Aug 2026· Mathematics· Vol 14, pp. 2870· 0 citations· 35 references
Abstract
A wide range of real-world multivariate time series encountered in practice exhibit two simultaneous and interacting seasonal cycles, for example hourly electricity demand, intraday financial prices, and sub-daily traffic volumes. Existing Bayesian frameworks for vector autoregressive (VAR) processes accommodate at most a single seasonal periodicity, leaving no established methodology for the double seasonal case commonly observed in high-frequency multivariate data. This paper bridges that gap by introducing the double seasonal VAR (DSVAR) models, which extend the univariate double seasonal literature to a coherent multivariate setting. These models are defined through a multiplicative triple autoregressive operator that naturally accommodates the second seasonal cycle. Under a Gaussian error assumption, we derive a comprehensive and analytically convenient Bayesian framework for both modeling and forecasting of DSVAR processes. We consider two prior families: a conjugate matrix normal-Wishart prior which yields exact closed-form inference, and a Jeffreys’ non-informative prior. Under each prior, we derive the marginal posterior distribution of the coefficient matrix as a matrix-t distribution and the marginal posterior of the precision matrix as a Wishart distribution. Moreover, we derive the predictive distribution of future observations as a multivariate-t with an exact analytic form, together with its highest predictive density regions. The methodology is validated through a Monte Carlo simulation experiment and applied to hourly electricity loads in Czech Republic and Germany, two physically interconnected markets with pronounced intraday and intraweek seasonal cycles. Benchmark comparisons against standard VAR, single-seasonal VAR, and univariate seasonal AR models confirm the substantial forecasting gains delivered by the proposed DSVAR framework at both short and long horizons.
Autoregressive models with exogenous variables (ARX) constitute a fundamental family of time series tools with broad applicability across engineering, environmental science, and finance. A persistent limitation of standard Bayesian treatments is the Gaussian error assumption, which frequently proves inadequate when dealing with real data displaying heavy tails or occasional extreme observations. To overcome this shortcoming, the present paper develops a complete Bayesian inferential framework for ARX models under the scale-mixtures of normal (SMN) error distribution, integrating model identification, parameter estimation, and multi-step-ahead prediction within a unified scheme. A stochastic search variable selection (SSVS) procedure is adapted to perform simultaneous selection of the autoregressive order and the active exogenous regressors by assigning binary latent indicators to each candidate coefficient. Mixture-of-normals priors are specified for the dynamic and exogenous coefficients and an inverse-gamma prior for the error scale, while Bernoulli priors govern the latent selection indicators. These choices yield tractable full conditional posterior distributions: multivariate normal for the complete coefficient vector, inverse-gamma for the scale, and Bernoulli for the indicators. The conditional predictive distribution of future observations is also multivariate normal. For SMN-specific mixing parameters whose conditionals lack standard forms, Metropolis–Hastings steps are embedded within the Gibbs sampler. An extensive simulation study evaluates recovery accuracy across three SMN distributions and several ARX configurations. The methodology is then applied to the forecasting of daily global horizontal irradiance (GHI) in Najran, southwestern Saudi Arabia, using clear-sky GHI as an exogenous covariate, demonstrating the practical value of the proposed framework in a renewable energy context.
Ayman A. Amin, Shuhrah A. Alghamdi· Mathematics· 0 citations
We propose an ARMA model that allows for multiple seasonal periods and time varying parameters in both regular and seasonal components, building upon previous work for pure AR processes and the conditional likelihood. The model is parameterized to ensure stability and invertibility at each time point. The parameter evolution is governed by dynamic shrinkage processes, enabling extended periods of essentially constant parameters, gradual changes, and abrupt shifts. The model includes a stochastic volatility component to account for potentially heterogeneous noise, also modeled by a dynamic shrinkage process. A Gibbs sampler is developed using the exact likelihood, with separate updating steps for the latent errors and the unobserved pre-sample history of the process. The time-varying AR and MA parameters are sampled jointly using a fast posterior sampler based on the extended Kalman filter. The model and the efficiency of the Gibbs sampler are evaluated using simulated and real data. A case study on monthly air passenger data in the US during 1990-2024 reveals significant changes in seasonality during the Covid-19 pandemic.
