Predictive Probability Density Research Articles

A multi-step probability density prediction model based on gaussian approximation of quantiles for offshore wind power

With the increasing utilization of offshore wind power, accurate prediction of offshore wind power is crucial for preventive control and scheduling. In this paper, a new hybrid probability density model is proposed for multi-step offshore wind power prediction, including time varying filter based empirical mode decomposition (TVFEMD), approximate entropy (AE), Yeo–Johnson transform quantile regression (YJQR) and gaussian approximation of quantile (GAQ). Firstly, TVFEMD decomposition and AE theory are used to preprocess the original data for reducing the complexity and modeling workload. Secondly, the 16-step ahead offshore wind power is predicted using YJQR, in which the model structures established for each component are selected by grid search for comprehensive optimization to ensure the best prediction performance. Finally, the GAQ method is adopted to construct probability density curves for the 16-step cumulative quantile prediction results. The variance of the probability density curves in each step is adjusted to optimize the interval prediction results, resulting in more robust and integrated prediction results. Taking the historical offshore wind power data collected by German transmission system operator 50 Hertz as an example, the model has higher prediction accuracy and stability on the basis of obtaining reasonable quantile estimation results in multi-step offshore wind power probabilistic prediction.

Probabilistic Forecast of Visibility at Gimpo, Incheon, and Jeju International Airports Using Weighted Model Averaging

In this study, weighted model averaging (WMA) was applied to calibrating ensemble forecasts generated using Limited-area ENsemble prediction System (LENS). WMA is an easy-to-implement post-processing technique that assigns a greater weight to the ensemble member forecast that exhibits better performance; it is used to provide probabilistic visibility forecasting in the form of a predictive probability density function for ensembles. The predictive probability density function is a mixture of discrete point mass and two-sided truncated normal distribution components. Observations were obtained at Gimpo, Incheon, and Jeju International Airports, and 13 ensemble member forecasts were obtained using LENS, for the period of December 2018 to June 2019. Prior to applying WMA, a reliability analysis was conducted using rank histograms and reliability diagrams to identify the statistical consistency between the ensembles and the corresponding observations. The WMA method was then applied to each raw ensemble model, and a weighted predictive probability density function was proposed. Performances were evaluated using the mean absolute error, the continuous ranked probability score, the Brier score, and the probability integral transform. The results showed that the proposed method provided improved performance compared with the raw ensembles, indicating that the raw ensembles were well calibrated using the predicted probability density function.

Calibrating the CAMS European multi-model air quality forecasts for regional air pollution monitoring

The CAMS air quality multi-model forecasts have been assessed and calibrated for PM10, PM2.5, O3, NO2, and CO against observations collected by the Regional Monitoring Network of the Liguria region (northwestern Italy) in the years 2019 and 2020. The calibration strategy used in the present work has its roots in the well-established Ensemble Model Output Statistics (EMOS) through which a raw ensemble forecast can be accurately transformed into a predictive probability density function, with a simultaneous correction of biases and dispersion errors. The strategy also provides a calibrated forecast of model uncertainties. As a result of our analysis, the key role of pollutant real-time observations to be ingested in the calibration strategy clearly emerge especially in the shorter look-ahead forecast hours. Our dynamic calibration strategy turns out to be superior with respect to its analogous where real-time data are not taken into account. The best calibration strategy we have identified makes the CAMS multi-model forecast system more reliable than other raw air quality models running at higher spatial resolution which exploit more detailed information from inventory emission. We expect positive impacts of our research for identifying and set up reliable and economic air pollution early warning systems.

A novel PM2.5 concentrations probability density prediction model combines the least absolute shrinkage and selection operator with quantile regression.

