Probability Distribution Model Research Articles

Robust optimal power flow considering uncertainty in wind power probability distribution

This paper proposes an optimal power flow model that takes into account the uncertainty in the probability distribution of wind power. The model can schedule controllable generators under any possible distribution of wind power to ensure the safe and economic operation of the system. Firstly, considering the incompleteness of historical wind power data, the paper models the uncertainty of wind power using second-order moments of probability distribution and their fluctuation intervals. Subsequently, a robust optimal power flow model based on probability distribution model and joint chance constraints is established. The Lagrangian duality theorem is then employed to eliminate random variables from the optimization model, transforming the uncertainty model into a deterministic linear matrix inequality problem. Finally, a convex optimization algorithm is used to solve the deterministic problem, and the results are compared with traditional chance-constrained optimal power flow model. The feasibility and effectiveness of the proposed method are validated through case study simulations.

Scalable hierarchical BayeSN inference: investigating dependence of SN Ia host galaxy dust properties on stellar mass and redshift

ABSTRACT We apply the hierarchical probabilistic spectral energy distribution (SED) model bayesn to analyse a sample of 475 type Ia supernovae (0.015 < z < 0.4) from Foundation, DES3YR and PS1MD to investigate the properties of dust in their host galaxies. We jointly infer the dust law RV population distributions at the SED level in high- and low-mass galaxies simultaneously with dust-independent, intrinsic differences. We find an intrinsic mass step of −0.049 ± 0.016 mag, at a significance of 3.1σ, when allowing for a constant intrinsic, achromatic magnitude offset. We additionally apply a model allowing for time- and wavelength-dependent intrinsic differences between SNe Ia in different mass bins, finding ∼2σ differences in magnitude and colour around peak and 4.5σ differences at later times. These intrinsic differences are inferred simultaneously with a difference in population mean RV of ∼2σ significance, demonstrating that both intrinsic and extrinsic differences may play a role in causing the host galaxy mass step. We also consider a model which allows the mean of the RV distribution to linearly evolve with redshift but find no evidence for any evolution – we infer the gradient of this relation ηR = −0.38 ± 0.70. In addition, we discuss in brief a new, GPU-accelerated python implementation of bayesn suitable for application to large surveys which is publicly available and can be used for future cosmological analyses; this code can be found here: https://github.com/bayesn/bayesn.

Appropriate indicator of modeling error for threshold-based model selection in statistics-based ultrasound tissue characterization

Analysis of the envelope statistics of ultrasound echo signals contributes to quantitative tissue characterization in medical ultrasound. Many probability distribution model functions have been studied, and the model function that should be used for tissue characterization depends on the type of disease, even in the same organ. Thus, an appropriate model selection is important for an accurate diagnosis. In this study, we aimed to select a model using threshold processing for modeling errors instead of a simple selection by minimizing the modeling error. For this purpose, we compared several indicators of modeling errors using random number simulations, ultrasonic simulation, and phantom experiment. The results validated that the Mahalanobis distance of moments is an appropriate indicator because it enables the use of a constant threshold value, regardless of the type of model function and data length.

Temporal Characterization of Fire Accidents in China

Based on the fire accident statistics, this study uses the seasonal index method to comprehensively analyze the temporal pattern of fire occurrence in China from 2010 to 2015. The study scientifically reflects that the number of fire occurrences has an extremely close relationship with time, and summarizes the temporal pattern and characteristics of fire occurrences. This study concludes that the high incidence of large fire accidents in China occurs in the time period of 3:00 a.m. to 5:00 a.m. The probability distribution model of the average total number of accidents in each time period was established and the Kolmogorov test was carried out, which shows that the time characteristics of fire accidents in China can be understood with the probability distribution model as a whole. The results of the analysis of the time characteristics of the fire accidents can provide decision-making information for combating fire accidents.

