Online Time Series Prediction by ARX(∞) Approximation Algorithm

Abstract

Time series prediction problems have a wide range of real-world applications and lots of study have been devoted on them. Most of the previous results need to determine the order for the model according to some information criterion such as Bayesian information criterion (BIC), Akaike information criterion (AIC) and so on. However, for online data stream scenario, keeping the order of the model fixed may not exploit the historical data adequately. Thus it is reasonable to consider a model with adaptive order. In this paper, we study the ARX (∞) approximation algorithm for ARX (autoregressive with exogenous inputs) model and allow the order of ARX model to increase with total data volume. Under the assumption that the noise terms are martingale difference sequence, we analyze the one-step prediction error and show that the growth rate of regret loss can be bounded by $O(\lceil\log T\rceil^{4})$. In addition, we demonstrate the performance of our approximation algorithm by comparing with ARX (p).