In the study of several real time series and other major fluctuations such as the trend,the cycle and the noise, the presence of seasonal fluctuations is one of the most important issues.
The investigation has an established practice provided that seasonal variations have been
regarded as a disruptive element and then must be eliminated. However, these fluctuations are
an integral part that must be studied in order to evaluate and forecast the studied model. A
well-known practice for modeling seasonal data is to utilise an autoregressive model that is able
to handle the presence of the seasonal patterns in the data. Autoregressive models are a kind
of time-series model that utilise lagged values of the target variable to make predictions about
future values. Notice that despite the fact that these data are obtainable in practice as sequences of discrete
observed values, they are basically approached as functions. Functional autoregressive models are well known for the analysis of time series
analysis. However, basic formulation is not suitable for investigating the seasonal behaviour
in functional time series data. Hence, we introduce seasonal functional autoregressive processes to model time series. For the autoregressive process of order one, we provide
conditions of stationarity and formulate limit theorems, and supply methods of estimation
and prediction. The worthiness of these models is displayed via algorithmic investigations.

Prediction of the stock market index is important for investors and financial analysts to mitigate risks and achieve profits. Multilayer perceptron is an efficient neural network to learn non-linear relationships for stock price prediction with high accuracy. FTSE Bursa Malaysia KLCI (FBM KLCI) represents the economic performance of Malaysia. Therefore, it is important to monitor the market sentiment based on FBM KLCI. Stock price prediction becomes a strident challenge for the investors and financial analysts to mitigate risks and achieve profits. This research aims to predict the closing prices of FBM KLCI with neural networks comprisingtwo hidden layers. The data consists of historical stock data, including volume, opening, closing, and low and high prices from January 2019 to May 2025. Model comparison is performed using random forest (RF). The model performances are evaluated with the coefficient of determination (R²), root mean squared error (RMSE), and mean absolute error (MAE). The result of this study shows that the neural network is more stable and consistent in predicting the next closing prices of FBM KLCI. This study is significant because it contributes to the field by offering an efficient method for predicting stock index prices, which is expected to guide investors and fund managers in their decision-making.

Sri Lanka’s pepper production and export industry is increasingly affected by climatic variability. This study examines the influence of temperature, rainfall, wind speed, and daylight duration on district-level pepper production (2018–2024) and national monthly pepper export volumes (2014–2024). Panel data analysis is employed to quantify the effects of climatic variables on pepper production, using Fixed Effect (FE) and Random Effect (RE) estimations, while the Difference Generalized Method of Moments (GMM) approach is applied to account for dynamic production persistence. Export behavior and forecasting performance are analyzed using the Autoregressive Distributed Lag–Error Correction Model (ARDL–ECM) and Seasonal Autoregressive Integrated Moving Average (SARIMA) model. The results show mean temperature has a positive and statistically significant contemporaneous effect on pepper production, while rainfall remains insignificant in static panel models. Difference GMM results confirm strong production persistence, with temperature emerging as the dominant positive driver and wind speed exerting a significant negative effect. Export modeling reveals short-run and long-run relationships between climatic variables and pepper exports. The ARDL–ECM framework confirms a stable long-run equilibrium, with about one-third of disequilibrium corrected within one month. Temperature and wind speed show significant lagged effects, while rainfall plays a limited role. Forecast comparisons indicate the ARDL model incorporating climatic variables outperforms SARIMA in predictive accuracy, although SARIMA better captures seasonal patterns. These findings show temperature and wind speed are key climatic drivers of Sri Lanka’s pepper sector. The positive and persistent temperature effect suggests the need for adaptive crop management to address heat stress, while the negative impact of wind speed highlights the importance of protective measures such as windbreak systems. The identified lagged climatic effects and long-run equilibrium indicate climate shocks influence exports with delays, supporting the integration of climate indicators into export forecasting and trade planning. Strengthening climate-resilient production and climate-informed export strategies is therefore essential for sustaining sector stability and competitiveness.

Introduction: The swap operator in Simulated Annealing (SA) typically exchanges nodes randomly without knowledge of the route structure. This randomness wastes computation and slows convergence in Vehicle Routing Problems (VRPs). To incorporate spatial information, we propose an Interpretable Position-Aware Statistical-Based Swap Operator (I-PASS) that makes every swap decision spatially informed.

Method: Instead of random pairwise swaps, I-PASS, as an interpretable mechanism, employs geometric median analysis. For a candidate pair from routes A and B, it measures each node's distance to the coordinate-wise median of the remaining nodes in its current route, compared with the median of the destination route. Processed symmetrically, the swap is accepted only if it decreases individual distances to the medians or reduces the total distance across both routes, thereby optimizing spatial clustering.

