Variational Bayesian (VB) methods produce posterior inference in a time frame considerably smaller than traditional Markov Chain Monte Carlo approaches. Although the VB posterior is an approximation, ...it has been shown to produce good parameter estimates and predicted values when a rich classes of approximating distributions are considered. In this paper, we propose the use of recursive algorithms to update a sequence of VB posterior approximations in an online, time series setting, with the computation of each posterior update requiring only the data observed since the previous update. We show how importance sampling can be incorporated into online variational inference allowing the user to trade accuracy for a substantial increase in computational speed. The proposed methods and their properties are detailed in two separate simulation studies. Additionally, two empirical illustrations are provided, including one where a Dirichlet Process Mixture model with a novel posterior dependence structure is repeatedly updated in the context of predicting the future behaviour of vehicles on a stretch of the US Highway 101.
Electricity spot prices exhibit strong time series properties, including substantial periodicity, both inter-day and intraday serial correlation, heavy tails and skewness. In this paper we capture ...these characteristics using a first order vector autoregressive model with exogenous effects and a skew t distributed disturbance. The vector is longitudinal, in that it comprises observations on the spot price at intervals during a day. A band two inverse scale matrix is employed for the disturbance, as well as a sparse autoregressive coefficient matrix. This corresponds to a parsimonious dependency structure that directly relates an observation to the two immediately prior, and the observation at the same time the previous day. We estimate the model using Markov Chain Monte Carlo, which allows for the evaluation of the complete predictive distribution of future spot prices. We apply the model to hourly Australian electricity spot prices observed over a three year period, with four different nested multivariate error distributions: skew t, symmetric t, skew normal and symmetric normal. The forecasting performance is judged over a 30 day forecast trial using the continuous ranked probability score, which accounts for both predictive bias and sharpness.
•Temporal Hierarchy is used to improve density forecasts.•Sampling methods and reconciliation methods are presented and compared.•Cross-validation reconciliation is proposed for hierarchical ...forecasting.•Methods are evaluated with forecasting case studies on wind power and electric load.
New methods are proposed for adjusting probabilistic forecasts to ensure coherence with the aggregation constraints inherent in temporal hierarchies. The different approaches nested within this framework include methods that exploit information at all levels of the hierarchy as well as a novel method based on cross-validation. The methods are evaluated using real data from two wind farms in Crete and electric load in Boston. For these applications, optimal decisions related to grid operations and bidding strategies are based on coherent probabilistic forecasts of energy power. Empirical evidence is also presented showing that probabilistic forecast reconciliation improves the accuracy of the probabilistic forecasts.
This paper presents a formal framework and proposes algorithms to extend forecast reconciliation to discrete-valued data, including low counts. A novel method is introduced based on recasting the ...optimisation of scoring rules as an assignment problem, which is solved using quadratic programming. The proposed framework produces coherent joint probabilistic forecasts for count hierarchical time series. Two discrete reconciliation algorithms are also proposed and compared against generalisations of the top-down and bottom-up approaches for count data. Two simulation experiments and two empirical examples are conducted to validate that the proposed reconciliation algorithms improve forecast accuracy. The empirical applications are forecasting criminal offences in Washington D.C. and product unit sales in the M5 dataset. Compared to benchmarks, the proposed framework shows superior performance in both simulations and empirical studies.
•Presents forecast reconciliation framework for discrete hierarchical time series.•Proposed DFR algorithm optimises the penalised Brier Score.•Optimisation cast as assignment solved using Quadratic Programming.•Second algorithm handles computational complexity of larger hierarchies.•Simulations and applications show the effectiveness of our framework.
Multivariate discrete response data can be found in diverse fields, including econometrics, finance, biometrics, and psychometrics. Our contribution, through this study, is to introduce a new class ...of models for multivariate discrete data based on pair copula constructions (PCCs) that has two major advantages. First, by deriving the conditions under which any multivariate discrete distribution can be decomposed as a PCC, we show that discrete PCCs attain highly flexible dependence structures. Second, the computational burden of evaluating the likelihood for an m -dimensional discrete PCC only grows quadratically with m . This compares favorably to existing models for which computing the likelihood either requires the evaluation of 2 ᵐ terms or slow numerical integration methods. We demonstrate the high quality of inference function for margins and maximum likelihood estimates, both under a simulated setting and for an application to a longitudinal discrete dataset on headache severity. This article has online supplementary material.
