonical variate form IVs extract 30% of the variance in the y variable.

Together they extract 70% of the variance in the DVs.

The first canonical variate from DVs extract 49% of the variance in the x variable.

The second canonical variate form DVs extract 24% of the variance in the x variable.

Together they extract 73% of the variance in the IVs.


Canonical correlation analysis cca

CCA

  • Rotation

    • A rotation does not influence the variance proportion or the redundancy.

= Loading matrix =


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