Modelling and forecasting based on recursive incomplete pseudoinverse matrices

Time series modelling has a wide spectrum of applications in several fields including engineering and finance. Most traditional modelling techniques rely on assumptions related to the input data and manual pre-processing, based on user observations, rendering them unsuitable for analysing time-series with varying characteristics automatically, while more general modelling techniques usually require increased computational work for application and tuning. Recently, a general modelling framework based on a recursive Schur complement technique, that utilizes an adaptively determined set of basis functions, has been proposed. Herewith, a novel modified approach based on recursive incomplete pseudoinverse matrices in conjunction with preconditioned iterative methods for large datasets, is proposed. This sparse approach greatly reduces storage requirements and the recursive nature of the procedure avoids recomputation of the preconditioner after addition of a new basis function. Moreover, update of the coefficients for a different window of data and predefined basis functions can be performed utilizing the incomplete pseudoinverse matrix as preconditioner. The case of sinusoidal basis functions is presented along with a novel adaptive frequency estimation technique. The stability of the resulting model is discussed with respect to the choice of basis functions. The case of basis derived from machine learning techniques is also discussed. Numerical results are given depicting the applicability, generality and effectiveness of the proposed technique. Comparative results with other methods show forecasting RMSE improvement between 7% to 80%, for the majority of the chosen time series.