A new maximum entropy method for estimation of multimodal probability density function

Highlights

• A new algorithm based on nonlinear transformation is proposed for estimation of multimodal probability density function.

• The original multimodal performance function can be converted to the suitable form through the transformation.

• Engineering examples show the applicability and accuracy of the proposed method in the real-life engineering setting.

Abstract

The high-precision estimation of a multimodal probability density function is a difficult problem in many engineering fields. We propose a new method to improve the estimation accuracy based on the fractional moment-based maximum entropy method with a nonlinear transformation and a multi-peak recognition method. For the translation parameters in the nonlinear transformation, three approaches, such as sample-based least square polynomial fitting technique, sample-based kernel density estimation and classical maximum entropy method, are presented to determine the parameters. By adjusting the translation parameter, the valley of the probability density function curve can be translated to the position with a larger slope, and the distance between adjacent peaks is enlarged to avoid the wrong fitting form of the probability density function curve with multiple peaks. After the parameters of the transformation are obtained, the fractional moment-based maximum entropy method is applied to predict the probability density function of the transformed performance function. Two numerical examples are used to verify the accuracy and stability of the proposed method. Two engineering examples are introduced to illustrate the applicability and efficiency of the proposed method in the real-life engineering setting. It is concluded that the proposed method uses fewer moments with less additional calculation costs, and has good computational efficiency and applicability for the modeling of multimodal probability density functions, compared with the classical fractional moment-based maximum entropy method.

Introduction

In 2013, Zhang and Pandey [1] proposed the fractional moment-based maximum entropy method (FM-MEM), which predicts the probability density function (PDF) of the performance function based on statistical moments. The principle of FM-MEM is to find the PDF with the maximum entropy from many PDFs satisfying the given moment constraints, because the maximum entropy indicates the prediction of PDF is an unbiased estimation with the highest degree of randomness. Without introducing prior assumptions, the FM-MEM overcomes the theoretical shortcomings of other classical moment-based methods. Compared with the classic integer moment maximum entropy method (IM-MEM) [2], FM-MEM is more flexible in the selection of moment order, which brings an improvement in accuracy and therefore, receives widespread attention. In recent years, many scholars have conducted in-depth research on various aspects of the FM-MEM.

For the determination of moment constraints, Xu and Kong [3] developed an adaptive scaled unscented transformation to obtain the fractional moments with only a few of sample evaluations. Based on the generalized dimension-reduction method, He et al. [4] proposed a hybrid dimension-reduction method, which has high calculation accuracy for cases where the variables have significant correlation. Xu et al. [5] presented a rotational quasi-symmetric point method (RQ-SPM) to renew the fractional moment constraints of MEM with high computational efficiency. To improve the efficiency and accuracy, Xu and Dang [6] proposed the good lattice point method based partially stratified sampling, which weighs the efficiency and accuracy of high-dimensional reliability problems. Li et al. [7] proposed to combine the Laplace transform with the MEM to improve the optimization strategy and considerably reduce the calculational cost. Xu et al. [8] determined the initial values of Lagrange multipliers and moment orders by solving nonlinear equations, and then searched more accurate solutions in its vicinity to improve the calculation accuracy. The method is also effective for high-dimensional reliability analysis. For different practical problems, Gzyl et al. [9] used FM-MEM to determine the probability of ultimate ruin as a function of the initial surplus. Yu et al. [10] combined the extreme moment method and an improved MEM to solve problems with multiple failure modes and time parameters. Zhang et al. [11] employed the FM-MEM to solve the problem of probabilistic lifetime modeling of the dynamical and discontinuous stochastic systems.

In fact, fitting the PDF accurately is very challenging when only limited moment information is available, especially for the multimodal PDF. Although many scholars have conducted extensive and in-depth research on the MEM, the method is mostly applied to solve the unimodal PDF. When the common MEM is used to model a multimodal PDF, the predicted failure probability may have significant errors. Actually, modeling multimodal PDFs is needed in many engineering fields. For example, the PDF of fatigue stress in a steel bridge structure is a bimodal distribution [12], the axle load of a heavy haul vehicle [13] and the vibrating load of turbine generator blades [14] are random variables with multimodal distributions. Therefore, some additional processing is required to obtain a more reasonable fitting result for those problems [15]. Jiang [16] proposed a bimodal estimation form of PDFs from normal distributions with unknown parameters when the distance between two peaks of PDF is appropriate. He et al. [14] proposed an uncertain propagation method combining the Laplace transform and the first-order second-moment method and solved the reliability analysis problem involving multimodal distributions when the first-order reliability method is applicable and accurate. Hu and Du [17] studied the saddle point approximation of the mean value from the bimodal distributed random function to estimate the failure probability. Zhang et al. [18] improved the MEM for the bimodal PDF estimation by introducing a new type of the maximum entropy convergence criterion with the high-order moments. In addition, the FM-MEM [19] has been used to solve some problems with bimodal PDFs profited from more statistical information of the fractional moment, although increasing the number of fractional moments leads to more computation efforts for an accurate prediction of a bimodal PDF with two close peaks.

To increase the efficiency and accuracy of FM-MEM for the problem of multimodal PDF estimation, we propose an improved FM-MEM by introducing a specified nonlinear transformation of the performance function. Through the transformation, the sampling density in the original space can be rearranged and the multimodal characteristics of the probability distribution of the samples become more obvious, which facilitates reducing the modeling difficulty and improving modeling accuracy. The proposed method significantly enhances the performance of multimodal recognition, without changing PDF estimation form or increasing the number of terms of its exponential polynomial; therefore, no additional undetermined parameters are needed. Moreover, some tricky problems, such as high-order moment variability and function oscillations can be also avoided. Three types of algorithms, including pre-computation with the MEM, the least squares polynomial fitting, and the kernel density estimation, are provided for different situations to determine the unknown parameter of the transformation. Several typical examples are tested to verify the validity of the proposed method, compared with the original FM-MEM and the Monte Carlo simulation.