PM2.5 has a significant negative impact on human health and atmospheric quality, and accurate prediction of its concentration is necessary. When using common point prediction models for PM2.5 concentration prediction, the influence of various uncertainties on PM2.5 concentrations makes the prediction results suffer from poor accuracy. To address this issue, this paper proposes the quantile regression neural network (QRNN) model based on the least absolute shrinkage and selection operator (LASSO), combined with kernel density estimation (KDE) for probabilistic density prediction of PM2.5 concentrations. The model uses LASSO regression to select the influential factors, and then the quartiles of daily PM2.5 concentrations calculated by the QRNN model are imported into the KDE model to obtain the probability density predictions of PM2.5 concentrations. In the paper, empirical analyses are carried out with the cities of Beijing and Jinan in China as well as six other datasets, and the prediction performance of the model is assessed by using evaluation criteria in both point prediction and interval prediction. The simulation reveals that the predictive performance of the LASSO-QRNN-KDE model is well, and the model is not only effective in filtering high-dimensional data, but also has a higher accuracy compared to common research models. In addition, the model is able to describe the uncertainty of PM2.5 concentration fluctuations and carry more information on the variation of PM2.5 concentrations, which can provide a novel and excellent PM2.5 concentration prediction tool for relevant policy makers.

Wind energy forecasting by fitting predicted probability density functions of wind speed measurements

Wind energy forecasting by fitting predicted probability density functions of wind speed measurements

Comparison of Gated Recurrent Unit vs. Mixture Density Network in Insulin Sensitivity Prediction

There are several methods for the prediction of future values of a time series, or in general, for the probability density prediction problem. In this paper two neural network prediction models are compared: Mixture Density Network (MDN) and Gated Recurrent Unit (GRU) in the prediction of the insulin sensitivity (SI) of a patient under intensive care.The basic difference between the two methods is that while MDN is a stateless network model the GRU is a stateful model. In the study, both of the models predict the probability density function of the future SI value, but the MDN considers the physiological process like a Markov-chain: the future SI value depends lowly on the previous value. In contrast, the GRU model attempts to model the longer-term dependencies between subsequent SI values in its internal states and improve the prediction accuracy based on the information derived from the previous SI values.In the study we have found that the GRU implemented stateful prediction improved the prediction accuracy by 10 percent, showing the prediction power of the GRU model in physiological problems and proving the existence of longer-term dependencies in the SI time series. These outcomes confirm previously published results showing that the insulin sensitivity prediction can be improved considering the longer-term history of the patients.

Analysis of wind farm participation in frequency regulation considering multi-market interests

For wind farms to participate in the combined energy-frequency regulation (E-FR) market, wind farms are considered as a combination of both power generation and frequency regulation capability, and wind farms bid in both the energy market and the FR market. In this paper, the impact of different bidding decisions on the distribution of wind farm revenues is analyzed in a process where the interest of two markets is played against each other. A wind power probability density prediction model of kernel extreme learning machine (KELM)-particle swarm optimization (PSO)-adaptive diffusion kernel density estimation (AKDE) is established using an improved extreme learning machine (KELM) with good fitting ability and the AKDE method, wind farm bidding is carried out on the premise of wind power probability prediction, which is optimally solved by the multi-objective quantum genetic algorithm, and the optimization results are filtered using entropy-fuzzy C-means clustering. Based on the actual wind farm operation data for simulation analysis, the model analyzes the benefits of wind farm participation in the joint market from different preference perspectives, which is a reference for wind farm participation in bidding decisions.

Day-ahead load probability density forecasting using monotone composite quantile regression neural network and kernel density estimation

Due to the aggregation of renewable resources and the fluctuation of demand side, exploring more advanced approaches to maintain the accuracy of short-term load forecasting has become a momentous task. The neural network-based quantile regression probability density forecasting can reveal the prediction uncertainty, which is critical in building energy-saving and reliable power grid. However, quantile curves estimated by previous quantile regression probability density forecasting model may cross each other, violating the basic properties of quantile and jeopardizing the flexibility of modeling. For counteracting this disadvantage, a probability density prediction method based on monotone composite quantile regression neural network and Gaussian kernel function (MCQRNNG) for day-ahead load is proposed. The proposed method guarantees non-crossing by simultaneously estimating multiple nonlinear quantile regression functions to obtain better estimation accuracy. Besides, three-dimensional grid search is adopted to adaptively determine the optimal network structure. Taking real load data carrying quantile crossing from Ottawa in Canada, Baden-Württemberg in Germany and a certain region in China, the performance of the MCQRNNG model is verified from three aspects of point, interval and probability prediction. Results show that the proposed model significantly outperforms the comparison model on the basis of effectively extricating the quantile crossing for power load forecasting.

