Forecasting

This study introduces a novel forecasting framework that identifies and predicts recurrently emerging structural patterns in stock trends using rising visibility graphs (RVGs) and the Weisfeiler–Lehman (WL) subtree kernel. The proposed method, RVGWL, addresses a key limitation of traditional visibility graphs, namely the structural indistinguishability between rising and falling trends, by selectively constructing edges only along upward price movements. This approach produces graph representations that capture direction-sensitive market dynamics and facilitate the extraction of meaningful topological features from price data. By applying the WL kernel, RVGWL quantifies structural similarities between graph-transformed time series, enabling the identification of structurally similar preceding patterns and the probabilistic forecasting of their subsequent trajectories based on nine canonical trend templates. Experiments on time series data from four major stock indices and their constituent stocks during the year 2023—characterized by diverse market regimes across the U.S., Japan, the U.K., and China—demonstrate that RVGWL consistently outperforms classical rule-based strategies. These results support the predictive value of recurring topological structures in financial time series and higight the potential of structure-aware forecasting methods in quantitative analysis. Full article

In this work, we present a novel approach for predicting short-term electrical energy consumption. Most energy consumption methods work well for their case study datasets. The proposed method utilizes a cloud computing platform that allows for integrating information from different sources, such as weather data and historical energy consumption, while employing machine learning techniques to achieve higher accuracy in forecasting. We collected detailed weather data from the “Weather Underground Company” website, known for its accurate records. Then, we studied past energy consumption data provided by PJM (focusing on DEO&K, which serves Cincinnati and northern Kentucky) and identified features that significantly impact energy consumption. We also introduced a processing step to ensure accurate predictions for holidays. Our goal is to predict the next 24 h of load consumption. We developed a hybrid, generalizable forecasting methodology with deviation correction. The methodology is characterized by fault tolerance due to distributed cloud deployment and an introduced voting mechanism. The proposed approach improved the accuracy of LSTM, SARIMAX, and SARIMAX + SVM, with MAPE values of 5.17%, 4.21%, and 2.21% reduced to 1.65%, 1.00%, and 0.88%, respectively, using our CM-LSTM-DC, CM-SARIMAX-DC, and CM-SARIMAX + SVM-DC models. Full article

Water temperature is a fundamental parameter influencing a range of biotic and abiotic processes occurring within various components of the hydrosphere. This study presents a multi-step, data-driven predictive modeling framework to estimate water temperatures for the period 2021–2100 in three aquatic environments in Central Europe: the Odra River, the Szczecin Lagoon, and the Baltic Sea. The framework integrates Bayesian Model Averaging (BMA), Random Sample Consensus (RANSAC) regression, Gradient Boosting Regressor (GBR), and Random Forest (RF) machine learning models. To assess the performance of the models, the coefficient of determination (R2), mean absolute error (MAE), and root mean square error (RMSE) were used. The results showed that the application of statistical downscaling methods improved the prediction of air temperatures with respect to the BMA. Moreover, the RF method was used to predict water temperature. The best model performance was obtained for the Baltic Sea and the lowest for the Odra River. Under the SSP2-4.5 and SSP5-8.5 scenario-based simulations, projected air temperature increases in the period 2021–2100 could range from 1.5 °C to 1.7 °C and 4.7 to 5.1 °C. In contrast, the increase in water temperatures by 2100 will be between 1.2 °C and 1.6 °C (SSP2-4.5 scenario) and between 3.5 °C and 4.9 °C (SSP5-8.5). Full article

Natural gas consumption in Europe has undergone substantial changes in recent years, driven by geopolitical tensions, economic dynamics, and the continent’s ongoing transition towards cleaner energy sources. Furthermore, as noted in the International Energy Agency’s Gas Market Report 2025, natural gas demand is becoming increasingly sensitive to fluctuations in weather patterns, including cold snaps and heatwaves. These factors make the task of forecasting future annual consumption particularly challenging from a statistical perspective and underscore the importance of accurately quantifying the uncertainty surrounding predictions. In this paper, we propose a simple yet flexible approach to issuing dynamic probabilistic forecasts based on an additive time series model. To capture long-term trends, the model incorporates a deterministic component based on the Guseo–Guidolin innovation diffusion framework. In addition, a stochastic innovation term governed by an ARIMAX process is used to describe year-over-year fluctuations, helping to account for the potential presence of variance nonstationarity over time. The proposed methodology is applied to forecast future annual consumption in six key European countries: Austria, France, Germany, Italy, the Netherlands, and the United Kingdom. Full article

