Canonical Correlation Analysis (CCA) can be conceptualized as a multivariate regression involving multiple outcome variables. CCA compares two sets of variables and is the second-most general. Canonical Correlation Analysis is used to test the correlation between two sets of variables. Available in Excel using the XLSTAT add-on statistical software. Canonical correlation is a method of modelling the relationship between two sets of variables. This post provides: (a) Examples of when canonical correlation can be useful; (b) Links to good online resources where you can learn about the technique; (c) Links to examples of running the analysis in R or SPSS; and (d) Examples of articles showing how to report a canonical correlation analysis.

1. Canonical Correlation Analysis Software
2. Canonical Correlation Analysis Pdf
3. Canonical Correlation Analysis Matlab
4. Canonical Correlation Analysis: An Overview With Application To Learning Methods

This tutorial will show you how to set up and interpret a Canonical Correlation Analysis in Excel using the XLSTAT statistical software. Not sure if t. Xerox phaser 4510. Jun 17, 2010  Canonical correlation is a method of modelling the relationship between two sets of variables. This post provides: (a) Examples of when canonical correlation can be useful; (b) Links to good online resources where you can learn about the technique; (c) Links to examples of running the analysis in R or SPSS; and (d) Examples of articles showing how to report a canonical correlation analysis. Canonical correlation analysis determines a set of canonical variates, orthogonal linear combinations of the variables within each set that best explain the variability both within and between sets. Please Note: The purpose of this page is to show how to use various data analysis commands.

In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors X = (X₁, .., X_n) and Y = (Y₁, .., Y_m) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of X and Y which have maximum correlation with each other.^[1] T. R. Knapp notes that 'virtually all of the commonly encountered parametric 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 by Harold Hotelling in 1936,^[3] although in the context of angles between flats the mathematical concept was published by Jordan in 1875.^[4]

• 2Computation

Definition[edit]

Given two column vectors and of random variables with finitesecond moments, one may define the cross-covariance${\displaystyle {\mathrm{\Sigma }}_{XY}=\mathrm{cov}(X,Y)}$ to be the ${\displaystyle n\times m}$matrix whose ${\displaystyle (i,j)}$ entry is the covariance${\displaystyle \mathrm{cov}(x_i,y_j)}$. 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 vectors ${\displaystyle a}$ (${\displaystyle a}$${\displaystyle \in {\mathbb{R}}^n}$ ) and ${\displaystyle b}$ (${\displaystyle b\in {\mathbb{R}}^m}$) such that the random variables ${\displaystyle a^{T}X}$ and ${\displaystyle b^{T}Y}$ maximize the correlation${\displaystyle \rho =\mathrm{corr}(a^{T}X,b^{T}Y)}$. The random variables ${\displaystyle U=a^{T}X}$ and ${\displaystyle V=b^{T}Y}$ 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 ${\displaystyle min\{m,n\}}$ times.

Computation[edit]

Derivation[edit]

Let ${\displaystyle {\mathrm{\Sigma }}_{XX}=\mathrm{cov}(X,X)}$ and ${\displaystyle {\mathrm{\Sigma }}_{YY}=\mathrm{cov}(Y,Y)}$. The parameter 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 a change of basis and define

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

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

And thus we have

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

Canonical Correlation Analysis Software

By the Cauchy–Schwarz inequality, we have

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

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

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 the eigenvector with the maximum eigenvalue for the matrix ${\displaystyle {\mathrm{\Sigma }}_{XX}^{-1/2}{\mathrm{\Sigma }}_{XY}{\mathrm{\Sigma }}_{YY}^{-1}{\mathrm{\Sigma }}_{YX}{\mathrm{\Sigma }}_{XX}^{-1/2}}$ (see Rayleigh quotient). The subsequent pairs are found by using eigenvalues 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 right singular vectors of the correlation matrix of X and Y corresponding to the highest singular value.

Solution[edit]

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:

Implementation[edit]

CCA can be computed using singular value decomposition on a correlation matrix.^[5] It is available as a function in^[6]

• MATLAB as canoncorr (also in Octave)
• R as the standard function cancor and several other packages. CCP for statistical hypothesis testing in canonical correlation analysis.
• SAS as proc cancorr
• Python in the libraryscikit-learn, as Cross decomposition and in statsmodels, as CanCorr.
• SPSS as macro CanCorr shipped with the main software
• Julia (programming language) in the MultivariateStats.jl package.

CCA computation using singular value decomposition on a correlation matrix is related to the cosine of the angles between flats. The cosine function is ill-conditioned for small angles, leading to very inaccurate computation of highly correlated principal vectors in finite precisioncomputer arithmetic. To fix this trouble, alternative algorithms^[7] are available in

• SciPy as linear-algebra function subspace_angles
• MATLAB as FileExchange function subspacea

Hypothesis testing[edit]

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 a chi-squared with ${\displaystyle (m-i+1)(n-i+1)}$degrees of freedom for large ${\displaystyle p}$.^[8] 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.^[9]

Practical uses[edit]

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.^[10] For example, in psychological testing, one could take two well established multidimensional personality tests such as the Minnesota Multiphasic Personality Inventory (MMPI-2) and the NEO. 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 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.^[11]

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.^[12]

Examples[edit]

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

Connection to principal angles[edit]

Assuming that and have zero expected values, i.e., ${\displaystyle \mathrm{E}(X)=\mathrm{E}(Y)=0}$, their covariance matrices and can be viewed as Gram matrices in an inner 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 the covariance${\displaystyle \mathrm{cov}(x_i,y_j)}$; see Covariance#Relationship to inner products.

