Interval Forecasts Research Articles

Statistical Inference of Uncertain Autoregressive Model via the Principle of Least Squares

In the study of uncertain autoregressive models, how to estimate the unknown parameters and uncertain disturbance term in the models is always a key problem. In view of this, this paper proposes a statistical inference method based on the principle of least squares to determine the unknown parameters and uncertain disturbance term in an uncertain autoregressive model, and designs a numerical algorithm to calculate the numerical solutions of the corresponding estimators. Then, the uncertain hypothesis test is used to verify the applicability of the estimated uncertain autoregressive model, and point forecast and interval forecast are also made for the time series of future moments. Finally, a case study of the Consumer Price Index for all items in U.S. cities is provided to illustrate the effectiveness of the approach proposed in this paper.

Forecasting age distribution of life-table death counts via α-transformation

We introduce a compositional power transformation, known as an α-transformation, to model and forecast a time series of life-table death counts, possibly with zero counts observed at older ages. As a generalisation of the isometric log-ratio transformation (i.e. α = 0 ), the α transformation relies on the tuning parameter α, which can be determined in a data-driven manner. Using the Australian age-specific period life-table death counts from 1921 to 2020, the α transformation can produce more accurate short-term point and interval forecasts than the log-ratio transformation. The improved forecast accuracy of life-table death counts is of great importance to demographers and government planners for estimating survival probabilities and life expectancy and actuaries for determining annuity prices and reserves for various initial ages and maturity terms.

Testing Forecasting Properties of Different Approaches to Interval Forecasting (Using the Example of Inflation in Russia)

The paper contains the main results of the study of the quality of interval forecasts based on quantile regression and smooth quantile regression models relative to interval forecasts based on OLS models for Russian inflation. All models in the work were built from the logic of the possibility of forecasting in real-time, in connection with which the authors used inflation that was not cleared of seasonality in the expression month to the same month of the previous year. For each forecast horizon, the authors developed a separate model from lagged values of inflation and regressors so that the known values of lagged variables were always used on the right side of the equation.The authors tested a wide range of specifications – from very compact autoregressive models with one or two lags to wide ones with three additional regressors in addition to inflation lags. The best quality from the point of view of CRPS (Continuous Ranked Probability Score) metrics typical for such a task were the specifications that included the only additional variable – the USD/RUB exchange rate. As the confidence interval narrowed and at longer horizons, quantile and smooth quantile models became increasingly better relative to OLS models according to the CRPS metric.According to the authors, in the presence of inflation targeting, qualitative inflation forecasts can be used by the Bank of Russia in conducting monetary policy when forecasting future inflation dynamics and stress-testing the Russian economy. In general, this can help increase the economic agents’ confidence in the Bank of Russia within the concept of rational expectations

Metamorphosing forex: advancements in volatility forecasting using a modified fuzzy time series framework

The interplay of exchange rates among nations significantly influences both international and domestic trade, underscoring the pivotal role of the foreign exchange market (Forex) in a country's economic landscape. Forex fluctuations have a significant impact on the everyday lives of both government agencies and the public population, directly influencing a country's prosperity or misfortune. This work proposes an advanced fuzzy time series model that incorporates domain universe sub-partitioning, parameter adjustment optimization methodologies, and interval forecasting methods. We utilized this model to examine annual exchange rate patterns between the Pakistani rupee (PKR) and the US dollar (US$), comparing its forecast accuracy to that of other models. Our proposed methodology outperformed existing methodologies in terms of forecasting precision, providing stakeholders with valuable insights for making informed, data-driven business decisions that benefit both individual firms and the country overall.