Analysis and Prediction of Airspace Availability for Urban Air Mobility Operations in the Sao Paulo Metropolitan Region

Urban Air Mobility (UAM) is an emerging form of transportation that is expected to introduce novel flight networks into already busy and complex airspace surrounding major cities and metropolitan regions. This paper studies the dynamics of urban airspace use by conventional aircraft over the Sao Paulo metropolitan region in order to identify and predict which airspace volumes are least constrained and best accessible for future UAM flights. Using historical flight tracking data, clustering analysis is first performed to identify departure and arrival trajectory patterns flown by conventional traffic at the two major airports – Sao Paulo/Guarulhos International airport and Sao Paulo/Congonhas airport. We then create a probabilistic model of the spatiotemporal distribution of air traffic under known meteorological conditions, which enables the prediction of active procedures, their spatial confidence regions and the resulting airspace availability for UAM in response to dynamic operational factors. The data-based approach allowed for a high-fidelity characterization of the Sao Paulo urban airspace use patterns as well as for accurate predictions of the available airspace for UAM, bringing novel insights and capabilities in support of dynamic and efficient urban airspace management.

Stochastic inversion of soil dynamic parameters based on non-intrusive data

The prediction and evaluation of environmental vibrations generated by traffic loads have been playing more and more important role during the planning and designing of high-quality traffic system. From the traffic source to sensitive building, the subsoil is the one of key propagation media, so the determination of soil dynamic parameters is very crucial in studying the transmitting and attenuation of environmental vibration. Due to the spatial variability of natural soil layers, the soil dynamic parameters are actually not certain but probabilistic with stochastic characteristics. However, most current inversion methods of soil parameters cannot consider the uncertainty. In this paper, an efficient inversion method is proposed to obtain the random distribution of soil dynamic parameters based on the combination of field experiment, forward soil modelling, and Bayesian theory. Take the shear modulus and damping ratio of subsoil as the typical parameters, the experimental dispersion curve and spatial attenuation curve can be obtained by field experiment, while the corresponding theoretical ones can be calculated by the thin-layered soil model with perfectly matched layer. The combining of experimental and theoretical curves generates the likelihood function. A prior probability distribution model is firstly established, and subsequently the posterior probability distribution model is deduced by genetic algorithm, Monte Carlo-Markov chain algorithm, and Bayesian theory. In order to verify the proposed inversion method, an on-site experiment based on Multichannel Analysis of Surface Waves(MASW) was conducted in an open field in Beijing. The results show the mean values of inversed soil parameters were close to the determined parameters provided by geological exploration. Compared to the traditional determined inversion, the proposed stochastic inversion method can consider the variety of soil parameters within small depth by providing probability distribution, mean and root-mean-square error, which has greater applicability in the further probabilistic prediction of environmental vibration.

Fragility Analysis of a Transmission Tower-Line System Subjected to Wind and Ice Loads Considering Fatigue Damage

Disasters such as ice and wind can pose a serious threat to the normal operation of power transmission lines. Wind-induced fatigue damage can further reduce the load-carrying capacity of transmission towers, increasing their fragility to ice and wind disasters. To assess the comprehensive capability of transmission towers to resist extreme ice and wind disasters, this paper proposes a failure probability evaluation framework for transmission towers considering wind-induced fatigue damage under the coupled effect of ice and wind. Taking a certain transmission line in Hunan as an example, the failure probability caused by ice and wind disasters is calculated for different years of service. First, based on the historical meteorological data in three cities in Hunan, a joint probability model of wind speed and wind direction considering their correlation is established using the copula function. Then, based on this probability model, the wind-induced fatigue damage of transmission towers is calculated using the Miner linear damage theory and the S-N curve. Subsequently, the fragility of the tower under the coupled load of ice and wind is calculated for different years of service. Finally, by combining the structural fragility function with the joint probability distribution model of ice thickness and wind speed, the collapse probability of the transmission tower under the action of ice and wind disasters is calculated. The results indicate that the influence of wind-induced fatigue damage cannot be ignored when transmission towers encounter ice and wind disasters. With increasing service time, the ability of transmission towers to resist ice and wind disasters gradually decreases, and the failure probability also increases under extreme ice and wind conditions.

