A new maximum entropy method for estimation of multimodal probability density function
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.
Section snippets
A new FM-MEM for multimodal PDF estimation
In this paper, we propose a new FM-MEM for multimodal PDF estimation by introducing a specified nonlinear transformation, which leads to a convenient way to capture the property of multiple peaks in the PDF curve. The estimation of the PDF can be completed with high accuracy by using fewer fractional moments. The algorithm of the new FM-MEM has three steps to estimate the multimodal PDF: nonlinear transformation of the performance function, determination of translation parameters, and PDF
Examples
In this section, four numerical examples and one engineering example are illustrated to verify the feasibility of the proposed method. The accuracy of PDF estimation is characterized by the cumulative error of the PDF, with the value of [0, 1], expressed by the formula as follows:where f(y) represents the estimated PDF from classic FM-MEM or the proposed nonlinear transformation-based FM-MEM (NTBFM-MEM), and ft(y) represents the PDF from Monte Carlo
Conclusion and remarks
Aiming at the limitation of classical FM-MEM in multimodal probabilistic modeling problem, we propose a new method for estimating a multimodal PDF, in which the performance function is converted with a specific nonlinear transformation function to make it more suitable for FM-MEM. The transformation with adjustable parameters can extract the multimodal feature in PDFs adaptively, thus reduce the difficulty of modeling. Three ways, such as sample-based least square polynomial fitting technique,
Acknowledgement
The work is supported by the National Key Research and Development Program (Grant No. 2019YFA0706803), the National Natural Science Foundation of China (Grant Nos. 11872142 and 11672052) and Fundamental Research Funds for the Central Universities of China (Grant No. DUT2019TD37).
References (30)
- et al.
Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method
Struct. Saf.
(2013) - et al.
Adaptive scaled unscented transformation for highly efficient structural reliability analysis by maximum entropy method
Struct. Saf.
(2019) - et al.
Maximum Entropy Method-Based Reliability Analysis With Correlated Input Variables via Hybrid Dimension-Reduction Method
J. Mech. Des.
(2019) - et al.
Efficient reliability analysis of structures with the rotational quasi-symmetric point-and the maximum entropy methods
Mech. Syst. Sig. Process.
(2017) - et al.
A novel fractional moments-based maximum entropy method for high-dimensional reliability analysis
Appl. Math. Modell.
(2019) - et al.
An efficient approach for high-dimensional structural reliability analysis
Mech. Syst. Sig. Process.
(2019) - et al.
Determination of the probability of ultimate ruin by maximum entropy applied to fractional moments
Insurance Math. Econ.
(2013) - et al.
An effective approach for probabilistic lifetime modelling based on the principle of maximum entropy with fractional moments
Appl. Math. Modell.
(2017) - et al.
A high-precision probabilistic uncertainty propagation method for problems involving multimodal distributions
Mech. Syst. Sig. Process.
(2019) - et al.
An efficient reliability method combining adaptive support vector machine and Monte Carlo simulation
Struct. Saf.
(2017)
New maximum entropy-based algorithm for structural design optimization
Appl. Math. Modell.
Three-term conjugate approach for structural reliability analysis
Appl. Math. Modell.
Bimodal Renewal Processes Models of Highway Vehicle Loads
Reliab. Eng. Syst. Saf.
An efficient algorithm to compute maximum entropy densities
Econ. Rev.
An improved maximum entropy method via fractional moments with Laplace transform for reliability analysis
Struct. Multidiscipl. Optim.
Cited by (19)
A multiple kernel-based kernel density estimator for multimodal probability density functions
2024, Engineering Applications of Artificial IntelligenceA novel maximum entropy method based on the B-spline theory and the low-discrepancy sequence for complex probability distribution reconstruction
2024, Reliability Engineering and System SafetyDimension reduction for constructing high-dimensional response distributions by accounting for unimportant and important variables
2024, Probabilistic Engineering MechanicsRegional reliability sensitivity analysis based on dimension reduction technique
2023, Probabilistic Engineering Mechanics