Improving the prediction and parsimony of ARX models using multiscale estimation

Multiscale wavelet-based data representation has been shown to be a powerful data analysis tool in various applications. In this paper, the advantages of multiscale representation are utilized to improve the prediction accuracy and parsimony of the auto-regressive with exogenous variable (ARX) model by developing a multiscale ARX (MSARX) modeling algorithm. The idea is to decompose the input-output data at multiple scales, construct an ARX model at each scale using the scaled signal approximations of the data, and then using cross validation, select among all MSARX models the one which best predicts the process response. The MSARX algorithm is shown to improve the parsimony of the estimated models, as ARX models with a fewer number of coefficients are needed at coarser scales. This advantage is attributed to the down-sampling used in multiscale representation. Another important advantage of the MSARX algorithm is that it inherently accounts for the presence of measurement noise through the application of low-pass filters in the multiscale decomposition of the data, which in turn improves the model robustness to measurement noise and enhances its prediction. These prediction and parsimony advantages of MSARX modeling are demonstrated through a simulated second order example, in which the MSARX algorithm outperformed the time-domain one at different noise contents, and the relative improvement of MSARX increased at higher levels of noise.