A novel time-power based grey model for nonlinear time series forecasting

To deal with various nonlinear issues in real applications, a novel time-power based grey model is put forward. However, in the original form of this model, the time-power parameter α normally equals to an integer, and then the analytical expression of the time response function will be obtained. Otherwise, if the parameter α equals to a non-integer, one cannot obtain the concrete time response function for future estimations. This situation may significantly restrict the applications of this grey model. To address such drawbacks, an optimized version is designed in this work. In the proposed model, a simplified solution to the differential equation is derived by using the definite integral technique. Furthermore, for improving accuracy, the time-power parameter α is optimized by utilizing the Particle Swarm Optimization algorithm based on the model parameter packages. Subsequently, the efficacy and practicality of this simplified function have been verified by numerical simulations and experimental studies. Moreover, the method of probability density prediction is employed for verifying the reliability and stability of the proposed model for the first time when predicting the settlement of the soft-clay subgrade on an expressway. The demonstration cases illustrate that the quantitative improvements over forecasts of the proposed model are even more pronounced with a level accuracy of 2.29% and 1.19% MAPE values in the fitted and predicted periods, respectively, which can significantly increase the predicting accuracy by more than 10% with respect to the other benchmarks. Therefore, the new proposed model not only has greater application fields and prospects but also achieves higher and more reliable predicting accuracy with the optimal α under the support of the Particle Swarm Optimization algorithm, compared with the competing models.

Assessing uncertainty for decision‐making in climate adaptation and risk mitigation

AbstractFuture water availability or crop yield studies, tied to statistics of river flow, precipitation, temperature or evaporation over medium to long‐term horizons, are becoming frequent in climate impact and risk analysis. During the last two decades, access to multi‐system integration of climate models has given rise to the concept of using model ensembles to issue probabilistic climatological projections. These probabilistic projections have not yet been exploited to the full extent in decision support, and are still used to mainly quantify uncertainty bands only for selected climate variables and indicators. One of the reasons of this limited use is the fact that the multi‐system ensemble dispersion is sub‐optimal and does not provide an accurate and reliable representation of the predictive probability density, which is essential for rational decision support under uncertain conditions. The aims of this paper are twofold. First, it seeks to highlight the potential benefits of using climate projections in conjunction with Bayesian paradigms towards educated decision‐making. Second, it discusses how to appropriately formulate probabilistic forecasts by coherently integrating information contained in climate projection ensembles with observations to improve the estimation of the probability density function of future climate states. The results show that the proposed Bayesian approach yields unbiased and sharper predictive distributions for temperature with respect to using the unprocessed ensemble distribution. It also yields improved predictive densities with respect to the Reliability Ensemble Averaging (REA) method.

A runoff probability density prediction method based on B-spline quantile regression and kernel density estimation

