A two-dimensional Poisson equation formulation of non-parametric statistical non-linear modeling

The present paper deals with a Poisson equation arising in statistical modeling of semi-deterministic non-linear systems with two independent (input) variables and one dependent (output) variable. Statistical modeling is formulated in terms of a differential equation that relates the second-order joint probability density functions of the model's input/output random variables with the sought nonlinear model transference. The discussed modeling procedure makes no prior assumptions on the functional structure of the model, except for monotonicity and continuity with respect to both input variables. In particular, the method is non-parametric. Results of numerical tests are presented and discussed in order to get an insight into the behavior of the devised statistical modeling procedure. The results of numerical tests confirm that the proposed statistical modeling approach is able to cope with both synthetic and real-world data sets and, in particular, with underlying systems and data that exhibit strong hidden nuisance variables and measurement disturbances.