سری زمانی تابعی مقید: کاربردهایی برای بازار گاز ایتالیایی   Constrained functional time series: Applications to the Italian gas market

1. Introduction Energy markets in general, and natural gas markets in particular, are emerging fields that pose a great variety of forecasting problems, including load forecasting (Hong, 2014), price forecasting (Weron, 2014), daily price curve profile forecasting (Chen & Li, 2015), consumption forecasting (Brabec, Konár, Pelikán, & Malı, 2008), and many others. Motivated by price prediction in the Italian natural gas balancing market, this paper proposes a model for forecasting the day-to-day evolution of supply and demand curves. The proposed model is innovative from both the methodological and applied perspectives. The supply and demand curves model is indeed a wellknown microeconomic model of price determination, buits application is typically descriptive and static rather than strategic and predictive, which clearly does not help gas traders with either the forecasting of future prices or decision making and bidding. At the same time, while the usual forecasting methods, such as classical time series analysis, produce useful predictions of scalar quantities of interest (e.g., prices), they do not provide the insights into the market that are given by the supply and demand model. Furthermore, in markets with a moderate number of traders, the effect of a single offer or demand cannot be incorporated directly into either the inferential procedure or what-if simulations. For all of these reasons, the prediction of the entire supply and demand curves, and hence of their intersection, can be of strong interest. We deal with this problem using a functional data analysis (FDA) approach. FDA is an extremely useful set of tools for dealing with data that can be modeled as functions, such as our demand and supply curves; for a quick introduction, refer to Ferraty and Vieu (2006), Ramsay and Silverman (2002, 2005), or Sørensen, Goldsmith, and Sangalli (2013). However, our approach differs from the most common FDA framework in two ways. First, we focus on functions that are constrained (i.e., monotonic and with an equality constraint on one edge of the domain and an inequality constraint on the other edge), and second, we embed such constraints for curves that are temporally dependent. The statistical literature has focused separately on (a) the problem of obtaining a constrained estimation of the underlying function given some point-wise evaluations of it, and (b) the problem of modeling functional data with temporal dependence (i.e., functional time series). To the best of our knowledge, the present work is the first to tackle the temporal dependence jointly with constraints pertaining to monotonicity, boundedness, and values of the function at the boundary of the domain. We refer to this joint framework henceforth as constrained functional time series.