Canonical correlation - AbsoluteAstronomy.com

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, canonical correlation analysis, introduced by Harold Hotelling

Harold Hotelling

Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...

, is a way of making sense of cross-covariance matrices. If we have two sets of variables,

and

, and there are correlations among the variables, then canonical correlation analysis will enable us to find linear combinations of the

's and the

's which have maximum correlation with each other.

Definition

Given two column vectors

and

of random variables

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

with finite second moments, one may define the cross-covariance

to be the

matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

whose

entry is the covariance

Covariance

In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

. In practice, we would estimate the covariance matrix based on sampled data from

and

(i.e. from a pair of data matrices).

Canonical correlation analysis seeks vectors

and

such that the random variables

and

maximize the correlation

Correlation

In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

. The random variables

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 may be continued up to

times.

Proof

Let

and

. The parameter to maximize is

The first step is to define a change of basis

Change of basis

In linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases.-Expression of a basis:...

and define

And thus we have

By the Cauchy-Schwarz inequality, we have

There is equality if the vectors

and

are collinear. 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

Reversing the change of coordinates, we have that

is an eigenvector of
is an eigenvector of
is proportional to
is proportional to

The canonical variables are defined by:

Hypothesis testing

Each row can be tested for significance with the following method. Since the correlations are sorted, saying that row

is zero implies all further correlations are also zero. If we have

independent observations in a sample and

is the estimated correlation for

. For the

th row, the test statistic is:

which is asymptotically distributed as a chi-squared with

degrees of freedom

Degrees of freedom (statistics)

In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...

for large

. Since all the correlations from

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

MMPI

MMPI may refer to:*Minnesota Multiphasic Personality Inventory, one of the most frequently used personality tests in mental health*Matrix metalloproteinase inhibitor, inhibits cell migration and has antiangiogenic effects...

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

Extraversion and introversion

The trait of extraversion-introversion is a central dimension of human personality theories.Extraverts tend to be gregarious, assertive, and interested in seeking out external stimulus. Introverts, in contrast, tend to be introspective, quiet and less sociable. They are not necessarily loners but...

or neuroticism

Neuroticism

Neuroticism is a fundamental personality trait in the study of psychology. It is an enduring tendency to experience negative emotional states. Individuals who score high on neuroticism are more likely than the average to experience such feelings as anxiety, anger, guilt, and depressed mood...

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

and

have zero expected value

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

s, i.e.,

, their covariance

Covariance

matrices

and

can be viewed as Gram matrices in an inner product, see Covariance#Relationship to inner products, for the columns of

and

, correspondingly. The definition of the canonical variables

and

is equivalent to the definition of principal vectors

Principal angles

In linear algebra , the principal angles, also called canonical angles, provide information about the relative position of two subspaces of a Euclidean space...

for the pair of subspaces spanned by the columns of

and

with respect to this inner product. The canonical correlations

is equal to the cosine of principal angles