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Main article: Multivariate analysis

In statistics, canonical correlation analysis, introduced by Harold Hotelling, is a way of making sense of cross-covariance matrices.

## Definition

Given two column vectors and of random variables with finite second moments, one may define the cross-covariance to be the matrix whose entry is the covariance .

Canonical correlation analysis seeks vectors and such that the random variables and maximize the correlation . The random vectors and are the first pair of canonical variables. Then one seeks vectors maximizing the same correlation subject to the constraint that they are to be uncorrelated with the first pair of canonical variables; this gives the second pair of canonical variables. This procedure continues times.

## Computation

### Proof

Let and . The parameter to maximize is

The first step is to define a change of basis and define

And thus we have

By the Cauchy-Schwarz inequality, we have

There is equality if the vectors and are colinear. In addition, the maximum of correlation is attained if is the eigenvector with the maximum eigenvalue for the matrix (see Rayleigh quotient). The subsequent pairs are found by using eigenvalues of decreasing magnitudes. Orthogonality is guaranteed by the symmetry of the correlation matrices.

### Solution

The solution is therefore:

• is an eigenvector of
• is proportional to

Reciprocally, there is also:

• is an eigenvector of
• is proportional to

The canonical variables are defined by:

## Hypothesis testing

Each row can be tested for significance with the following method. If we have independent observations in a sample and is the estimated correlation for . For the th row, the test statistic is:

which is distributed as a chi-square with degrees of freedom.

## Practical Uses

A typical use for canonical correlation in the psychological context is to take a two sets of variables and see what is common amongst the two tests. For example you could take two well established multidimensional personality tests such as the MMPI and the NEO. By seeing how the MMPI factors relate to the NEO factors, you could gain insight into what dimensions were common between the tests and how much variance was shared. For example you might find that an extraversion or neuroticism dimension accounted for a substantial amount of shared variance between the two tests.

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