Specific Probability Density Function Research Articles

New typical power curves generation approach for accurate renewable distributed generation placement in the radial distribution system

<span lang="EN-US">This paper investigates, for the first time, the accuracy of normalized power curves (NPCs), often used to incorporate uncertainties related to wind and solar power generation, when integrating renewable distributed generation (RDG), in the radial distribution system (RDS). In this regard, the present study proposes a comprehensive, simple, and more accurate model, for estimating the expected hourly solar and wind power generation, by adopting a purely probabilistic approach. Actually, in the case of solar RDG, the proposed model allows the calculation of the expected power, without going through a specific probability density function (PDF). The validation of this model is performed through a case study comparing between the classical and the proposed model. The results show that the proposed model generates seasonal NPCs in a less complex and more relevant way compared to the discrete classical model. Furthermore, the margin of error of the classical model for estimating the expected supplied energy is about 12.6% for the photovoltaic (PV) system, and 9% for the wind turbine (WT) system. This introduces an offset of about 10% when calculating the total active losses of the RDS after two RDGs integration.</span>

Four-compartment epidemic model with retarded transition rates.

We study an epidemic model for a constant population by taking into account four compartments of the individuals characterizing their states of health. Each individual is in one of the following compartments: susceptible S; incubated, i.e., infected yet not infectious, C; infected and infectious I; and recovered, i.e., immune, R. An infection is visible only when an individual is in state I. Upon infection, an individual performs the transition pathway S→C→I→R→S, remaining in compartments C, I, and R for a certain random waiting time t_{C}, t_{I}, and t_{R}, respectively. The waiting times for each compartment are independent and drawn from specific probability density functions (PDFs) introducing memory into the model. The first part of the paper is devoted to the macroscopic S-C-I-R-S model. We derive memory evolution equationsinvolving convolutions (time derivatives of general fractional type). We consider several cases. The memoryless case is represented by exponentially distributed waiting times. Cases of long waiting times with fat-tailed waiting-time distributions are considered as well where the S-C-I-R-S evolution equationstake the form of time-fractional ordinary differential equations. We obtain formulas for the endemic equilibrium and a condition of its existence for cases when the waiting-time PDFs have existing means. We analyze the stability of healthy and endemic equilibria and derive conditions for which the endemic state becomes oscillatory (Hopf) unstable. In the second part, we implement a simple multiple-random-walker approach (microscopic model of Brownian motion of Z independent walkers) with random S-C-I-R-S waiting times in computer simulations. Infections occur with a certain probability by collisions of walkers in compartments I and S. We compare the endemic states predicted in the macroscopic model with the numerical results of the simulations and find accordance of high accuracy. We conclude that a simple random-walker approach offers an appropriate microscopic description for the macroscopic model. The S-C-I-R-S-type models open a wide field of applications allowing the identification of pertinent parameters governing the phenomenology of epidemic dynamics such as extinction, convergence to a stable endemic equilibrium, or persistent oscillatory behavior.

Used (Lived) versus Offered (Plain) densities of human settlement in space: An instance of the probabilistic consumption model

To people living in areas, the denser is the area, the more numerous are the opportunities of interpersonal and social interaction, of employment and of amenities of all kinds. The spatial density of human settlement is basically studied according to places, that is, area weighted. The notion of population-weighted density, or lived density, puts human density in the perspective of the people that experience it. Considering, respectively, the land units and the people as statistical populations of their own, the article provides a probabilistic model of human density in a geographical space, as a random variable in each statistical population, with specific probability density functions (PDFs) and cumulative distribution functions. The PDF of lived, “Used density” is derived from that of the plain, “Offered density” through a consumption model: Thus, their relationship is a specific instance of a well-established probabilistic model. The average used density is systematically larger than its offered counterpart: The ratio amounts to one plus the squared coefficient of variation of offered density. The relation between the two statistical distributions is illustrated using a Lorenz curve; the associated Gini index constitutes an indicator of population heterogeneity in a geographical space. A case of France’s population as of 2019 is studied to demonstrate the methodology.

