Canonical correlation

In statistics, canonical correlation analysis, introduced by Harold Hotelling, is a way of making sense of cross-covariance matrices. If we have two sets of variables, $x_1, \dots, x_n$ and $y_1, \dots, y_m$, and there are correlations among the variables, then canonical correlation analysis will enable us to find linear combinations of the $x \$'s and the $y \$'s which have maximum correlation with each other.

Definition

Given two column vectors $X = (x_1, \dots, x_n)'$ and $Y = (y_1, \dots, y_m)'$ of random variables with finite second moments, one may define the cross-covariance $\Sigma _{XY} = \operatorname{cov}(X, Y)$ to be the $n \times m$ matrix whose (i,j) entry is the covariance $\operatorname{cov}(x_i, y_j)$. In practice, we would estimate the covariance matrix based on sampled data from X and Y (i.e. from a pair of data matrices).

Canonical correlation analysis seeks vectors a and b such that the random variables a'X and b'Y maximize the correlation $\rho = \operatorname{cor}(a' X, b' Y)$. The random variables U = a'X and V = b'Y 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 may be continued up to min{m,n} times.

Computation

Proof

Let $\Sigma _{XX} = \operatorname{cov}(X, X)$ and $\Sigma _{YY} = \operatorname{cov}(Y, Y)$. The parameter to maximize is

$\rho = \frac{a' \Sigma _{XY} b}{\sqrt{a' \Sigma _{XX} a} \sqrt{b' \Sigma _{YY} b}}.$

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

$c = \Sigma _{XX} ^{1/2} a,$

$d = \Sigma _{YY} ^{1/2} b.$

And thus we have

$\rho = \frac{c' \Sigma _{XX} ^{-1/2} \Sigma _{XY} \Sigma _{YY} ^{-1/2} d}{\sqrt{c' c} \sqrt{d' d}}.$

By the Cauchy-Schwarz inequality, we have

$c' \Sigma _{XX} ^{-1/2} \Sigma _{XY} \Sigma _{YY} ^{-1/2} d \leq \left(c' \Sigma _{XX} ^{-1/2} \Sigma _{XY} \Sigma _{YY} ^{-1/2} \Sigma _{YY} ^{-1/2} \Sigma _{YX} \Sigma _{XX} ^{-1/2} c \right)^{1/2} \left(d' d \right)^{1/2},$

$\rho \leq \frac{\left(c' \Sigma _{XX} ^{-1/2} \Sigma _{XY} \Sigma _{YY} ^{-1} \Sigma _{YX} \Sigma _{XX} ^{-1/2} c \right)^{1/2}}{\left(c' c \right)^{1/2}}.$

There is equality if the vectors d and $\Sigma _{YY} ^{-1/2} \Sigma _{YX} \Sigma _{XX} ^{-1/2} c$ are collinear. In addition, the maximum of correlation is attained if c is the eigenvector with the maximum eigenvalue for the matrix $\Sigma _{XX} ^{-1/2} \Sigma _{XY} \Sigma _{YY} ^{-1} \Sigma _{YX} \Sigma _{XX} ^{-1/2}$ (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:

• c is an eigenvector of $\Sigma _{XX} ^{-1/2} \Sigma _{XY} \Sigma _{YY} ^{-1} \Sigma _{YX} \Sigma _{XX} ^{-1/2}$
• d is proportional to $\Sigma _{YY} ^{-1/2} \Sigma _{YX} \Sigma _{XX} ^{-1/2} c$

Reciprocally, there is also:

• d is an eigenvector of $\Sigma _{YY} ^{-1/2} \Sigma _{YX} \Sigma _{XX} ^{-1} \Sigma _{XY} \Sigma _{YY} ^{-1/2}$
• c is proportional to $\Sigma _{XX} ^{-1/2} \Sigma _{XY} \Sigma _{YY} ^{-1/2} d$

