Probabilistic forecasting of functional time seriesapplication to scenario-generation of residual demand curves in electricity markets

Abstract

Due to recent technological advances in science and computing, the collection and analysis of high-dimensional data has become a relevant topic in the field of applied statistics. Many of these observations arise from continuous processes observed over time, which can be naturally expressed as functions. Thus, the name functional data is commonly given to such observations. Statistical methodologies that deal with the analysis of functional data belong to the Functional Data Analysis (FDA) framework, a field of research devoted to the exploration and modelization of these data. If these functional data are collected sequentially over time, it is natural to expect a time dependence between functional observations. The term Functional Time Series (FTS) is often used to refer to these processes. These series of functional observations appear frequently in numerous fields, such as medical research, finance and ecology. Nevertheless, this thesis is concerned with the applications in electricity markets that involve FTS. Particularly, the bidding information of market auctions can be used to define Residual Demand Curves (RDC), functions that model the market-clearing price as a function of the energy that agents are willing to trade. Having accurate estimations of the hourly series of RDCs is of utmost importance for market agents, as it would provide them with an accurate description of the bidding strategy of its competitors. RDCs often exhibit complex dynamics, as the bidding behaviour of the market agents is often influenced by external factors and often exhibits a strong seasonal dependence. The SARMAHX model is a generalization of the scalar ARMAX model to the functional data framework, which has proven to be an adequate model to obtain accurate short-term forecasts of FTS that appear in electricity markets. The work described in this thesis is aimed at developing a novel functional probabilistic model that is able to generate coherent scenarios of functional time series. Probabilistic forecasts of the series are obtained by quantifying the uncertainty associated with the forecasts of a fitted SARMAHX model. Hence, a forecasting methodology for functional time series has been developed, which involves the identification and diagnosis of the SARMAHX model fitted to the series. Once deterministic forecasts of the series have been obtained, the conditional distribution of the residuals is characterized by the proposed probabilistic model, which then is used to generate future scenarios of the curves. The proposed probabilistic forecasting methodology has been successfully applied to the short-term estimation of the series of hourly Residual Demand Curves in different electricity markets, outperforming other reference models commonly found in the functional data literature.