Forecasting with temporal hierarchies Athanasopoulos, George; Hyndman, Rob J.; Kourentzes, Nikolaos ...
European journal of operational research,
10/2017, Volume:
262, Issue:
1
Journal Article
Peer reviewed
Open access
•Temporal hierarchies can be used for any time series to improve forecasting.•The proposed methodology is independent of forecasting models.•It results in temporally reconciled, accurate and robust ...forecasts.•The implied combination mitigates modelling uncertainty.•We demonstrate empirically that it offers superior accuracy and decision making.
This paper introduces the concept of Temporal Hierarchies for time series forecasting. A temporal hierarchy can be constructed for any time series by means of non-overlapping temporal aggregation. Predictions constructed at all aggregation levels are combined with the proposed framework to result in temporally reconciled, accurate and robust forecasts. The implied combination mitigates modelling uncertainty, while the reconciled nature of the forecasts results in a unified prediction that supports aligned decisions at different planning horizons: from short-term operational up to long-term strategic planning. The proposed methodology is independent of forecasting models. It can embed high level managerial forecasts that incorporate complex and unstructured information with lower level statistical forecasts. Our results show that forecasting with temporal hierarchies increases accuracy over conventional forecasting, particularly under increased modelling uncertainty. We discuss organisational implications of the temporally reconciled forecasts using a case study of Accident & Emergency departments.
Forecasting competitions are now so widespread that it is often forgotten how controversial they were when first held, and how influential they have been over the years. I briefly review the history ...of forecasting competitions, and discuss what we have learned about their design and implementation, and what they can tell us about forecasting. I also provide a few suggestions for potential future competitions, and for research about forecasting based on competitions.
Global methods that fit a single forecasting method to all time series in a set have recently shown surprising accuracy, even when forecasting large groups of heterogeneous time series. We provide ...the following contributions that help understand the potential and applicability of global methods and how they relate to traditional local methods that fit a separate forecasting method to each series: •Global and local methods can produce the same forecasts without any assumptions about similarity of the series in the set.•The complexity of local methods grows with the size of the set while it remains constant for global methods. This result supports the recent evidence and provides principles for the design of new algorithms.•In an extensive empirical study, we show that purposely naïve algorithms derived from these principles show outstanding accuracy. In particular, global linear models provide competitive accuracy with far fewer parameters than the simplest of local methods.
•Bootstrapping with aggregation (bagging) performs well for time series forecasting.•Bagging handles three sources of uncertainty: data, model and parameter uncertainty.•We decompose the performance ...benefits of bagging for each source of uncertainty.•Tackling model uncertainty brings the most benefits, even outperforming bagging.•We propose a strategy for weighted combinations of the suitable models.
In a recent study, Bergmeir, Hyndman and Benítez (2016) successfully employed a bootstrap aggregation (bagging) technique for improving the performance of exponential smoothing. Each series is Box-Cox transformed, and decomposed by Seasonal and Trend decomposition using Loess (STL); then bootstrapping is applied on the remainder series before the trend and seasonality are added back, and the transformation reversed to create bootstrapped versions of the series. Subsequently, they apply automatic exponential smoothing on the original series and the bootstrapped versions of the series, with the final forecast being the equal-weight combination across all forecasts. In this study we attempt to address the question: why does bagging for time series forecasting work? We assume three sources of uncertainty (model uncertainty, data uncertainty, and parameter uncertainty) and we separately explore the benefits of bagging for time series forecasting for each one of them. Our analysis considers 4004 time series (from the M- and M3-competitions) and two families of models. The results show that the benefits of bagging predominantly originate from the model uncertainty: the fact that different models might be selected as optimal for the bootstrapped series. As such, a suitable weighted combination of the most suitable models should be preferred to selecting a single model.
Large collections of time series often have aggregation constraints due to product or geographical groupings. The forecasts for the most disaggregated series are usually required to add-up exactly to ...the forecasts of the aggregated series, a constraint we refer to as "coherence." Forecast reconciliation is the process of adjusting forecasts to make them coherent.
The reconciliation algorithm proposed by Hyndman et al. (
2011
) is based on a generalized least squares estimator that requires an estimate of the covariance matrix of the coherency errors (i.e., the errors that arise due to incoherence). We show that this matrix is impossible to estimate in practice due to identifiability conditions.
We propose a new forecast reconciliation approach that incorporates the information from a full covariance matrix of forecast errors in obtaining a set of coherent forecasts. Our approach minimizes the mean squared error of the coherent forecasts across the entire collection of time series under the assumption of unbiasedness. The minimization problem has a closed-form solution. We make this solution scalable by providing a computationally efficient representation.
