Causal regression for online estimation of highly nonlinear parametrically varying models

This paper presents an online estimation method for highly nonlinear systems. This method fits the data to a nonlinear in the parameters regression model, where each parameter is an unknown function of known variables that we call causal regressors. We define this model as an Autoregressive with eXogenous inputs and Causal-regressors-Dependent Parameters (ARX-CDP) model. We are interested on modeling problems where the causal regressors can vary rapidly, as well as the parameters, to the point that the latter cannot be estimated by standard methods. Since the parameters causally depend on the causal regressor but do not depend directly on time, we propose to model each parameter evolution as a random walk process in its causal regressors domain, rather than in the time variable domain, as usual. Therefore, the parameter prediction depends on some past estimates at instants when the causal regressors are similar to the present one. The absolute difference between the current and past causal regressors is used to measure this similarity. We call this prediction method as causal prediction, which, together with a standard recursive least square algorithm, used to update the model, makes up the online causal estimator. Finally, a finite impulse response and a system with multi-dependent parameters are modeled using this proposal. We show experimentally that the causal regression method has a similar performance with other offline methods from literature, with the important advantage of operating in online mode.