Coherent Forecasts Research Articles

Cross-Temporal Hierarchical Forecast Reconciliation of Natural Gas Demand

Local natural gas distribution companies (LDCs) require accurate demand forecasts across various time periods, geographic regions, and customer class hierarchies. Achieving coherent forecasts across these hierarchies is challenging but crucial for optimal decision making, resource allocation, and operational efficiency. This work introduces a method that structures the gas distribution system into cross-temporal hierarchies to produce accurate and coherent forecasts. We apply our method to a case study involving three operational regions, forecasting at different geographical levels and analyzing both hourly and daily frequencies. Trained on five years of data and tested on one year, our model achieves a 10% reduction in hourly mean absolute scaled error and a 3% reduction in daily mean absolute scaled error.

Bivariate first-order random coefficient integer-valued autoregressive processes based on modified negative binomial operator

In this paper, a new bivariate random coefficient integer-valued autoregressive process based on modified negative binomial operator with dependent innovations is proposed. Basic probabilistic and statistical properties of this model are derived. To estimate unknown parameters, Yule-Walker, conditional least squares and conditional maximum likelihood methods are considered and evaluated by Monte Carlo simulations. Asymptotic properties of the estimators are derived. Moreover, coherent forecasting and possible extension of the proposed model is provided. Finally, the proposed model is applied to the monthly crime datasets and compared with other models.

A systematic vector autoregressive framework for modeling and forecasting mortality

AbstractRecently, there is a new stream of mortality forecasting research using the vector autoregressive model with different sparse model specifications. They have been shown to be able to overcome some of the limitations of the more traditional factor models such as the Lee–Carter model. In this paper, we propose a more generalized systematic vector autoregressive framework for modeling and forecasting mortality. Under this framework, we progressively increase the sophistication of the diagonal parameters in the autoregressive matrix and formulate a range of model structures in a systematic fashion. They offer much flexibility for capturing the mortality patterns of different populations. The resulting models produce age coherent forecasts, and their parameters are reasonably interpretable for modelers, demographers, and industry practitioners. Using the mortality data of Australia, Japan, New Zealand, and Taiwan, we demonstrate that the proposed approach generates appropriate forecasts of mortality rates and life expectancies and produces very good performance in the fitting and out‐of‐sample analysis.

Forecast reconciliation: A review

Collections of time series formed via aggregation are prevalent in many fields. These are commonly referred to as hierarchical time series and may be constructed cross-sectionally across different variables, temporally by aggregating a single series at different frequencies, or even generalised beyond aggregation as time series that respect linear constraints. When forecasting such time series, a desirable condition is for forecasts to be coherent: to respect the constraints. The past decades have seen substantial growth in this field with the development of reconciliation methods that ensure coherent forecasts and improve forecast accuracy. This paper serves as a comprehensive review of forecast reconciliation and an entry point for researchers and practitioners dealing with hierarchical time series. The scope of the article includes perspectives on forecast reconciliation from machine learning, Bayesian statistics and probabilistic forecasting, as well as applications in economics, energy, tourism, retail demand and demography.

Point and probabilistic forecast reconciliation for general linearly constrained multiple time series

Forecast reconciliation is the post-forecasting process aimed to revise a set of incoherent base forecasts into coherent forecasts in line with given data structures. Most of the point and probabilistic regression-based forecast reconciliation results ground on the so called “structural representation” and on the related unconstrained generalized least squares reconciliation formula. However, the structural representation naturally applies to genuine hierarchical/grouped time series, where the top- and bottom-level variables are uniquely identified. When a general linearly constrained multiple time series is considered, the forecast reconciliation is naturally expressed according to a projection approach. While it is well known that the classic structural reconciliation formula is equivalent to its projection approach counterpart, so far it is not completely understood if and how a structural-like reconciliation formula may be derived for a general linearly constrained multiple time series. Such an expression would permit to extend reconciliation definitions, theorems and results in a straightforward manner. In this paper, we show that for general linearly constrained multiple time series it is possible to express the reconciliation formula according to a “structural-like” approach that keeps distinct free and constrained, instead of bottom and upper (aggregated), variables, establish the probabilistic forecast reconciliation framework, and apply these findings to obtain fully reconciled point and probabilistic forecasts for the aggregates of the Australian GDP from income and expenditure sides, and for the European Area GDP disaggregated by income, expenditure and output sides and by 19 countries.

Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues

Forecast reconciliation is a post-forecasting process that involves transforming a set of incoherent forecasts into coherent forecasts which satisfy a given set of linear constraints for a multivariate time series. In this paper, we extend the current state-of-the-art cross-sectional probabilistic forecast reconciliation approach to encompass a cross-temporal framework, where temporal constraints are also applied. Our proposed methodology employs both parametric Gaussian and non-parametric bootstrap approaches to draw samples from an incoherent cross-temporal distribution. To improve the estimation of the forecast error covariance matrix, we propose using multi-step residuals, especially in the time dimension where the usual one-step residuals fail. To address high-dimensionality issues, we present four alternatives for the covariance matrix, where we exploit the two-fold nature (cross-sectional and temporal) of the cross-temporal structure, and introduce the idea of overlapping residuals. We assess the effectiveness of the proposed cross-temporal reconciliation approaches through a simulation study that investigates their theoretical and empirical properties and two forecasting experiments, using the Australian GDP and the Australian Tourism Demand datasets. For both applications, the optimal cross-temporal reconciliation approaches significantly outperform the incoherent base forecasts in terms of the continuous ranked probability score and the energy score. Overall, the results highlight the potential of the proposed methods to improve the accuracy of probabilistic forecasts and to address the challenge of integrating disparate scenarios while coherently taking into account short-term operational, medium-term tactical, and long-term strategic planning.

Coherent Mortality Forecasting with a Model Averaging Approach: Evidence from Global Populations

Accurate forecasts and analyses of mortality rates are essential to many economic and finance practices, such as the designing of pension schemes. Recent studies have proposed advanced mortality models with age coherent forecasts, such that long-term predictions will not diverge infinitely among ages. Despite their effectiveness, individual models are inevitably misspecified in the empirical analysis, which reduces the reliableness. In this article, we propose a model averaging approach (MAA) that allows for age-specific weights and asymptotically achieves age coherence. Relevant technical details are also provided. The proposed method balances both the in-sample fitting and out-of-sample forecasting, with a uniquely designed smoothness penalty to resolve the potential overfitting and thus avoid abrupt changes in the long-term mortality forecasting. Using a large empirical dataset of 10 countries including 0 to 100 ages and spanning 1950 to 2018, the outstanding forecasting performance of MAA is presented. This robustly holds in various sensitivity analyses and is supported by simulation evidence. A case study is further conducted to discuss the improved forecasting efficiency of MAA, as well as its usefulness in economic and finance applications such as the annuity pricing. The proposed MAA approach is therefore a useful tool in forecasting mortality data for other practices.

Probabilistic Forecast Reconciliation under the Gaussian Framework

Forecast reconciliation of multivariate time series maps a set of incoherent forecasts into coherent forecasts to satisfy a given set of linear constraints. Available methods in the literature either follow a projection matrix-based approach or an empirical copula-based reordering approach to revise the incoherent future sample paths to obtain reconciled probabilistic forecasts. The projection matrices are estimated either by optimizing a scoring rule such as energy or variogram score or simply using a projection matrix derived for point forecast reconciliation. This article proves that (a) if the incoherent predictive distribution is jointly Gaussian, then MinT (minimum trace) minimizes the logarithmic scoring rule for the hierarchy; and (b) the logarithmic score of MinT for each marginal predictive density is smaller than that of OLS (ordinary least squares). We illustrate these theoretical results using a set of simulation studies and the Australian domestic tourism dataset. The estimation of MinT needs to estimate the covariance matrix of the base forecast errors. We have evaluated the performance using the sample covariance matrix and shrinkage estimator. It was observed that the theoretical properties noted above are greatly impacted by the covariance matrix used and highlighted the importance of estimating it reliably, especially with high dimensional data.

Comparison of multivariate post‐processing methods using global ECMWF ensemble forecasts

AbstractAn influential step in weather forecasting was the introduction of ensemble forecasts in operational use owing to their capability to account for the uncertainties in the future state of the atmosphere. However, ensemble weather forecasts are often underdispersive and might also contain bias, which calls for some form of post‐processing. A popular approach to calibration is the ensemble model output statistics approach resulting in a full predictive distribution for a given weather variable. However, this form of univariate post‐processing may ignore the prevailing spatial and/or temporal correlation structures among different dimensions. Since many applications call for spatially and/or temporally coherent forecasts, multivariate post‐processing aims to capture these possibly lost dependencies. We compare the forecast skill of different non‐parametric multivariate approaches to modeling temporal dependence of ensemble weather forecasts with different forecast horizons. The focus is on two‐step methods, where, after univariate post‐processing, the ensemble model output statistics predictive distributions corresponding to different forecast horizons are combined to a multivariate calibrated prediction using an empirical copula. Based on global ensemble predictions of temperature, wind speed, and precipitation accumulation of the European Centre for Medium‐Range Weather Forecasts from January 2002 to March 2014, we investigate the forecast skill of different versions of ensemble copula coupling (ECC) and Schaake shuffle. In general, compared with the raw and independently calibrated forecasts, multivariate post‐processing substantially improves the forecast skill. Although even the simplest ECC approach with low computational cost provides a powerful benchmark method, recently proposed advanced extensions of the ECC and the Schaake shuffle are found to not provide any significant improvements over their basic counterparts.

