System identification: Signal causality and feedback determination
The problem of determining a bivariate innovations model of two linear stationary stochastic processes in the context of their causal relationships is considered. It is approached in two ways which lead to dual identification algorithms. One method is the direct approach which directly uses the observation data, while the other one is the indirect approach which uses the whitened versions of the observation data instead. Crosscorrelation estimators are employed to extract the causal information in the joint system. As a result of applying the two approaches, closed-form formulae are derived that clearly indicate the dependence of the coupling parameters in the bivariate innovations model on the crosscorrelations of its member signals or their univariate innovations. These formulae assume the general feedback structure and includes those special structures in which only one causal relation exists or no causal relation exists in the joint system. Simulation examples are presented to illustrate the effectiveness of the two methods. Monte Carlo experiments are also conducted to evaluate and compare the statistical properties of the estimates of the coupling parameters of several simulated bivariate systems as the two dual algorithms are applied to them.