Continuous Probability Density Function Research Articles

프로젝트 네트워크에서 사업완성시간의 적률 추정

프로젝트 네트워크 분석에서 사업완성시간의 분포를 추정하는 것은 매우 기본적이다. 본 논문에서는 활동시간이 상호 독립적이고 정규분포를 따른다는 가정 하에서 사업완성시간의 적률(평균, 분산, 왜도, 첨도)을 추정하기 위한 방법을 제안한다. 제안된 방법은 연속형의 활동시간 분포를 이산형 분포로 근사화하기 위한 이산화 기법과 난수발생을 이용한다. 제안된 방법은 대규모 네트워크에 대해서도 쉽게 적용 가능하며, 그리고 제안된 방법에 의한 결과는 몬테칼로 시뮬레이션에 의해 얻어진 결과와 비교할 때 매우 정확함을 보여준다. For a project network analysis, a fundamental problem is to estimate the distribution function of the project completion time. In this paper, we propose a method for evaluating moments(mean, variance, skewness, kurtosis) of the project completion time under the assumption that the durations of activities are independently and normally distributed. The proposed method utilizes the technique of discretization to replace the continuous probability density function(pdf) of activity duration with its discrete pdf and a random number generation. The proposed method is easy to use for large-sized project networks, and the computational results of the proposed method indicate that the accuracy is comparable to that of direct Monte Carlo simulation.

Computing interval-valued reliability measures: application of optimal control methods

The paper describes an approach to deriving interval-valued reliability measures given partial statistical information on the occurrence of failures. We apply methods of optimal control theory, in particular, Pontryagin’s principle of maximum to solve the non-linear optimisation problem and derive the probabilistic interval-valued quantities of interest. It is proven that the optimisation problem can be translated into another problem statement that can be solved on the class of piecewise continuous probability density functions (pdfs). This class often consists of piecewise exponential pdfs which appear as soon as among the constraints there are bounds on a failure rate of a component under consideration. Finding the number of switching points of the piecewise continuous pdfs and their values becomes the focus of the approach described in the paper. Examples are provided.

Estimating reliability of degraded system based on the probability density evolution with multi-parameter

System degradation was usually caused by multiple-parameter degradation. The assessment result of system reliability by universal generating function was low accurate when compared with the Monte Carlo simulation. And the probability density function of the system output performance cannot be got. So the reliability assessment method based on the probability density evolution with multi-parameter was presented for complexly degraded system. Firstly, the system output function was founded according to the transitive relation between component parameters and the system output performance. Then, the probability density evolution equation based on the probability conservation principle and the system output function was established. Furthermore, probability distribution characteristics of the system output performance was obtained by solving differential equation. Finally, the reliability of the degraded system was estimated. This method did not need to discrete the performance parameters and can establish continuous probability density function of the system output performance with high calculation efficiency and low cost. Numerical example shows that this method is applicable to evaluate the reliability of multi-parameter degraded system.

A general approximation to quantiles

ABSTRACTFor many continuous distributions, a closed-form expression for their quantiles does not exist. Numerical approximations for their quantiles are developed on a distribution-by-distribution basis. This work develops a general approximation for quantiles using the Taylor expansion. Our method only requires that the distribution has a continuous probability density function and its derivatives can be derived to a certain order (usually 3 or 4). We demonstrate our unified approach by approximating the quantiles of the normal, exponential, and chi-square distributions. The approximation works well for these distributions.