Spatio-temporal reconciliation of solar forecasts

In recent works by Yang et al. (2017a, 2017b), and Yang et al. (2019), geographical, temporal, and sequential deterministic reconciliation of hierarchical photovoltaic (PV) power generation have been considered for a simulated PV dataset in California. In the first two cases, the reconciliations were carried out in spatial and temporal domains separately. To further improve forecasting accuracy, in the third case these two reconciliation approaches were applied sequentially. During the replication of the forecasting experiment, some issues emerged about non-negativity and coherency (in space and/or in time) of the sequentially reconciled forecasts. Furthermore, while the accuracy improvement of the considered approaches over the benchmark persistence forecasts is clearly visible at any data granularity, we argue that an even better performance may be obtained by a thorough exploitation of spatio-temporal hierarchies. To this end, in this paper the spatio-temporal point forecast reconciliation approach is applied to generate non-negative, fully coherent (both in space and time) forecasts. New spatio-temporal reconciliation approaches are adopted, exploiting for the first time some relationships between two-step, iterative and simultaneous spatio-temporal reconciliation procedures. Non-negativity issues of the final reconciled forecasts are discussed and correctly dealt with in a simple and effective way. The spatio-temporal reconciliation procedures are applied to the base forecasts with forecast horizon of 1 day, of PV generated power at different time granularities (1 h to 1 day), of a geographical hierarchy consisting of 324 series along 3 levels. The normalized Root Mean Square Error (nRMSE) and the normalized Mean Bias Error are used to measure forecasting accuracy, and a statistical multiple comparison procedure is performed to rank the approaches. In addition to assuring full coherence and non-negativity of the reconciled forecasts, the results show that for the considered dataset, spatio-temporal forecast reconciliation significantly improves on the sequential procedures proposed by Yang et al. (2019), at any level of the spatial hierarchy and for any temporal granularity. For example, the forecasted hourly PV generated power by the new spatio-temporal forecast reconciliation approaches improve on the NWP 3TIER forecasts in a range from 4.7% to 18.4% in terms of nRMSE.

Optimal reconciliation with immutable forecasts

The practical importance of coherent forecasts in hierarchical forecasting has inspired many studies on forecast reconciliation. Under this approach, base forecasts are produced for every series in the hierarchy and are subsequently adjusted to be coherent in a second reconciliation step. Reconciliation methods have been shown to improve forecast accuracy but will generally adjust the base forecast of every series. However, in an operational context, it is sometimes necessary or beneficial to keep forecasts of some variables unchanged after forecast reconciliation. In this paper, we formulate a reconciliation methodology that keeps forecasts of a pre-specified subset of variables unchanged or “immutable”. In contrast to existing approaches, these immutable forecasts need not all come from the same level of a hierarchy, and our method can also be applied to grouped hierarchies. We prove that our approach preserves unbiasedness in base forecasts. Our method can also account for correlations between base forecasting errors and ensure the non-negativity of forecasts. We also perform empirical experiments, including an application to a large-scale online retailer’s sales, to assess our proposed methodology’s impacts.

Extensions of the Lee–Carter model to project the data‐driven rotation of age‐specific mortality decline and forecast coherent mortality rates

AbstractAccurate forecasts and analyses of mortality rates are essential to many operational practices, such as the designing of pension schemes. Recent studies have focused on the age coherent forecasts, such that long‐term predictions will not diverge infinitely among ages. Although intuitively and biologically reasonable, this age coherence property is lost when the seminal Lee–Carter (LC) model and almost all its existing extensions are employed. Those include the famous Li–Lee (LL) model, which is argued to produce coherent forecasts of mortality rates across populations, but not ages. In this paper, we progressively explore two effective frameworks with three extensions, by applying the LL specification to the single‐population model (LC‐S) and by introducing time‐dependent age patterns of mortality declines in the forecasting steps with fixed (LC‐VF) and variant (LC‐V) intercepts when modeling those patterns. In particular, the recommended LC‐V model is motivated from the intermediate LC‐S and LC‐VF and resolves all remaining issues of both models. Also, the final LC‐V model is easy to implement, incorporates the dynamic and rotating age‐specific patterns of mortality decline, provides age‐coherent forecasts of mortality rates in the long run, and is straightforwardly extensible to multi‐population cases. Using a large sample of 15 countries, we show that the LC‐V model and its multipopulation counterpart can consistently improve the forecasting accuracy of the competing LC, LL and other extensions in both single‐population and multipopulation scenarios. Long‐term analyses up to 2100 further demonstrate the existence of age coherence when forecasting with the proposed LC‐V model.

