Instatistics,canonical-correlation analysis (CCA), also calledcanonical variates analysis, is a way of inferring information fromcross-covariance matrices. If we have two vectorsX = (X₁, ..., X_n) andY = (Y₁, ..., Y_m) ofrandom variables, and there arecorrelations among the variables, then canonical-correlation analysis will findlinear combinations ofX andY that have a maximum correlation with each other.^[1] T. R. Knapp notes that "virtually all of the commonly encounteredparametric tests of significance can be treated as special cases of canonical-correlation analysis, which is the general procedure for investigating the relationships between two sets of variables."^[2] The method was first introduced byHarold Hotelling in 1936,^[3] although in the context ofangles between flats the mathematical concept was published byCamille Jordan in 1875.^[4]

CCA is now a cornerstone of multivariate statistics and multi-view learning, and a great number of interpretations and extensions have been proposed, such as probabilistic CCA, sparse CCA, multi-view CCA, deep CCA,^[5] and DeepGeoCCA.^[6] Unfortunately, perhaps because of its popularity, the literature can be inconsistent with notation, we attempt to highlight such inconsistencies in this article to help the reader make best use of the existing literature and techniques available.

Like its sister methodPCA, CCA can be viewed inpopulation form (corresponding to random vectors and their covariance matrices) or insample form (corresponding to datasets and their sample covariance matrices). These two forms are almost exact analogues of each other, which is why their distinction is often overlooked, but they can behave very differently in high dimensional settings.^[7] We next give explicit mathematical definitions for the population problem and highlight the different objects in the so-calledcanonical decomposition - understanding the differences between these objects is crucial for interpretation of the technique.

Given twocolumn vectors${\displaystyle X=(x_1,\dots ,x_n{)}^T}$ and${\displaystyle Y=(y_1,\dots ,y_m{)}^T}$ ofrandom variables withfinitesecond moments, one may define thecross-covariance${\displaystyle {\mathrm{\Sigma }}_{XY}=\mathrm{cov}(X,Y)}$ to be the${\displaystyle n\times m}$matrix whose${\displaystyle (i,j)}$ entry is thecovariance${\displaystyle \mathrm{cov}(x_i,y_j)}$. In practice, we would estimate the covariance matrix based on sampled data from${\displaystyle X}$ and${\displaystyle Y}$ (i.e. from a pair of data matrices).

Canonical-correlation analysis seeks a sequence of vectors${\displaystyle a_k}$ (${\displaystyle a_k\in {\mathbb{R}}^n}$) and${\displaystyle b_k}$ (${\displaystyle b_k\in {\mathbb{R}}^m}$) such that the random variables${\displaystyle a_k^{T}X}$ and${\displaystyle b_k^{T}Y}$ maximize thecorrelation${\displaystyle \rho =\mathrm{corr}(a_k^{T}X,b_k^{T}Y)}$. The (scalar) random variables${\displaystyle U=a_1^{T}X}$ and${\displaystyle V=b_1^{T}Y}$ are thefirst 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 thesecond pair of canonical variables. This procedure may be continued up to${\displaystyle min\{m,n\}}$ times.

$${\displaystyle (a_k,b_k)=\underset{a,b}{\mathrm{argmax}}\mathrm{corr}(a^{T}X,b^{T}Y)\text{ subject to }\mathrm{cov}(a^{T}X,a_j^{T}X)=\mathrm{cov}(b^{T}Y,b_j^{T}Y)=0\text{ for }j=1,\dots ,k-1}$$

The sets of vectors${\displaystyle a_k,b_k}$ are calledcanonical directions orweight vectors or simplyweights. The 'dual' sets of vectors${\displaystyle {\mathrm{\Sigma }}_{XX}{a}_k,{\mathrm{\Sigma }}_{YY}{b}_k}$ are calledcanonical loading vectors or simplyloadings; these are often more straightforward to interpret than the weights.^[8]

Let${\displaystyle {\mathrm{\Sigma }}_{XY}}$ be thecross-covariance matrix for any pair of (vector-shaped) random variables${\displaystyle X}$ and${\displaystyle Y}$. The target function to maximize is

${\displaystyle \rho =\frac{a^T{\mathrm{\Sigma }}_{XY}b}{\sqrt{a^T{\mathrm{\Sigma }}_{XX}a}\sqrt{b^T{\mathrm{\Sigma }}_{YY}b}}.}$

The first step is to define achange of basis and define

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

${\displaystyle d={\mathrm{\Sigma }}_{YY}^{1/2}b,}$

where${\displaystyle {\mathrm{\Sigma }}_{XX}^{1/2}}$ and${\displaystyle {\mathrm{\Sigma }}_{YY}^{1/2}}$ can be obtained from the eigen-decomposition (or bydiagonalization):