Estimation of wind speed distribution with time window and new kernel function

Wind speed distribution analysis is important for selecting the potential wind farm and improving wind energy utilization efficiency. However, due to the randomness, intermittence, and chaos of wind speed, it is difficult to accurately estimate its probability distribution. Many studies use specific probability density function to estimate wind speed distribution, but only a few analyze the changes of wind speed distribution in the same region. The purpose of this study is to improve the accuracy of wind speed distribution estimation and analyze the changes in wind speed distribution under different time windows. Three new kernel functions are proposed to improve the accuracy of distribution estimation. A new point-to-point comparison method is proposed to evaluate the goodness-of-fit of wind distribution. A new time window analysis method is applied to analyze the monthly, quarterly, and semiannual wind speed distribution. The results show that (a) under different time windows, the wind speed distribution in Hexi Corridor is different; (b) the performance of kernel function is affected by its peak value and shape; and (c) one of the three new kernel functions has the smallest error, in which mean square errors in monthly, quarterly, and semiannual time windows are 0.0057, 0.0061, and 0.0056, respectively.

A Simplified Approach to the Evaluation of the Influences of Key Factors on Agricultural Tractor Fuel Consumption during Heavy Drawbar Tasks under Field Conditions

The optimization of energy input on agricultural farms, such as through improved fuel consumption, is currently under investigation in agricultural mechanization research with the aim of achieving economic and environmental goals. In previous research, we developed a simplified algorithm focused on defining the most efficient tractor–implement combination considering the factors that most influence this aspect. The ASABE (American Society of Agricultural and Biological Engineers) equation for calculating the drawbar pull force was adopted to fit the results to the soil conditions. Agricultural tires of different sizes were tested at different pressure settings under field conditions to assess differences in drawbar force. The resulting algorithm underwent a linear regression analysis to achieve a simplified equation for assessing the optimal wheel-slip, mass, engine power, and tire pull force properties during drawbar works that result in optimal fuel consumption with a minimal tractor efficiency impairment. Using a specific probability density function, the Monte Carlo Simulation method introduced randomness into the input and runs a sufficiently large number of trials to identify the most probable output. The result is a simplified algorithm that can be used to investigate the effects of certain parameters on fuel consumption; however, it can be adapted to evaluate the effects of different implements, tires, engine settings, or fleet management methods on fuel consumption.

Uncertainty quantification of nonlinear system stochastic response estimates based on the Wiener path integral technique: A Bayesian compressive sampling treatment

The Wiener path integral (WPI) technique for determining the stochastic response of diverse nonlinear systems is enhanced herein based on a Bayesian compressive sampling (CS) treatment. Specifically, first, sparse expansions for the system response joint probability density function (PDF) are employed. Next, exploiting the localization capabilities of the WPI technique for direct evaluation of specific PDF points leads to an underdetermined linear system of equations for the expansion coefficients. Further, relying on a Bayesian CS solution formulation yields a posterior distribution for the expansion coefficient vector. In this regard, a significant advantage of the herein developed methodology relates to the fact that the uncertainty of the response PDF estimates obtained by the WPI technique is quantified. Furthermore, an adaptive scheme is proposed based on the quantified uncertainty of the estimates for optimal selection of PDF sample points. This yields considerably fewer boundary value problems to be solved as part of the WPI technique, and thus, the associated computational cost is significantly reduced. Two indicative numerical examples pertaining to a Duffing nonlinear oscillator and to an oscillator with asymmetric nonlinearities are considered for demonstrating the capabilities of the developed technique. Comparisons with pertinent Monte Carlo simulation data are included as well.

