Dependent component analysis

Dependent component analysis (DCA) is a blind signal separation (BSS) method. It is used to unmix signals to individual sets of signals that are dependent on signals within their own set. Independent component analysis (ICA) is similar but is unmixing of signals to individual signals. DCA can be ICA if all sets of signals only contain a single signal within their own set.

Mathematical Representation

For simplicity, we will assume that all individual sets of signals are the same size, k, and total N sets. Building off the basic equations of BSS as seen below instead of source signals being independent, we have independent sets of signals, s(t) = ({s₁(t),...,s_k(t)},...,{s_kN-k+1(t)...,s_kN(t)})^T, is mixed by coefficients, A=[a_ij]εR^mxkN, that produces set of mixed signals, x(t)=(x₁(t),...,x_m(t))^T. The signals can multidimension.

$x(t)=A*s(t)$

The following equation BSS separates the set of mixed signals, x(t) by finding and using coefficients, B=[B_ij]εR^kNxm, to separate and getting the set of approximation of the original signals, y(t)=({y₁(t),...,y_k(t)},...,{y_kN-k+1(t)...,y_kN(t)})^T.

$y(t)=B*x(t)$