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

Whitening and probabilistic canonical correlation analysis[edit]

Canonical Correlation Analysis Pdf

CCA can also be viewed as special whitening transformation where 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.^[13]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.

See also[edit]

References[edit]

1. ^Härdle, Wolfgang; Simar, Léopold (2007). 'Canonical Correlation Analysis'. Applied Multivariate Statistical Analysis. pp. 321–330. CiteSeerX10.1.1.324.403. doi:10.1007/978-3-540-72244-1_14. ISBN978-3-540-72243-4.
2. ^Knapp, T. R. (1978). 'Canonical correlation analysis: A general parametric significance-testing system'. Psychological Bulletin. 85 (2): 410–416. doi:10.1037/0033-2909.85.2.410.
3. ^Hotelling, H. (1936). 'Relations Between Two Sets of Variates'. Biometrika. 28 (3–4): 321–377. doi:10.1093/biomet/28.3-4.321. JSTOR2333955.
4. ^Jordan, C. (1875). 'Essai sur la géométrie à ${\displaystyle n}$ dimensions'. Bull. Soc. Math. France. 3: 103.
5. ^Hsu, D.; Kakade, S. M.; Zhang, T. (2012). 'A spectral algorithm for learning Hidden Markov Models'(PDF). Journal of Computer and System Sciences. 78 (5): 1460. arXiv:0811.4413. doi:10.1016/j.jcss.2011.12.025.
6. ^Huang, S. Y.; Lee, M. H.; Hsiao, C. K. (2009). 'Nonlinear measures of association with kernel canonical correlation analysis and applications'(PDF). Journal of Statistical Planning and Inference. 139 (7): 2162. doi:10.1016/j.jspi.2008.10.011.
7. ^Knyazev, A.V.; Argentati, M.E. (2002), 'Principal Angles between Subspaces in an A-Based Scalar Product: Algorithms and Perturbation Estimates', SIAM Journal on Scientific Computing, 23 (6): 2009–2041, CiteSeerX10.1.1.73.2914, doi:10.1137/S1064827500377332
8. ^Kanti V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis. Academic Press.
9. ^Yang Song, Peter J. Schreier, David Ram´ırez, and Tanuj Hasija Canonical correlation analysis of high-dimensionaldata with very small sample supporthttps://arxiv.org/pdf/1604.02047.pdf
10. ^Sieranoja, S.; Sahidullah, Md; Kinnunen, T.; Komulainen, J.; Hadid, A. (July 2018). 'Audiovisual Synchrony Detection with Optimized Audio Features'(PDF). IEEE 3rd Int. Conference on Signal and Image Processing (ICSIP 2018).
11. ^Tofallis, C. (1999). 'Model Building with Multiple Dependent Variables and Constraints'. Journal of the Royal Statistical Society, Series D. 48 (3): 371–378. arXiv:1109.0725. doi:10.1111/1467-9884.00195.
12. ^Degani, A.; Shafto, M.; Olson, L. (2006). 'Canonical Correlation Analysis: Use of Composite Heliographs for Representing Multiple Patterns'(PDF). Diagrammatic Representation and Inference. Lecture Notes in Computer Science. 4045. p. 93. CiteSeerX10.1.1.538.5217. doi:10.1007/11783183_11. ISBN978-3-540-35623-3.
13. ^Jendoubi, T.; Strimmer, K. (2018). 'A whitening approach to probabilistic canonical correlation analysis for omics data integration'. BMC Bioinformatics. 20 (1): 15. arXiv:1802.03490. doi:10.1186/s12859-018-2572-9. PMC6327589. PMID30626338.

Canonical Correlation Analysis Matlab

External links[edit]

• 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. CiteSeerX10.1.1.14.6452. doi:10.1162/0899766042321814. PMID15516276.
• A note on the ordinal canonical-correlation analysis of two sets of ranking scores (Also provides a FORTRAN 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 a FORTRAN program)- in Journal of Applied Economic Sciences 4(1), 2009, pp. 115–124

1. ^Haghighat, Mohammad; Abdel-Mottaleb, Mohamed; Alhalabi, Wadee (2016). 'Discriminant Correlation Analysis: Real-Time Feature Level Fusion for Multimodal Biometric Recognition'. IEEE Transactions on Information Forensics and Security. 11 (9): 1984–1996. doi:10.1109/TIFS.2016.2569061.

Canonical Correlation Analysis: An Overview With Application To Learning Methods