When are predictions useful? A new method for evaluating epidemic forecasts

BackgroundCOVID-19 will not be the last pandemic of the twenty-first century. To better prepare for the next one, it is essential that we make honest appraisals of the utility of different responses to COVID. In this paper, we focus specifically on epidemiologic forecasting. Characterizing forecast efficacy over the history of the pandemic is challenging, especially given its significant spatial, temporal, and contextual variability. In this light, we introduce the Weighted Contextual Interval Score (WCIS), a new method for retrospective interval forecast evaluation.MethodsThe central tenet of the WCIS is a direct incorporation of contextual utility into the evaluation. This necessitates a specific characterization of forecast efficacy depending on the use case for predictions, accomplished via defining a utility threshold parameter. This idea is generalized to probabilistic interval-form forecasts, which are the preferred prediction format for epidemiological modeling, as an extension of the existing Weighted Interval Score (WIS).ResultsWe apply the WCIS to two forecasting scenarios: facility-level hospitalizations for a single state, and state-level hospitalizations for the whole of the United States. We observe that an appropriately parameterized application of the WCIS captures both the relative quality and the overall frequency of useful forecasts. Since the WCIS represents the utility of predictions using contextual normalization, it is easily comparable across highly variable pandemic scenarios while remaining intuitively representative of the in-situ quality of individual forecasts.ConclusionsThe WCIS provides a pragmatic utility-based characterization of probabilistic predictions. This method is expressly intended to enable practitioners and policymakers who may not have expertise in forecasting but are nevertheless essential partners in epidemic response to use and provide insightful analysis of predictions. We note that the WCIS is intended specifically for retrospective forecast evaluation and should not be used as a minimized penalty in a competitive context as it lacks statistical propriety. Code and data used for our analysis are available at https://github.com/maximilian-marshall/wcis.

Interval Forecasting of Carbon Price With a Novel Hybrid Multiscale Decomposition and Bootstrap Approach

ABSTRACTThis paper proposes a novel hybrid multiscale decomposition and bootstrap approach for carbon price interval forecasting, aiming to overcome the limitations of traditional carbon price point forecasting. The original carbon price is decomposed into simple modes using the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), and various bootstrap methods are applied to perform a random sampling with a replacement on each mode, generating pseudo datasets forecasted using extreme gradient boosting (XGB). The forecasting values of all modes are then integrated into the original carbon price interval forecasting values. The empirical results, based on samples from China's Guangdong and Hubei carbon markets, provide compelling evidence of the effectiveness of our model. It achieves higher forecasting accuracy, higher interval coverage, and narrower forecasting intervals than currently popular prediction models, instilling confidence in its potential to enhance carbon price forecasting.

Point and interval forecasting approach for short-term urban subway passenger flow based on residual term decomposition and fuzzy information granulation

Accurate forecasting information of short-term subway passenger flow is an important scientific reference for daily operations and urban management. The rapid time-varying nature of subway passenger flow caused by various factors that affect travel behavior poses a huge challenge to accurate forecasting. The complexity and uncertainty of data mainly focus on residual terms that deeply reflect the fluctuations. Reasonably mining the residual terms has become the key to achieving accurate forecasting. Therefore, this paper proposes a sophisticated decomposition strategy to extract and analyze the useful information hidden in the residual terms. Firstly, the potential trend, seasonal and residual terms from the original data are extracted by seasonal-trend decomposition procedure based on loess. Secondly, the residual term is decomposed into a series of subcomponents with different frequency features by symplectic geometric mode decomposition. Thirdly, these subcomponents are classified into three clusters based on fuzzy C-means clustering (FCM), and then the corresponding forecasting models are matched to the three obtained clusters and trend and seasonal terms. Finally, based on a decomposition-ensemble framework and information granulation for high-frequency components, we have established point and interval forecasting approach, respectively. Three experiments on real data sets in Beijing, Guangzhou and Shenzhen are conducted to verify the performance of the proposed approach, and the experimental results show that our approach is superior to all benchmark models and contributes to improving operational management and service quality.

Multiple yield curve modeling and forecasting using deep learning

AbstractThis manuscript introduces deep learning models that simultaneously describe the dynamics of several yield curves. We aim to learn the dependence structure among the different yield curves induced by the globalization of financial markets and exploit it to produce more accurate forecasts. By combining the self-attention mechanism and nonparametric quantile regression, our model generates both point and interval forecasts of future yields. The architecture is designed to avoid quantile crossing issues affecting multiple quantile regression models. Numerical experiments conducted on two different datasets confirm the effectiveness of our approach. Finally, we explore potential extensions and enhancements by incorporating deep ensemble methods and transfer learning mechanisms.