Exact and dependable runoff forecasting plays a vital role in water resources management and utilization. This paper proposes a B-spline quantile regression probability density prediction method to predict future runoff and quantify the uncertainty of prediction. The method includes three steps. First, the B-spline function is used to perform spline processing on the runoff data. Secondly, the spline-processed training data is input into the quantile regression model to calculate the parameters of the B-spline quantile regression model, and the successfully constructed B-spline quantile regression model is combined with kernel density estimation to construct a probability density prediction method. Finally, the constructed B-spline quantile regression probability density prediction method is used to forecast future runoff flow, and quantitatively analyze the relevant prediction uncertainty. Six evaluation indicators are constructed, among which the root mean square error, the deterministic coefficient, the pass rate are the evaluation metrics of point predictions based on probability mean, median and mode; the prediction interval coverage probability, prediction interval normalized average width and continuous ranked probability score are the evaluation criteria of interval predictions and probabilistic forecasting. The presented method is able to depict the probability density curve of future runoff flow, and obtain more comprehensive information than point forecasting and interval predictions. As a case study, this method is applicable to the Shigu station of the Jinsha River in China. The results show that the results of the proposed method are better than that of existing some state-of-the-art methods. From the perspective of application, the pass rate and the deterministic coefficient of the method have reached the grade A of accuracy. Therefore, the B-spline quantile regression model provides an alternative scheme to runoff prediction.

Trajectory Generation by Chance-Constrained Nonlinear MPC With Probabilistic Prediction.

Continued great efforts have been dedicated toward high-quality trajectory generation based on optimization methods; however, most of them do not suitably and effectively consider the situation with moving obstacles; and more particularly, the future position of these moving obstacles in the presence of uncertainty within some possible prescribed prediction horizon. To cater to this rather major shortcoming, this work shows how a variational Bayesian Gaussian mixture model (vBGMM) framework can be employed to predict the future trajectory of moving obstacles; and then with this methodology, a trajectory generation framework is proposed which will efficiently and effectively address trajectory generation in the presence of moving obstacles, and incorporate the presence of uncertainty within a prediction horizon. In this work, the full predictive conditional probability density function (PDF) with mean and covariance is obtained and, thus, a future trajectory with uncertainty is formulated as a collision region represented by a confidence ellipsoid. To avoid the collision region, chance constraints are imposed to restrict the collision probability, and subsequently, a nonlinear model predictive control problem is constructed with these chance constraints. It is shown that the proposed approach is able to predict the future position of the moving obstacles effectively; and, thus, based on the environmental information of the probabilistic prediction, it is also shown that the timing of collision avoidance can be earlier than the method without prediction. The tracking error and distance to obstacles of the trajectory with prediction are smaller compared with the method without prediction.

Uncertainty analysis of wind power probability density forecasting based on cubic spline interpolation and support vector quantile regression

Accurate forecasting of wind power plays an important role in an effective and reliable power system. However, the fact of non-schedulability and fluctuation of wind power significantly increases the uncertainty of power systems. The output power of a wind farm is usually mixed with uncertainties, which reduce the effectiveness and accuracy of wind power forecasting. In order to handle the uncertainty of wind power, this paper first proposes to conduct outlier detection and reconstruct data before the prediction. Then, a wind power probability density forecasting method is proposed, based on cubic spline interpolation and support vector quantile regression (CSI-SVQR), which can better estimate the whole wind power probability density curve. However, the probability density prediction method can not acquire the optimal point prediction and interval prediction results at the same time. In order to analyze the uncertainty of wind power, the present study considers the prediction results from the perspective of probabilistic point prediction and interval prediction respectively. Three sets of real-world wind power data from Canada and China are used to validate the CSI-SVQR method. The results show that the proposed method not only efficiently eliminates the outliers of wind power but also provides the probability density function, offering a complete description of wind power generation fluctuation. Furthermore, more accurate point prediction and prediction interval (PI) can be obtained compared to existing methods. Wilcoxon signed rank test is used to verify that CSI can improve the performance of forecasting methods.

