TableOfReal: To CCA...

TableOfReal: To CCA...

A command that creates a CCA object from the selected TableOfReal object.

Settings

Dimension of dependent variate (ny): defines the partition of the table into the two parts whose correlations will be determined. The first ny columns should be the dependent part, the rest of the columns will be interpreted as the independent part (nx columns). In general nx should be larger than or equal to ny.

Behaviour

Calculates canonical correlations between the dependent and the independent parts of the table. The corresponding canonical coefficients are also determined.

Algorithm

The canonical correlation equations for two data sets T_y [n × p] and T_x [n × q] are:

(1) (S_yx S_xx^-1 S_yx′ -λ S_yy)y = 0

(2) (S_yx′ S_yy^-1 S_yx -λ S_xx)x = 0

where S_yy [p × p] and S_xx [q × q] are the covariance matrices of data sets T_y and T_x, respectively, S_yx [p × q] is the matrix of covariances between data sets T_y and T_x, and the vectors y and x are the canonical weights or the canonical function coefficients for the dependent and the independent data, respectively. In terms of the (dependent) data set T_y and the (independent) data set T_x, these covariances can be written as:

S_yy = T_y′ T_y, S_yx = T_y′ T_x and S_xx = T_x′ T_x.

The following singular value decompositions

T_y = U_y D_y V_y′ and T_x = U_x D_x V_x′

transform equation (1) above into:

(3) (V_y D_y U_y′U_x U_x′ U_y D_y V_y′ - λ V_y D_y D_y V_y′)y = 0

where we used the fact that:

S_xx^-1 = V_x D_x^-2 V_x′.

Equation (3) can be simplified by multiplication from the left by D_y^-1 V_y' to:

(4) ((U_x′ U_y)′ (U_x′ U_y) - λ I)D_y V_y′ y = 0

This equation can, finally, be solved by a substitution of the s.v.d of U_x′ U_y = U D V′ into (4). This results in

(5) (D² - λ I) V′ D_y V_y′ y = 0

In an analogous way we can reduce eigenequation (2) to:

(6) (D² - λ I) U′ D_x V_x′ x = 0

From (5) and (6) we deduce that the eigenvalues in both equations are equal to the squared singular values of the product matrix U_x′U_y. These singular values are also called canonical correlation coefficients. The eigenvectors y and x can be obtained from the columns of the following matrices Y and X:

Y = V_y D_y^-1 V

X = V_x D_x^-1 U

For example, when the vector y equals the first column of Y and the vector x equals the first column of X, then the vectors u = T_yy and v = T_xx are the linear combinations from T_y and T_x that have maximum correlation. Their correlation coefficient equals the first canonical correlation coefficient.

Settings

Behaviour

Algorithm

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