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.

翻译：函数型时间序列是由取值于某函数空间上的相依随机元组成的序列。该领域的大部分研究聚焦于构建预测器，以便在观测到部分序列后预测下一个函数值。为此，自回归希尔伯特过程是一种合适的框架。本文针对平稳函数型时间序列的同时预测置信带构建问题展开研究。该方法基于随机过程（尤其是函数型时间序列）的熵度量。为构建预测带，我们采用函数型自助法，通过伪预测值估计预测分布律。每个伪实现随后被投影到与函数基相关联的有限维空间中。我们利用再生核希尔伯特空间表示函数，并考虑与再生核相关联的基函数。通过简单决策规则，将投影空间中的点划分为属于最小熵集和不属于最小熵集的类别。我们将最小熵集推回至函数空间，并利用RKHS的正则性构建预测带。通过人工数据集和真实数据集验证了所提方法的有效性。