Canonical correlation

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Short description: Way of inferring information from cross-covariance matrices

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, ..., Xn) and Y = (Y1, ..., Ym) of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of X and Y that have a 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 Camille 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 method PCA, CCA can be viewed in population form (corresponding to random vectors and their covariance matrices) or in sample 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-called canonical decomposition \- understanding the differences between these objects is crucial for interpretation of the technique.

## Contents

* 1 Population CCA definition via correlations
* 2 Computation
* 2.1 Derivation
* 2.2 Solution
* 2.3 Implementation
* 3 Hypothesis testing
* 4 Practical uses
* 5 Examples
* 6 Connection to principal angles
* 7 Whitening and probabilistic canonical correlation analysis
* 8 See also
* 9 References
* 10 External links

## Population CCA definition via correlations

Given two column vectors X=(x1,…,xn)T and Y=(y1,…,ym)T of random variables with finite second moments, one may define the cross-covariance ΣXY=cov⁡(X,Y) to be the n×m matrix whose (i,j) entry is the covariance cov⁡(xi,yj). 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 a sequence of vectors ak (ak∈ℝn) and bk (bk∈ℝm) such that the random variables akTX and bkTY maximize the correlation ρ=corr⁡(akTX,bkTY). The (scalar) random variables U=a1TX and V=b1TY 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.

(ak,bk)=argmaxa,bcorr⁡(aTX,bTY) subject to cov⁡(aTX,ajTX)=cov⁡(bTY,bjTY)=0 for j=1,…,k−1

The sets of vectors ak,bk are called canonical directions or weight vectors or simply weights. The 'dual' sets of vectors ΣXXak,ΣYYbk are called canonical loading vectors or simply loadings; these are often more straightforward to interpret than the weights.[8]

## Computation

### Derivation

Let ΣXY be the cross-covariance matrix for any pair of (vector-shaped) random variables X and Y. The target function to maximize is

ρ=aTΣXYbaTΣXXabTΣYYb.

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

c=ΣXX1/2a,

d=ΣYY1/2b,

where ΣXX1/2 and ΣYY1/2 can be obtained from the eigen-decomposition (or by diagonalization):

ΣXX1/2=VXDX1/2VX⊤,VXDXVX⊤=ΣXX,

and

ΣYY1/2=VYDY1/2VY⊤,VYDYVY⊤=ΣYY.

Thus

ρ=cTΣXX−1/2ΣXYΣYY−1/2dcTcdTd.

By the Cauchy–Schwarz inequality,

(cTΣXX−1/2ΣXYΣYY−1/2)(d)≤(cTΣXX−1/2ΣXYΣYY−1/2ΣYY−1/2ΣYXΣXX−1/2c)1/2(dTd)1/2,

ρ≤(cTΣXX−1/2ΣXYΣYY−1ΣYXΣXX−1/2c)1/2(cTc)1/2.

There is equality if the vectors d and ΣYY−1/2ΣYXΣXX−1/2c are collinear. In addition, the maximum of correlation is attained if c is the eigenvector with the maximum eigenvalue for the matrix ΣXX−1/2ΣXYΣYY−1ΣYXΣXX−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 c and d are the left and right singular vectors of the correlation matrix of X and Y corresponding to the highest singular value.

### Solution

The solution is therefore:

* c is an eigenvector of ΣXX−1/2ΣXYΣYY−1ΣYXΣXX−1/2
* d is proportional to ΣYY−1/2ΣYXΣXX−1/2c

Reciprocally, there is also:

* d is an eigenvector of ΣYY−1/2ΣYXΣXX−1ΣXYΣYY−1/2
* c is proportional to ΣXX−1/2ΣXYΣYY−1/2d

Reversing the change of coordinates, we have that

* a is an eigenvector of ΣXX−1ΣXYΣYY−1ΣYX,
* b is proportional to ΣYY−1ΣYXa;
* b is an eigenvector of ΣYY−1ΣYXΣXX−1ΣXY,
* a is proportional to ΣXX−1ΣXYb.

The canonical variables are defined by:

U=cTΣXX−1/2X=aTX

V=dTΣYY−1/2Y=bTY

### Implementation

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

* MATLAB as canoncorr (also in Octave)
* R as the standard function cancor and several other packages, including candisc, CCA and vegan. CCP for statistical hypothesis testing in canonical correlation analysis.
* SAS as proc cancorr
* Python in the library scikit-learn, as cross decomposition and in statsmodels, as CanCorr. 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 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 precision computer arithmetic. To fix this trouble, alternative algorithms[12] are available in

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

## 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 ρ^i is the estimated correlation for i=1,…,min⁡{m,n}. For the ith row, the test statistic is:

χ2=−(p−1−12(m+n+1))ln⁡∏j=imin⁡{m,n}(1−ρ^j2),

which is asymptotically distributed as a chi-squared with (m−i+1)(n−i+1) degrees of freedom for large p.[13] 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.

Note that in the small sample size limit with p<n+m then we are guaranteed that the top m+n−p correlations will be identically 1 and hence the test is meaningless.[14]

## Practical uses

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 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.[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]

## Examples

Let X=x1 with zero expected value, i.e., E⁡(X)=0.

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

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

## Connection to principal angles

Assuming that X=(x1,…,xn)T and Y=(y1,…,ym)T have zero expected values, i.e., E⁡(X)=E⁡(Y)=0, their covariance matrices ΣXX=Cov⁡(X,X)=E⁡[XXT] and ΣYY=Cov⁡(Y,Y)=E⁡[YYT] can be viewed as Gram matrices in an inner product for the entries of X and Y, correspondingly. In this interpretation, the random variables, entries xi of X and yj of Y are treated as elements of a vector space with an inner product given by the covariance cov⁡(xi,yj); see Covariance.