Transfer, persistence, contamination and background levels of inorganic gunshot residues

This paper summarises the available literature data for the evidential evaluation topics of transfer (196 experiments), persistence (63 time series), contamination (1515 samples), and background prevalence (2158 samples) of inorganic gunshot residues (IGSR). In-house data on IGSR transfer, and the prevalence and persistence of IGSR on different types of glove are also contributed.Combining new and previously published data in a meta-analysis, we report the following findings: The median transfer rate of IGSR was 11 % and the probability distribution of contact transfer was modelled using a Beta distribution. The half-life of IGSR on hands was estimated at 52 min. On gloves, decay followed a two-phase process with the slower process proceeding at a decreased rate compared to on hands (t1/2,slow = 77 min). The occurrence of characteristic IGSR on the hands of police officers was modelled using a generalised Pareto model (GPM). Combining the prevalence and the transfer probability models, a product probability distribution model was established. The product model estimates the probability of finding any amount of IGSR post-arrest on previously clean hands, estimating 8 % probability of non-zero transfer. Characteristic IGSR detected on the general public (1 % positives), on at-risk individuals (2 % positives), and in public places (0 % positives) showed low background levels. The likelihood of finding any amount of IGSR on the general public (at-risk included) was modelled using a GPM giving a 1.3 % probability of finding at least one, and 0.2 % probability to find more than three characteristic IGSR on the general public.

Tight Lower Bound on Differential Entropy for Mixed Gaussian Distributions

In this paper, a tight lower bound for the differential entropy of the Gaussian mixture model is presented. First, the probability model of mixed Gaussian distribution that is created by mixing both discrete and continuous random variables is investigated in order to represent symmetric bimodal Gaussian distribution using the hyperbolic cosine function, on which a tighter upper bound is set. Then, this tight upper bound is used to derive a tight lower bound for the differential entropy of the Gaussian mixture model introduced. The proposed lower bound allows to maintain its tightness over the entire range of the model's parameters and shows more tightness when compared with other bounds that lose their tightness over certain parameter ranges. The presented results are then extended to introduce a more general tight lower bound for asymmetric bimodal Gaussian distribution, in which the two modes have a symmetric mean but differ in terms of their weights.

The total factor characteristics evaluation of photovoltaic power by coarse-fine-grained method

The study of photovoltaic(PV) power uncertainty is of great significance to promote the collaborative optimization of source-grid-load-storage. On the one hand, using a single probability distribution model to describe the fluctuation, intermittency and randomness of PV power is limited and controversial; On the other hand, qualitative analysis oriented to only one or two of them is characterized with poor interpretability and one-sidedness. So that, a evaluation framework for the total factor characteristics named "dual-DMU" is established corresponding to the discrete, mutation and unordered phenomenon of PV power, and a coarse-fine-grained evaluation method is proposed, which can realize the comprehensive description and all-sided evaluation of PV power. Combined with the historical data of a rooftop PV power plant, the analysis reveals the variability of total factor characteristics in different seasons, weather types, and typical time periods, and the methodology can provide a theoretical basis and decision-making support for the construction of PV power forecasting models, energy storage capacity optimization, and adjustable load resource allocation.

Unbiasing Enhanced Sampling on a High-Dimensional Free Energy Surface with a Deep Generative Model.

Biased enhanced sampling methods that utilize collective variables (CVs) are powerful tools for sampling conformational ensembles. Due to their large intrinsic dimensions, efficiently generating conformational ensembles for complex systems requires enhanced sampling on high-dimensional free energy surfaces. While temperature-accelerated molecular dynamics (TAMD) can trivially adopt many CVs in a simulation, unbiasing the simulation to generate unbiased conformational ensembles requires accurate modeling of a high-dimensional CV probability distribution, which is challenging for traditional density estimation techniques. Here we propose an unbiasing method based on the score-based diffusion model, a deep generative learning method that excels in density estimation across complex data landscapes. We demonstrate that this unbiasing approach, tested on multiple TAMD simulations, significantly outperforms traditional unbiasing methods and can generate accurate unbiased conformational ensembles. With the proposed approach, TAMD can adopt CVs that focus on improving sampling efficiency and the proposed unbiasing method enables accurate evaluation of ensemble averages of important chemical features.

