Nonlinear Canonical Correlation Analysis

Nonlinear Canonical Correlation Analysis is a very general procedure with many different applications. The goal of nonlinear canonical correlation analysis is to analyze the relationships between two or more sets of variables instead of between the variables themselves, as in principal components analysis. For example, you may have two sets of variables, where one set of variables might be demographic background items on a set of respondents and a second set might be responses to a set of attitude items. The scaling levels in the analysis can be any mix of nominal, ordinal, and numerical. Optimal scaling canonical correlation analysis determines the similarity among the sets by simultaneously comparing the canonical variables from each set to a compromise set of scores assigned to the objects.

Relation to other Categories procedures. If there are two or more sets of variables with only one variable per set, optimal scaling canonical correlation analysis is equivalent to optimal scaling principal components analysis. If all variables in a one-variable-per-set analysis are multiple nominal, optimal scaling canonical correlation analysis is equivalent to multiple correspondence analysis. If there are two sets of variables, one of which contains only one variable, optimal scaling canonical correlation analysis is equivalent to categorical regression with optimal scaling.

Relation to standard techniques. Standard canonical correlation analysis is a statistical technique that finds a linear combination of one set of variables and a linear combination of a second set of variables that are maximally correlated. Given this set of linear combinations, canonical correlation analysis can find subsequent independent sets of linear combinations, referred to as canonical variables, up to a maximum number equal to the number of variables in the smaller set.

If there are two sets of variables in the analysis and all variables are defined to be numerical, optimal scaling canonical correlation analysis is equivalent to a standard canonical correlation analysis. Although IBM® SPSS® Statistics does not have a canonical correlation analysis procedure, many of the relevant statistics can be obtained from multivariate analysis of variance.

Optimal scaling canonical correlation analysis has various other applications. If you have two sets of variables and one of the sets contains a nominal variable declared as single nominal, optimal scaling canonical correlation analysis results can be interpreted in a similar fashion to regression analysis. If you consider the variable to be multiple nominal, the optimal scaling analysis is an alternative to discriminant analysis. Grouping the variables in more than two sets provides a variety of ways to analyze your data.