Stochastic analysis of neural network modeling and identification of nonlinear memoryless MIMO systems

Neural network (NN) approaches have been widely applied for modeling and identification of nonlinear multiple-input multiple-output (MIMO) systems. This paper proposes a stochastic analysis of a class of these NN algorithms. The class of MIMO systems considered in this paper is composed of a set of single-input nonlinearities followed by a linear combiner. The NN model consists of a set of single-input memoryless NN blocks followed by a linear combiner. A gradient descent algorithm is used for the learning process. Here we give analytical expressions for the mean squared error (MSE), explore the stationary points of the algorithm, evaluate the misadjustment error due to weight fluctuations, and derive recursions for the mean weight transient behavior during the learning process. The paper shows that in the case of independent inputs, the adaptive linear combiner identifies the linear combining matrix of the MIMO system (to within a scaling diagonal matrix) and that each NN block identifies the corresponding unknown nonlinearity to within a scale factor. The paper also investigates the particular case of linear identification of the nonlinear MIMO system. It is shown in this case that, for independent inputs, the adaptive linear combiner identifies a scaled version of the unknown linear combining matrix. The paper is supported with computer simulations which confirm the theoretical results.