Joint Probability Density Prediction for Multiperiod Thermal Ratings of Overhead Conductors

For an overhead conductor, meteorological correlations exist among the meteorological elements that dominantly determine its thermal rating, and temporal correlations exist among the thermal ratings in sequential time periods. It is necessary to exploit these correlations to improve the performance of the probabilistic prediction of thermal ratings. To this end, a copula-based method of joint probability density prediction for multiperiod thermal ratings (JPDP-MPTR) is presented in this paper. In this method, the probability density functions (PDFs) of the thermal ratings for every 15 minutes over a 1-hour horizon are first predicted individually, considering the correlations among meteorological elements. Then, the joint probability density function (JPDF) of the multiperiod thermal ratings is further formulated based on copula theory. Finally, the probability distributions of the thermal ratings in the predicted time periods are estimated via joint sampling based on the JPDF. Numerical simulations based on actual meteorological data collected around an overhead conductor show that the proposed method can significantly improve prediction results through the integration of meteorological and temporal correlations into the probabilistic prediction of the thermal rating.

Stochastic distributed microgrid energy management based on over‐relaxed alternative direction method of multipliers

This study aims to design a stochastic distributed energy management framework taking into account the information privacy of autonomous agents. To develop this framework, first, the application of a scenario-based method on the predicted probability density function (PDF) is suggested to deal with uncertainties of the low-scale loads, renewable generations, i.e. wind and photovoltaic generations, and electricity market prices. Then, an alternative direction method of multipliers (ADMM) based method, namely over-relaxed ADMM, is presented to optimise the operational set points considering each agent benefits and technical constraints. In this framework, the agents participate in scheduling programs without sharing influential information and corresponding historical data. The presented framework is tested on a realistic small-scale microgrid (MG) system and real historical data. The performance and efficiency are verified by comparison of the proposed over-relaxed ADMM method application with the application of standard ADMM and analytic targeting cascading in terms of accuracy and convergence speed. Furthermore, higher accuracy and lower computational complexity of predictive PDF-based scenario generation techniques in distributed MG energy management are verified by comparison with distributed energy management based on predefined PDF and historical data.

Probability density forecasting of wind power based on multi-core parallel quantile regression neural network

The large-scale utilization of wind energy brings severe challenges to the dispatching operation of power systems. Currently, the probability density prediction method combining quantile regression neural network (QRNN) with Epanechnikov kernel function is an excellent algorithm for wind power prediction, which can give the comprehensive probability distribution of future wind power and effectively quantify the uncertainty of wind power generation. However, existing probability density prediction methods process data sequentially in different quantiles, and computational time costs multiply with the increase of training data. It affects the practicality of the probability density prediction method. To overcome this issue, this paper proposes a multi-core parallel quantile regression neural network (MPQRNN) based on parallel master–slave (MS) model. The algorithm divides the complex prediction tasks at all quantiles into multiple parallel sub-tasks, which are independently run on different cores, so that performance advantages of the multi-core CPU can be fully utilized for improving the computational efficiency of the joint operation model. We compare four different scale sample sets under different process numbers. The influence of different CPU core numbers on the parallel performance of MPQRNN are analyzed by the algorithmic nature of speedup and parallel efficiency. To demonstrate the effectiveness of the proposed model, comparative experiments of other four traditional models are carried out on data sets. The simulation results demonstrate that the MPQRNN can not only improve the training efficiency of QRNN, but also obtain precise results of wind power forecasting, showing potential value and utility for complex power system.

A Bayesian Representation of the Storm Approach Probability Based on Operational Track Forecast Errors

AbstractThis study provides a statistical review on the forecast errors of tropical storm tracks and suggests a Bayesian procedure for updating the uncertainty about the error. The forecast track errors are assumed to form an axisymmetric bivariate normal distribution on a two-dimensional surface. The parameters are a mean vector and a covariance matrix, which imply the accuracy and precision of the operational forecast. A Bayesian method improves quantifying the varying parameters in the bivariate normal distribution. A normal-inverse-Wishart distribution is employed to determine the posterior distribution (i.e., the weights on the parameters). Based on the posterior distribution, the predictive probability density of track forecast errors is obtained as the marginal distribution. Here, “storm approach” is defined for any location within a specified radius of a tropical storm. Consequently, the storm approach probability for each location is derived through partial integration of the marginal distribution within the forecast storm radius. The storm approach probability is considered a realistic and effective representation of storm warning for communicating the threat to local residents since the location-specific interpretation is available on a par with the official track forecast.

