ARXARMAX

ARXARMAX is a hybrid framework in time-series modeling and system identification that combines features of autoregressive with exogenous input (ARX) models and autoregressive moving average with exogenous input (ARMAX) models. It is designed to describe linear dynamical systems driven by external inputs while accounting for colored disturbances in the noise process. The model includes an autoregressive part for the output, a term for past exogenous inputs, and a noise term that may follow its own autoregressive or moving-average structure. By allowing both an MA component in the noise and a structured input response, ARXARMAX can capture a wider range of dynamics than ARX or ARMAX alone, and can collapse to either family when certain parameters are constrained to zero. The standard formulation expresses y_t as a linear combination of past outputs y_{t-1}, y_{t-2}, ..., past inputs u_{t-1}, u_{t-2}, ..., plus an error term e_t, with e_t following an ARMA process driven by white noise w_t. Estimation typically uses prediction-error or least-squares methods, sometimes with regularization or Bayesian techniques to manage model order selection and identifiability. Practical use requires careful handling of identifiability, multicollinearity, and data length. ARXARMAX is applied in process control, econometrics, environmental and engineering modeling, and any domain requiring interpretable input-output dynamics under correlated disturbances. See also ARX model, ARMAX model, system identification.