Whitening

Whitening (also named Sphering) is a standardized version of PCA. Possibly it is the most used diagonalization procedure among all. Like PCA, it corresponds to the situation $m=1$ (one dataset) and $k=1$ (one observation).

Let $X$ be a $n⋅t$ data matrix, where $n$ is the number of variables and $t$ the number of samples and let $C$ be its $n⋅n$ covariance matrix. Being $C$ a positive semi-definite matrix, its eigenvector matrix $U$ diagonalizes $C$ by rotation, as

$U^{H}CU=Λ$. $\hspace{1cm}$ [whitening.1]

The eigenvalues in the diagonal matrix $Λ$ are all non-negative. They are all real and positive if $C$ is positive definite, which is assumed in the remaining of this exposition. The linear transformation $Λ^{-1/2}U^{H}X$ yields uncorrelated data with unit variance at all $n$ components, that is,

$\frac{1}{T}Λ^{-1/2}U^{H}XX^{H}UΛ^{-1/2}=I$. $\hspace{1cm}$ [whitening.2]

Whitened data remains whitened after whatever further rotation. That is, for any orthogonal matrix $V$, it holds

$V^H\big(\frac{1}{T}Λ^{-1/2}U^{H}XX^{H}UΛ^{-1/2}\big)V=I$. $\hspace{1cm}$ [whitening.3]

Hence there exist an infinite number of possible whitening matrices with general form $V^HΛ^{-1/2}U^{H}$. Because of this property whitening plays a fundamental role as a first step in many two-steps diagonalization procedures (e.g., for the CSP, CSTP and CCA). Particularly important among the infinite family of whitening matrices is the only symmetric one (or Hermitian, if complex), which is the inverse of the principal square root of $C$ and is found repeatedly in computations on the manifold of positive definite matrices (see for example intro to Riemannian geometry).

For setting the subspace dimension $p$ manually, set the eVar optional keyword argument of the Whitening constructors either to an integer or to a real number, this latter establishing $p$ in conjunction with argument eVarMeth using the arev vector, which is defined as for the PCA filter, see Eq. [pca.6] therein and subspace dimension for details. By default, eVar is set to 0.999.

Solution

As for PCA, the solution is given by the eigenvalue-eigenvector decoposition of $C$

$\textrm{EVD}(C)=UΛU^{H}$.

Constructors

Three constructors are available (see here below), which use exactly the same syntax as for PCA. The constructed LinearFilter object holding the Whitening will have fields:

.F: matrix $\widetilde{U}\widetilde{Λ}^{-1/2}$ with scaled orthonormal columns, where $\widetilde{Λ}$ is the leading $p⋅p$ block of $Λ$ and $\widetilde{U}=[u_1 \ldots u_p]$ holds the first $p$ eigenvectors in $U$. If $p=n$, .F is just $UΛ^{-1/2}$.

.iF: $\widetilde{Λ}^{1/2}\widetilde{U}^H$, the left-inverse of .F.

.D: $\widetilde{Λ}$, i.e., the eigenvalues associated to .F in diagonal form.

.eVar: the explained variance for the chosen value of $p$. This is the same as for the PCA, see Eq. [pca.4] therein.

.ev: the vector diag(Λ) holding all $n$ eigenvalues.

.arev: the accumulated regularized eigenvalues. This is the same as for the PCA, see Eq. [pca.6] therein.