Reversing the change of coordinates, we have that

• a is an eigenvector of $\Sigma _{XX} ^{-1} \Sigma _{XY} \Sigma _{YY} ^{-1} \Sigma _{YX}$
• b is an eigenvector of $\Sigma _{YY} ^{-1} \Sigma _{YX} \Sigma _{XX} ^{-1} \Sigma _{XY}$
• a is proportional to $\Sigma _{XX} ^{-1} \Sigma _{XY} b$
• b is proportional to $\Sigma _{YY} ^{-1} \Sigma _{YX} a$

The canonical variables are defined by:

$U = c' \Sigma _{XX} ^{-1/2} X = a' X$

$V = d' \Sigma _{YY} ^{-1/2} Y = b' Y$

Hypothesis testing

Each row can be tested for significance with the following method. Since the correlations are sorted, saying that row i is zero implies all further correlations are also zero. If we have p independent observations in a sample and $\widehat{\rho}_i$ is the estimated correlation for $i = 1,\dots, \min\{m,n\}$. For the ith row, the test statistic is:

$\chi ^2 = - \left( p - 1 - \frac{1}{2}(m + n + 1)\right) \ln \prod _ {j = i} ^p (1 - \widehat{\rho}_j^2),$

which is asymptotically distributed as a chi-squared with (m − i + 1)(n − i + 1) degrees of freedom for large p. Since all the correlations from min{m,n} to p are logically zero (and estimated that way also) the product for the terms after this point is irrelevant.

Practical uses

A typical use for canonical correlation in the experimental context is to take two sets of variables and see what is common amongst the two sets. For example in psychological testing, 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.

One can also use canonical correlation analysis to produce a model equation which relates two sets of variables, for example a set of performance measures and a set of explanatory variables, or a set of outputs and set of inputs. Constraint restrictions can be imposed on such a model to ensure it reflects theoretical requirements or intuitively obvious conditions. This type of model is known as a maximum correlation model.

Visualization of the results of canonical correlation is usually through bar plots of the coefficients of the two sets of variables for the pairs of canonical variates showing significant correlation. Some authors suggest that they are best visualized by plotting them as heliographs, a circular format with ray like bars, with each half representing the two sets of variables.

Connection to principal angles

Assuming that $X = (x_1, \dots, x_n)'$ and $Y = (y_1, \dots, y_m)'$ have zero expected values, i.e., $\operatorname{E}(X)=\operatorname{E}(Y)=0$, their covariance matrices $\Sigma _{XX} =\operatorname{Cov}(X,X) = \operatorname{E}[X X']$ and $\Sigma _{YY} =\operatorname{Cov}(Y,Y) = \operatorname{E}[Y Y']$ can be viewed as Gram matrices in an inner product, see Covariance#Relationship_to_inner_products, for the columns of X and Y, correspondingly. The definition of the canonical variables U and V is equivalent to the definition of principal vectors for the pair of subspaces spanned by the columns of X and Y with respect to this inner product. The canonical correlations $\operatorname{cor}(U,V)$ is equal to the cosine of principal angles.

See also

• Regularized canonical correlation analysis
• Generalized Canonical Correlation
• RV coefficient
• Principal angles

External links

• FactoMineR (free exploratory multivariate data analysis software linked to R)
• Understanding canonical correlation analysis (Concepts and Techniques in Modern Geography)
• Canonical correlation analysis — An overview with application to learning methods http://eprints.ecs.soton.ac.uk/9225/01/tech_report03.pdf, pages 5–9 give a good introduction Neural Computation (2004) version
• A note on the ordinal canonical correlation analysis of two sets of ranking scores (Also provides a FORTRAN program)- in J. of Quantitative Economics 7(2), 2009, pp. 173-199
• Representation-Constrained Canonical Correlation Analysis: A Hybridization of Canonical Correlation and Principal Component Analyses (Also provides a FORTRAN program)- in J. of Applied Economic Sciences 4(1), 2009, pp. 115-124

References

• Johnson, Richard A.; Wichern, Dean W. (2007). Applied Multivariate Statistical Analysis (Sixth ed.). Prentice Hall. ISBN 0131877151, ISBN 978-0131877153.