Canonical correlation

In statistics, canonical-correlation analysis (CCA) is a way of making sense of 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 the X_i and Y_j which have maximum correlation with each other. T. R. Knapp notes "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." The method was first introduced by Harold Hotelling in 1936.

Given two column vectors ${\displaystyle X=(x_{1},\dots ,x_{n})'}$ and ${\displaystyle Y=(y_{1},\dots ,y_{m})'}$ of random variables with second moments, one may define the cross-covariance ${\displaystyle \Sigma _{XY}=\operatorname {cov} (X,Y)}$ to be the ${\displaystyle n\times m}$ matrix whose ${\displaystyle (i,j)}$ entry is the covariance ${\displaystyle \operatorname {cov} (x_{i},y_{j})}$. 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).

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