Bayesian estimation of complicated distributions

In a previous paper (Zong Z, Lam KY. Estimation of complicated disributions using B-spline functions. Structural safety 1998; 20(4): 323–32), we used a linear combination of B-spline functions to approximate complicated distributions. The method works well for large samples. In this paper, we extend the method to small samples. We still use a linear combination of B-spline functions to approximate a complicated probability density function (p.d.f). Strongly influenced by statistical fluctuations, the combination coefficients (unknown parameters) estimated from a small sample are highly irregular. Useful information is, however, still contained in these irregularities, and likelihood function is used to pool the information. We then introduce smoothness restriction, based on which the so-called smooth prior distribution is constructed. By combining the sample information (likelihood function) and the smoothness information (smooth prior distribution) in the Bayes' theorem, the influence of statistical fluctuations is effectively removed, and greatly improved estimation, which is close to the true distribution, can be obtained by maximizing the posterior probability. Moreover, an entropy analysis is employed to find the most suitable prior distribution in an “objective” way. Numerical experiments have shown that the proposed method is useful to identify an appropriate p.d.f. for a continuous random variable directly from a sample without using any prior knowledge of the distribution form. Especially, the method applies to large or small samples.     

Stochastic fatigue crack growth in steel structures subject to random loading

Uncertainty in fatigue crack growth under service load conditions arises from the statistical characteristics of crack growth under constant amplitude loading and from random variable amplitude loading. This study generalizes previous stochastic fatigue crack growth models by incorporating a time-dependent noise term described by arbitrary marginal distributions and autocorrelations to model the uncertainty in the crack growth under constant amplitude loading. A computationally efficient approach for handling wide-band random loadings based on the rainflow method of stress cycle identification also is developed. The method is illustrated with a fatigue reliability analysis of a steel miter gate at a lock and dam facility operated by the US Army Corps of Engineers.     

Direct estimation of quantile functions using the maximum entropy principle

The paper presents a distribution free method for estimating the quantile function of a non-negative random variable using the principle of maximum entropy (MaxEnt) subject to constraints specified in terms of the probability-weighted moments estimated from observed data. Traditionally, MaxEnt is used for estimating the probability density function under specified moment constraints. The density function is then integrated to obtain the cumulative distribution function, which needs to be inverted to obtain a quantile corresponding to some specified probability. For correct modelling of the distribution tail, higher order moments must be considered in the analysis. However, the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain. The difficulty in obtaining accurate moment estimates from small samples has been the main impediment to the application of the MaxEnt Principle in extreme quantile estimation. The present paper is an attempt to overcome this problem by the use of probability-weighted moments (PWMs), which are essentially the expectations of order statistics. In contrast with ordinary statistical moments, higher order PWMs can be accurately estimated from small samples. By interpreting the PWM as the moment of quantile function, the paper derives an analytical form of quantile function using MaxEnt principle. Monte Carlo simulations are performed to assess the accuracy of MaxEnt quantile estimates computed from small samples.     

Role of non-destructive evaluation in time-dependent reliability analysis

Uncertainties exist in the ability of nondestructive evaluation (NDE) methods to detect and measure flaws. Probabilistic methods can be used for characterizing these uncertainties and for updating flaw sizes predicted from stochastic fatigue crack growth analysis. The impact of NDE on condition assessment and reliability of structures, in which fatigue crack growth is occurring as a result of cyclic random service loads, is examined. Examples of probability-based condition assessment using magnetic particle or ultrasonic inspection and its potential impact on time-dependent fatigue reliability analysis are illustrated for a steel miter gate at a lock operated by the US Army Corps of Engineers.     

Estimation of the maximum entropy quantile function using fractional probability weighted moments

In a previous study, the conventional or integral-order probability weighted moments (IPWM) and the principle of maximum entropy were combined to derive an analytical form of the quantile function of a random variable [Pandey MD. Direct estimation of quantile functions using the maximum entropy principle. Struct Safety 2000;22(1):61–79]. This method is extended and improved in the present paper by utilizing the concept of fractional probability weighted moments (FPWMs). A general estimation method is proposed in which the Monte Carlo simulations and optimization algorithms are combined to estimate fractionals of FPWM that would lead to the best-fit quantile function. The numerical examples presented in the paper illustrate that the accuracy of the proposed FPWM based quantile function is superior to that estimated from the use of conventional IPWMs.     

Fatigue life estimation of steel girder of Yangpu cable-stayed Bridge due to buffeting

As the main span of modern cable-stayed bridges becomes longer and longer, the buffeting-induced fatigue damage problem of steel girders located in strong wind regions may have to be taken into consideration in the design of the bridge. This paper presents a method in the mixed frequency–time domain for estimating the fatigue life of steel girders of the Yangpu cable-stayed Bridge due to buffeting. In the suggested method, the joint probability density function of wind speed and wind direction at the deck level of the bridge is first established. The power spectra of the critical stress of the girder are then derived from the power spectra of the generalized coordinates of the bridge for different wind speeds and wind directions. The derived stress spectra are no longer a narrow spectrum when the background component of stress response is included. Thus, the time histories of the critical stress are simulated from their power spectra and the stress cycle distributions are estimated in terms of rainflow count method. The formulae derived based on the modified Miner law and the random vibration theory are finally used for estimating the fatigue life of the bridge girder. The results show that the effects of wind direction on the fatigue life of the Yangpu Bridge are significant. The predicted fatigue life due to buffeting is much longer than the design life of the bridge.