Periodic Regression in the Principal Component Space for Multivariate, Multi‐Horizon, Probabilistic Forecasting

This article develops a novel computationally efficient methodology for joint probabilistic forecasting of multiple, numerical sequences. This approach to forecasting is relevant for many important applications such as transportation planning, electrical power grid operation, weather forecasting, etc. Three important characteristics of the proposed methodology are (a) extracting features (principal components) of fixed‐length output sequences from the training data, (b) multivariate forecasting in the principal component space, and c) accounting for the periodicity in the original space, and hence the principal component space, when choosing the forecasting model form. The structure of output sequences of a fixed length is exploited to devise a procedure for building the quantities of interest (QoI) matrix using the training data. The most informative features of this QoI matrix are extracted using principal component analysis (PCA). To account for the periodicity in the original time series as well as their principal component (PCs), separate probabilistic regression models are trained to simultaneously predict all the PCs for each time step of a periodic cycle (termed as the “periodic trick”). This modeling approach is illustrated by generating forecasts for the regional wind generation, load demand, and solar generation time series of the French electricity grid (RTE) and for the ambient temperature in two cities. It is shown, using various deterministic and probabilistic model validation metrics, that the proposed approach performs better than sequence‐to‐sequence (machine learning‐based) forecasting methods. A computationally efficient and accurate forecast can thus be obtained by exploiting the problem‐specific structure (fixed‐length input and output), leveraging feature extraction techniques, and employing a meaningful treatment of the periodicity in the data.