A Stratified Sampling Approach for Improved Sampling from a Calibrated Ensemble Forecast Distribution

Abstract Before using the Schaake shuffle or empirical copula coupling (ECC) to reconstruct the dependence structure for postprocessed ensemble meteorological forecasts, a necessary step is to sample discrete samples from each postprocessed continuous probability density function (pdf), which is the focus of this paper. In addition to the equidistance quantiles (EQ) and independent random (IR) sampling methods commonly used at present, the stratified sampling (SS) method is proposed. The performance of the three sampling methods is compared using calibrated GFS ensemble precipitation reforecasts over the Xixian basin in China. The ensemble reforecasts are first calibrated using heteroscedastic extended logistic regression (HELR), and then the three sampling methods are used to sample calibrated pdfs with a varying number of discrete samples. Finally, the effect of the sampling method on the reconstruction of ensemble members with preserved space dependence structure is analyzed by using EQ, IR, and SS in ECC for reconstructing postprocessed ensemble members for four stations in the Xixian basin. There are three main results. 1) The HELR model has a significant improvement over the raw ensemble forecast. It clearly improves the mean and dispersion of the predictive distribution. 2) Compared to EQ and IR, SS can better cover the tails of the calibrated pdfs and a better dispersion of calibrated ensemble forecasts is obtained. In terms of probabilistic verification metrics like the ranked probability skill score (RPSS), SS is slightly better than EQ and clearly better than IR, while in terms of the deterministic verification metric, root-mean-square error, EQ is slightly better than SS. 3) ECC-SS, ECC-EQ, and ECC-IR all calibrate the raw ensemble forecast, but ECC-SS shows a better dispersion than ECC-EQ and ECC-IR in this study.

A new nonparametric density estimation for probabilistic security-constrained economic dispatch

Security-constrained economic dispatch (SCED) which is used to minimize the operation cost of the committed units with the constraints of power balance, ramp rate, and unit capacity is one of the routine challenges in power system operation. In this paper, a nonparametric estimation method based on kernel density and linear diffusion is proposed to obtain continuous probability density functions for probabilistic SCED outcomes. It is assumed that the probabilistic SCED problem is the second stage of a two-stage problem, while stochastic security-constrained unit commitment is the first stage. To evaluate the efficacy of the proposed method, a 6-bus test system and IEEE 118-bus system are used as case studies. Implementing the proposed method on these case studies demonstrates the accuracy of the proposed method for large scale power systems.

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.

A New Family of Distributions Based on the Generalized Pearson Differential Equation with Some Applications

Recently, a generalization of the Pearson differential equation has appeared in the literature, from which a vast majority of continuous probability density functions (pdf’s) can be generated, known as the generalized Pearson system of continuous probability distributions. This paper derives a new family of distributions based on the generalized Pearson differential equation, which is a natural generalization of the generalized inverse Gaussian distribution. Some characteristics of the new distribution are obtained.Plots for the cumulative distribution function, pdf and hazard function, tableswith percentiles and with values of skewness and kurtosis are provided. Itis observed that the new distribution is skewed to the right and bears mostof the properties of skewed distributions. As a motivation, the statistical applicationsof the results to a problem of forestry have been provided. It is found that our newly proposed model fits better than gamma, log-normal and inverse Gaussian distributions. Since many researchers have studied the use of the generalized inverse Gaussian distributions in the fields of biomedicine, demography, environmental and ecological sciences, finance, lifetime data, reliability theory, traffic data, etc., we hope the findings of the paper will be useful for the practitioners in various fields of theoretical and applied sciences.

A Probabilistic Framework for Joint Pedestrian Head and Body Orientation Estimation

We present a probabilistic framework for the joint estimation of pedestrian head and body orientation from a mobile stereo vision platform. For both head and body parts, we convert the responses of a set of orientation-specific detectors into a (continuous) probability density function. The parts are localized by means of a <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>pictorial structure</i></b> approach, which balances part-based detector responses with spatial constraints. Head and body orientations are estimated jointly to account for anatomical constraints. The joint single-frame orientation estimates are integrated over time by particle filtering. The experiments involved data from a vehicle-mounted stereo vision camera in a realistic traffic setting; 65 pedestrian tracks were supplied by a state-of-the-art pedestrian tracker. We show that the proposed joint probabilistic orientation estimation framework reduces the mean absolute head and body orientation error up to 15° compared with simpler methods. This results in a mean absolute head/body orientation error of about 21°/19°, which remains fairly constant up to a distance of 25 m. Our system currently runs in near real time (8–9 Hz).