This article describes a robust Gaussian Prior process state space modeling (GPSSM) approach to assess the impact of an intervention in a time series. Numerous applications can benefit from this approach. Examples include: (1) time series could be the stock price of a company and the intervention could be the acquisition of another company; (2) the time series under concern could be the noise coming out of an engine, and the intervention could be a corrective step taken to reduce the noise; (3) the time series could be the number of visits to a web service, and the intervention is changes done to the user interface; and so on. The approach we describe in this article applies to any times series and intervention combination. It is well known that Gaussian process (GP) prior models provide flexibility by placing a non-parametric prior on the functional form of the model. While GPSSMs enable us to model a time series in a state space framework by placing a Gaussian Process (GP) prior over the state transition function, probabilistic recurrent state space models (PRSSM) employ variational approximations for handling complicated posterior distributions in GPSSMs. The robust PRSSMs (R-PRSSMs) discussed in this article assume a scale mixture of normal distributions instead of the usually proposed normal distribution. This assumption will accommodate heavy-tailed behavior or anomalous observations in the time series. On any exogenous intervention, we use R-PRSSM for Bayesian fitting and forecasting of the IoT time series. By comparing forecasts with the future internal temperature observations, we can assess with a high level of confidence the impact of an intervention. The techniques presented in this paper are very generic and apply to any time series and intervention combination. To illustrate our techniques clearly, we employ a concrete example. The time series of interest will be an Internet of Things (IoT) stream of internal temperatures measured by an insurance firm to address the risk of pipe-freeze hazard in a building. We treat the pipe-freeze hazard alert as an exogenous intervention. A comparison of forecasts and the future observed temperatures will be utilized to assess whether an alerted customer took preventive action to prevent pipe-freeze loss. Full article

The security challenges associated with maritime migratory incidents in the Mediterranean Sea since the onset of the 21st century are considerable. Reports of such incidents are generated almost daily, leading to significant scientific interest, including that of this manuscript. This article introduces a forecasting model specifically designed to equip maritime security stakeholders around the Mediterranean Sea with a technical instrument for estimating the frequency of maritime migratory incidents. The proposed model, the SIFM, encompasses five methodological steps: Tessellation: The initial step involves partitioning the maritime area affected by these incidents into distinct cells. Subsidiary process: In this phase, the cells are classified according to the year in which incidents were recorded. Containment index: This index quantifies the magnitude of incidents within the designated cells. Incidence growth index: This metric further refines the forecasting methodology. Maritime migration incident forecasting: The concluding step establishes a forecast interval for the anticipated quantity of maritime migratory incidents. This systematic approach aims to enhance the understanding and prediction of maritime migratory incidents within the Mediterranean region. Full article

This article presents a mathematical model of cyclical economic processes, formulated as the sum of a deterministic polynomial function and a cyclic random process that simultaneously captures trend, stochasticity, cyclicity, and rhythm variability. Building on this stochastic framework, we propose rhythm-adaptive statistical techniques for estimating the probabilistic characteristics of the cyclic component; by adjusting to rhythm changes, these techniques improve estimation accuracy. We also introduce a forecasting procedure that constructs a system of rhythm-adaptive confidence intervals for future cycles. The effectiveness of the model and associated methods is demonstrated through a series of computational experiments using Federal Reserve Economic Data. Results show that the rhythm-adaptive forecasting approach achieves mean absolute errors less than half of those produced by a comparable non-adaptive method, underscoring its practical advantage for the analysis and prediction of cyclic economic phenomena. Full article

