CSTP

In the CSP one assumes that the multiplicity of data have a common structure along one dimension of the input data matrices. For example, in electroencephalography (EEG) a data matrix $X$, which is comprised of $n$ variables corresponding to the spatial locations for the electrodes on the scalp and $t$ temporal samples, this is the spatial dimension. The assumption holds because the same brain source engenders a fixed spatial pattern on the scalp, whereas, in general, the temporal pattern is arbitrary.

The common spatio-temporal pattern (CSTP) extends the CSP to situations when the multiplicity of data have a common structure along both dimensions. In EEG, for example, this is the case of event-related potentials (ERPs). The assumption holds because, again, the same brain source engenders a fixed spatial pattern on the scalp and furthermore ERPs have a quasi-fixed temporal pattern. As the CSP, the CSTP corresponds to the situation $m=1$ (one dataset) and $k=2$ (two observation).

Given a set of $k$ data matrices $\{X_1 \ldots X_k\}$ of dimension $n⋅t$, with mean $\bar{X}=\frac{1}{k}\sum_{i=1}^kX_i$, the goal of the CSTP is to find two matrices $B_{(1)}$ and $B_{(2)}$ verifying

$\left \{ \begin{array}{rl}B_{(1)}^TC_{(2)}B_{(1)}=I\\B_{(2)}^TC_{(1)}B_{(2)}=I\\B_{(1)}^T\bar{X}B_{(2)}=Λ \end{array} \right.$, $\hspace{1cm}$ [cstp.1],

where

$\left \{ \begin{array}{rl}C_{(1)}=\sum_{i=1}^k\frac{1}{t}(X_i^TX_i)\\C_{(2)}=\sum_{i=1}^k\frac{1}{n}(X_iX_i^T) \end{array} \right.$, $\hspace{1cm}$ [cstp.2]

are the mean covariance matrices along the first and second dimension of the $X_i$ matrices and $Λ$ is a diagonal matrix.

In words, the CSTP maximizes the ratio of the variance of the transformed $\bar{X}$ over the transformed mean covariance matrices $C_{(1)}$ and $C_{(2)}$. The CSTP can threfore be used to enhance the signal-to-noise ratio of data matrices mean estimation. For doing so, we retain the filters $\widetilde{B}_{(1)}=[b_{(1)1} \ldots b_{(1)p}]$ and $\widetilde{B}_{(2)}=[b_{(2)1} \ldots b_{(2)p}]$ holding the first $p$ vectors of $B_{(1)}$ and $B_{(2)}$ corresponding to the highest values of the variance ratio $Λ$.

For the CSTP we define the total variance ratio as

$λ_{TOT}=\sum_{i=1}^nλ_i$,

where the $λ_i$ are the diagonal elements of $Λ$ [cstp.1] and we define the explained variance for dimension $p$ such as

$σ_p=\frac{\sum_{i=1}^pλ_i}{λ_{TOT}}$. $\hspace{1cm}$ [cstp.3]

The .arev field of the CSTP filter is defined as the vector of accumulated variance ratios

$[σ_1≤\ldots≤σ_n]$, $\hspace{1cm}$ [cstp.4]

where $σ_j$ is defined in [cstp.3].

For setting the subspace dimension $p$ manually, set the eVar optional keyword argument of the CSTP 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

The CSTP solutions $B_{(1)}$ and $B_{(2)}$ can be found by a two-step procedure (Congedo et al., 2016)🎓:

1. get two whitening matrices $\hspace{0.1cm}W_{(1)}\hspace{0.1cm}$ and $\hspace{0.1cm}W_{(2)}\hspace{0.1cm}$ such that $\left \{ \begin{array}{rl}W_{(1)}^TC_{(1)}W_{(1)}=I\\W_{(2)}^TC_{(2)}W_{(2)}=I \end{array} \right.$
2. do $\hspace{0.1cm}\textrm{SVD}(W_{(2)}^T\bar{X}W_{(1)})=UΛV^{T}$

The solutions are $\hspace{0.1cm}B_{(1)}=W_{(2)}U\hspace{0.1cm}$ and $\hspace{0.1cm}B_{(2)}=W_{(1)}V$.

Constructors

Two constructors are available (see here below). The constructed LinearFilter object holding the CSTP will have fields:

.F[1]: matrix $\widetilde{B}_{(1)}=[b_{(1)1} \ldots b_{(1)p}]$. This is the whole matrix $B_{(1)}$ if $p=n$.

.F[2]: matrix $\widetilde{B}_{(2)}=[b_{(2)1} \ldots b_{(2)p}]$. This is the whole matrix $B_{(2)}$ if $p=n$

.iF[1]: the left-inverse of .F[1]

.iF[2]: the left-inverse of .F[2]

.D: the leading $p⋅p$ block of $Λ$ in [cstp.1].

.eVar: the explained variance for the chosen value of $p$, given by the $p^{th}$ value of [cstp.4].

.ev: the vector diag(Λ) holding all $n$ diagonal elements of matrix $Λ$ in [cstp.1].

.arev: the accumulated regularized eigenvalues, defined in [cstp.4].