Configuration of Bayesian Model Averaging Training Window to Improve Seasonal Rainfall Ensemble Prediction

Bayesian Model Averaging (BMA) is a statistical post-processing method for producing probabilistic forecasts from ensemble prediction in the form of predictive Probability Density Function (PDF). It is known that BMA is able to improve the reliability of the probabilistic forecast of short-and medium-range rainfall forecasts. This study aims to develop the application of BMA to calibrate long-range forecast in order to improve the quality of the seasonal forecast in Indonesia. The seasonal forecast that has been used is monthly rainfall from the output of the ensemble prediction European Center for Medium-Range Weather Forecasts (ECMWF) system 4 model (ECS4). This model was calibrated against observational data at 26 stations of Agency for Meteorology, Climatology and Geophysics of Republic of Indonesia (BMKG) over Java Island in 1981-2018. BMA predictive PDF was generated with a Gamma distribution which was obtained based on Training Window Sequential (TWS) and Conditional (TWC) training windows. Output of BMA-TWC was slightly better than BMA-TWS. Nevertheless, both of them were superior to raw model ECS4. BMA-TWC or BMA-TWS output was varying depend on spatial and temporal, but in general, the best result was in the dry season and during the El Nino phase. BMA was able to improve the distribution characteristics of ensemble prediction. BMA also increased the skill, resolution, and reliability of probability forecast of Below Normal (BN) and Above Normal (AN). Furthermore, the reliability of BN and AN of BMA output were also have the categories of “still very useful” and “perfect” compared to raw model ECS4 that were in the “dangerous” and “not useful” categories. The reliability “still very useful” and “perfect” show that the probabilistic forecast of BN and AN event can be used for making decisions related to seasonal forecast especially over Java island.

A Novel Robust Student’s t-Based Cubature Information Filter with Heavy-Tailed Noises

In this paper, a novel robust Student’s t-based cubature information filter is proposed for a nonlinear multisensor system with heavy-tailed process and measurement noises. At first, the predictive probability density function (PDF) and the likelihood PDF are approximated as two different Student’s t distributions. To avoid the process uncertainty induced by the heavy-tailed process noise, the scale matrix of the predictive PDF is modeled as an inverse Wishart distribution and estimated dynamically. Then, the predictive PDF and the likelihood PDF are transformed into a hierarchical Gaussian form to obtain the approximate solution of posterior PDF. Based on the variational Bayesian approximation method, the posterior PDF is approximated iteratively by minimizing the Kullback-Leibler divergence function. Based on the posterior PDF of the auxiliary parameters, the predicted covariance and measurement noise covariance are modified. And then the information matrix and information state are updated by summing the local information contributions, which are computed based on the modified covariance. Finally, the state, scale matrix, and posterior densities are estimated after fixed point iterations. And the simulation results for a target tracking example demonstrate the superiority of the proposed filter.

On the distribution of fluxes of gamma-ray blazars: hints for a stochastic process?

ABSTRACT We examine a model for the observed temporal variability of powerful blazars in the γ-ray band in which the dynamics is described in terms of a stochastic differential equation, including the contribution of a deterministic drift and a stochastic term. The form of the equation is motivated by the current astrophysical framework, accepting that jets are powered through the extraction of the rotational energy of the central supermassive black hole mediated by magnetic fields supported by a so-called magnetically arrested accretion disc. We apply the model to the γ-ray light curves of several bright blazars and we infer the parameters suitable to describe them. In particular, we examine the differential distribution of fluxes (dN/dFγ) and we show that the predicted probability density function for the assumed stochastic equation naturally reproduces the observed power-law shape at large fluxes $\mathrm{ d}N/\mathrm{ d}F_{\gamma } \propto F_{\gamma }^{-\alpha }$ with α > 2.