Simultaneous predictive bands for functional time series using minimum entropy sets

Functional Time Series are sequences of dependent random elements taking values on some functional space. Most of the research on this domain is focused on producing a predictor able to forecast the value of the next function having observed a part of the sequence. For this, the Autoregresive Hilbertian process is a suitable framework. We address here the problem of constructing simultaneous predictive confidence bands for a stationary functional time series. The method is based on an entropy measure for stochastic processes, in particular functional time series. To construct predictive bands we use a functional bootstrap procedure that allow us to estimate the prediction law through the use of pseudo-predictions. Each pseudo-realisation is then projected into a space of finite dimension, associated to a functional basis. We use Reproducing Kernel Hilbert Spaces (RKHS) to represent the functions, considering then the basis associated to the reproducing kernel. Using a simple decision rule, we classify the points on the projected space among those belonging to the minimum entropy set and those that do not. We push back the minimum entropy set to the functional space and construct a band using the regularity property of the RKHS. The proposed methodology is illustrated through artificial and real-world data sets.