We develop a Bayesian approach for the selection of skew in multivariate skew
t
distributions constructed through hidden conditioning in the manners suggested by either
Azzalini and Capitanio (2003) ...or
Sahu et al. (2003). We show that the skew coefficients for each margin are the same for the standardized versions of both distributions. We introduce binary indicators to denote whether there is symmetry, or skew, in each dimension. We adopt a proper beta prior on each non-zero skew coefficient, and derive the corresponding prior on the skew parameters. In both distributions we show that as the degrees of freedom increases, the prior smoothly bounds the non-zero skew parameters away from zero and identifies the posterior. We estimate the model using Markov chain Monte Carlo (MCMC) methods by exploiting the conditionally Gaussian representation of the skew
t
distributions. This allows for the search through the posterior space of all possible combinations of skew and symmetry in each dimension. We show that the proposed method works well in a simulation setting, and employ it in two multivariate econometric examples. The first involves the modeling of foreign exchange rates and the second is a vector autoregression for intra-day electricity spot prices. The approach selects skew along the original coordinates of the data, which proves insightful in both examples.
This paper extends the technique of gradient boosting with a focus on using domain-specific models instead of trees. The domain of mortality forecasting is considered as an application. The two novel ...contributions are to use well-known stochastic mortality models as weak learners in gradient boosting rather than trees, and to include a penalty that shrinks mortality forecasts in adjacent age groups and nearby geographical regions closer together. The proposed method demonstrates superior forecasting performance based on US male mortality data from 1969 to 2019. The proposed approach also enables us to interpret and visualize the results. The boosted model with age-based shrinkage yields the most accurate national-level mortality forecast. For state-level forecasts, spatial shrinkage provides further improvement in accuracy in addition to the benefits of age-based shrinkage. This improvement can be attributed to data sharing across states with large and small populations in adjacent regions and states with common risk factors.
Model combinations through revised base rates Petropoulos, Fotios; Spiliotis, Evangelos; Panagiotelis, Anastasios
International journal of forecasting,
07/2023, Volume:
39, Issue:
3
Journal Article
Peer reviewed
Open access
Standard selection criteria for forecasting models focus on information that is calculated for each series independently, disregarding the general tendencies and performance of the candidate models. ...In this paper, we propose a new way to perform statistical model selection and model combination that incorporates the base rates of the candidate forecasting models, which are then revised so that the per-series information is taken into account. We examine two schemes that are based on the precision and sensitivity information from the contingency table of the base rates. We apply our approach on pools of either exponential smoothing or ARMA models, considering both simulated and real time series, and show that our schemes work better than standard statistical benchmarks. We test the significance and sensitivity of our results, discuss the connection of our approach to other cross-learning approaches, and offer insights regarding implications for theory and practice.
Optimal reconciliation with immutable forecasts Zhang, Bohan; Kang, Yanfei; Panagiotelis, Anastasios ...
European journal of operational research,
07/2023, Volume:
308, Issue:
2
Journal Article
Peer reviewed
Open access
•We present a method to keep forecasts unchanged (immutable) after reconciliation.•Immutable series need not come from the same level and we prove unbiasedness.•Our method applies to grouped ...hierarchies.•An application to large-scale retail data demonstrates improved forecast accuracy.
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.
•A novel forecast reconciliation framework using Bayesian state-space methods.•Introduces judgmental adjustments into the reconciliation procedure.•Joint reconciliation of all forecasting ...horizons.•Comprehensive forecasting study using Swiss merchandise exports.
This paper proposes a novel forecast reconciliation framework using Bayesian state-space methods. It allows for the joint reconciliation at all forecast horizons and uses predictive distributions rather than past variation of forecast errors. Informative priors are used to assign weights to specific predictions, which makes it possible to reconcile forecasts such that they accommodate specific judgmental predictions or managerial decisions. The reconciled forecasts adhere to hierarchical constraints, which facilitates communication and supports aligned decision-making at all levels of complex hierarchical structures. An extensive forecasting study is conducted on a large collection of 13,118 time series that measure Swiss merchandise exports, grouped hierarchically by export destination and product category. We find strong evidence that in addition to producing coherent forecasts, reconciliation also leads to substantial improvements in forecast accuracy. The use of state-space methods is particularly promising for optimal decision-making under conditions with increased model uncertainty and data volatility.
PDF
