Parameter estimation of multiple input‐output time series models: Application to rainfall‐runoff processes

In time series modeling of hydrologic systems the model structure either is determined a priori from physical considerations or is identified statistically. This paper sets forth a maximum likelihood procedure for estimating the parameters of a class of statistical models (linear time‐invariant state‐space) once a suitable member of the class has been identified. Using the innovation form of the state‐space model, the parameters of the transition, input weighting, gain and output, or measurement matrices are estimated as well as the innovation covariance matrix. Procedures for estimating process and measurement covariances in the state‐space model, and the parameters of the equivalent multivariate autoregressive moving average with exogenous inputs (ARMAX) model are also developed. A convergent and asymptotically efficient on‐line method of estimation is derived from the off‐line algorithm. Four examples are presented: daily rainfall‐runoff forecasting, four‐site monthly streamflow, seasonal model, and river flow input‐output model with a tributary.