Using Copula functions to predict climatic change impacts on floods in river source regions

Flood frequency in river source regions is significantly affected by rainfall and snowmelt as part of climatic changes. A traditional univariate flood frequency analysis cannot reflect the complexity of floods, and when used in isolation, it can only underestimate flood risk. For effective flood prevention and mitigation, it is essential to consider the combined effects of precipitation and snowmelt. Copula functions can effectively quantify the joint distribution relationship between floods and their associated variables without restrictions on their distribution characteristics. This study uses copula functions to consider a multivariate probability distribution model of flood peak flow (Q) with cumulative snowmelt (CSm) and cumulative precipitation (CPr) for the Hutubi River basin located in northern Xinjiang, China. The joint frequencies of rainfall and snowmelt floods are predicted using copula models based on the Coupled Model Intercomparison Project Phase 6 data. The results show that Q has a significant positive correlation with 24-d CSm (r = 0.559, p = 0.002) and 23-d CPr (r = 0.965, p < 0.05). Flood frequency will increase in the future, and mid- (2050–2074) and long-term (2075–2099) floods will be more severe than those in the near-term (2025–2049). The probability of flood occurrence is higher under the SSP2-4.5 and SSP1-2.6 scenarios than under SSP5-8.5. Precipitation during the historical period (1990–2014) led to extreme floods, and increasing future precipitation trends are found to be insignificant. Snowmelt increases with rising temperatures and occurs earlier than estimated, leading to an earlier flood period in the basin and more frequent snowmelt floods. The Q under the joint return period is larger than that during the same univariate return period. This difference indicates that neglecting the interaction between precipitation and snowmelt for floods leads to an underestimation of the flood risk (with underestimations ranging from 0.3% to 22%). The underestimations decrease with an increase in the return period. The joint risks of rainfall or snowmelt according to various flood periods should be considered for rivers with multi-source runoff recharge in flood control design. This study reveals the joint impact of precipitation and snowmelt on extreme floods under climate change in river source regions. This study also provides a scientific basis for regional flood prevention and mitigation strategies, as well as for the rational allocation of water resources.

Evaluation of survival weighted Pareto distribution: Analytical properties and applications to industrial and aeronautics data

In statistical modeling, lifetime datasets play an important and significant role. Weighted distributions can be useful to attain the essential purposes relating to the developed distributions’ flexibility and to improve further compliances in data forming; they provide the extension of distributions. The current study suggests a new best-fit probability distribution model based on Pareto distribution. The new model is called Survival Weighted Pareto (SWP) distribution model. Furthermore, the statistical expressions for different properties, which include moments, inverse moments, quantile function, and order statistics, are reported. Moreover, to check the performance and the behavior of different estimators using different sample sizes, a simulation study, which is based on SWP distribution, is performed. According to the simulation results, it is established that the proposed model is a most useful fitted probability model. In addition, an application of the SWP model for three datasets is provided to check the goodness of fit measures of the model.

Unit Maxwell-Boltzmann Distribution and Its Application to Concentrations Pollutant Data

In the vast statistical literature, there are numerous probability distribution models that can model data from real-world phenomena. New probability models, nevertheless, are still required in order to represent data with various spread behaviors. It is a known fact that there is a great need for new models with limited support. In this study, a flexible probability model called the unit Maxwell-Boltzmann distribution, which can model data values in the unit interval, is derived by selecting the Maxwell-Boltzmann distribution as a base-line model. The important characteristics of the derived distribution in terms of statistics and mathematics are investigated in detail in this study. Furthermore, the inference problem for the mentioned distribution is addressed from the perspectives of maximum likelihood, method of moments, least squares, and maximum product space, and different estimators are obtained for the unknown parameter of the distribution. The derived distribution outperforms competitive models according to different fit tests and information criteria in the applications performed on four actual air pollutant concentration data sets, indicating that it is an effective model for modeling air pollutant concentration data.

