Large-scale streaming data are common in modern machine learning applications and have led to the development of online learning algorithms. Many fields, such as supply chain management, weather and meteorology, energy markets, and finance, have pivoted towards using probabilistic forecasts, which yields the need not only for accurate learning of the expected value but also for learning the conditional heteroskedasticity. Against this backdrop, we present a methodology for online estimation of regularized linear distributional models for conditional heteroskedasticity. The proposed algorithm is based on a combination of recent developments for the online estimation of LASSO models and the well-known GAMLSS framework. We provide a case study on day-ahead electricity price forecasting, in which we show the competitive performance of the adaptive estimation combined with strongly reduced computational effort. Our algorithms are implemented in a computationally efficient Python package.

Accurate mid-term (weeks to one year) hourly electricity load forecasts are essential for strategic decision-making in power plant operation, ensuring supply security and grid stability, and energy trading. While numerous models effectively predict short-term (hours to a few days) hourly load, mid-term forecasting solutions remain scarce. In mid-term load forecasting, besides daily, weekly, and annual seasonal and autoregressive effects, capturing weather and holiday effects, as well as socio-economic non-stationarities in the data, poses significant modeling challenges. To address these challenges, we propose a novel forecasting method using Generalized Additive Models (GAMs) built from interpretable P-splines and enhanced with autoregressive post-processing. This model uses smoothed temperatures, Error-Trend-Seasonal (ETS) modeled non-stationary states, a nuanced representation of holiday effects with weekday variations, and seasonal information as input. The proposed model is evaluated on load data from 24 European countries. This analysis demonstrates that the model not only has significantly enhanced forecasting accuracy compared to state-of-the-art methods but also offers valuable insights into the influence of individual components on predicted load, given its full interpretability. Achieving performance akin to day-ahead TSO forecasts in fast computation times of a few seconds for several years of hourly data underscores the model's potential for practical application in the power system industry.

This paper presents a new method for combining (or aggregating or ensembling) multivariate probabilistic forecasts, considering dependencies between quantiles and marginals through a smoothing procedure that allows for online learning. We discuss two smoothing methods: dimensionality reduction using Basis matrices and penalized smoothing. The new online learning algorithm generalizes the standard CRPS learning framework into multivariate dimensions. It is based on Bernstein Online Aggregation (BOA) and yields optimal asymptotic learning properties. The procedure uses horizontal aggregation, i.e., aggregation across quantiles. We provide an in-depth discussion on possible extensions of the algorithm and several nested cases related to the existing literature on online forecast combination. We apply the proposed methodology to forecasting day-ahead electricity prices, which are 24-dimensional distributional forecasts. The proposed method yields significant improvements over uniform combination in terms of continuous ranked probability score (CRPS). We discuss the temporal evolution of the weights and hyperparameters and present the results of reduced versions of the preferred model. A fast C++ implementation of the proposed algorithm will be made available in connection with this paper as an open-source R-Package on CRAN.

Aggregated curves are common structures in economics and finance, and the most prominent examples are supply and demand curves. In this study, we exploit the fact that all aggregated curves have an intrinsic hierarchical structure, and thus hierarchical reconciliation methods can be used to improve the forecast accuracy. We provide an in-depth theory on how aggregated curves can be constructed or deconstructed, and conclude that these methods are equivalent under weak assumptions. We consider multiple reconciliation methods for aggregated curves, including previously established bottom-up, top-down, and linear optimal reconciliation approaches. We also present a new benchmark reconciliation method called 'aggregated-down' with similar complexity to bottom-up and top-down approaches, but it tends to provide better accuracy in this setup. We conducted an empirical forecasting study on the German day-ahead power auction market by predicting the demand and supply curves, where their equilibrium determines the electricity price for the next day. Our results demonstrate that hierarchical reconciliation methods can be used to improve the forecasting accuracy of aggregated curves.

We study the prices of European Emission Allowances (EUA), whereby we analyze their uncertainty and dependencies on related energy prices (natural gas, coal, and oil). We propose a probabilistic multivariate conditional time series model with a VECM-Copula-GARCH structure which exploits key characteristics of the data. Data are normalized with respect to inflation and carbon emissions to allow for proper cross-series evaluation. The forecasting performance is evaluated in an extensive rolling-window forecasting study, covering eight years out-of-sample. We discuss our findings for both levels- and log-transformed data, focusing on time-varying correlations, and in view of the Russian invasion of Ukraine.