Results: Experiments were run across 30 clustered VRP instances (20 customer nodes, 4 vehicles, 600 SA iterations). Results confirm that I-PASS significantly outperforms random-swap SA. The mean total route cost was reduced from 305.96 to 217.80, representing a 28.82% improvement. Additionally, the standard deviation dropped from 62.75 to 42.01, indicating more consistent solutions. Convergence curves show that I-PASS reaches lower cost values faster and maintains this advantage throughout the search.

Conclusion: Embedding spatial reasoning into a swap operator delivers substantial and consistent gains over blind random selection. I-PASS improves solution quality, reduces variance, and naturally clusters nodes into geographically coherent routes without needing a separate clustering step.

Petroleum distribution is an essential component of national infrastructure, directly influencing transportation, industry, and energy security. Network-based methods enable the analysis of structural properties of distribution networks compared with conventional routing approaches, such as point-point or standard vehicle routing problems (VRP). The study highlights the importance of network analysis using graph theory to optimize petroleum distribution systems and determine the optimal locations for establishing new depots by minimizing travel distance. This study focuses on the Gampaha District located in the western province of Sri Lanka, a region with high population density. A total of 66 fuel stations, including one terminal, were modeled as a weighted graph, where nodes represent stations and edges denote routes weighted by shortest-path distances obtained from Google Maps. While this approach reflects the road structure, it does not explicitly account for traffic congestion or road weight limits. Four centrality measures were used to identify structural properties of the network. Degree centrality revealed highly connected hubs, and closeness centrality identified the locations that can be approached within the shortest time. Betweenness centrality measures the frequency with which a particular node or edge is included in the shortest path. Eigenvector centrality determined how central the depot is to the system. To enhance regional efficiency, community structures within the network were detected using spectral clustering techniques. The eigengap heuristic was used to initially determine the optimal number of clusters. Within each cluster, the most suitable node for depot establishment was selected based on normalized centrality rankings, and its stability was measured using sensitivity analysis. The sensitivity analysis showed that the degree centrality was the most stable measurement. Four optimal clusters were identified, and the most influential nodes within each cluster were determined. Accordingly, Welisara, Gampaha, Katunayake, and Ja-Ela are the most suitable cities for establishing new depots. Overall, the findings demonstrate that the analysis is helpful for the strategic planning of petroleum distribution systems.

Introduction: Real-world time series often arise from mixtures of multiple causal generating processes, each subject to sudden emergence or parameter changes—phenomena that challenge classical models assuming a single dynamic process. While recent work introduces the Adaptive Logistic Model (ALM) to address this using nonlinear least squares, it lacks uncertainty quantification and formal convergence guarantees. This paper develops a Bayesian adaptive mixture framework that detects emergent dynamics, quantifies uncertainty, and provides theoretical convergence guarantees.

Methods: We propose a hierarchical Bayesian model where observations are generated by a mixture of nonlinear dynamic processes (e.g., logistic growth/decay) with unknown regime allocations and change points. Dirichlet process priors automatically infer the number of regimes, while Gaussian process priors capture parameter evolution. We derive a Markov chain Monte Carlo sampler and prove its geometric ergodicity under regularity conditions, establishing rates of posterior convergence. The framework outputs posterior probabilities for change-point locations and regime parameters, enabling uncertainty-aware forecasting and causal interpretation.

Results: Simulation studies demonstrate that the Bayesian framework accurately recovers true change points and regime parameters, with coverage rates matching nominal levels—an improvement over point-estimate methods. In empirical applications, the model detects known regime shifts in U.S. GDP growth (recessions), river flow data (flood events), and COVID-19 case counts (new variants), with change points aligning closely with documented external events. Forecast accuracy matches or exceeds the original ALM while providing uncertainty intervals.

Conclusions: This paper provides the first Bayesian extension of the Adaptive Logistic Model with formal convergence guarantees, enabling reliable detection of causal emergent features in multi-regime time series. The framework offers a mathematically rigorous tool for economists, hydrologists, and epidemiologists to understand when and why dynamics change.

The non-homogeneous Poisson process (NHPP) is the most widely used stochastic counting model in software reliability. The maximum likelihood estimation (MLE) is useful when the likelihood function of NHPP is available. However, it is well-known that the MLE is bound to fail in the J-shaped distributions, such as in the Weibull and gamma distribution when the shape parameter is less than 1. The maximum likelihood estimator also does not exist inside the parameter space with positive probability. The maximum likelihood equations of the NHPP-based SRM cannot be solved, if and only if the observed time to last software failure is less than twice the mean observed-time-to-failure. Furthermore, in some generalized software reliability models, it is quite hard to obtain the likelihood function in a closed form. Therefore, we apply a likelihood-free estimation approach on NHPP-based software reliability models with finite mean value function. Our method is motivated by the maximum product of spacing estimation which provides the parameter estimation without the likelihood function and intensity function. In contrast to existing likelihood-free parameter estimation methods, such as least squares estimation, our method can yield an estimator that is consistent with the underlying NHPP probability law. We have demonstrated the predictive performance of our method through real-data analysis.