${\displaystyle {\mathrm{\Sigma }}_{XX}^{1/2}=V_X{D}_X^{1/2}{V}_X^{\mathrm{\top }},V_X{D}_X{V}_X^{\mathrm{\top }}={\mathrm{\Sigma }}_{XX},}$

and

${\displaystyle {\mathrm{\Sigma }}_{YY}^{1/2}=V_Y{D}_Y^{1/2}{V}_Y^{\mathrm{\top }},V_Y{D}_Y{V}_Y^{\mathrm{\top }}={\mathrm{\Sigma }}_{YY}.}$

Thus

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

By theCauchy–Schwarz inequality,

${\displaystyle \left(c^T{\mathrm{\Sigma }}_{XX}^{-1/2}{\mathrm{\Sigma }}_{XY}{\mathrm{\Sigma }}_{YY}^{-1/2}\right)(d)\le {\left(c^T{\mathrm{\Sigma }}_{XX}^{-1/2}{\mathrm{\Sigma }}_{XY}{\mathrm{\Sigma }}_{YY}^{-1/2}{\mathrm{\Sigma }}_{YY}^{-1/2}{\mathrm{\Sigma }}_{YX}{\mathrm{\Sigma }}_{XX}^{-1/2}c\right)}^{1/2}{\left(d^{T}d\right)}^{1/2},}$

${\displaystyle \rho \le \frac{{\left(c^T{\mathrm{\Sigma }}_{XX}^{-1/2}{\mathrm{\Sigma }}_{XY}{\mathrm{\Sigma }}_{YY}^{-1}{\mathrm{\Sigma }}_{YX}{\mathrm{\Sigma }}_{XX}^{-1/2}c\right)}^{1/2}}{{\left(c^{T}c\right)}^{1/2}}.}$

There is equality if the vectors${\displaystyle d}$ and${\displaystyle {\mathrm{\Sigma }}_{YY}^{-1/2}{\mathrm{\Sigma }}_{YX}{\mathrm{\Sigma }}_{XX}^{-1/2}c}$ are collinear. In addition, the maximum of correlation is attained if${\displaystyle c}$ is theeigenvector with the maximum eigenvalue for the matrix${\displaystyle {\mathrm{\Sigma }}_{XX}^{-1/2}{\mathrm{\Sigma }}_{XY}{\mathrm{\Sigma }}_{YY}^{-1}{\mathrm{\Sigma }}_{YX}{\mathrm{\Sigma }}_{XX}^{-1/2}}$ (seeRayleigh quotient). The subsequent pairs are found by usingeigenvalues of decreasing magnitudes. Orthogonality is guaranteed by the symmetry of the correlation matrices.

Another way of viewing this computation is that${\displaystyle c}$ and${\displaystyle d}$ are the left and rightsingular vectors of the correlation matrix of X and Y corresponding to the highest singular value.

The solution is therefore:

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

Reciprocally, there is also:

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

Reversing the change of coordinates, we have that

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

The canonical variables are defined by:

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

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

CCA can be computed usingsingular value decomposition on a correlation matrix.^[9] It is available as a function in^[10]

• MATLAB ascanoncorr (also inOctave)
• R as the standard functioncancor and several other packages, includingcandisc,CCA andvegan.CCP for statistical hypothesis testing in canonical correlation analysis.
• SAS asproc cancorr
• Python in the libraryscikit-learn, ascross decomposition and in statsmodels, asCanCorr. The CCA-Zoo library^[11] implements CCA extensions, such as probabilistic CCA, sparse CCA, multi-view CCA, and deep CCA.
• SPSS as macro CanCorr shipped with the main software
• Julia (programming language) in theMultivariateStats.jl package.

CCA computation usingsingular value decomposition on a correlation matrix is related to thecosine of theangles between flats. Thecosine function isill-conditioned for small angles, leading to very inaccurate computation of highly correlated principal vectors in finiteprecisioncomputer arithmetic. Tofix this trouble, alternative algorithms^[12] are available in

• SciPy aslinear-algebra function subspace_angles
• MATLAB asFileExchange function subspacea

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

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

which is asymptotically distributed as achi-squared with${\displaystyle (m-i+1)(n-i+1)}$degrees of freedom for large${\displaystyle p}$.^[13] Since all the correlations from${\displaystyle min\{m,n\}}$ to${\displaystyle p}$ are logically zero (and estimated that way also) the product for the terms after this point is irrelevant.

