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Canonical Correlation Analysis

Canonical Correlation Analysis(CCA) is a statistical analysis technique to identify correlations between two sets of variables. Given two vector variables $X$ and $Y$, it finds two projections, one for each, to transform them to a common space with maximum correlations.

The package defines a CCA type to represent a CCA model, and provides a set of methods to access the properties.

Let M be an instance of CCA, dx be the dimension of X, dy the dimension of Y, and p the output dimension (i.e the dimension of the common space).

StatsAPI.fit — Method
fit(CCA, X, Y; ...)

Perform CCA over the data given in matrices X and Y. Each column of X and Y is an observation.

X and Y should have the same number of columns (denoted by n below).

This method returns an instance of CCA.

Keyword arguments:

  • method: The choice of methods:
    • :cov: based on covariance matrices
    • :svd: based on SVD of the input data (default)
  • outdim: The output dimension, i.e dimension of the common space (default: min(dx, dy, n))
  • mean: The mean vector, which can be either of:
    • 0: the input data has already been centralized
    • nothing: this function will compute the mean (default)
    • a pre-computed mean vector

Notes: This function calls ccacov or ccasvd internally, depending on the choice of method.

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Base.size — Method
size(M:CCA)

Return a tuple with the dimension of X, Y, and the output dimension.

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Statistics.mean — Method
mean(M::CCA, c::Symbol)

Get the mean vector for the component c of the model M. The component parameter can be :x or :y.

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MultivariateStats.projection — Method
projection(M::CCA, c::Symbol)

Get the projection matrix for the component c of the model M. The component parameter can be :x or :y.

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Statistics.cor — Method
cor(M::CCA)

The correlations of the projected components (a vector of length p).

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StatsAPI.predict — Method
predict(M::CCA, Z::AbstractVecOrMat{<:Real}, c::Symbol)

Given a CCA model, one can transform observations into both spaces into a common space, as

\[\mathbf{z}_x = \mathbf{P}_x^T (\mathbf{x} - \boldsymbol{\mu}_x) \\ \mathbf{z}_y = \mathbf{P}_y^T (\mathbf{y} - \boldsymbol{\mu}_y)\]

Here, $\mathbf{P}_x$ and $\mathbf{P}_y$ are projection matrices for $X$ and $Y$; $\boldsymbol{\mu}_x$ and $\boldsymbol{\mu}_y$ are mean vectors.

Parameter Z can be either a vector of length dx, dy, or a matrix where each column is an observation. The component parameter c can be :x or :y.

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Auxiliary functions:

MultivariateStats.ccacov — Function
ccacov(Cxx, Cyy, Cxy, xmean, ymean, p)

Compute CCA based on analysis of the given covariance matrices, using generalized eigenvalue decomposition, and return CCA model.

Parameters:

  • Cxx: The covariance matrix of X.
  • Cyy: The covariance matrix of Y.
  • Cxy: The covariance matrix between X and Y.
  • xmean: The mean vector of the original samples of X, which can be

a vector of length dx, or an empty vector indicating a zero mean.

  • ymean: The mean vector of the original samples of Y, which can be

a vector of length dy, or an empty vector indicating a zero mean.

  • p: The output dimension, i.e the dimension of the common space.
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MultivariateStats.ccasvd — Function
ccasvd(Zx, Zy, xmean, ymean, p)

Compute CCA based on singular value decomposition of centralized sample matrices Zx and Zy, and return CCA model[1].

Parameters:

  • Zx: The centralized sample matrix for X.
  • Zy: The centralized sample matrix for Y.
  • xmean: The mean vector of the original samples of X, which can be

a vector of length dx, or an empty vector indicating a zero mean.

  • ymean: The mean vector of the original samples of Y, which can be

a vector of length dy, or an empty vector indicating a zero mean.

  • p: The output dimension, i.e the dimension of the common space.
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References

  • 1David Weenink, Canonical Correlation Analysis, Institute of Phonetic Sciences, Univ. of Amsterdam, Proceedings 25, 81-99, 2003.