Hierarchical container throughput forecasting: The value of coherent forecasts in the management of ports operations

A port's overall container throughput (CT) data can be arranged into diverse hierarchical dimensions (cross-sectional or temporal). Wherein the forecasts are generated using the “horses-for-courses” approach to achieve maximum accuracy at the base level. Generally, these base forecasts do not add up across temporal or cross-sectional hierarchies and give rise to the issue of forecast “incoherence”. Accordingly, the present study extends the idea of utilising optimal forecast reconciliation techniques in CT forecasting. However, the frameworks of these techniques do not allow for their interaction across different dimensions of hierarchy present in CT data. Hence, the paper addresses this problem by proposing a cross-temporal forecast reconciliation framework.Further, monthly CT data from the Port of Los Angeles is used for empirical analysis. It is observed that the proposed framework, with exponential smoothing as the base forecasting method, can provide coherent forecasts at all levels and hierarchical dimensions while simultaneously improving the accuracy of the forecasts. Further, the study also devises a mechanism to evaluate the efficacy of the proposed framework in providing decision support for various port operations (the requirement for terminal ground slots and gantry cranes). The results indicate that improvements could be made in port operations decisions using the proposed framework.

Hierarchical mortality forecasting with EVT tails: An application to solvency capital requirement

In this paper, we propose a new framework to coherently produce probabilistic mortality forecasts by exploiting techniques in seasonal time-series models, extreme value theory (EVT), and hierarchical forecast reconciliation. In a hierarchical setting, coherent forecasts are those forecasts that add up in a manner consistent with the aggregation structure of the collection of time series. We are amongst the first to model and analyze U.S. monthly death data during the period from 1968–2019 to explore the seasonality and the age–gender dependence structure of mortality. Our results indicate that incorporating EVT and hierarchical forecast reconciliation greatly improves the overall forecast accuracy, which has important implications for life insurers in terms of rate making, reserve setting, and capital adequacy compliance. Using the solvency capital requirement (SCR) under Solvency II as an example, we show that the SCR calculated by our approach is considerably higher than those calculated by alternative models, suggesting that failing to account for extreme mortality risk and mortality dependence in the hierarchy can result in significantly underfunded problems for life insurers. We also find that our model can yield death forecasts that are largely consistent with actual observations in most of months in 2021 when death tolls surged due to COVID-19. This provides additional evidence of the effectiveness of our model for practical uses.

Coherent Forecasting for a Mixed Integer-Valued Time Series Model

In commerce, economics, engineering and the sciences, quantitative methods based on statistical models for forecasting are very useful tools for prediction and decision. There is an abundance of papers on forecasting for continuous-time series but relatively fewer papers for time series of counts which require special consideration due to the integer nature of the data. A popular method for modelling is the method of mixtures which is known for its flexibility and thus improved prediction capability. This paper studies the coherent forecasting for a flexible stationary mixture of Pegram and thinning (MPT) process, and develops the likelihood-based asymptotic distribution. Score functions and the Fisher information matrix are presented. Numerical studies are used to assess the performance of the forecasting methods. Also, a comparison is made with existing discrete-valued time series models. Finally, the practical application is illustrated with two sets of real data. It is shown that the mixture model provides good forecasting performance.

Spatial panel count data: modeling and forecasting of urban crimes

The steadily growing access to high-quality spatio-temporal crime count data with a high level of spatial detail allows to uncover interesting relationships between crime types within and between small regional units. Data coherent forecasting of such counts has to take the integer and non-negative nature of the data into account. Spatial panel data models that meet the criterion of coherency are relatively sparse. This paper proposes a new spatial panel regression framework with fixed effects to overcome these shortcomings. Depending on whether time dynamic effects are included in the model specification, estimation and inference are based either on a pseudo maximum likelihood method or on quasi-differenced generalized methods of moments. The models’ usefulness is demonstrated in a forecasting exercise of monthly crime counts at census tract level from Pittsburgh, Pennsylvania.