Ganna Fagerberg, Mattias Villani, Robert Kohn· 0 citations
Autocorrelated bivariate count data frequently arise in criminal, environmental and financial studies, where capturing both serial dependence and cross‐series interaction is essential for statistical modelling and inference. In many applications, such dynamics are further influenced by exogenous covariates such as policy interventions or environmental factors, leading to time‐varying dependence structures that are not adequately captured by standard models. Existing bivariate integer‐valued autoregressive (BINAR) models mainly rely on constant or observation‐driven coefficients and rarely incorporate covariate information, which restricts their ability to represent evolving dependence in multivariate count processes. To address this limitation, we propose a covariate‐driven doubly stochastic bivariate integer‐valued autoregressive process, in which the thinning mechanism evolves jointly with past observations and exogenous covariates. This formulation extends the classical BINAR framework by allowing the dependence structure to vary dynamically under both internal and external driving mechanisms. The basic statistical properties of the proposed process are derived, and two estimation methods are developed, including an EM‐based algorithm. Monte Carlo simulations and a real data application are conducted to assess finite‐sample performance and robustness under different settings.
A suite of R packages for macroeconomic forecasting that leverages advanced Bayesian, structural, multivariate, dynamic, hierarchical, hierarchical, non-linear, and non-Gaussian models is presented, which enables both structural and predictive analyses.
Fei Shang, Xiao-Lei Wang, Tomasz Woźniak· 0 citations
Accurate forecasting in volatile and structurally complex demand environments requires models capable of capturing evolving trends, seasonality, nonlinear behavior and externally driven fluctuations. This study aims to propose the iterative ensemble forecasting with residual integration (IEFRI) framework, a structured ensemble methodology that improves forecast accuracy by explicitly and iteratively learning from residual errors rather than treating them as random noise. Unlike conventional one-stage ensemble approaches, IEFRI uses a multistage refinement process in which forecasts from heterogeneous base models are progressively corrected through successive neural and regression-based residual-learning cycles, allowing both nonlinear and linear error structures to be captured systematically. The framework integrates Holt–Winters exponential smoothing, seasonal auto regressive integrated moving average (SARIMA), artificial neural networks (ANNs) and multiple linear regression (MLR) within a decomposition-driven architecture. IEFRI is empirically validated using 25 years of monthly demand data (April 1998–March 2024) covering all major petroleum products in India – petrol, diesel, Liquified Petroleum Gas (LPG), jet fuel and kerosene – each exhibiting strong seasonality, trend shifts, structural change and sensitivity to macroeconomic and policy factors. Comparative evaluation using standard accuracy measures shows that IEFRI consistently outperforms individual models and widely used ensemble techniques such as bagging, boosting, random forests and stacked regression. While demonstrated using petroleum demand, the framework is generalizable to other complex and uncertainty-prone time-series forecasting applications.
This study develops and evaluates the IEFRI framework for complex demand time series. The approach combines heterogeneous base forecasting models – Holt–Winters exponential smoothing, SARIMA, ANNs and MLR – within a decomposition-driven ensemble structure. Initial forecasts are aggregated and subsequently refined through a multistage residual-learning process that integrates neural-network-based nonlinear correction and iterative regression-based linear adjustment. Model performance is assessed using rolling-origin out-of-sample evaluation and standard accuracy measures mean absolute percentatge error (MAPE), symmetric mean absolute percentage error (SMAPE), root mean square error (RMSE). The framework is empirically validated using 25 years of monthly demand data across five petroleum products, enabling robust comparison with conventional ensemble methods.
