Independent Component Analysis Method via Maximum Negentropy Optimization

Independent Component Analysis (ICA) is a classical blind source separation technique whose core objective is to recover mutually independent source signals from mixed observations. The negentropy-maximization-based ICA approach achieves this by maximizing the non-Gaussianity of signals, enabling perfect separation of all independent components under noise-free conditions.

Core Concept and Implementation Steps: Preprocessing Phase: First, observed data undergoes centering and whitening to eliminate second-order statistical dependencies between signals, establishing a foundation for subsequent orthogonalization. Whitened data features normalized variance across dimensions with zero correlation between components.

Negentropy Computation: Utilizes higher-order statistics (such as kurtosis) or approximate entropy as non-Gaussianity measures. Negentropy, defined as the entropy difference between a Gaussian random variable and the current signal, reaches maximum when non-Gaussian characteristics are enhanced. In MATLAB implementation, this typically involves calculating kurtosis using 'kurtosis()' function or developing custom entropy approximation algorithms.

Orthogonalization Constraint: During each iterative update of the separation matrix, symmetric orthogonalization (e.g., Gram-Schmidt method) or projection techniques maintain orthogonality of separation vectors. This prevents different components from converging to identical extremum points, enabling simultaneous extraction of all components. Code implementation often employs 'orth()' function or Gram-Schmidt orthogonalization routines.

Optimization Algorithm: Commonly uses fixed-point iteration (FastICA) or natural gradient methods. The algorithm iteratively adjusts separation vector directions until negentropy reaches extremum values. Orthogonalization steps ensure vectors maintain orthogonality after each iteration, ultimately yielding a complete separation matrix. FastICA implementation typically involves while-loop iterations with convergence checks using tolerance thresholds.

Technical Advantages: Precisely recovers source signals under noiseless assumptions Orthogonalization enhances algorithm stability and convergence speed No prerequisite knowledge of source signal distribution required (only non-Gaussianity assumption)

This method holds significant application value in EEG signal processing, financial time series analysis, and related fields. Its MATLAB implementation crucially involves whitening matrix computation using 'zscore' and 'cov' functions, nonlinear function selection (tanh, pow3), and iteration termination condition settings via maximum iteration counts or convergence thresholds.