Dynamic Boltzmann machines (DyBMs) are recently developed generative models of a time series. They are designed to learn a time series by efficient online learning algorithms, whilst taking long-term dependencies into account with help of eligibility traces, recursively updatable memory units storing descriptive statistics of all the past data. The current DyBMs assume a finite-dimensional time series and cannot be applied to a functional time series, in which the dimension goes to infinity (e.g., spatiotemporal data on a continuous space). In this paper, we present a functional dynamic Boltzmann machine (F-DyBM) as a generative model of a functional time series. A technical challenge is to devise an online learning algorithm with which F-DyBM, consisting of functions and integrals, can learn a functional time series using only finite observations of it. We rise to the above challenge by combining a kernel-based function approximation method along with a statistical interpolation method and finally derive closed-form update rules. We design numerical experiments to empirically confirm the effectiveness of our solutions. The experimental results demonstrate consistent error reductions as compared to baseline methods, from which we conclude the effectiveness of F-DyBM for functional time series prediction.
A Plug-in Bandwidth Selection Procedure for Long-Run Covariance Estimation with Stationary Functional Time Series