Classification of Emotional Speech of Children Using Probabilistic Neural Network

&lt;p&gt;Child emotions are highly flexible and overlapping. The recognition is a difficult task when single emotion conveys multiple informations. We analyze the relevance and importance of these features and use that information to design classifier architecture. Designing of a system for recognition of children emotions with reasonable accuracy is still a challenge specifically with reduced feature set. In this paper, Probabilistic neural network (PNN) has been designed for such task of classification. PNN has faster training ability with continuous class probability density functions. It provides better classification even with reduced feature set. LP_VQC and pH vectors are used as the features for the classifier. It has been attempted to design the PNN classifier with these features. Various emotions like angry, bore, sad and happy have been considered for this piece of work. All these emotions have been collected from children in three different languages as English, Hindi, and Odia. Result shows remarkable classification accuracy for these classes of emotions. It has been verified in standard databse EMO-DB to validate the result.&lt;/p&gt;

Lane Detection Method of Statistical Hough Transform Based on Gradient Constraint

A lane detection method of statistical Hough transform based on gradient constraint is proposed to solve the problem of computational cost and grid quantization precision of classical Hough transform. Statistical Hough transform uses the Gaussian kernel function to model each pixel in the image .The size of initial data set is limited by using the method of gradient constraint. Eventually lane parameters’ continuous probability density function is given. The results of the experimentation show that under highway circumstance the provided method can rapidly and robustly detect the lane.

Deformable multi-modal image registration by maximizing Rényi's statistical dependence measure

A novel variational model for deformable multi-modal image registration is presented in this work. As an alternative to the models based on maximizing mutual information, the Rényi's statistical dependence measure of two random variables is proposed as a measure of the goodness of matching in our objective functional. The proposed model does not require an estimation of the continuous joint probability density function. Instead, it only needs observed independent instances. Moreover, the theory of reproducing kernel Hilbert space is used to simplify the computation. Experimental results and comparisons with several existing methods are provided to show the effectiveness of the model.

Quantification and classification of neuronal responses in kernel-smoothed peristimulus time histograms.

Peristimulus time histograms are a widespread form of visualizing neuronal responses. Kernel convolution methods transform these histograms into a smooth, continuous probability density function. This provides an improved estimate of a neuron's actual response envelope. We here develop a classifier, called the h-coefficient, to determine whether time-locked fluctuations in the firing rate of a neuron should be classified as a response or as random noise. Unlike previous approaches, the h-coefficient takes advantage of the more precise response envelope estimation provided by the kernel convolution method. The h-coefficient quantizes the smoothed response envelope and calculates the probability of a response of a given shape to occur by chance. We tested the efficacy of the h-coefficient in a large data set of Monte Carlo simulated smoothed peristimulus time histograms with varying response amplitudes, response durations, trial numbers, and baseline firing rates. Across all these conditions, the h-coefficient significantly outperformed more classical classifiers, with a mean false alarm rate of 0.004 and a mean hit rate of 0.494. We also tested the h-coefficient's performance in a set of neuronal responses recorded in humans. The algorithm behind the h-coefficient provides various opportunities for further adaptation and the flexibility to target specific parameters in a given data set. Our findings confirm that the h-coefficient can provide a conservative and powerful tool for the analysis of peristimulus time histograms with great potential for future development.

Physical context for theoretical approaches to sediment transport magnitude‐frequency analysis in alluvial channels