The main aim of this study is to compare the accuracy of econometrics and deep learning models in forecasting the exchange rate volatility of the US dollar and Sri Lankan rupee in a multivariate framework. Foreign currency movements in a country mainly arise from export and import transactions recorded in the current account, as well as from Foreign Direct Investments, bank borrowings, and other capital inflows recorded in the capital account. In addition, foreign exchange flows are influenced by economic, political, and financial uncertainties within the country. Accordingly, as crude oil price is the largest import item representing the country’s current account, the 3-Month Dollar Sri Lankan Rupee Forward Rate, All-Share Price Index Returns, Deutscher Aktien Index Returns, Dow Jones Industrial Average Returns and Hang Seng Returns are selected as representations of the capital account via covered interest arbitrage conditions, the S&P 20 Sri Lanka Index and MSI Sri Lanka Index. These represent the relative uncertainty in the Sri Lankan market, where foreign markets use a set of independent variables with the USD/LKR exchange rate as the dependent variable. Daily data was collected from December 2011 to April 2023, and DCC-GARCH, LSTM and DCC-GARCH-LSTM hybrid models were employed for the comparison. According to the model accuracy results measured by MSE, RMSE and MAE, the USD/LKR exchange rate is best predicted using LSTM, followed by DCC-GARCH and the hybrid model. Sri Lankan exchange rates operate under managed conditions and are subject to policy changes, causing non-linear patterns that can be more accurately captured by LSTM models than by GARCH models, as the structure is primarily volatility-driven in such models. Furthermore, DCC-GARCH may not react instantly to sudden changes, but it effectively captures a change that has occurred. Hybrid models combine this slow adjustment with the faster reactions of an LSTM, which can create conflicting signals, making prediction difficult.

The environmentally constrained Profit-Based Unit Commitment Problem (PBUCP) is a large-scale, nonlinear, mixed-integer, multi-objective optimization problem essential for operational planning in deregulated power systems. It determines optimal unit commitment and dispatch schedules to simultaneously maximize profit and minimize emissions under technical and market constraints. With increasing renewable penetration and stricter environmental regulations, conventional approaches that treat emissions merely as constraints fail to support comprehensive multi-objective decision-making. Renewable variability further introduces stochasticity and nonlinearity, making traditional optimization techniques less effective. This paper proposes a hybrid Binary Artificial Bee Colony–NSGA-II (BABC–NSGA-II) framework for a renewable-integrated multi-objective PBUCP. The BABC component efficiently handles discrete ON/OFF scheduling using adaptive neighborhood exploration, while NSGA-II optimizes continuous dispatch variables to generate a well-distributed Pareto front. The model explicitly considers profit maximization and emission minimization under constraints including minimum up/down time, ramp rate limits, spinning reserve, renewable uncertainty, power balance, and generation limits. The proposed method is validated on three benchmark systems: a 10-unit system with 20% renewable penetration, a 26-unit system with 30% wind–solar integration, and a 54-unit large-scale system with 40% renewable penetration over a 24-hour horizon. Comparative analysis against Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Multi-Objective Artificial Bee Colony (MOABC), and standard NSGA-II demonstrates superior performance. The proposed approach achieves 6.8%–9.4% higher profit, 11.2%–15.7% lower emissions, and approximately 12% improvement in hypervolume index, confirming enhanced convergence and diversity. The framework provides a scalable and robust decision-support tool for environmentally sustainable power system operation.

Numerous classical probability distributions have shown little adaptability in recent decades when successfully representing the complexity of real-world data sets, including complex, symmetric, asymmetric, and skewed data sets. This has encouraged researchers to add more parameters to traditional models to create more flexible families of distributions. In this study, we introduced a novel extension of the exponential Rayleigh distribution utilizing a weighted general class of distributions (WG-ERD). The cumulative distribution function (CDF) and probability distribution function (PDF) of the proposed WG-ERD are given respectively as

G(w)=log(1+(1-exp(-aw²-cw))^b)/log(2),

and

g(w)=b(2aw+c)exp(-aw²-cw)(1-exp(-aw²-cw))^b-1/log(2)(1+(1-exp(-aw²-cw))^b).

About its density and hazard functions, the suggested model is highly adaptable and provides a variety of behaviors. The statistical and reliability properties of the proposed distribution are thoroughly established, including the quantile function, moments, skewness, kurtosis, and order statistics. Parameter estimation employs both Bayesian and classical methods, including maximum likelihood, maximum product spacing, and Bayes estimators based on various loss functions. A simulation experiment illustrating the efficacy of the suggested distribution reveals that the Bayes estimator based on the square error loss function generates more precise parameter estimates than other approaches. Finally, two data sets are analyzed to investigate the new distribution's advantage and adaptability. When the suggested model is applied to the various probability distributions currently in use, we find that it provides the best fit compared to competing distributions based on specific evaluation criteria.