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

## Whitening and probabilistic canonical correlation analysis

CCA can also be viewed as a special whitening transformation where the random vectors X and Y are simultaneously transformed in such a way that the cross-correlation between the whitened vectors XCCA and YCCA is diagonal.[18] The canonical correlations are then interpreted as regression coefficients linking XCCA and YCCA 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]

## See also

* 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

## References

1. ↑ Härdle, Wolfgang; Simar, Léopold (2007). "Canonical Correlation Analysis". Applied Multivariate Statistical Analysis. pp. 321–330. doi:10.1007/978-3-540-72244-1_14. ISBN 978-3-540-72243-4.
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.
4. ↑ Jordan, C. (1875). "Essai sur la géométrie à n dimensions". Bull. Soc. Math. France 3: 103. https://www.numdam.org/item?id=BSMF_1875__3__103_2.
5. ↑ Andrew, Galen; Arora, Raman; Bilmes, Jeff; Livescu, Karen (2013-05-26). "Deep Canonical Correlation Analysis" (in en). Proceedings of the 30th International Conference on Machine Learning (PMLR): 1247–1255. https://proceedings.mlr.press/v28/andrew13.html.
6. ↑ Ju, Ce; Kobler, Reinmar J; Tang, Liyao; Guan, Cuntai; Kawanabe, Motoaki (2024). Deep Geodesic Canonical Correlation Analysis for Covariance-Based Neuroimaging Data. The Twelfth International Conference on Learning Representations (ICLR 2024, spotlight). https://openreview.net/pdf?id=PnR1MNen7u.
7. ↑ "Statistical Learning with Sparsity: the Lasso and Generalizations". https://hastie.su.domains/StatLearnSparsity/.
8. ↑ Gu, Fei; Wu, Hao (2018-04-01). "Simultaneous canonical correlation analysis with invariant canonical loadings" (in en). Behaviormetrika 45 (1): 111–132. doi:10.1007/s41237-017-0042-8. ISSN 1349-6964. https://doi.org/10.1007/s41237-017-0042-8.
9. ↑ Hsu, D.; Kakade, S. M.; Zhang, T. (2012). "A spectral algorithm for learning Hidden Markov Models". Journal of Computer and System Sciences 78 (5): 1460. doi:10.1016/j.jcss.2011.12.025. https://www.cs.mcgill.ca/~colt2009/papers/011.pdf.
10. ↑ Huang, S. Y.; Lee, M. H.; Hsiao, C. K. (2009). "Nonlinear measures of association with kernel canonical correlation analysis and applications". Journal of Statistical Planning and Inference 139 (7): 2162. doi:10.1016/j.jspi.2008.10.011. http://www.stat.sinica.edu.tw/syhuang/papersdownload/KCCA-080906.pdf. Retrieved 2015-09-04.
11. ↑ Chapman, James; Wang, Hao-Ting (2021-12-18). "CCA-Zoo: A collection of Regularized, Deep Learning based, Kernel, and Probabilistic CCA methods in a scikit-learn style framework" (in en). Journal of Open Source Software 6 (68): 3823. doi:10.21105/joss.03823. ISSN 2475-9066. Bibcode: 2021JOSS....6.3823C.
12. ↑ 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, doi:10.1137/S1064827500377332, Bibcode: 2002SJSC...23.2008K
13. ↑ Kanti V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis. Academic Press.
14. ↑ Yang Song, Peter J. Schreier, David Ram´ırez, and Tanuj Hasija Canonical correlation analysis of high-dimensional data with very small sample support arXiv:1604.02047
15. ↑ Sieranoja, S.; Sahidullah, Md; Kinnunen, T.; Komulainen, J.; Hadid, A. (July 2018). "Audiovisual Synchrony Detection with Optimized Audio Features". 2018 IEEE 3rd International Conference on Signal and Image Processing (ICSIP). pp. 377–381. doi:10.1109/SIPROCESS.2018.8600424. ISBN 978-1-5386-6396-7. http://cs.joensuu.fi/pages/tkinnu/webpage/pdf/audiovisual_synchrony_2018.pdf.
16. ↑ Tofallis, C. (1999). "Model Building with Multiple Dependent Variables and Constraints". Journal of the Royal Statistical Society, Series D 48 (3): 371–378. doi:10.1111/1467-9884.00195.
17. ↑ Degani, A.; Shafto, M.; Olson, L. (2006). "Canonical Correlation Analysis: Use of Composite Heliographs for Representing Multiple Patterns". Diagrammatic Representation and Inference. Lecture Notes in Computer Science. 4045. pp. 93. doi:10.1007/11783183_11. ISBN 978-3-540-35623-3. http://ti.arc.nasa.gov/m/profile/adegani/Composite_Heliographs.pdf.
18. ↑ Jendoubi, T.; Strimmer, K. (2018). "A whitening approach to probabilistic canonical correlation analysis for omics data integration". BMC Bioinformatics 20 (1): 15. doi:10.1186/s12859-018-2572-9. PMID 30626338.
19. ↑ Jendoubi, Takoua; Strimmer, Korbinian (9 January 2019). "A whitening approach to probabilistic canonical correlation analysis for omics data integration" (in en). BMC Bioinformatics 20 (1): 15. doi:10.1186/s12859-018-2572-9. ISSN 1471-2105. PMID 30626338.

## External links

* 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. doi:10.1162/0899766042321814. PMID 15516276.
* 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

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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. https://zenodo.org/record/889881.

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