The empirical results show that the proposed IEFRI framework consistently improves forecast accuracy across all five petroleum products – petrol, diesel, LPG, jet fuel and kerosene – each characterized by strong seasonality, structural change and nonlinear dynamics. Compared with individual base models and conventional ensemble techniques such as bagging, boosting, random forests and stacked regression, IEFRI achieves lower forecasting errors under rolling-origin evaluation. The iterative residual-learning mechanism is found to be effective in systematically reducing both nonlinear and linear forecast errors across successive refinement stages. Performance gains are stable across products with differing demand characteristics, demonstrating the robustness of the framework. These findings indicate that explicitly modeling and reintegrating residual information enhances ensemble forecasting accuracy in complex, volatile demand environments.
Despite its strengths, this study has several limitations. First, the analysis relies on secondary data compiled from multiple sources, requiring interpolation and harmonization, which may introduce minor measurement bias. Second, the IEFRI framework is validated using historical monthly data from a single domain, and its generalizability to other sectors and higher-frequency data has not been empirically tested. Third, while the iterative residual-learning process improves accuracy, it increases computational complexity and may be sensitive to data quality and parameter settings. Finally, the study does not incorporate probabilistic or uncertainty forecasting, which could further enhance decision support in practice.
The proposed IEFRI framework provides practitioners with a systematic and robust approach for improving forecast accuracy in environments characterized by volatility, nonlinearity and structural change. By explicitly learning from residual errors, the model supports more reliable demand forecasts that can enhance operational planning, capacity allocation, inventory management and medium-term strategic decision-making. The framework is flexible and can be integrated with existing forecasting systems that already use statistical or machine learning models, allowing organizations to improve performance without fully replacing current tools. For policymakers and managers, more accurate forecasts enable better resource planning, risk mitigation and policy evaluation in sectors where demand uncertainty has significant economic and operational consequences.
This study is based on original research and contributes original value by introducing the IEFRI framework, which advances ensemble forecasting beyond conventional one-stage aggregation. Unlike standard ensemble methods that treat residuals as random noise, IEFRI explicitly and iteratively models residual errors through a structured combination of neural-network-based nonlinear learning and regression-based linear correction. This multistage residual integration provides a systematic mechanism for continuous forecast refinement. The study has been subjected to extensive empirical validation using the demand for various petroleum products which exhibit multiple, structurally diverse demand series. The framework offers a reusable, domain-agnostic methodology for improving forecast accuracy in complex, volatile time-series environments.
Ramesh Murthy· Journal of Modelling in Mana...· 0 citations
Vector autoregressive moving-average (VARMA) models have long been considered impractical beyond moderate dimensions: the likelihood is non-convex, the parametrization is identified only up to equivalence, and every evaluation costs a pass over the entire series. Yet their moving-average term captures with a few parameters what a pure autoregression matches only with many lags. We introduce an estimation framework that removes this computational barrier: each optimization iteration is independent of the series length $T$. The framework combines a partial-autocorrelation reparametrization that guarantees stationarity and invertibility by construction, Gaussian priors on the reparametrized coefficients with separate scales for diagonal and off-diagonal entries, and losses that depend on the data only through fixed-size sufficient statistics, evaluated by a Parseval (Fourier) identity at near-linear cost in the truncation length. This yields two point estimators: a regularized least-squares fit and a covariance-marginalized maximum-a-posteriori estimator. We prove that both recover the infinite-autoregressive representation of the true process at a near-parametric rate in fixed dimension, so the truncation introduces no asymptotic bias. The same machinery extends, at the same leading cost, to seasonal dynamics, exogenous regressors (VARMAX), and rolling-window refits. Empirically, the estimators stay close to the oracle forecast error from $d=10$ to $d=40$ (where classical conditional MLE returns non-invertible fits whose forecasts diverge) and match or beat VAR, Bayesian-VAR, component-wise ARMA, and sparse-VARMA baselines on retail-demand, meteorological, and air-quality data. This brings likelihood-based VARMA estimation, at a per-iteration cost independent of the series length, to the problem sizes where practitioners have so far relied on VAR models.
D. Paulin, V. Elvira· 0 citations
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