Investigation on Bubble Diameter Distribution in Upward Flow by the Two-Fluid and Multi-Fluid Models

Bubble flow can be simulated by the two-fluid model and the multi-fluid model based on the Eulerian method. In this paper, the gas phase was further divided into several groups of dispersed phases according to the diameter by using the Eulerian-Eulerian (E-E) multi-fluid model. The diameters of bubbles in each group were considered to be the same, and their distributions were reorganized according to a specific probability density function. The experimental data of two kinds of bubble flow with different characteristics were used to verify the model. With the help of the open-source CFD software, OpenFOAM-7.x (OpenFOAM-7.0, produced by OpenFOAM foundation, Reading, England), the influences of the group number, the probability distribution function, and the parameters of different bubble diameters on the calculation results were studied. Meanwhile, the numerical simulation results were compared with the two-fluid model and the experimental data. The results show that for the bubble flow with the unimodal distribution, both the multi-fluid model and the two-fluid model can obtain the distribution of gas volume fraction along the pipe radius. The calculation results of the multi-fluid model agree with the experimental data, while those of the two-fluid model differ greatly from the experimental data, which verifies the advantage of the multi-fluid model in calculating the distribution of gas volume fraction in the polydisperse bubble flow. Meanwhile, the multi-fluid model can be used to accurately predict the distribution of the parameters of each phase of the bubble flow if the reasonable bubble diameter distribution is provided and the appropriate interphase force calculation model is determined.

Quantized Control of Networked Control Systems Under Stochastic Clock Offsets

In this paper, the quantized control of networked control systems under stochastic clock offsets is considered. We assume that the clock offsets are caused by asynchronous clocks between sensors and controllers. A stochastic variable with a specific probability density function is used to describe the stochastic clock offsets. A quantized controller is designed to deal with quantization error such that the corresponding systems are stochastically stable. Finally, two numerical examples are used to show the validity of the proposed design method.

Probability Density Forecasting of Wind Speed Based on Quantile Regression and Kernel Density Estimation

Based on quantile regression (QR) and kernel density estimation (KDE), a framework for probability density forecasting of short-term wind speed is proposed in this study. The empirical mode decomposition (EMD) technique is implemented to reduce the noise of raw wind speed series. Both linear QR (LQR) and nonlinear QR (NQR, including quantile regression neural network (QRNN), quantile regression random forest (QRRF), and quantile regression support vector machine (QRSVM)) models are, respectively, utilized to study the de-noised wind speed series. An ensemble of conditional quantiles is obtained and then used for point and interval predictions of wind speed accordingly. After various experiments and comparisons on the real wind speed data at four wind observation stations of China, it is found that the EMD-LQR-KDE and EMD-QRNN-KDE generally have the best performance and robustness in both point and interval predictions. By taking conditional quantiles obtained by the EMD-QRNN-KDE model as the input, probability density functions (PDFs) of wind speed at different times are obtained by the KDE method, whose bandwidth is optimally determined according to the normal reference criterion. It is found that most actual wind speeds lie near the peak of predicted PDF curves, indicating that the probabilistic density prediction by EMD-QRNN-KDE is believable. Compared with the PDF curves of the 90% confidence level, the PDF curves of the 80% confidence level usually have narrower wind speed ranges and higher peak values. The PDF curves also vary with time. At some times, they might be biased, bimodal, or even multi-modal distributions. Based on the EMD-QRNN-KDE model, one can not only obtain the specific PDF curves of future wind speeds, but also understand the dynamic variation of density distributions with time. Compared with the traditional point and interval prediction models, the proposed QR-KDE models could acquire more information about the randomness and uncertainty of the actual wind speed, and thus provide more powerful support for the decision-making work.

Inference of COVID-19 epidemiological distributions from Brazilian hospital data.

Knowing COVID-19 epidemiological distributions, such as the time from patient admission to death, is directly relevant to effective primary and secondary care planning, and moreover, the mathematical modelling of the pandemic generally. We determine epidemiological distributions for patients hospitalized with COVID-19 using a large dataset (N = 21 000 − 157 000) from the Brazilian Sistema de Informação de Vigilância Epidemiológica da Gripe database. A joint Bayesian subnational model with partial pooling is used to simultaneously describe the 26 states and one federal district of Brazil, and shows significant variation in the mean of the symptom-onset-to-death time, with ranges between 11.2 and 17.8 days across the different states, and a mean of 15.2 days for Brazil. We find strong evidence in favour of specific probability density function choices: for example, the gamma distribution gives the best fit for onset-to-death and the generalized lognormal for onset-to-hospital-admission. Our results show that epidemiological distributions have considerable geographical variation, and provide the first estimates of these distributions in a low and middle-income setting. At the subnational level, variation in COVID-19 outcome timings are found to be correlated with poverty, deprivation and segregation levels, and weaker correlation is observed for mean age, wealth and urbanicity.

