Narx

NARX stands for nonlinear autoregressive with exogenous inputs. It is a class of discrete-time dynamic models used to describe systems in which the current output depends nonlinearly on past outputs and on past external inputs. NARX models are widely employed in time-series forecasting, system identification, and control applications because they can capture complex, nonlinear relationships influenced by external factors.

Mathematical form and structure: In its common form, the current output y(t) is modeled as a nonlinear function of a finite set of past outputs and past exogenous inputs: y(t) = f(y(t-1), y(t-2), ..., y(t-na), u(t-1), u(t-2), ..., u(t-nb)) + e(t), where na and nb are the chosen lag orders and e(t) represents noise. The model can be linear or nonlinear depending on the chosen function f. Exogenous inputs u(t) are external signals that affect the system.

Implementation: The function f can be approximated by various nonlinear models, including neural networks (commonly a NARX network with tapped-delay lines for y and u), polynomial models, radial basis functions, or other nonlinear regressors. In neural-network implementations, separate input delay lines for the target output and the exogenous inputs feed a nonlinear processor to predict the next value. Training can be conducted with backpropagation through time or with standard backpropagation using open-loop or teacher-forced strategies to mitigate error accumulation in feedback.

Applications and considerations: NARX models are used for system identification, forecasting, and control in engineering, economics, and signal processing. They offer flexibility to capture nonlinear dynamics and input effects but require careful selection of lag orders and model complexity to avoid overfitting and ensure robust generalization. NARX is related to NARMAX, which includes moving-average terms for noise modeling.