Dependent component analysis

Dependent component analysis (DCA) is a blind signal separation (BSS) method and a extension of Independent component analysis (ICA). ICA is the separating of mixed signals to individual signals without knowing anything about source signals. DCA is used to separating of mixed signals to individual sets of signals that are dependent on signals within their own set without knowing anything about the original signals. DCA can be ICA if all sets of signals only contain a single signal within their own set.^[1]

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.

${\displaystyle x(t)=A*s(t)}$

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.^[1]

${\displaystyle y(t)=B*x(t)}$

Methods

Sub-Band Decomposition ICA (SDICA) is based on that wideband source signals are dependent but there are other subbands that are independent. It uses adaptive filter by choosing subbands using minimum of mutual information (MI) to separate mixed signals. After finding subband signals, you will use ICA to reconstruct based off subband signals. reconstruct using ICA. Below is formula to find MI based on entropy were H is entropy.^[2]

${\displaystyle {\widehat {I_{H}}}(y)=\sum _{n=1}^{N}{\widehat {H}}(y_{n})-{\widehat {H}}(y)}$

${\displaystyle {\widehat {H}}(y_{n})=-{\frac {1}{T}}\sum _{t=1}^{T}log{\widehat {P}}_{yn}(y_{n}(t))}$

${\displaystyle {\widehat {H}}(y)=-{\frac {1}{T}}\sum _{t=1}^{T}log{\widehat {P}}_{y}(y_{n}(t))}$

References

1. Rui Li, Hongwei Li, and Fasong Wang. “Dependent Component Analysis: Concepts and Main Algorithms” http://www.jcomputers.us/vol5/jcp0504-13.pdf
2. Ivica Kopriva and Damir Sersic "Robust Blind Separation of Statistically Dependent Sources using Dual Tree Wavelets"https://pdfs.semanticscholar.org/5ffe/a962dc8b612a637a608cb77de8a4b1025c44.pdf