Modeling and Stochastic Learning for Forecasting in High Dimensions

The estimation of baseline electricity consumptions for energy efficiency and load management measures is an essential issue. When implementing real-time energy management platforms for Automatic Monitoring and Targeting (AMT) of energy consumption, baselines shall be calculated previously and must be adaptive to sudden changes. Short Term Load Forecasting (STLF) techniques can be a solution to determine a pertinent frame of reference. In this study, two different forecasting methods are implemented and assessed: a first method based on load curve clustering and a second one based on signal decomposition using Principal Component Analysis (PCA) and Multiple Linear Regression (MLR). Both methods were applied to three different sets of data corresponding to three different industrial sites from different sectors across France. For the evaluation of the methods, a specific criterion adapted to the context of energy management is proposed. The obtained results are satisfying for both of the proposed approaches but the clustering based method shows a better performance. Perspectives for exploring different forecasting methods for these applications are considered for future works, as well as their application to different load curves from diverse industrial sectors and equipments.

We present a compact review of methods for constructing tests and confidence intervals in high-dimensional models. Links to theory, finite sample performance results and software allows to obtain a “quick” but sufficiently deep overview for applying the procedures.

In this paper, we discuss the problem of short-term electricity load forecasting by regarding electricity load on each day as a curve. The dependence between successive daily loads and other relevant factors such as temperature, is modelled via curve linear regression where both the response and the regressor are functional (curves). The key ingredient of the proposed method is the dimension reduction based on the singular value decomposition in a Hilbert space, which reduces the curve linear regression problem to several ordinary (i.e. scalar) linear regression problems. This method has previously been adopted in the hybrid approach proposed by Cho et al. (J Am Stat Assoc 108:7–21, 2013) for the same purpose, where the curve linear regression modelling was applied to the data after the trend and the seasonality were removed by a generalised additive model fitted at the weekly level. We show that classifying the successive daily loads prior to curve linear regression removes the necessity of such a two-stage approach as well as resulting in reducing the forecasting error by a great margin. The proposed methodology is illustrated using the electricity load dataset collected between 2007 and mid-2012, on which it is compared to the hybrid approach and other available competitors. Finally, various ways for improving the forecasting performance of the curve linear regression technique are discussed.

A focused information criterion is developed to estimate undirected graphical models where for each node in the graph a generalized linear model is put forward conditioned upon the other nodes in the graph. The proposed method selects a graph with a small estimated mean squared error for a user-specified focus, which is a function of the parameters in the generalized linear models, by selecting an appropriate model at each node. For situations where the number of nodes is large in comparison with the number of cases, the procedure performs penalized estimation with quadratic approximations to several popular penalties. To show the procedure’s applicability and usefulness we have applied it to two datasets involving voting behavior of U.S. senators and to a clinical dataset on psychopathology.

Electricity Transmission System Operators (TSO) are responsible for operating, maintaining and developing the high and extra high voltage network. They guarantee the reliability and proper operation of the power network. Anticipating electricity demand helps to guarantee the balance between generation and consumption at all times, and directly influences the reliability of the power system. In this paper, we focus on predicting short term electricity consumption in France. Several competitors such as iterative bias reduction, functional nonparametric model or non-linear additive autoregressive approach are compared to the actual SARIMA method. Our results show that iterative bias reduction approach outperforms all competitors both on Mean Absolute Percentage Error and on the percentage of forecast errors higher than 2,000 MW.

Short-term electricity forecasting has been studied for years at EDF and different forecasting models were developed from various fields of statistics or machine learning (functional data analysis, time series, non-parametric regression, boosting, bagging). We are interested in the forecasting of France’s daily electricity load consumption based on these different approaches. We investigate in this empirical study how to use them to improve prediction accuracy. First, we show how combining members of the original set of forecasts can lead to a significant improvement. Second, we explore how to build various and heterogeneous forecasts from these models and analyze how we can aggregate them to get even better predictions.

