Conditional Joint Probability Density Function Research Articles

Probability distributions for dynamic and extreme responses of linear elastic structures under quasi-stationary harmonizable loads

The non-stationary load models based on the evolutionary power spectral density (EPSD) may lead to ambiguous structural responses. Quasi-stationary harmonizable processes with non-negative Wigner-Ville spectra are suitable for modeling non-stationary loads and analyzing their induced structural responses. In this study, random environmental loads are modeled as quasi-stationary harmonizable processes. The Loève spectrum of a harmonizable load process contains several random physical parameters. An explicit approach to calculate the probability distributions for the dynamic and extreme responses of a linear elastic structure subjected to a quasi-stationary harmonizable load is proposed. Conditioned on the specific values of the load spectral parameters, the harmonizable load process is assumed to be Gaussian. The conditional joint probability density function (PDF) of structural dynamic responses at any finite time instants and the conditional cumulative distribution function (CDF) of the structural extreme response are provided. By multiplying these two conditional probability distributions with the joint PDF of the load spectral parameters, and then integrating these two products over the parameter sample space, the joint PDF of structural dynamic responses at any finite time instants and the CDF of the structural extreme response can be calculated. The efficacy of the proposed approach is numerically validated using two linear elastic systems, which are subjected to non-stationary and non-Gaussian wind and seismic loads, respectively. The merit of the harmonizable load process model is highlighted through a comparative analysis with the EPSD load model.

An efficient method based on AK-MCS for estimating failure probability function

The function of failure probability varying with distribution parameters of random inputs is referred as failure probability function (FPF), it is often required in reliability-based design optimization. However, it is a computational challenging task since large number of expensive model evaluations are needed to estimate the failure probability at every distribution parameter. A method combining adaptive Kriging with Monte Carlo simulation is employed by this paper to efficiently estimate the FPF. Based on the augmented reliability theory and Bayes’ rule, the estimation of FPF is firstly transformed into that of the augmented failure probability and conditional joint probability density function (PDF) of distribution parameters on failure domain. Then, a Kriging model for the actual performance function is iteratively trained by the U-learning function until the stopping criterion is satisfied. The well-trained Kriging model can efficiently and accurately recognize the failure samples in the sample pool, on which the augmented failure probability and conditional joint PDF of distribution parameters on failure domain can be simultaneously estimated without extra model evaluations. The results of test examples illustrate that the method used in this work is more efficient than the existing methods, but its accuracy depends on the PDF approximation algorithms.

Bayes theorem–based and copula-based estimation for failure probability function

The failure probability function (FPF), aiming to decouple the nested-loop reliability-based design optimization solution into a single-loop optimization problem, has attracted a great deal of interest from designers and researchers. It is defined as a function of failure probability with respect to the design parameter. Among the estimation methods for the FPF, the Bayes theorem based on probability distribution function methods is competitive. It transforms the FPF as the estimation of the conditional joint probability density function (PDF) of design parameters on the failure event and the augmented failure probability. The augmented failure probability can be estimated by Monte Carlo simulation, while for the joint multi-dimensional PDF, the existing estimation methods are unavailable. To alleviate this issue, a novel FPF estimation method is proposed by Bayes theorem and copula. In the proposed method, the FPF is derived as a product of the conditional copula density and the augmented failure probability, in which the vine copula is employed to disassemble the multi-dimensional conditional copula density into several bivariate copula density functions, and they can be completed by the existing PDF estimation methods. In contrast to the existing Bayes theorem–based estimation methods for the FPF, the proposed method interprets the FPF as the dependence function between design parameters and the augmented failure probability in terms of copula, which involuntarily breaks the limitation of the multi-dimensional design parameters. In addition, the adaptive Kriging surrogate model is embedded in the proposed method to improve the efficiency of the proposed method. The presented examples demonstrate the efficiency and accuracy of the proposed method.

An innovative estimation of failure probability function based on conditional probability of parameter interval and augmented failure probability

Abstract The failure probability function (FPF) is defined as a function of failure probability varying with the design parameters of random inputs, and it is usually estimated in advance to transform reliability-based design optimization into ordinary one. The existing method on Bayes formula can transform the FPF as the estimation of the augmented failure probability (AFP) and conditional joint probability density function (PDF) of design parameters on the failure event. Although the existing method can estimate the FPF by one Monte Carlo simulation (MCS), it suffers the limited capability of the existing PDF estimation techniques in capturing the tail characteristic and being extended to multi-dimensional problem. To alleviate this issue, an innovative strategy is presented by introducing a small interval of parameters to approximate the parameter, on which the FPF can be transformed into the estimation of the AFP and the conditional probability of the parameter interval by the Bayes formula. After the criterion of selecting the small intervals of the parameters is determined to satisfy the approximation precision, a single MCS and the adaptive Kriging nested MCS (AK-MCS) are organized to estimate AFP and the conditional probability of the parameter interval simultaneously. By introducing a small interval to approximate the parameter and replacing the conditional PDF estimation with the conditional probability one, the applicability and accuracy of the proposed method are greatly improved to the existing method. Two examples are employed to verify the accuracy and efficiency of the proposed method. The results show that the proposed method is more accurate than the existing method, and the AK-MCS is more efficient than the direct MCS, the small enough computational cost helps the proposed method be used in actual engineering application.

Region-Level Motion-Based Foreground Segmentation Under a Bayesian Network

This paper presents a probabilistic approach for automatically segmenting foreground objects from a video sequence. In order to save computation time and be robust to noise effects, a region detection algorithm incorporating edge information is first proposed to identify the regions of interest, within which the spatial relationships are represented by a region adjacency graph. Next, we consider the motion of the foreground objects and, hence, utilize the temporal coherence property in the regions detected. Thus, the foreground segmentation problem is formulated as follows. Given two consecutive image frames and the segmentation result priorly obtained, we simultaneously estimate the motion vector field and the foreground segmentation mask in a mutually supporting manner by maximizing the conditional joint probability density function of these two elements. To represent the conditional joint probability density function in a compact form, a Bayesian network is adopted, which is derived to model the interdependency of these two elements. Experimental results for several video sequences are provided to demonstrate the effectiveness of the proposed approach.

Planktonic Contact Rates in Homogeneous Isotropic Turbulence: Theoretical Predictions and Kinematic Simulations

The key role played by turbulence in the environment of plankton and larval fish populations has become appreciated in recent years. In particular, the turbulent enhancement of encounter rates between different species of microorganisms, either swimming or passively advected by the flow, is well established. However, most of the current modelling approaches are rather ad hoc, giving rise to ambiguities in the specification of certain key parameters. In this paper, the encounter problem in a turbulent flow of large Reynolds number is re-examined from first principles and a number of new formulae will be established for different swimming strategies. The key innovation is the proposal of a model form for the conditional joint probability density function of predator and prey velocities when the organisms are separated by their given contact radius, R. Particular attention will be paid to the case when a microorganism follows a random trajectory, due to a combination of its own swimming and the action of the flow. The theoretical predictions are subsequently tested against corresponding quantities derived from a series of kinematic simulations of a turbulent-like flow field. Good agreement is demonstrated between the predictions and simulations.