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Canonical correlation

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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 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]

DefinitionEdit

Given two column vectors   and   of random variables with finite second moments, one may define the cross-covariance   to be the   matrix whose   entry is the covariance  . 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  . 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.

 

ComputationEdit

DerivationEdit

Let   and  . The parameter to maximize is

 

The first step is to define a change of 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.

Another way of viewing this computation is that   and   are the left and right singular vectors of the correlation matrix of X and Y corresponding to the highest singular value.

SolutionEdit

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 proportional to  
  •   is an eigenvector of  
  •   is proportional to  .

The canonical variables are defined by:

 
 

ImplementationEdit

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

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[7] are available in

Hypothesis testingEdit

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 for large  .[8] Since all the correlations from   to   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   correlations will be identically 1 and hence the test is meaningless.[9]

Practical usesEdit

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]

ExamplesEdit

Let   with zero expected value, i.e.,  . If  , i.e.,   and   are perfectly correlated, then, e.g.,   and  , so that the first (and only in this example) pair of canonical variables is   and  . If  , i.e.,   and   are perfectly anticorrelated, then, e.g.,   and  , so that the first (and only in this example) pair of canonical variables is   and  . We notice that in both cases  , which illustrates that the canonical-correlation analysis treats correlated and anticorrelated variables similarly.

Connection to principal anglesEdit

Assuming that   and   have zero expected values, i.e.,  , their covariance matrices   and   can be viewed as Gram matrices in an inner product for the entries of   and  , correspondingly. In this interpretation, the random variables, entries   of   and   of   are treated as elements of a vector space with an inner product given by the covariance  ; see Covariance#Relationship to inner products.

The definition of the canonical variables   and   is then equivalent to the definition of principal vectors for the pair of subspaces spanned by the entries of   and   with respect to this inner product. The canonical correlations   is equal to the cosine of principal angles.

Whitening and probabilistic canonical correlation analysisEdit

CCA can also be viewed as special whitening transformation where random vectors   and   are simultaneously transformed in such a way that the cross-correlation between the whitened vectors   and   is diagonal.[13] The canonical correlations are then interpreted as regression coefficients linking   and   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 alsoEdit

ReferencesEdit

  1. ^ Härdle, Wolfgang; Simar, Léopold (2007). "Canonical Correlation Analysis". Applied Multivariate Statistical Analysis. pp. 321–330. CiteSeerX 10.1.1.324.403. 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. JSTOR 2333955.
  4. ^ Jordan, C. (1875). "Essai sur la géométrie à   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, CiteSeerX 10.1.1.73.2914, 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-dimensional data with very small sample support https://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. CiteSeerX 10.1.1.538.5217. doi:10.1007/11783183_11. ISBN 978-3-540-35623-3.
  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. PMC 6327589. PMID 30626338.

External linksEdit

  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.