Analytic Sensor Rules for Optimal Distributed Decision Given $K$-Out-of-$L$ Fusion Rule Under Monte Carlo Approximation

With the development of large-scale sensor networks, there is an increasing interest for high dimensional distributed decision fusion problem in statistical decision, machine learning, control theory, etc. In this article, we investigate the optimal sensor quantized rules of the distributed decision under a given fusion rule for conditionally dependent and high dimensional sensor observations. We provide a Monte Carlo importance sampling method to reduce the computational complexity of the distributed decision fusion. For the K-out-of-L rule, we derive a group of analytic sensor rules based on the necessary and sufficient condition of the optimal sensor rules. In term of efficiency, it is significantly better than the traditional iterative search algorithms, such as the Riemann sum approximation iteration algorithm. The analytic solution is also a general result since it does not rely on the specific K value of the K-out-of-L rule, the specific conditional probability density function and the dimension of sensor observations. Thus, it can be applied to multisensor decision fusion problems with high dimensional dependent sensor observations. For the general fusion rules rather than the K-out-of-L rule, the Monte Carlo Gauss-Seidel algorithm is also provided. The numerical examples demonstrate the effectiveness of the analytic solution and the Monte Carlo Gauss-Seidel algorithm.

Impacts of Atmospheric Mercury Deposition on Human Multimedia Exposure: Projection from Observations in the Pearl River Delta Region, South China.

A preliminary projection was performed to determine human multimedia exposure to mercury (Hg) based on deposition flux observations and to identify the impacts of atmospheric Hg deposition in Pearl River Delta (PRD) region, South China. The Monte Carlo technique was used to propagate the variability throughout the projection. The regional specific probability density functions (PDFs) of the studied parameters were regressed from the provincial/national published data, except when the data were deficient. The atmospheric Hg deposition flux ranged from 43.70 to 321.19 μg/m2/year and did not significantly contribute to Hg accumulation in the regional topsoil, freshwater bodies, and most food items except fish. The consumption of fish and milk/dairy products was the major contributor to the total exposure for adults (>18 years)/6- to 12-year children and 0- to 6-year children, respectively. The projected concentrations and exposure levels were the results combining MeHg and inorganic Hg (Hg2+). Under the 30-year projection, the probability of risks caused by Hg deposition (combining Hg2+ and MeHg) was the highest for 0- to 6-year children, followed by 6- to 12-year children and adults. The ground effects driven by precipitation had a significantly greater effect relative to the mass transport effects in this region.

A Fuzzy Trustworthiness System with Probability Presentation Based on Center-of-gravity Method

Fuzzy methods are widely used in the study of trustworthiness. Based on this fact, the paper researches the fuzzy trustworthiness system and probability presentation theory based on bounded product implication and Larsen square implication. Firstly, we convert a group of single-input and single-output data into fuzzy inference rules and generate fuzzy relation by selecting the appropriate fuzzy implication operator, then calculate joint probability density function of two-dimensional random variables by using of this fuzzy relation. Two specific probability density functions can be obtained by selecting the fuzzy implication as bounded product implication or Larsen square implication. Secondly, we study the marginal distribution and numerical characteristics of these two kinds probability distributions and point out that two of these probability distributions have the same mathematical expectation and nearly the same variance and covariance. Finally, we study of the center-of-gravity fuzzy trustworthiness system based on these two probability distributions. We gave the sufficient conditions of universal approximations for those fuzzy trustworthiness systems.