A combined model based on decomposition and reorganization, weight optimization algorithms for carbon price point and interval prediction

With the increasing environmental and climate problems caused by global warming, more and more countries are paying attention to reducing carbon emissions. Accurately predicting carbon prices can help create a stable carbon market and mitigate global warming. Therefore, a combined model which has four modules is proposed, they are the decomposition of carbon price series, reorganization of subsequences, point forecasting, and interval forecasting. For data preprocessing of decomposition, it can reduce the non-linearity of sequences. Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) decomposes the original series into multiple Intrinsic Mode Functions (IMFs). Reorganization uses SampEn (SE) and T-test to reorganize the decomposed sequence into three new sequences according to actual meanings. For point prediction, the combined prediction model uses Particle Swarm Optimization (PSO) to obtain the optimal weights of Recurrent Neural Network (RNN), Long Short-Term Memory (LSTM), and Gated Recurrent Unit (GRU), which can integrate their strengths to improve prediction accuracy. Interval prediction is based on fitting the optimal distribution function by Maximum Likelihood Estimate (MLE). Different significance intervals can be obtained from the point prediction results and the distribution function. Empirical and comparative experiments show this combined model has excellent point and interval predictions. Besides, the significance and stability are demonstrated. Finally, some valuable references are provided for policymakers.

A Deep Learning Quantile Regression Photovoltaic Power-Forecasting Method under a Priori Knowledge Injection

Accurate and reliable PV power probabilistic-forecasting results can help grid operators and market participants better understand and cope with PV energy volatility and uncertainty and improve the efficiency of energy dispatch and operation, which plays an important role in application scenarios such as power market trading, risk management, and grid scheduling. In this paper, an innovative deep learning quantile regression ultra-short-term PV power-forecasting method is proposed. This method employs a two-branch deep learning architecture to forecast the conditional quantile of PV power; one branch is a QR-based stacked conventional convolutional neural network (QR_CNN), and the other is a QR-based temporal convolutional network (QR_TCN). The stacked CNN is used to focus on learning short-term local dependencies in PV power sequences, and the TCN is used to learn long-term temporal constraints between multi-feature data. These two branches extract different features from input data with different prior knowledge. By jointly training the two branches, the model is able to learn the probability distribution of PV power and obtain discrete conditional quantile forecasts of PV power in the ultra-short term. Then, based on these conditional quantile forecasts, a kernel density estimation method is used to estimate the PV power probability density function. The proposed method innovatively employs two ways of a priori knowledge injection: constructing a differential sequence of historical power as an input feature to provide more information about the ultrashort-term dynamics of the PV power and, at the same time, dividing it, together with all the other features, into two sets of inputs that contain different a priori features according to the demand of the forecasting task; and the dual-branching model architecture is designed to deeply match the data of the two sets of input features to the corresponding branching model computational mechanisms. The two a priori knowledge injection methods provide more effective features for the model and improve the forecasting performance and understandability of the model. The performance of the proposed model in point forecasting, interval forecasting, and probabilistic forecasting is comprehensively evaluated through the case of a real PV plant. The experimental results show that the proposed model performs well on the task of ultra-short-term PV power probabilistic forecasting and outperforms other state-of-the-art deep learning models in the field combined with QR. The proposed method in this paper can provide technical support for application scenarios such as energy scheduling, market trading, and risk management on the ultra-short-term time scale of the power system.