We present a novel approach to probabilistic electricity price forecasting which utilizes distributional neural networks. The model structure is based on a deep neural network that contains a so-called probability layer. The network's output is a parametric distribution with 2 (normal) or 4 (Johnson's SU) parameters. In a forecasting study involving day-ahead electricity prices in the German market, our approach significantly outperforms state-of-the-art benchmarks, including LASSO-estimated regressions and deep neural networks combined with Quantile Regression Averaging. The obtained results not only emphasize the importance of higher moments when modeling volatile electricity prices, but also -- given that probabilistic forecasting is the essence of risk management -- provide important implications for managing portfolios in the power sector.

During the last years, European intraday power markets have gained importance for balancing forecast errors due to the rising volumes of intermittent renewable generation. However, compared to day-ahead markets, the drivers for the intraday price process are still sparsely researched. In this paper, we propose a modelling strategy for the location, shape and scale parameters of the return distribution in intraday markets, based on fundamental variables. We consider wind and solar forecasts and their intraday updates, outages, price information and a novel measure for the shape of the merit-order, derived from spot auction curves as explanatory variables. We validate our modelling by simulating price paths and compare the probabilistic forecasting performance of our model to benchmark models in a forecasting study for the German market. The approach yields significant improvements in the forecasting performance, especially in the tails of the distribution. At the same time, we are able to derive the contribution of the driving variables. We find that, apart from the first lag of the price changes, none of our fundamental variables have explanatory power for the expected value of the intraday returns. This implies weak-form market efficiency as renewable forecast changes and outage information seems to be priced in by the market. We find that the volatility is driven by the merit-order regime, the time to delivery and the closure of cross-border order books. The tail of the distribution is mainly influenced by past price differences and trading activity. Our approach is directly transferable to other continuous intraday markets in Europe.

This paper covers predicting high-resolution electricity peak demand features given lower-resolution data. This is a relevant setup as it answers whether limited higher-resolution monitoring helps to estimate future high-resolution peak loads when the high-resolution data is no longer available. That question is particularly interesting for network operators considering replacing high-resolution monitoring predictive models due to economic considerations. We propose models to predict half-hourly minima and maxima of high-resolution (every minute) electricity load data while model inputs are of a lower resolution (30 minutes). We combine predictions of generalized additive models (GAM) and deep artificial neural networks (DNN), which are popular in load forecasting. We extensively analyze the prediction models, including the input parameters' importance, focusing on load, weather, and seasonal effects. The proposed method won a data competition organized by Western Power Distribution, a British distribution network operator. In addition, we provide a rigorous evaluation study that goes beyond the competition frame to analyze the models' robustness. The results show that the proposed methods are superior to the competition benchmark concerning the out-of-sample root mean squared error (RMSE). This holds regarding the competition month and the supplementary evaluation study, which covers an additional eleven months. Overall, our proposed model combination reduces the out-of-sample RMSE by 57.4\% compared to the benchmark.

Forecasting has always been at the forefront of decision making and planning. The uncertainty that surrounds the future is both exciting and challenging, with individuals and organisations seeking to minimise risks and maximise utilities. The large number of forecasting applications calls for a diverse set of forecasting methods to tackle real-life challenges. This article provides a non-systematic review of the theory and the practice of forecasting. We provide an overview of a wide range of theoretical, state-of-the-art models, methods, principles, and approaches to prepare, produce, organise, and evaluate forecasts. We then demonstrate how such theoretical concepts are applied in a variety of real-life contexts. We do not claim that this review is an exhaustive list of methods and applications. However, we wish that our encyclopedic presentation will offer a point of reference for the rich work that has been undertaken over the last decades, with some key insights for the future of forecasting theory and practice. Given its encyclopedic nature, the intended mode of reading is non-linear. We offer cross-references to allow the readers to navigate through the various topics. We complement the theoretical concepts and applications covered by large lists of free or open-source software implementations and publicly-available databases.

The M5 competition uncertainty track aims for probabilistic forecasting of sales of thousands of Walmart retail goods. We show that the M5 competition data faces strong overdispersion and sporadic demand, especially zero demand. We discuss resulting modeling issues concerning adequate probabilistic forecasting of such count data processes. Unfortunately, the majority of popular prediction methods used in the M5 competition (e.g. lightgbm and xgboost GBMs) fails to address the data characteristics due to the considered objective functions. The distributional forecasting provides a suitable modeling approach for to the overcome those problems. The GAMLSS framework allows flexible probabilistic forecasting using low dimensional distributions. We illustrate, how the GAMLSS approach can be applied for the M5 competition data by modeling the location and scale parameter of various distributions, e.g. the negative binomial distribution. Finally, we discuss software packages for distributional modeling and their drawback, like the R package gamlss with its package extensions, and (deep) distributional forecasting libraries such as TensorFlow Probability.