gMCA

Generalized Maximum Covariance Analysis (gMCA) is a multiple approximate joint diagonalization prodedure generalizing the maximum covariance analysis (MCA) to the situation $m>2$ (number of datasets), as for MCA with $k=1$ (one observation).

Let ${X_1,...,X_m}$ be a set of $m$ data matrices of dimension $n⋅t$, where $n$ is the number of variables and $t$ the number of samples, both common to all datasets. From these data matrices let us estimate

$C_{ij}=\frac{1}{t}X_iX_j^H$, for all $i,j∈[1...m]$, $\hspace{1cm}$ [gmca.1]

i.e., all covariance ($i=j$) and cross-covariance ($i≠j$) matrices.

The gMCA seeks $m$ matrices $F_1,...,F_m$ diagonalizing as much as possible all products

$F_i^H C_{ij} F_j$, for all $i≠j∈[1...m]$. $\hspace{1cm}$ [gmca.2]

If the MCA ($m=2$) diagonalizes the cross-covariance, this generalized model ($m>2$) diagonalizes all cross-covariance matrices.

alternative model for gMCA

The gMCA constructors also allow to seeks $m$ matrices $F_1,...,F_m$ diagonalizing as much as possible all products

$F_i^H C_{ij} F_j$, for all $i,j∈[1...m]$. $\hspace{1cm}$ [gmca.3]

As compared to model [gmca.2], this model diagonalizes the covariance matrices in addition to the cross-covariance matrices.

permutation for gMCA

As usual, the approximate diagonalizers $F_1,...,F_m$ are arbitrary up to a scale and permutation. In gMCA scaling is fixed by appropriate constraints. For the remaining sign and permutation ambiguities, Diagonalizations.jl attempts to solve them by finding signed permutation matrices for $F_1,...,F_m$ so as to make all diagonal elements of [gmca.2] or [gmca.3] positive and sorted in descending order.

Let

$λ=[λ_1...λ_n]$$\hspace{1cm}$ [gmca.4]

be the diagonal elements of

$\frac{1}{m^2-m}\sum_{i≠j=1}^m(F_i^H C_{ij} F_j)$$\hspace{1cm}$ [gmca.5]

and $σ_{TOT}=\sum_{i=1}^nλ_i$ be the total covariance.

We denote $\widetilde{F}_i=[f_{i1} \ldots f_{ip}]$ the matrix holding the first $p<n$ column vectors of $F_i$, for $i∈[1...m]$, where $p$ is the subspace dimension. The explained variance is given by

$σ_p=\frac{\sum_{i=1}^pλ_i}{σ_{TOT}}$, $\hspace{1cm}$ [gmca.6]

and the accumulated regularized eigenvalues (arev) by

$σ_j=\sum_{i=1}^j{σ_i}$, for $j=[1 \ldots n]$, $\hspace{1cm}$ [gmca.7]

where $σ_i$ is given by Eq. [gmca.6].

For setting the subspace dimension $p$ manually, set the eVar optional keyword argument of the gMCA constructors either to an integer or to a real number, this latter establishing $p$ in conjunction with argument eVarMeth using the arev vector (see subspace dimension). By default, eVar is set to 0.999.

Solution

There is no closed-form solution to the AJD problem in general. See Algorithms.

Note that the solution of the MCA are orthogonal matrices. In order to mimic this in gMCA use OJoB. Using NoJoB will constraint the solution only in the general linear group.

Constructors

One constructor is available (see here below). The constructed LinearFilter object holding the gMCA will have fields:

.F: vector of matrices $\widetilde{F}_1,...,\widetilde{F}_m$ with columns holding the first $p$ eigenvectors in $F_1,...,F_m$, or just $F_1,...,F_m$ if $p=n$

.iF: the vector of the left-inverses of the matrices in .F

.D: the leading $p⋅p$ block of $Λ$, i.e., the elements [gmca.4] associated to the matrices in .F in diagonal form.

.eVar: the explained variance [gmca.6] for the chosen value of $p$.

.ev: the vector $λ$ [gmca.4].

.arev: the accumulated regularized eigenvalues, defined by Eq. [gmca.7].