Mass Conservative Time-Series GAN for Synthetic Extreme Flood-Event Generation: Impact on Probabilistic Forecasting Models

The lack of data on flood events poses challenges in flood management. In this paper, we propose a novel approach to enhance flood-forecasting models by utilizing the capabilities of Generative Adversarial Networks (GANs) to generate synthetic flood events. We modified a time-series GAN by incorporating constraints related to mass conservation, energy balance, and hydraulic principles into the GAN model through appropriate regularization terms in the loss function and by using mass conservative LSTM in the generator and discriminator models. In this way, we can improve the realism and physical consistency of the generated extreme flood-event data. These constraints ensure that the synthetic flood-event data generated by the GAN adhere to fundamental hydrological principles and characteristics, enhancing the accuracy and reliability of flood-forecasting and risk-assessment applications. PCA and t-SNE are applied to provide valuable insights into the structure and distribution of the synthetic flood data, highlighting patterns, clusters, and relationships within the data. We aimed to use the generated synthetic data to supplement the original data and train probabilistic neural runoff model for forecasting multi-step ahead flood events. t-statistic was performed to compare the means of synthetic data generated by TimeGAN with the original data, and the results showed that the means of the two datasets were statistically significant at 95% level. The integration of time-series GAN-generated synthetic flood events with real data improved the robustness and accuracy of the autoencoder model, enabling more reliable predictions of extreme flood events. In the pilot study, the model trained on the augmented dataset with synthetic data from time-series GAN shows higher NSE and KGE scores of NSE = 0.838 and KGE = 0.908, compared to the NSE = 0.829 and KGE = 0.90 of the sixth hour ahead, indicating improved accuracy of 9.8% NSE in multistep-ahead predictions of extreme flood events compared to the model trained on the original data alone. The integration of synthetic training datasets in the probabilistic forecasting improves the model’s ability to achieve a reduced Prediction Interval Normalized Average Width (PINAW) for interval forecasting, yet this enhancement comes with a trade-off in the Prediction Interval Coverage Probability (PICP).

Seasonal solar irradiance forecasting using artificial intelligence techniques with uncertainty analysis

Renewable integration in utility grid is crucial in the current energy scenario. Optimized utilization of renewable energy can minimize the energy consumption from the grid. This demands accurate forecasting of renewable contribution and planning. Most of the researches aim to find a suitable forecasting model in terms of accuracy and error metrics. However, the uncertainty and variability in these forecasts are also significant. This work combines point forecast with interval forecast to provide comprehensive information about the forecast uncertainty. In this work, solar irradiance forecasting is carried out using artificial intelligence (AI) techniques. Forecasting is done using seasonal auto-regressive moving average with exogenous factors (SARIMAX), support vector regression (SVR), long short term memory (LSTM) techniques and performance is evaluated. SVR model exhibited the best performance with R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^2$$\\end{document} values of 0.97 and 0.96 for winter and summer respectively and 0.85 for monsoon and post-monsoon seasons. This is followed by forecast error distribution studies and uncertainty analysis. For this, SVR forecast error data is fitted using laplace distribution. Uncertainty study is carried out using confidence intervals and coverage rates. Excellent coverage rates are obtained for various confidence levels for all seasons, indicating the appropriate fitting of error distribution. For the narrow 85% confidence band, coverage rates of 89%, 95%, 90%, and 88% are obtained for winter, summer, monsoon and post-monsoon respectively. The work emphasizes the need for error-distribution studies, modeling of forecast errors and their application in providing reliable forecast intervals with the perspective of enhancing system reliability.

Forecasting Age- and Sex-Specific Survival Functions: Application to Annuity Pricing

We introduce the function principal component regression (FPCR) forecasting method to model and forecast age-specific survival functions observed over time. The age distribution of survival functions is an example of constrained data whose values lie within a unit interval. Because of the constraint, such data do not reside in a linear vector space. A natural way to deal with such a constraint is through an invertible logit transformation that maps constrained onto unconstrained data in a linear space. With a time series of unconstrained data, we apply a functional time-series forecasting method to produce point and interval forecasts. The forecasts are then converted back to the original scale via the inverse logit transformation. Using the age- and sex-specific survival functions for Australia, we investigate the point and interval forecast accuracies for various horizons. We conclude that the functional principal component regression (FPCR) provides better forecast accuracy than the Lee–Carter (LC) method. Therefore, we apply FPCR to calculate annuity pricing and compare it with the market annuity price.