In this chapter we first review recent developments in the use of copulas for studying dependence structures between variables. We discuss and illustrate the concepts of unconditional and conditional copulas and association measures, in a bivariate setting. Statistical inference for conditional and unconditional copulas is discussed, in various modeling settings. Modeling the dynamics in a dependence structure between time series is of particular interest. For this we present a semiparametric approach using local polynomial approximation for the dynamic time parameter function. Throughout the chapter we provide some illustrative examples. The use of the proposed dynamical modeling approach is demonstrated in the analysis and forecast of wind speed data.

Demand response levers, as tariff incentive or direct load control on residential electrical appliances, are potential solutions to efficiently manage peak consumption and aid in grid security. The major objective is to estimate the consumption that would have been used in the absence of demand reduction: the baseline. This is an important issue to enhance demand response for electricity markets and to allow the grid operators to efficiently manage the grid. For these reasons, baseline estimation methods have to satisfy the following operational objectives: highly accurate, computationally efficient, cost-effective and flexible to the demand response customer turnover. In general, methods using available data from the control group give the best results, but current control group methods do not satisfy the aforementioned operational objectives. Having a real control group is highly costly because it requires to meter thousands customers who will not be used in the demand response offer. So there is a need to find new methods to select a control group. The advancement of smart meters can now provide a wealth of data to construct this group. This paper proposes the use of individual smart meter loads to select a control group. The method satisfies the aforementioned operational objectives since the selected control group is adaptable in operations even to demand response customers changes. The methodology developed is based on a selection algorithm and constraint regression approach. These new methods have been successfully tested in an online environment.

In this paper we provide a prediction method, the prediction box, based on a sparse learning process elaborated on very high dimensional information, which will be able to include new – potentially high dimensional – influential variables and adapt to different contexts of prediction. We elaborate and test this method in the setting of predicting the national French intra day load curve, over a period of time of 7 years on a large data basis including daily French electrical consumptions as well as many meteorological inputs, calendar statements and functional dictionaries. The prediction box incorporates a huge contextual information coming from the past, organizes it in a manageable way through the construction of a smart encyclopedia of scenarios, provides experts elaborating strategies of prediction by comparing the day at hand to referring scenarios extracted from the encyclopedia, and then harmonizes the different experts. More precisely, the prediction box is built using successive learning procedures: elaboration of a data base of historical scenarios organized on a high dimensional and functional learning of the intra day load curves, construction of expert forecasters using a retrieval information task among the scenarios, final aggregation of the experts. The results on the national French intra day load curves strongly show the benefits of using a sparse functional model to forecast the electricity consumption. They also appear to meet quite well with the business knowledge of consumption forecasters and even shed new lights on the domain.

Time series that arise on a graph or network arises in many scientific fields. In this paper we discuss a method for modelling and prediction of such time series with potentially complex characteristics. The method is based on the lifting scheme first proposed by Sweldens, a multiscale transformation suitable for irregular data with desirable properties. By repeated application of this algorithm we can transform the original network time series data into a simpler, lower dimensional time series object which is easier to forecast. The technique is illustrated with a data set arising from an energy time series application.

The emergence of Smart Grids is posing a wide range of challenges for electric utility companies and network operators: Integration of non-dispatchable power from renewable energy sources (e.g., photovoltaics, hydro and wind), fundamental changes in the way energy is consumed (e.g., due to dynamic pricing, demand response and novel electric appliances), and more active operations of the networks to increase efficiency and reliability. A key in managing these challenges is the ability to forecast network loads at low levels of locality, e.g., counties, cities, or neighbourhoods. Accurate load forecasts improve the efficiency of supply as they help utilities to reduce operating reserves, act more efficiently in the electricity markets, and provide more effective demand-response measures. In order to prepare for the Smart Grid era, there is a need for a scalable simulation environment which allows utilities to develop and validate their forecasting methodology under various what-if-scenarios. This paper presents a massive-scale simulation platform which emulates electrical load in an entire electrical network, from Smart Meters at individual households, over low- to medium-voltage network assets, up to the national level. The platform supports the simulation of changes in the customer portfolio and the consumers’ behavior, installment of new distributed generation capacity at any network level, and dynamic reconfigurations of the network. The paper explains the underlying statistical modeling approach based on Generalized Additive Models, outlines the system architecture, and presents a number of realistic use cases that were generated using this platform.

