Canonical Correlation Analysis

Canonical Correlation Analysis Model

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.

size(M:CCA)

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

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.

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.

cor(M::CCA)

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

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.

WilksLambdaTest(cca; n=-1, k=1)

Use Wilks Lambda to test the dimension of a CCA. The null hypothesis of the test is that canonical correlations k, k+1, ... are zero. If the CCA was fit with a covariance matrix then the sample size n must be provided.

PillaiTraceTest(cca; n=-1, k=1)

Use Pillai's trace to test the dimension of a CCA. The null hypothesis of the test is that canonical correlations k, k+1, ... are zero. If the CCA was fit with a covariance matrix then the sample size n must be provided.

LawleyHotellingTest(cca; n=-1, k=1)

Use the Lawley Hotelling statistics to test the dimension of a CCA. The null hypothesis of the test is that canonical correlations k, k+1, ... are zero. If the CCA was fit with a covariance matrix then the sample size n must be provided.

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.

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.