Application of Quantum Neural Network for Solar Irradiance Forecasting: A Case Study Using the Folsom Dataset, California

Merging machine learning with the power of quantum computing holds great potential for data-driven decision making and the development of powerful models for complex datasets. This area offers the potential for improving the accuracy of the real-time prediction of renewable energy production, such as solar irradiance forecasting. However, the literature on this topic is sparse. Addressing this knowledge gap, this study aims to develop and evaluate a quantum neural network model for solar irradiance prediction up to 3 h in advance. The proposed model was compared with Support Vector Regression, Group Method of Data Handling, and Extreme Gradient Boost classical models. The proposed framework could provide competitive results compared to its competitors, considering forecasting intervals of 5 to 120 min ahead, where it was the fourth best-performing paradigm. For 3 h ahead predictions, the proposed model achieved the second-best results compared with the other approaches, reaching a root mean squared error of 77.55 W/m2 and coefficient of determination of 80.92% for global horizontal irradiance forecasting. The results for longer forecasting horizons suggest that the quantum model may process spatiotemporal information from the input dataset in a manner not attainable by the current classical approaches, thus improving forecasting capacity in longer predictive windows.

A hybrid model for point and interval forecasting of agricultural price based on the decomposition-ensemble and KDE

A hybrid model for point and interval forecasting of agricultural price based on the decomposition-ensemble and KDE

Сравнение годовых грузооборотов железнодорожного и автомобильного транспортов РФ: моделирование, верификация, точечный и интервальный прогноз

Purpose: a study was conducted comparing the freight turnover of rail and road transport in the Russian Federation over the last twenty years and a forecast assessment was made. The comparative assessment was carried out due to the excess of freight turnover by rail over road transport. The study includes the following main stages: modeling based on the observed annual values of cargo turnover; model verification by comparing model and experimental values in years not involved in the construction of the model; model point and interval forecasting; analysis of the forecast trend of changes. The mathematical model of the excess of freight turnover by rail over road transport from the year of cargo delivery is based on a sample of statistical data from Rosstat from 2004 to 2021. The model was verified based on the results of cargo turnover in 2022 and 2023. Methods: to build a mathematical model, test it and carry out forecasting, a number of theoretical provisions of econometrics were used, including dynamic series, regression analysis, the least squares method, point and interval forecasting. The applied calculation apparatus was used using the Excel PPP. Results: the quality of the constructed model was evaluated by the following indicators: approximation errors, correlation and determination coefficients, and the Fisher criterion. A comparison of the observed and model values of excess cargo turnover showed that in the period from 2004 to 2021, according to the above indicators, they have minor deviations. The verification performed confirmed the satisfactory simulation. Practical significance: the conducted research has shown that it is possible to predict the results of transport operations according to the described algorithm and it is justified to make management decisions at the beginning of their implementation. Recommendations: to continue improving the modeling and forecasting apparatus, taking into account the assessment of risk indicators.

Leveraging the ETAS model to forecast mining microseismicity

SUMMARY Mining operations result in changes of the subsurface stress field that can lead to the occurrence of microseismic events. The development of strategies for forecasting and avoidance of significant events is crucial for safe and efficient operations of mines. One such example, discussed here is the observed induced microseismicity in soft rock potash mines. It is primarily driven by the rock excavations but can also be triggered by preceding events or can result from the delayed effects of plastic creep of soft rocks. Therefore, it is important from seismic hazard assessment and risk mitigation points of view to understand the statistical aspects of microseismicity in potash or other types of mines. In this study, the temporal evolution of the induced microseismicity from a potash mine in Saskatchewan is analysed and modelled. Specifically, the epidemic type aftershock sequence model is used to approximate the occurrence rate of the induced mining microseismicity. The estimated parameters signify that the microseismicity displays swarm-type characteristics with limited inter-event triggering. Moreover, the Bayesian predictive framework is used to compute the probabilities of the occurrences of the largest expected events above a certain magnitude for prescribed forecasting time intervals during the evolution of the sequence. This approach for computing the probabilities allows one to incorporate fully the uncertainties of the model parameters. The Markov Chain Monte Carlo sampling of the posterior distribution are used to generate parameter chains to quantify their variability. Furthermore, several statistical tests are conducted to assess the credibility of the obtained retrospective forecasts compared to the observed microseismicity. The obtained results show that the developed approach can accurately forecast the number of events and intensity of the sequence. It also provides a framework for computing the probabilities for the largest expected events.