Modeling Network Interference in the Angular Domain: Interference Azimuth Spectrum

The performance of wireless networks is fundamentally limited by interference [or, equivalently, the signal-to-interference ratio (SIR)]. In an attempt to characterize the interference as a direction-selective quantity and motivated by the useful analogy between classical propagation channels and wireless networks, we propose a novel network description framework, namely the interference azimuth spectrum (IAS). The IAS represents the distribution of interference in the angular domain and is parallel to the conventional power azimuth spectrum (PAS) used in propagation channels. We also extend this concept to the directional characterization of average achievable rate, assuming that interference is treated as noise. Provided with this analytical framework, we present the notion of local area outage, defined as the probability that a receiver is in the state of outage within a local area where both interference and desired signal are assumed to be wide-sense stationary (WSS). We further propose the geometry-based stochastic models (GBSMs) as a part of the IAS framework, where the interfering terminals are randomly distributed according to a specific probability density function (pdf) of their positions. The GBSMs are applicable to a wide variety of wireless network environments without and with interferer clustering. The proposed methodology would provide useful insight on the design and performance assessment of future networks, featured by opportunistic, randomized, and dense placement of nodes.

A probability model for key analysis in music

This paper presents a novel method to analyse the tonal behaviour of a music piece. The method is based on the development of a novel probability model of the predominant key using the well known Pitch Class Profile (PCP or chroma) descriptor. This feature represents the importance of each note of the chromatic scale within the spectral content of the audio signal.Making use of PCP feature, a main novel contribution presented is the development of new profile models for the characterization of major and minor keys. Most of the key profile models found in the literature assign a single value to each pitch class for a particular key. This value represents the salience of this pitch class in a specific key. The new key profiles, that will be described in this paper, are based on the identification of specific probability density functions (PDFs), defined after the analysis of the presence of the pitch classes of the chromatic scale in the different keys.

Sensitivity analysis of vertically loaded pile reliability

The purpose of this paper is to examine the influence of geotechnical uncertainties on the reliability of vertically loaded pile foundations and the use of this information in decision-making support, especially when gathering the information necessary for reliability analyses. Two case studies of single pile foundations were selected, and each uncertainty source was investigated to identify which are the most important and influential in the evaluation of vertical pile resistance under axial loading. Reliability sensitivity analyses were conducted using FORM (the first-order reliability method) and MCS (Monte Carlo simulations). The characterisation of uncertainties is not an easy task in geotechnical engineering. The aim of the analyses described in this paper is to optimise resources and investments in the investigation of the variables in pile reliability. The physical uncertainties of actions, the inherent variability of soil and model error were assessed by experimental in situ standard penetration tests (SPT) or from information available in the literature. For the cases studied, the sensitivity analysis results show that, in spite of the high variability of the soils involved, model error also plays a very important role in geotechnical pile reliability and was considerably more important than soil variability in both case studies. From a comparison of the two reliability methods (FORM and MCS), it was concluded that FORM is applicable in simple cases and as a first approach because it is an approximate method and sometimes does not have the capability to incorporate every detail of the problem, namely a specific probability density function or more specific limit conditions.

Organ sample generator for expected treatment dose construction and adaptive inverse planning optimization