The fresh produce supply chain sector is a vital pillar of any society and an indispensable part of the national economic structure. As a significant segment of the agricultural market, accurately forecasting vegetable prices holds significant importance. Vegetable market pricing is subject to a myriad of complex influences, resulting in nonlinear patterns that conventional time series methodologies often struggle to decode. Future planning for commodity pricing is achievable by forecasting the future price anticipated by the current circumstances. This paper presents a price forecasting methodology for tomatoes which uses price and production data taken from 2008 to 2021 and analyzed by means of advanced deep learning-based Long Short-Term Memory (LSTM) networks. A comparative analysis of three models based on Root Mean Square Error (RMSE) identifies LSTM as the most accurate model, achieving the lowest RMSE (0.2818), while SARIMA performs relatively well. The proposed deep learning-based method significantly improved the results versus other conventional machine learning and statistical time series analysis methods. Full article

Accurate Day-Ahead Energy Price (DAEP) forecasting is essential for optimizing energy market operations. This study introduces a machine learning framework to predict the DAEP with a 24 h lead time, leveraging historical data and forecasts available at the prediction time. Hourly DAEP data from the California Independent System Operator (January 2017 to July 2023) were integrated with exogenous and engineered endogenous features. A custom rolling window cross-validation, with 24 h validation blocks sliding daily across 2372 folds, evaluates an Extreme Gradient Boosting (XGBoost) model’s performance under diverse market conditions, achieving a median mean absolute error of 6.26 USD/MWh and root mean squared error of 8.27 USD/MWh, with variability reflecting market volatility. The feature importance analysis using Shapley additive explanations highlighted the dominance of engineered endogenous features in driving the 24 h lead time forecasts under relatively stable market conditions. Forecasting the DAEP at a runtime of 10 AM on the prior day was used to assess model uncertainty. This involved training random forest, support vector regression, XGBoost, and feed forward neural network models, followed by stacking and voting ensembles. The results indicate the need for ensemble forecasting and evaluation beyond a static train–test split to ensure the practical utility of machine learning for DAEP forecasting across varied market dynamics. Finally, operationalizing the forecast model for bidding decisions by forecasting the DAEP and real-time prices at runtime is presented and discussed. Full article

In numerous domains of finance and economics, modelling and predicting stock market volatility is essential. Predicting stock market volatility is widely used in the management of portfolios, analysis of risk, and determination of option prices. This study is about volatility modelling of the daily Johannesburg Stock Exchange All Share Index (JSE ALSI) stock price data between 1 January 2014 and 29 December 2023. The modelling process incorporated daily log returns derived from the JSE ALSI. The following volatility models were presented for the period: sGARCH(1, 1) and fGARCH(1, 1). The models for volatility were fitted using five unique error distribution assumptions, including Student’s t, its skewed version, the generalized error and skewed generalized error distributions, and the generalized hyperbolic distribution. Based on information criteria such as Akaike, Bayesian, and Hannan–Quinn, the ARMA(0, 0)-fGARCH(1, 1) model with a skewed generalized error distribution emerged as the best fit. The chosen model revealed that the JSE ALSI prices are highly persistent with the leverage effect. JSE ALSI price volatility was notably influenced during the COVID-19 pandemic. The forecast over the next 10 days shows a rise in volatility. A comparative study was then carried out with the JSE Top 40 and the S&P500 indices. Comparison of the FTSE/JSE Top 40, S&P 500, and JSE ALLSI return indices over the COVID-19 pandemic indicated higher initial volatility in the FTSE/JSE Top 40 and S&P 500, with the JSE ALLSI following a similar trend later. The S&P 500 showed long-term reliability and high rolling returns in spite of short-run volatility, the FTSE/JSE Top 40 showed more pre-pandemic risk and volatility but reduced levels of rolling volatility after the pandemic, similar in magnitude for each index with low correlations among them. These results provide important insights for risk managers and investors navigating the South African equity market. Full article

