CCA tutorial

Canonical Correlation a Tutorial Magnus Borga January 12, 2001 Contents 1 About this tutorial 1 2 Introduction 2 3 ...

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Canonical Correlation a Tutorial Magnus Borga January 12, 2001

Contents 1 About this tutorial

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2 Introduction

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3 Definition

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4 Calculating canonical correlations

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5 Relating topics 5.1 The difference between CCA and ordinary correlation analysis 5.2 Relation to mutual information . . . . . . . . . . . . . . . . . 5.3 Relation to other linear subspace methods . . . . . . . . . . . 5.4 Relation to SNR . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Equal noise energies . . . . . . . . . . . . . . . . . . 5.4.2 Correlation between a signal and the corrupted signal .

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A Explanations A.1 A note on correlation and covariance matrices A.2 Affine transformations . . . . . . . . . . . . A.3 A piece of information theory . . . . . . . . . A.4 Principal component analysis . . . . . . . . . A.5 Partial least squares . . . . . . . . . . . . . . A.6 Multivariate linear regression . . . . . . . . . A.7 Signal to noise ratio . . . . . . . . . . . . . .

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1 About this tutorial This is a printable version of a tutorial in HTML format. The tutorial may be modified at any time as will this version. The latest version of this tutorial is available at http://people.imt.liu.se/˜magnus/cca/. 1

2 Introduction Canonical correlation analysis (CCA) is a way of measuring the linear relationship between two multidimensional variables. It finds two bases, one for each variable, that are optimal with respect to correlations and, at the same time, it finds the corresponding correlations. In other words, it finds the two bases in which the correlation matrix between the variables is diagonal and the correlations on the diagonal are maximized. The dimensionality of these new bases is equal to or less than the smallest dimensionality of the two variables. An important property of canonical correlations is that they are invariant with respect to affine transformations of the variables. This is the most important difference between CCA and ordinary correlation analysis which highly depend on the basis in which the variables are described. CCA was developed by H. Hotelling [10]. Although being a standard tool in statistical analysis, where canonical correlation has been used for example in economics, medical studies, meteorology and even in classification of malt whisky, it is surprisingly unknown in the fields of learning and signal processing. Some exceptions are [2, 13, 5, 4, 14], For further details and applications in signal processing, see my PhD thesis [3] and other publications.

3 Definition Canonical correlation analysis can be defined as the problem of finding two sets of basis vectors, one for x and the other for y, such that the correlations between the projections of the variables onto these basis vectors are mutually maximized. Let us look at the case where only one pair of basis vectors are sought, namely the ones corresponding to the largest canonical correlation: Consider the linear combinations x = xT w ^ x and y = yT w ^ y of the two variables respectively. This means that the function to be maximized is

E [w ^ xT xyT w ^ y] E [xy] =q 2 2 E [x ]E [y ] E [w ^ xT xxT w ^ x]E [w ^ yT yyT w ^ y]

= p =q

wxT Cxy wy wxT CxxwxwyT Cyy wy

(1)

:

The maximum of  with respect to wx and wy is the maximum canonical correlation. The subsequent canonical correlations are uncorrelated for different solutions, i.e. 8 > : E [xi yj ]

T xxT w ] = wT C w = 0 = E [wxi xj xi xx xj T yyT w ] = wT C w = 0 = E [wyi yj yi yy yj T xyT w ] = = E [wxi yj

2

TC w =0 wxi xy yj

for i 6= j:

(2)

The projections onto wx and wy , i.e. x and y , are called canonical variates.

4 Calculating canonical correlations Consider two random variables x and y with zero mean. The total covariance matrix "    #   Cxx Cxy x x T C= =E (3)

Cyx Cyy

y

y

is a block matrix where Cxx and Cxx are the within-sets covariance matrices of x and y respectively and Cxy = CTyx is the between-sets covariance matrix. The canonical correlations between x and y can be found by solving the eigenvalue equations ( Cxx1 Cxy Cyy1 Cyx w ^ x = 2 w ^x (4) 1 1 2 Cyy CyxCxx Cxy w ^y =  w ^y where the eigenvalues 2 are the squared canonical correlations and the eigenvectors w ^ x and w ^ y are the normalized canonical correlation basis vectors. The number of non-zero solutions to these equations are limited to the smallest dimensionality of x and y. E.g. if the dimensionality of x and y is 8 and 5 respectively, the maximum number of canonical correlations is 5. Only one of the eigenvalue equations needs to be solved since the solutions are related by 8