Online Learning for Time Series Prediction of AR Model with Missing Data

Abstract

Recently online learning algorithm is applied to time series prediction with missing data without the strict assumption on the noise terms. The existing algorithm only uses the observed data to predict time series, which does not impute the missing data and costs most time to count the number of common missing observation in any two sub-sequences of time series with constant length. In this paper, we consider online learning algorithm for time series prediction with missing data, which use the estimated values to impute the missing data. We firstly propose an online algorithm for the standard autoregressive (AR) model with missing data. In addition, the length of coefficients vector is added to utilize more previous time series data under an improper setting. On this basis, two sparse online learning algorithms are proposed, one is for $$ \alpha $$-exp-concave loss functions and the other is for general loss functions. Our theorem guarantees the algorithms to approach the performance of the best AR coefficients vector in hindsight. Furthermore, we conduct a set of experiments on real data to show that our algorithm can achieve much the same error compared to the state-of-the-art algorithm and need less time.