This paper presents a model that combines mode decomposition approaches with a bi-directional long short-term memory (BiLSTM) attention mechanism and a transformer (AMT) to predict the concentration level of ozone (O₃) in Johannesburg, South Africa. Johannesburg is a densely populated city and the industrial and economic hub of South Africa. Being the industrial hub, air pollution is a major concern as it affects human health. Using air pollutants and meteorological datasets, a model was proposed that uses a mode decomposition approach to address the nonlinear nature of O₃ concentration. This nonlinearity is one of the most challenging issues in air quality prediction, and this study proposed a model to decompose input data and identify the most relevant features and leverage attention mechanisms to produce weighted parameters that can enhance the model’s performance. The model’s performance enhancement approach was aimed at ensuring an effective model that easily adapts to frequently changing pollutant data in air quality prediction. The performance was evaluated statistically with root mean squared error (RMSE), mean absolute error (MAE), and mean square error (MSE). The proposed EEMD-CEEMDAN-BiLSTM-AMT model produced the most optimal result with MSE (4.80 × 10⁻⁶), RMSE (0.002), and MAE (0.001). When compared with the other similar models, the proposed model was best in terms of MSE value. Future work seeks to enhance the proposed model to fine-tune its performance on different air pollutant concentrations in South Africa. Full article

One of the biggest challenges in designing a logistics network is predicting the demand flows between all pairs of points in the network. Currently, the gravity model is mainly used for estimating the demand flow between points. However, the gravity model uses historical data to estimate values for its multiple parameters and distance between pairs to forecast the demand flow. Distance values close to zero and unprecedented changes in demand flow data create numerical instability for the gravity model’s output. Hence, the proximity factor, a single parameter model that uses the relative ordering of pairs instead of distance, was developed. In this paper, we systematically compare the proximity factor and the gravity model. It is shown that the proximity factor is a robust in terms of reliability and competitive alternative to the gravity model. According to our analysis, the proximity factor model can replace the gravity model in some applications when no historical data are available to adjust the parameters of the latter. Full article

Wind energy in Brazil has been steadily growing, influenced significantly by climate change. To enhance wind energy generation, it is essential to incorporate external climatic variables into wind speed modeling to reduce uncertainties. Periodic Autoregressive Models with Exogenous Variables (PARX), which include the exogenous variable ENSO, are effective for this purpose. This study modeled wind speed series in Rio Grande do Norte, Paraíba, Pernambuco, Alagoas, Sergipe, Rio Grande do Sul, and Santa Catarina, considering the spatial correlation between these states through PARX-Cov modeling. Additionally, the correlation with ENSO indicators was used for out-of-sample prediction of climatic variables, aiding in wind speed scenario simulation. The proposed PARX and PARX-Cov models outperformed the current model used in the Brazilian electric sector for simulating future wind speed series. Specifically, the PARX-Cov model with the Cumulative ONI index is most suitable for Pernambuco, Rio Grande do Sul, and Santa Catarina, while the PARX-Cov with the SOI index is more appropriate for Rio Grande do Norte. For Alagoas and Sergipe, the PARX with the Cumulative ONI index is the best fit, and the PARX with the Cumulative Niño 4 index is most suitable for Paraíba. Full article

The global demand for clean and sustainable energy has driven the rapid growth of wind power. However, wind farm managers face the challenge of forecasting wind power for efficient power generation and management. Accurate wind speed forecasting (WSF) is vital for predicting wind power; yet, the variability and intermittency of the wind make forecasting wind speeds difficult. Consequently, WSF remains a challenging area of wind research, driving continuous improvement in the field. This study aimed to enhance the optimization of multifeature-driven short multistep WSF. The primary contributions of this research include the integration of ReliefF feature selection (RFFS), a novel approach to variational mode decomposition for multifeature decomposition (NAMD), and a recursive non-linear autoregressive with exogenous inputs (NARXR) neural network. In particular, RFFS aids in identifying meteorological features that significantly influence wind speed variations, thus ensuring the selection of the most impactful features; NAMD improves the accuracy of neural network training on historical data; and NARXR enhances the overall robustness and stability of the wind speed forecasting results. The experimental results demonstrate that the predictive accuracy of the proposed NAMD–NARXR hybrid model surpasses that of the models used for comparison, as evidenced by the forecasting error and statistical metrics. Integrating the strengths of RFFS, NAMD, and NARXR enhanced the forecasting performance of the proposed NAMD–NARXR model, highlighting its potential suitability for applications requiring multifeature-driven short-term multistep WSF. Full article

