Canonical correlation
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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.
Contents |
[edit] 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.
[edit] Computation
[edit] 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.
[edit] 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:
[edit] 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.
[edit] 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.
[edit] External links
- See also generalized canonical correlation.
- Applied Multivariate Statistical Analysis, Fifth Edition, Richard Johnson and Dean Wichern
| This page uses content from the English-language version of Wikipedia. The original article was at Canonical correlation. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License. |