Model selection in reconciling hierarchical time series

Model selection has been proven an effective strategy for improving accuracy in time series forecasting applications. However, when dealing with hierarchical time series, apart from selecting the most appropriate forecasting model, forecasters have also to select a suitable method for reconciling the base forecasts produced for each series to make sure they are coherent. Although some hierarchical forecasting methods like minimum trace are strongly supported both theoretically and empirically for reconciling the base forecasts, there are still circumstances under which they might not produce the most accurate results, being outperformed by other methods. In this paper we propose an approach for dynamically selecting the most appropriate hierarchical forecasting reconciliation method and leading to more accurate coherent forecasts. The approach, which we call conditional hierarchical forecasting, is based on machine learning classification methods that use time series features to select the reconciliation method for each hierarchy. Moreover, it allows the selection to be tailored according to the accuracy measure of preference and the hierarchical level(s) of interest. Our results suggest that conditional hierarchical forecasting can lead to significantly more accurate forecasts than standard approaches, especially at lower hierarchical levels.

The performance of the global bottom-up approach in the M5 accuracy competition: A robustness check

The M5 accuracy competition has presented a large-scale hierarchical forecasting problem in a realistic grocery retail setting in order to evaluate an extended range of forecasting methods, particularly those adopting machine learning. The top ranking solutions adopted a global bottom-up approach, by which is meant using global forecasting methods to generate bottom level forecasts in the hierarchy and then using a bottom-up strategy to obtain coherent forecasts for aggregate levels. However, whether the observed superior performance of the global bottom-up approach is robust over various test periods or only an accidental result, is an important question for retail forecasting researchers and practitioners. We conduct experiments to explore the robustness of the global bottom-up approach, and make comments on the efforts made by the top-ranking teams to improve the core approach. We find that the top-ranking global bottom-up approaches lack robustness across time periods in the M5 data. This inconsistent performance makes the M5 final rankings somewhat of a lottery. In future forecasting competitions, we suggest the use of multiple rolling test sets to evaluate the forecasting performance in order to reward robustly performing forecasting methods, a much needed characteristic in any application.

Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives

Forecast reconciliation is a post-forecasting process aimed to improve the quality of the base forecasts for a system of hierarchical/grouped time series. Cross-sectional and temporal hierarchies have been considered in the literature, but generally, these two features have not been fully considered together. The paper presents two new results by adopting a notation that simultaneously deals with both forecast reconciliation dimensions. (i) The closed-form expression of the optimal (in the least squares sense) point forecasts fulfilling both contemporaneous and temporal constraints. (ii) An iterative procedure that produces cross-temporally reconciled forecasts by alternating forecast reconciliation along one single dimension (either cross-sectional or temporal) at each iteration step. The feasibility of the proposed procedures, along with first evaluations of their performance as compared to the most performing ‘single dimension’ (either cross-sectional or temporal) forecast reconciliation procedures, is studied through a forecasting experiment on the 95 quarterly time series of the Australian Gross Domestic Product from Income and Expenditure sides. For this dataset, the new procedures, in addition to providing fully coherent forecasts in both cross-sectional and temporal dimensions, improve the forecast accuracy of the state-of-the-art point forecast reconciliation techniques.

Integer-valued autoregressive processes with prespecified marginal and innovation distributions: a novel perspective

Integer-valued autoregressive (INAR) processes are generally defined by specifying the thinning operator and either the innovations or the marginal distributions. The major limitations of such processes include difficulties in deriving the marginal properties and justifying the choice of the thinning operator. To overcome these drawbacks, we propose a novel approach for building an INAR model that offers the flexibility to prespecify both marginal and innovation distributions. Thus, the thinning operator is no longer subjectively selected but is rather a direct consequence of the marginal and innovation distributions specified by the modeler. Novel INAR processes are introduced following this perspective; these processes include a model with geometric marginal and innovation distributions (Geo-INAR) and models with bounded innovations. We explore the Geo-INAR model, which is a natural alternative to the classical Poisson INAR model. The Geo-INAR process has interesting stochastic properties, such as MA() representation, time reversibility, and closed forms for the -order transition probabilities, which enables a natural framework to perform coherent forecasting. To demonstrate the real-world application of the Geo-INAR model, we analyze a count time series of criminal records in sex offenses using the proposed methodology and compare it with existing INAR and integer-valued generalized autoregressive conditional heteroscedastic models.