Canonical Correlation

Canonical correlation was developed by H.Hotelling. Canonical correlation analysis is a method of measuring the linear relationship between two variables that are multidimensional in nature. The canonical correlation analysis locates the bases for each variable that are most favorable with respect to the association. The canonical correlation analysis also determines the corresponding associations. In other words, the canonical correlation analysis determines the two bases in which the association matrix between the variables is diagonal and the associations on the diagonals are maximized. The dimensionality of these new bases in canonical correlation is equal to or less than the smallest dimensionality of the two variables.

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One of the crucial properties of canonical correlation is that they are independent in relation to the transformation of the variables. This property of canonical correlation indicates the difference between the canonical correlation and the ordinary types of correlation.

Canonical correlation is a standard tool in statistical analysis which is used in the fields of economics, medical studies, etc.

In statistical language, the canonical correlation can be defined as the problem of finding two sets of basis vectors in such a manner that associations between the projections of the variables into the basis vectors are mutually maximized.

The canonical correlation between the two random vectors can be obtained by calculating the Eigen value equations. The Eigen values are nothing, but are equivalent to the square of the canonical correlation.

The canonical correlation is a member of the multiple general linear hypothesis family and contributes most of the assumptions of multiple regression, such as the linearity of relationships, homoscedasticity, interval level of data, proper specification of the model, lack of high multicollinearity, etc. The canonical correlation is also called a characteristic root.

The maximum number of canonical correlation between the two sets of variables is the number of variables in the smaller set.

The pooled canonical correlation is the sum of squares of all the canonical coefficients and it represents all the orthogonal dimensions in the solution by which the two sets of variables are associated. The pooled canonical correlation is used to extract the extent to which one set of variables can be forecasted by the other set of variables.

The canonical weights are nothing but the canonical coefficient in canonical correlation, which is used to assess the comparative importance that is contributed by the individual variable to a given canonical correlation.

The canonical scores in the canonical correlation are the values that are assigned to the canonical variable for a particular case. This score of canonical correlation is based on the value of the canonical coefficients for that variable. The canonical coefficients in canonical correlation are multiplied by the scores that are standardized and are then summed to yield canonical scores.

The structure correlation coefficients in canonical correlation are also called canonical factor loadings. It is defined as the canonical correlation of a canonical variable with an original variable in its set. The squared structure correlations in canonical correlation depict the contribution of a variable to the explanatory power of the canonical variate based on the set of variables.