Editing Canonical correlation (section)

==Population CCA definition via correlations==
Given two [[column vectors]] <math>X = (x_1, \dots, x_n)^T</math> and <math>Y = (y_1, \dots, y_m)^T</math> of [[random variable]]s with [[Wikt:finite|finite]] [[second moments]], one may define the [[cross-covariance]] <math>\Sigma _{XY} = \operatorname{cov}(X, Y) </math> to be the <math> n \times m</math> [[matrix (mathematics)|matrix]] whose <math>(i, j)</math> entry is the [[covariance]] <math>\operatorname{cov}(x_i, y_j)</math>. In practice, we would estimate the covariance matrix based on sampled data from <math>X</math> and <math>Y</math> (i.e. from a pair of data matrices).

Canonical-correlation analysis seeks a sequence of vectors <math>a_k</math>  (<math>a_k \in\mathbb R^n</math>) and <math>b_k</math> (<math>b_k \in\mathbb R^m</math>) such that the random variables <math>a_k^T X</math> and <math>b_k^T Y</math> maximize the [[correlation]] <math>\rho = \operatorname{corr}(a_k^T X, b_k^T Y)</math>. The (scalar) random variables <math>U = a_1^T X</math> and <math>V = b_1^T Y</math> 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 <math>\min\{m,n\}</math> times.

: <math display="block">(a_k,b_k) = \underset{a,b}\operatorname{argmax} \operatorname{corr}(a^T X, b^T Y) \quad\text{ subject to } \operatorname{cov}(a^T X, a_j^T X) = \operatorname{cov}(b^T Y, b_j^T Y) = 0 \text{ for } j=1, \dots, k-1</math>

The sets of vectors <math>a_k, b_k</math> are called '''''canonical directions''''' or '''''weight vectors''''' or simply '''''weights'''''. The 'dual' sets of vectors <math>\Sigma_{XX}a_k, \Sigma_{YY} b_k</math> are called '''''canonical loading vectors''''' or simply '''''loadings'''''; these are often more straightforward to interpret than the weights.<ref>{{Cite journal |last1=Gu |first1=Fei |last2=Wu |first2=Hao |date=2018-04-01 |title=Simultaneous canonical correlation analysis with invariant canonical loadings |url=https://doi.org/10.1007/s41237-017-0042-8 |journal=Behaviormetrika |language=en |volume=45 |issue=1 |pages=111–132 |doi=10.1007/s41237-017-0042-8 |issn=1349-6964|url-access=subscription }}</ref>