AbstractTheoretical approaches to magnitude‐frequency analysis (MFA) of sediment transport in channels couple continuous flow probability density functions (PDFs) with power law flow‐sediment transport relations (rating curves) to produce closed‐form equations relating MFA metrics such as the effective discharge, Qeff, and fraction of sediment transported by discharges greater than Qeff, f+, to statistical moments of the flow PDF and rating curve parameters. These approaches have proven useful in understanding the theoretical drivers behind the magnitude and frequency of sediment transport. However, some of their basic assumptions and findings may not apply to natural rivers and streams with more complex flow‐sediment transport relationships or management and design scenarios, which have finite time horizons. We use simple numerical experiments to test the validity of theoretical MFA approaches in predicting the magnitude and frequency of sediment transport. Median values of Qeff and f+ generated from repeated, synthetic, finite flow series diverge from those produced with theoretical approaches using the same underlying flow PDF. The closed‐form relation for f+ is a monotonically increasing function of flow variance. However, using finite flow series, we find that f+ increases with flow variance to a threshold that increases with flow record length. By introducing a sediment entrainment threshold, we present a physical mechanism for the observed diverging relationship between Qeff and flow variance in fine and coarse‐bed channels. Our work shows that through complex and threshold‐driven relationships sediment transport mode, channel morphology, flow variance, and flow record length all interact to influence estimates of what flow frequencies are most responsible for transporting sediment in alluvial channels.

Epistemic uncertainties in RANS model free coefficients

The epistemic uncertainty in the free coefficients of two popular Reynolds-averaged Navier–Stokes (RANS) eddy-viscosity turbulence models is quantified. In particular, the Launder–Sharma low-Reynolds K-ε and the Menter K-ω SST models are considered. The free coefficients present in turbulence models are retrieved from some properties of benchmark turbulent flows, viz. the energy power law exponent for decaying homogeneous isotropic turbulence, the value of the Von Karman constant, the turbulence production over dissipation rate estimated in the asymptotic regime of a homogeneous shear flow and the dimensionless turbulent kinetic energy in the logarithmic layer. The values presented in literature for these quantities, obtained from experiments or direct numerical simulations (DNS), show a significant dispersion, indicating the presence of an epistemic uncertainty. Starting from the data collected in literature, realistic continuous probability density functions of the basic flow properties, and hence of the RANS model coefficients, are obtained through generalized Polynomial Chaos (gPC). The impact of this uncertainty on the results of RANS simulations of the turbulent channel flow is then investigated for different Reynolds numbers through comparison with DNS data. The solution over the continuous multi-dimensional uncertainty space of the considered random variables is reconstructed through the application of a surrogate model (response surface) obtained by means of gPC. In general, the predictions of the K-ω SST model are less sensitive to the uncertainty in the model parameters than those of the low-Reynolds K-ε model. For both models and for any combination of the coefficients, the predictions of the turbulent kinetic energy profile are not satisfactory, while low errors can be obtained for friction and mean velocity. Evaluation of optimal values of free parameters is a Data Assimilation (DA) problem. An example of the use of the gPC response surface for an efficient calibration of the model coefficients in order to minimize the error in the prediction of these two last variables is provided. The result accuracy estimated through the gPC surrogate model well agrees with that obtained in deterministic simulations carried out with the calibrated values of the model constants.

Evaluating the influence of defects on the young's modulus of carbon nanotubes using stochastic modeling

The main goal of this research is to investigate the influence of structural defects on the mechanical properties of single-walled carbon nanotubes (CNTs). Two different types of the structural defects called Stone-Wales and vacancy defect are studied. While the former is categorized under the process-induced defect and it appears during the growth process of CNT, the later is caused when chemical functionalization is applied to CNT for fabrication carbon nanotube reinforced nanocomposite providing better bonding between CNT and surrounding resin. The number of broken C-C bonds, distributions and their orientations are all taken into account as random variables accounting for full stochastic analysis. Therefore, a computer code is provided for the stochastic modeling. The finite element (FE) model of the CNT is built using nanoscale continuum mechanics approach and then structural defects are applied randomly to the CNT. The Young's modulus of defected CNTs are obtained and compared with non-defected ones. It is revealed that the importance of vacancy defect is considerably higher than that of Stone-Wales defects implying on the drawback of chemical functionalization process. A detailed study is carried out on the topology of the defect and also continuous probability density functions of defect CNT Young's modulus are characterized.