To create an organ sample generator (OSG) for expected treatment dose construction and adaptive inverse planning optimization. The OSG generates random samples of organs of interest from a distribution obeying the patient specific organ variation probability density function (PDF) during the course of adaptive radiotherapy. Principle component analysis (PCA) and a time-varying least-squares regression (LSR) method were used on patient specific geometric variations of organs of interest manifested on multiple daily volumetric images obtained during the treatment course. The construction of the OSG includes the determination of eigenvectors of the organ variation using PCA, and the determination of the corresponding coefficients using time-varying LSR. The coefficients can be either random variables or random functions of the elapsed treatment days depending on the characteristics of organ variation as a stationary or a nonstationary random process. The LSR method with time-varying weighting parameters was applied to the precollected daily volumetric images to determine the function form of the coefficients. Eleven h&n cancer patients with 30 daily cone beam CT images each were included in the evaluation of the OSG. The evaluation was performed using a total of 18 organs of interest, including 15 organs at risk and 3 targets. Geometric variations of organs of interest during h&n cancer radiotherapy can be represented using the first 3 ∼ 4 eigenvectors. These eigenvectors were variable during treatment, and need to be updated using new daily images obtained during the treatment course. The OSG generates random samples of organs of interest from the estimated organ variation PDF of the individual. The accuracy of the estimated PDF can be improved recursively using extra daily image feedback during the treatment course. The average deviations in the estimation of the mean and standard deviation of the organ variation PDF for h&n cancer radiotherapy were less than 2 and 1 mm, respectively, for most organs after the second week of treatment. After the first three weeks of treatment, the mean discrepancy of the dose estimation accuracy was within 1% for most of organs, the corresponding standard deviation was within 2.5% for parotids, the brain stem and the cochleae, and within 1% for other organs. A patient specific OSG is feasible and can be used to generate random samples of organs of interest for the expected treatment dose construction and adaptive inverse planning. The accuracy of the OSG can be improved continuously and recursively during the adaptive treatment course using daily volumetric image feedback.

Inferences on weather extremes and weather-related disasters: a review of statistical methods

Abstract. The study of weather extremes and their impacts, such as weather-related disasters, plays an important role in research of climate change. Due to the great societal consequences of extremes – historically, now and in the future – the peer-reviewed literature on this theme has been growing enormously since the 1980s. Data sources have a wide origin, from century-long climate reconstructions from tree rings to relatively short (30 to 60 yr) databases with disaster statistics and human impacts. When scanning peer-reviewed literature on weather extremes and its impacts, it is noticeable that many different methods are used to make inferences. However, discussions on these methods are rare. Such discussions are important since a particular methodological choice might substantially influence the inferences made. A calculation of a return period of once in 500 yr, based on a normal distribution will deviate from that based on a Gumbel distribution. And the particular choice between a linear or a flexible trend model might influence inferences as well. In this article, a concise overview of statistical methods applied in the field of weather extremes and weather-related disasters is given. Methods have been evaluated as to stationarity assumptions, the choice for specific probability density functions (PDFs) and the availability of uncertainty information. As for stationarity assumptions, the outcome was that good testing is essential. Inferences on extremes may be wrong if data are assumed stationary while they are not. The same holds for the block-stationarity assumption. As for PDF choices it was found that often more than one PDF shape fits to the same data. From a simulation study the conclusion can be drawn that both the generalized extreme value (GEV) distribution and the log-normal PDF fit very well to a variety of indicators. The application of the normal and Gumbel distributions is more limited. As for uncertainty, it is advisable to test conclusions on extremes for assumptions underlying the modelling approach. Finally, it can be concluded that the coupling of individual extremes or disasters to climate change should be avoided.

An Exploration of the Exact Distribution and Probabilities for Sample Means for Small Random Samples

The computer algebra system, MathematicaTM, is used to determine the exact distributions for sums and means of small random samples taken from a specific probability density function. The method used is the Inverse Laplace Transform for real-valued functions. These distributions are used to compare exact probabilities for probability interval statements for sums and means with normal approximations for these probabilities using the Central Limit Theorem. The maximum normal approximation errors are determined for probability intervals for various sample sizes.

Generation of Non-Gaussian Wide-Sense Stationary Random Processes with Desired PSDs and PDFs

This paper describes a new method to generate discrete signals with arbitrary power spectral density (PSD) and first order probability density function (PDF) without any limitation on PDFs and PSDs. The first approximation has been achieved by using a nonlinear transform function. At the second stage the desired PDF was approximated by a number of symmetric PDFs with defined variance. Each one provides a part of energy from total signal with different ratios of remained desired PSD. These symmetric PDFs defined by sinusoidal components with random amplitude, frequency and phase variables. Both analytic results and examples are included. The proposed scheme has been proved to be useful in simulations involving non-Gaussian processes with specific PSDs and PDFs.