Note that in the small sample size limit with then we are guaranteed that the top${\displaystyle m+n-p}$ correlations will be identically 1 and hence the test is meaningless.^[14]

A typical use for canonical correlation in the experimental context is to take two sets of variables and see what is common among the two sets.^[15] For example, in psychological testing, one could take two well established multidimensionalpersonality tests such as theMinnesota Multiphasic Personality Inventory (MMPI-2) and theNEO. By seeing how the MMPI-2 factors relate to the NEO factors, one could gain insight into what dimensions were common between the tests and how much variance was shared. For example, one might find that anextraversion orneuroticism 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.^[16]

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.^[17]

Let${\displaystyle X=x_1}$ with zeroexpected value, i.e.,${\displaystyle \mathrm{E}(X)=0}$.

1. If${\displaystyle Y=X}$, i.e.,${\displaystyle X}$ and${\displaystyle Y}$ are perfectly correlated, then, e.g.,${\displaystyle a=1}$ and${\displaystyle b=1}$, so that the first (and only in this example) pair of canonical variables is${\displaystyle U=X}$ and${\displaystyle V=Y=X}$.
2. If${\displaystyle Y=-X}$, i.e.,${\displaystyle X}$ and${\displaystyle Y}$ are perfectly anticorrelated, then, e.g.,${\displaystyle a=1}$ and${\displaystyle b=-1}$, so that the first (and only in this example) pair of canonical variables is${\displaystyle U=X}$ and${\displaystyle V=-Y=X}$.

We notice that in both cases${\displaystyle U=V}$, which illustrates that the canonical-correlation analysis treats correlated and anticorrelated variables similarly.

Assuming that${\displaystyle X=(x_1,\dots ,x_n{)}^T}$ and${\displaystyle Y=(y_1,\dots ,y_m{)}^T}$ have zeroexpected values, i.e.,${\displaystyle \mathrm{E}(X)=\mathrm{E}(Y)=0}$, theircovariance matrices${\displaystyle {\mathrm{\Sigma }}_{XX}=\mathrm{Cov}(X,X)=\mathrm{E}[X{X}^T]}$ and${\displaystyle {\mathrm{\Sigma }}_{YY}=\mathrm{Cov}(Y,Y)=\mathrm{E}[Y{Y}^T]}$ can be viewed asGram matrices in aninner product for the entries of${\displaystyle X}$ and${\displaystyle Y}$, correspondingly. In this interpretation, the random variables, entries${\displaystyle x_i}$ of${\displaystyle X}$ and${\displaystyle y_j}$ of${\displaystyle Y}$ are treated as elements of a vector space with an inner product given by thecovariance${\displaystyle \mathrm{cov}(x_i,y_j)}$; seeCovariance#Relationship to inner products.

The definition of the canonical variables${\displaystyle U}$ and${\displaystyle V}$ is then equivalent to the definition ofprincipal vectors for the pair of subspaces spanned by the entries of${\displaystyle X}$ and${\displaystyle Y}$ with respect to thisinner product. The canonical correlations${\displaystyle \mathrm{corr}(U,V)}$ is equal to thecosine ofprincipal angles.

CCA can also be viewed as a specialwhitening transformation where the random vectors${\displaystyle X}$ and${\displaystyle Y}$ are simultaneously transformed in such a way that the cross-correlation between the whitened vectors${\displaystyle X^{CCA}}$ and${\displaystyle Y^{CCA}}$ is diagonal.^[18]The canonical correlations are then interpreted as regression coefficients linking${\displaystyle X^{CCA}}$ and${\displaystyle Y^{CCA}}$ and may also be negative. The regression view of CCA also provides a way to construct a latent variable probabilistic generative model for CCA, with uncorrelated hidden variables representing shared and non-shared variability.^[19]

• Generalized canonical correlation
• RV coefficient
• Angles between flats
• Principal component analysis
• Linear discriminant analysis
• Regularized canonical correlation analysis
• Singular value decomposition
• Partial least squares regression

• Discriminant Correlation Analysis (DCA)^[1] (MATLAB)
• Hardoon, D. R.; Szedmak, S.; Shawe-Taylor, J. (2004). "Canonical Correlation Analysis: An Overview with Application to Learning Methods".Neural Computation.16 (12):2639–2664.CiteSeerX 10.1.1.14.6452.doi:10.1162/0899766042321814.PMID 15516276.S2CID 202473.
• A note on the ordinal canonical-correlation analysis of two sets of ranking scores (Also provides aFORTRAN program)- in Journal 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 aFORTRAN program)- in Journal of Applied Economic Sciences 4(1), 2009, pp. 115–124

• Covariance and correlation

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