Dimensionality Reduction

The variance (power) of the associated time courses is given by the diagonal matrix $A_P^T A_P = D^T U^T U D = D^2$ which are the squares of the singular values. It is then possible to threshold the power so that only components that represent significant amounts of signal (assumedly not noise) are retained.

Keeping the first $M$ components (or sources) leaves a matrix $A_1$ of $M$ time courses and $S_1$ of the corresponding $M$ spatial maps. That is

$$A_P = \left[ \, A_1 \; : \; A_{reject} \, \right] \qquad \mathrm{and} \qquad S_P^T = \left[ \, S_1^T \; : \; S_{reject}^T \, \right].$$

The reconstructed data, which has an appropriately reduced number of sources, is

$$Y_1 = A_1 S_1.$$ (3)

Equivalently, the dimension of the data can now be effectively reduced by pre-multiplying by the pseudo-inverse of $A_1$. That is

$$Y_R = A_1^{-} Y_1 = S_1$$ (4)

where $A_1^{-} = (A_1^T A_1)^{-1} A_1^T$. This data matrix, $Y_R$, is now $M \times N$ (it has fewer time points).

Note that in each case the spatial maps are still orthogonal: that is, $S_1 S_1^T = I$.