Rainfall Frequency Analysis of Some Cities in Niger Delta Region of Nigeria

Rainfall frequency analyses for four cities namely Benin City, Port Harcourt, Calabar and Uyo in Nigeria's Niger Delta area were carried out utilizing daily rainfall data from the yearly maxima series for 48 years (1965 – 2012) at each location. The study's goal was to identify the probability distribution model that fit the data the best and applicable to each location from among six candidate probability distribution models namely: Pearson Type III (PIII), log Normal (LN), log Pearson Type III, Generalized Extreme value (GEV), Extreme value type I (EVI), and Generalized Pareto (GPA). The method of moments (MOM) was used to estimate the distributions' parameters applying the outcomes of seven goodness of fit tests, the most optimal fit distribution model was chosen for each site namely Root Mean Squared Error (RMSE), Relative Root Mean Square Error (RRMSE), Maximum Absolute Derivation Index (MADI), Chi–Square etc. The best distribution model at each location was utilized to predict rainfall of desired return periods. Based on our findings, PIII for Benin City, GEV for Port Harcourt, GEV for Calabar and EVI and LN for Uyo are the distribution models that suit the data the best., The Best Fit Probability Distribution Model predicted rainfall return values (RT) at 200 years return period ranging from192.92mm at Uyo, 185mm at Port Harcourt; 218mm for Benin City; and 245 mm at Calabar.. The study's findings are helpful in the planning, designing, and maintaining of hydraulic structures for preventing flood damage and mitigating floods at the locations

Quantifying regional low flows under data scarce conditions

Estimating low flow quantiles is essential for regulating minimum flow requirements and ensuring water availability, ecological health, and overall system sustainability. This study aims to quantify regional low flows by analyzing low-flow time series in data-limited environments. We utilized annual minimum 7-day instantaneous streamflow data collected from gaging stations. Discordancy measure assessments revealed that all sites were homogeneous, forming a single region. We employed Easy-Fit Statistical Software to select the best-fit probability distribution model and determine estimation parameter values. Through goodness-of-fit tests (GOFs), the Generalized Pareto model emerged as the most suitable, predicting low flow quantiles for 100-year returns. Rigorous application of GOFs ensures the statistical soundness of the model, capturing underlying data patterns. A correlation coefficient determination (R2) of 0.989 demonstrates the high satisfaction of the selected distribution model. The developed regression line for the region exhibited strong agreement between predicted low flows and catchment area. Thus, accurate estimation proves valuable in environmental and human-influenced decision-making processes, providing insights into low-flow behavior and mitigating drought effects on aquatic ecosystems.

One at a Time: Progressive Multi-Step Volumetric Probability Learning for Reliable 3D Scene Perception

Numerous studies have investigated the pivotal role of reliable 3D volume representation in scene perception tasks, such as multi-view stereo (MVS) and semantic scene completion (SSC). They typically construct 3D probability volumes directly with geometric correspondence, attempting to fully address the scene perception tasks in a single forward pass. However, such a single-step solution makes it hard to learn accurate and convincing volumetric probability, especially in challenging regions like unexpected occlusions and complicated light reflections. Therefore, this paper proposes to decompose the complicated 3D volume representation learning into a sequence of generative steps to facilitate fine and reliable scene perception. Considering the recent advances achieved by strong generative diffusion models, we introduce a multi-step learning framework, dubbed as VPD, dedicated to progressively refining the Volumetric Probability in a Diffusion process. Specifically, we first build a coarse probability volume from input images with the off-the-shelf scene perception baselines, which is then conditioned as the basic geometry prior before being fed into a 3D diffusion UNet, to progressively achieve accurate probability distribution modeling. To handle the corner cases in challenging areas, a Confidence-Aware Contextual Collaboration (CACC) module is developed to correct the uncertain regions for reliable volumetric learning based on multi-scale contextual contents. Moreover, an Online Filtering (OF) strategy is designed to maintain representation consistency for stable diffusion sampling. Extensive experiments are conducted on scene perception tasks including multi-view stereo (MVS) and semantic scene completion (SSC), to validate the efficacy of our method in learning reliable volumetric representations. Notably, for the SSC task, our work stands out as the first to surpass LiDAR-based methods on the SemanticKITTI dataset.