We evaluate the performance of the proposed method compared to alternative methods using a series of simulation designs which take into account various features of the collected time series. This is followed by an empirical application using Australian domestic tourism data. The results indicate that the proposed method works well with artificial and real data. Supplementary materials for this article are available online.
One of the most widely used standard procedures for model evaluation in classification and regression is K-fold cross-validation (CV). However, when it comes to time series forecasting, because of ...the inherent serial correlation and potential non-stationarity of the data, its application is not straightforward and often replaced by practitioners in favour of an out-of-sample (OOS) evaluation. It is shown that for purely autoregressive models, the use of standard K-fold CV is possible provided the models considered have uncorrelated errors. Such a setup occurs, for example, when the models nest a more appropriate model. This is very common when Machine Learning methods are used for prediction, and where CV can control for overfitting the data. Theoretical insights supporting these arguments are presented, along with a simulation study and a real-world example. It is shown empirically that K-fold CV performs favourably compared to both OOS evaluation and other time-series-specific techniques such as non-dependent cross-validation.
Short-term load forecasting is an essential instrument in power system planning, operation, and control. Many operating decisions are based on load forecasts, such as dispatch scheduling of ...generating capacity, reliability analysis, and maintenance planning for the generators. Overestimation of electricity demand will cause a conservative operation, which leads to the start-up of too many units or excessive energy purchase, thereby supplying an unnecessary level of reserve. On the other hand, underestimation may result in a risky operation, with insufficient preparation of spinning reserve, causing the system to operate in a vulnerable region to the disturbance. In this paper, semi-parametric additive models are proposed to estimate the relationships between demand and the driver variables. Specifically, the inputs for these models are calendar variables, lagged actual demand observations, and historical and forecast temperature traces for one or more sites in the target power system. In addition to point forecasts, prediction intervals are also estimated using a modified bootstrap method suitable for the complex seasonality seen in electricity demand data. The proposed methodology has been used to forecast the half-hourly electricity demand for up to seven days ahead for power systems in the Australian National Electricity Market. The performance of the methodology is validated via out-of-sample experiments with real data from the power system, as well as through on-site implementation by the system operator.
The energy industry has been going through a significant modernization process over the last decade. Its infrastructure is being upgraded rapidly. The supply, demand and prices are becoming more ...volatile and less predictable than ever before. Even its business model is being challenged fundamentally. In this competitive and dynamic environment, many decision-making processes rely on probabilistic forecasts to quantify the uncertain future. Although most of the papers in the energy forecasting literature focus on point or single-valued forecasts, the research interest in probabilistic energy forecasting research has taken off rapidly in recent years. In this paper, we summarize the recent research progress on probabilistic energy forecasting. A major portion of the paper is devoted to introducing the Global Energy Forecasting Competition 2014 (GEFCom2014), a probabilistic energy forecasting competition with four tracks on load, price, wind and solar forecasting, which attracted 581 participants from 61 countries. We conclude the paper with 12 predictions for the next decade of energy forecasting.
Exponential smoothing is one of the most popular forecasting methods. We present a technique for the bootstrap aggregation (bagging) of exponential smoothing methods, which results in significant ...improvements in the forecasts. The bagging uses a Box–Cox transformation followed by an STL decomposition to separate the time series into the trend, seasonal part, and remainder. The remainder is then bootstrapped using a moving block bootstrap, and a new series is assembled using this bootstrapped remainder. An ensemble of exponential smoothing models is then estimated on the bootstrapped series, and the resulting point forecasts are combined. We evaluate this new method on the M3 data set, and show that it outperforms the original exponential smoothing models consistently. On the monthly data, we achieve better results than any of the original M3 participants.
The explosion of time series data in recent years has brought a flourish of new time series analysis methods, for forecasting, clustering, classification and other tasks. The evaluation of these new ...methods requires either collecting or simulating a diverse set of time series benchmarking data to enable reliable comparisons against alternative approaches. We propose GeneRAting TIme Series with diverse and controllable characteristics, named GRATIS, with the use of mixture autoregressive (MAR) models. We simulate sets of time series using MAR models and investigate the diversity and coverage of the generated time series in a time series feature space. By tuning the parameters of the MAR models, GRATIS is also able to efficiently generate new time series with controllable features. In general, as a costless surrogate to the traditional data collection approach, GRATIS can be used as an evaluation tool for tasks such as time series forecasting and classification. We illustrate the usefulness of our time series generation process through a time series forecasting application.
PDF