We develop further the spot volatility estimator introduced in Hoffmann et al. (Ann Inst H Poincaré (B) Probab Stat 48(4):1186–1216, 2012) from a practical point of view and make it applicable to the analysis of high-frequency financial data. In a first part, we adjust the estimator substantially in order to achieve good finite sample performance and to overcome difficulties arising from violations of the additive microstructure noise model (e.g. jumps, rounding errors). These modifications are justified by simulations. The second part is devoted to investigate the behavior of volatility in response to macroeconomic events. We give evidence that the spot volatility of Euro-BUND futures is considerably higher during press conferences of the European Central Bank. As an outlook, we present an estimator for the spot covolatility of two different prices.

We address the problem of forecasting a time series meeting the Causal Bernoulli Shift model, using a parametric set of predictors. The aggregation technique provides a predictor with well established and quite satisfying theoretical properties expressed by an oracle inequality for the prediction risk. The numerical computation of the aggregated predictor usually relies on a Markov chain Monte Carlo method whose convergence should be evaluated. In particular, it is crucial to bound the number of simulations needed to achieve a numerical precision of the same order as the prediction risk. In this direction we present a fairly general result which can be seen as an oracle inequality including the numerical cost of the predictor computation. The numerical cost appears by letting the oracle inequality depend on the number of simulations required in the Monte Carlo approximation. Some numerical experiments are then carried out to support our findings.

In hierarchical time series (HTS) forecasting, the hierarchical relation between multiple time series is exploited to make better forecasts. This hierarchical relation implies one or more aggregate consistency constraints that the series are known to satisfy. Many existing approaches, like for example bottom-up or top-down forecasting, therefore attempt to achieve this goal in a way that guarantees that the forecasts will also be aggregate consistent. We propose to split the problem of HTS into two independent steps: first one comes up with the best possible forecasts for the time series without worrying about aggregate consistency; and then a reconciliation procedure is used to make the forecasts aggregate consistent. We introduce a Game-Theoretically OPtimal (GTOP) reconciliation method, which is guaranteed to only improve any given set of forecasts. This opens up new possibilities for constructing the forecasts. For example, it is not necessary to assume that bottom-level forecasts are unbiased, and aggregate forecasts may be constructed by regressing both on bottom-level forecasts and on other covariates that may only be available at the aggregate level. We illustrate the benefits of our approach both on simulated data and on real electricity consumption data.

The BAGIDIS (semi-) distance of Timmermans and von Sachs (BAGIDIS: statistically investigating curves with sharp local patterns using a new functional measure of dissimilarity. Under revision. http://​www.​uclouvain.​be/​en-369695.​html. ISBA Discussion Paper 2013-31, Université catholique de Louvain, 2013) is the central building block of a nonparametric method for comparing curves with sharp local features, with the subsequent goal of classification or prediction. This semi-distance is data-driven and highly adaptive to the curves being studied. Its main originality is its ability to consider simultaneously horizontal and vertical variations of patterns. As such it can handle curves with sharp patterns which are possibly not well-aligned from one curve to another. The distance is based on the signature of the curves in the domain of a generalised wavelet basis, the Unbalanced Haar basis. In this note we give insights on the problem of stability of our proposed algorithm, in the presence of observational noise. For this we use theoretical investigations from Timmermans, Delsol and von Sachs (J Multivar Anal 115:421–444, 2013) on properties of the fractal topology behind our distance-based method. Our results are general enough to be applicable to any method using a distance which relies on a fractal topology.