Carbon price forecasting using leaky integrator echo state networks with the framework of decomposition-reconstruction-integration

Carbon trading is one of the strategies to achieve the goal of dual carbon. However, carbon prices exhibit non-stationary and nonlinear characteristics, posing significant challenges for accurate forecasting. Inspired by the ideas of “division and conquest” and “granularity reconstruction”, a decomposition-reconstruction-integration framework is built for carbon price forecasting. The proposed model leverages a comprehensive ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and sample entropy (SE) to extract features from two perspectives of time and complexity, and thus the one-dimensional time series of carbon price is reconstructed into different items. Subsequently, leaky integrator echo state networks (LiESNs) are utilized to forecast each of these items individually. These individual forecasts are then integrated to generate the final point forecast. To evaluate the uncertainty of the point forecast, a Gaussian process model is applied to interval forecast. The proposed model comprehensively considers the time series features of carbon price including the temporal modal features in the decomposition stage, the entropy probability features in the reconstruction stage, and the essential temporal dependencies via dynamic reservoir of interconnected neurons in the integration forecasting stage. Experimental results demonstrate the model's exceptional forecasting performance, surpassing the effectiveness of alternative approaches.

Quantifying and reducing flood forecast uncertainty by the CHUP-BMA method

Abstract. The Bayesian model averaging (BMA), hydrological uncertainty processor (HUP), and HUP-BMA methods have been widely used to quantify flood forecast uncertainty. This study proposes the copula-based hydrological uncertainty processor BMA (CHUP-BMA) method by introducing a copula-based HUP in the framework of BMA to bypass the need for a normal quantile transformation of the HUP-BMA method. The proposed ensemble forecast scheme consists of eight members (two forecast precipitation inputs; two advanced long short-term memory, LSTM, models; and two objective functions used to calibrate parameters) and is applied to the interval basin between the Xiangjiaba and Three Gorges Reservoir (TGR) dam sites. The ensemble forecast performance of the HUP-BMA and CHUP-BMA methods is explored in the 6–168 h forecast horizons. The TGR inflow forecasting results show that the two methods can improve the forecast accuracy over the selected member with the best forecast accuracy and that the CHUP-BMA performs much better than the HUP-BMA. Compared with the HUP-BMA method, the forecast interval width and continuous ranked probability score metrics of the CHUP-BMA method are reduced by a maximum of 28.42 % and 17.86 % within all forecast horizons, respectively. The probability forecast of the CHUP-BMA method has better reliability and sharpness and is more suitable for flood ensemble forecasts, providing reliable risk information for flood control decision-making.

Short-term wind farm cluster power point-interval prediction based on graph spatio-temporal features and S-Stacking combined reconstruction

Wind energy is becoming increasingly competitive, Accurate and reliable multi-engine wind power forecasts can reduce power system operating costs and improve wind power consumption capacity. Existing research on wind power forecasting has neglected the importance of interval forecasting using clusters of wind farms to capture spatial characteristics and the objective selection of forecasting sub-learners, leading to increased uncertainty and risk in system operation. This paper proposes a new “decomposition-aggregation-multi-model parallel prediction” method. The data set is pre-processed by a decomposition-aggregation strategy and spatial feature extraction, and then a Stacking model with multiple parallel sub-learners selected by bootstrap method is used for point and interval forecasting. Experiments and discussions are conducted based on 15-min resolution wind power data from a cluster dataset of a wind farm in northwest China. The experimental results indicate that the method achieves higher accuracy and reliability in both point prediction and interval prediction than other comparative models, with a root mean square error value of 7.47 and an average F value of 1.572, which can provide a reliable reference for power generation planning from wind farm clusters.