Modeling of Soil Water Distribution in a Small Mid-Latitude Watershed on the British Isle for Short Term Landslide and Flood Risk Assessment

The R-TopModel hydrological model coupled with two landslide and flood probability distribution models was applied to simulate the daily hydrological conditions of a small catchment in the Midlands of the British Isles, throughout 2017. Originally, the methodology was applied to a risk area in the tropical region. In this work, the application was extended to mid-latitude watersheds. The hydrographic basin around the Carsington Water dam (located in the Midlands of the Great Britain) is chosen because it presents risks. The model Nash-Sutcliffe Efficiency for the upstream discharge from the Carsington Water dam reached 50% with a correlation coefficient of the order of 70%, an acceptable value considering the seasonal effects of the dam on evapotranspiration and higher soil permeability. Modeling the distribution of soil moisture and excess surface water allowed obtaining the spatial distribution of the maximum conditional probability of landslides and floods in the Carsington Water catchment. These probability maps obtained are consistent with long-term susceptibility maps for Great Britain.

FORECASTING OF FLOOD EVENTS FOR WATER RESOURCES MANAGEMENT IN GUCHA-MIGORI RIVER BASIN, KENYA

Floods are devastating natural disasters frequently occurring in many river basins such as Gucha-Migori River Basin, Kenya. However, non-structural countermeasures such as hydrologic modelling, flood proofing, and continuous forecasting have not been fully explored and implemented to reduce risk, damage, and vulnerabilities of flood events. The major challenge in flood forecasting is the selection of the relevant probability distribution. The main objective of the study was to forecast flood events in Gucha-Migori River Basin for integrated water resource management. Daily hydrological datasets between 1969 and 2015 were obtained from Water Resources Authority. After the ranking of independent flood events had been achieved, return periods and flood frequencies were computed. The relationship between flood magnitudes and their respective frequencies was through modelling using probability distribution functions (PDFs). The evaluated PDFs include Normal Distribution, Log-Normal Distribution, Gumbel Distribution, and Log-Person Type III Distribution. Coefficient of Determination (R2), Goodness of Fit test, and Best-Fit Distribution Curve guided selection of the suitable distribution model. The suitable probability distribution model out of the tested four was then applied to predict P-percent annual exceedance probabilities corresponding to specific flood magnitudes. From the results, out of the 132 flood events that occurred in the period 1969 to 2015, only 42 were identified as independent flood events. Based on the frequency analysis, the highest and lowest, and the average and standard deviation of the recorded independent flood events were 423.90, 240.8, 284.80 m3/s, and 45.54 respectively. It was revealed that the probability of a flood event of a magnitude equal to or exceeding the lowest (240.8 m3/s) and the highest (423.90 m3/s) occurring for a particular year were 0.98 and 0.02 respectively. Gumbel distribution curve was selected to be the best fit for the Gucha-Migori River Basin. According to the derived Gumbel's formula, the estimated magnitude for the return period 1.05, 1.11, 1.25, 2, 5, 10, 25, 50, 100, and 200 years were 221.38, 231.92, 245.77, 278.01, 321.39, 350.11, 386.39, 413.31, 440.03, and 466.65 m3/s. This research provides useful information for enhanced determination of probability occurrence of flood events and thus proper water resources planning and management strategies in the Gucha-Migori river basin.

Extreme load extrapolation and long-term fatigue assessment of offshore wind turbine tower based on monitoring data

Extreme load and fatigue damage are important parameters in the structural design of offshore wind turbines (OWTs). The purpose of this study is to obtain extreme loads and long-term fatigue damage assessment of OWT structures based on measured data. In the present study, a utility-scale OWT was monitored with a dynamic response monitoring system and long-term strains of the OWT tower were collected. Based on the monitoring strain and the operational conditions of the OWT, the load and fatigue characteristics of the OWT tower were discussed. Furthermore, statistical extrapolation of the extreme load of the OWT tower was performed and the applicability of different statistical extrapolation methods was studied. The results indicate that the same probability distribution model does not apply to all wind speeds. The maximum extreme load is extrapolated in the vicinity of the rated wind speed. Finally, the relationship between short-term fatigue damage and long-term fatigue damage of the OWT structure was discussed. Superimposing fatigue from a short calculation period underestimates the long-term fatigue of OWT structures due to significant calculation period impact. Therefore, an evaluation method for OWT structures' long-term fatigue was established to improve the accuracy of long-term fatigue calculation by superimposing short calculation periods.