A measure of asymmetry based on a new necessary and sufficient condition for symmetry

It is common practice to make assertions about the symmetry or asymmetry of a probability density function based on coefficients of skewness. Since most coefficients of skewness are designed to be zero for a symmetric density, they do, overall, provide an indication of symmetry. However, skewness, as opposed to asymmetry, is primarily influenced by the tail behavior of a density function. Therefore, coefficients of skewness do not reliably calibrate asymmetry in the density curve. Our goal in this paper is to present a new measure of asymmetry. With this purpose, we first provide a new necessary and sufficient condition for a continuous probability density function to be symmetric. This condition is then used to produce an asymmetry measure - a coefficient of asymmetry- for a continuous probability density function on the scale of −1 to 1. We show through examples that the proposed measure does an admirable job of capturing the visual impression of asymmetry of a continuous density function. Further, we discuss its implementation in practice and conclude by demonstrating its usefulness via an application to a real world dataset.

An Improved Approach of Terrain-Aided Navigation Based on RBPF

Rao-Blackwellized Particle Filter (RBPF) is suitable for solving the linear/nonlinear mixed Terrain-Aided Navigation (TAN) problem. But the Particle Filter (PF) part of RBPF is Standard Particle Filter (SPF), causing particle diversity reduction and even filters divergence under extreme conditions. To get a better estimation of the errors of INS, this paper proposes an improved approach called Regularized Rao-Blackwellized Particle Filter (RRBPF). After updating the nonlinear state and corresponding importance weights, RRBPF resamples from the Epanechnikov kernel and then get the resampled particles through a linear transition process. Theoretically, the resampling part of RRBPF is equivalent to resampling from the approximated continuous posterior probability density function. Shuttle Radar Topography Mission (SRTM) terrain data is used in simulations to investigate the performance of RRBPF. Results show that RRBPF can provide more accurate estimation of TAN and bear larger initial position error than Sandia Inertial Terrain Aided Navigation (SITAN).

The distribution of the maximum of a first order moving average: the continuous case

We give the cumulative distribution function of \(M_n\), the maximum of a sequence of n observations from a first order moving average. Solutions are first given in terms of repeated integrals and then for the case, where the underlying independent random variables have an absolutely continuous probability density function. When the correlation is positive, $P( M_n \leq x ) \ =\ \sum\limits _{j=1}^{\infty } \beta _{j, x} \ \nu _{j, x}^{n},$where \(\{\nu _{j, x}\}\) are the eigenvalues (singular values) of a Fredholm kernel and \(\beta _{j, x}\) are some coefficients determined later. A similar result is given when the correlation is negative. The result is analogous to large deviations expansions for estimates, since the maximum need not be standardized to have a limit. For the continuous case the integral equations for the left and right eigenfunctions are converted to first order linear differential equations. The eigenvalues satisfy an equation of the form $\sum\limits _{i=1}^{\infty } w_i ( \lambda -\theta _i )^{-1}=\lambda -\theta _0$for certain known weights \(\{ w_i\}\) and eigenvalues \(\{ \theta _i\}\) of a given matrix. This can be solved by truncating the sum to an increasing number of terms.

Generalized Order Statistics from Generalized Exponential Distributions in Explicit Forms

The generalized order statistics which introduced by [1] are studied in the present paper. The Gompertz distribution is widely used to describe the distribution of adult deaths, and some related models used in the economic applications [2]. Previous works concentrated on formulating approximate relationships to characterize it [3-5]. The main aim of this paper is to obtain the distribution of single, two, and all generalized order statistics from Gompertz distribution with some special cases. In addition the conditional distribution of two generalized order statistics from the same distribution is obtained. The Gompertz distribution has a continuous probability density function with location parameter a and shape parameter b, , where x restricted by the interval . The nth moment generated function of the Gompertz distributed random variable X is given on the form: where, is the generalized integro-exponential function [6]. In this paper we shall obtain joint distribution, distribution of product of two generalized order statistics from the Gompertz distribution, and then derive some useful formulas of these distributions as special cases.