This paper investigates the dynamics of Hurricane-Induced Failure (HIF) by developing a probabilistic framework using a Dynamic Bayesian Network (DBN) model. The model captures the complex interplay of factors influencing Hurricane Wind Speed Intensity (HWSI) and its impact on asset failures. In the proposed DBN model, the pole failure mechanism is represented using Bayesian probabilistic principles, encompassing bending elasticity endurance and the foundational strength of the system poles. To characterize the stochastic properties of HIF, Monte Carlo simulation (MCS) is employed in conjunction with fragility curves (FC) and the scenario reduction (SCENRED) algorithm. The proposed DBN model evaluates the probability of asset failure and compares the results using stochastic Monte Carlo simulation based on the fragility curve scenario reduction algorithm (FC-MCS-SCENRED) model. The results are validated on a standard IEEE 15 bus and IEEE 33 bus radial distribution system as a case study. The DBN results show that they are consistent with the data obtained using the FC-MCS-SCENRED model. The results also reveal that the HWSI plays a critical role in determining HIF rates and the likelihood of asset failures. These findings hold significant implications for the inspection and maintenance scheduling of distribution overhead power lines susceptible to hurricane-induced impacts. Full article

In the past, South African monetary policy aimed to protect the external value of the domestic currency (Rand); however, these efforts failed. Later, its monetary policy approach changed to allow the foreign exchange rate market to determine the exchange rates. In such a change, the South African Reserve Bank (SARB) aimed to stabilize the demand for the Rand in the foreign exchange market by providing information to stabilize market expectations and create favorable market conditions. However, South African policymakers have struggled with currency depreciation since the early 60s, increasing the uncertainty of South African exports. This study aims to examine the effect of currency depreciation on exports using the Threshold Autoregressive (TAR) model. Additionally, this study created and validated the supply chain activity/disruption index to capture the sea trade activity. The sample period for the analysis is 2009 to 2023. The study finds that currency depreciation does not improve trade between South Africa and its trading partners over time. Furthermore, the currency depreciation was found to be asymmetric to the effect of international trade across the different regimes. The supply chain activity index shows that the effect of supply chain activity/disruption on exports is regime-dependent. This implies that the effect on exports is dependent on the economic environment. Full article

Uncertainty quantification (UQ) is critical for modeling complex dynamic systems, ensuring robustness and interpretability. This study extends Physics-Guided Bayesian Neural Networks (PG-BNNs) to enhance model robustness by integrating physical laws into Bayesian frameworks. Unlike Artificial Neural Networks (ANNs), which provide deterministic predictions, and Bayesian Neural Networks (BNNs), which handle uncertainty probabilistically but struggle with generalization under sparse and noisy data, PG-BNNs incorporate the laws of physics, such as governing equations and boundary conditions, to enforce physical consistency. This physics-guided approach improves generalization across different noise levels while reducing data dependency. The effectiveness of PG-BNNs is validated through a one-degree-of-freedom vibration system with multiple noise levels, serving as a representative case study to compare the performance of Monte Carlo (MC) dropout ANNs, BNNs, and PG-BNNs across interpolation and extrapolation domains. Model accuracy is assessed using Mean Squared Error (MSE), Mean Absolute Percentage Error (MAE), and Coefficient of Variation of Root Mean Square Error (CVRMSE), while UQ is evaluated through 95% Credible Intervals (CIs), Mean Prediction Interval Width (MPIW), the Quality of Confidence Intervals (QCI), and Coverage Width-based Criterion (CWC). Results demonstrate that PG-BNNs can achieve high accuracy and good adherence to physical laws simultaneously, compared to MC dropout ANNs and BNNs, which confirms the potential of PG-BNNs in engineering applications related to dynamic systems. Full article

This paper contributes significantly to time series analysis by discussing the empirical properties of white noise and their implications for model selection. This paper illustrates the ways in which the standard assumptions about white noise typically fail in practice, with a special emphasis on striking differences in sample ACF and PACF. Such findings prove particularly important when assessing model adequacy and discerning between residuals of different models, especially ARMA processes. This study addresses issues involving testing procedures, for instance, the Ljung–Box test, to select the correct time series model determined in the review. With the improvement in understanding the features of white noise, this work enhances the accuracy of modeling diagnostics toward real forecasting practice, which gives it applied value in time series analysis and signal processing. Full article