Mathematical Problems in Engineering

Volume 2018, Article ID 3124048, 9 pages

https://doi.org/10.1155/2018/3124048

## Parameter Estimation Method of Mixture Distribution for Construction Machinery

School of Mechanical Science and Engineering, Jilin University, Changchun 130025, China

Correspondence should be addressed to Jinshi Chen; nc.ude.ulj@gnidaerps

Received 18 May 2018; Accepted 14 August 2018; Published 10 September 2018

Academic Editor: Surajit Kumar Paul

Copyright © 2018 Xinting Zhai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to the harsh working environment of the construction machinery, a simple distribution cannot be used to approximate the shape of the rainflow matrix. In this paper, the Weibull-normal (W-n) mixture distribution is used. The lowest Akaike information criterion (AIC) value is employed to determine the components number of the mixture. A parameter estimation method based on the idea of optimization is proposed. The method estimates parameters of the mixture by maximizing the log likelihood function (LLF) using an intelligent optimization algorithm (IOA), genetic algorithm (GA). To verify the performance of the proposed method, one of the already existing methods is applied in the simulation study and the practical case study. The fitting effects of the fitted distributions are compared by calculating the AIC and chi-square () value. It can be concluded that the proposed method is feasible and effective for parameter estimation of the mixture distribution.

#### 1. Introduction

The construction machinery, such as the wheel loader, often operates under complex and changeable operating conditions, which leads to the random characteristics of the load time history [1, 2]. To extract load cycles for extrapolation from load time history, the rainflow counting method is used. The rainflow matrix contains the number of load cycles ordered by range and mean is obtained. As a result, the rainflow matrix also presents complex and uncertain features which cannot be modelled by a simple distribution [3]. The finite mixture distribution is available as it is a flexible and powerful probabilistic modelling tool for univariate and multivariate data [4].

The mixture distribution model is widely acknowledged in the area which involves the statistical modelling of data [4]. The fitting effect of the load data can be improved when the mixture distribution replaces a poor single distribution. Nagode M [5] and Tovo R [6] suggested that the load range probability density function (PDF) should be defined as a linear combination of Weibull distributions. Klemenc J [7] thought the PDFs of both load mean and load range are the multivariate normal PDF. Ni Yiqing [8] used three types of finite mixture distributions (normal, lognormal, and Weibull) to model the stress range. For the load range in time domain, Wang Jixin [9] adopted the normal mixture distribution. In order to have an accurate mixture distribution, the parameters estimation methods have been used. For the rainflow matrix, [7] shows the advantages of mixture distribution with the Expectation Maximum (EM) algorithm on which the correlation between ranges and means is considered. According to the nonnegative characteristics of the load ranges of the rainflow matrix, Nagode M [3] proposed a mixture of joint Weibull-normal (W-n) distributions to avoid the shortcomings by the multivariate normal PDF. Reference [10] reviewed that load range of the rainflow matrix of the construction machinery could be defined as the Weibull distribution with 3 parameters. In the present paper, the mixture of joint Weibull with 3 parameters and normal distributions are used to model the rainflow matrix for load extrapolation.

The EM algorithm is proved to be the most appropriate algorithm to estimate parameters of the mixtures [8, 9, 11], but it also has a strong sensitivity to the initial values, which is the cause that the solution does not converge. Some studies are aimed on the initial values [12]. Laird proposed a grid search for setting the initial values [13]. Leroux used the means of clusters as the initial values [14]. McLachlan suggested the use of principal component analysis for selecting initial values [15]. Nagode and Fajdiga proposed an alternative algorithm for the parameter estimation of the finite mixture distribution [16–18]. Bučar T used the results of the algorithm proposed by Nagode and Fajdiga as initial values to reduce the effects of the drawback [11]. Although there are many improvements, the main idea of EM, which is the maximization of the conditional expectation, is unchanged. Consequently the results do not necessarily converge, especially for a great number of components [3]. In order to avoid the above problem, the intelligent optimization algorithm (IOA) is employed to convert the problem into an optimization problem. The genetic algorithm (GA) [19], particle swarm optimization [20], and ant colony optimization [21] belong to the IOA. In this paper, we prefer GA for its ability to maximize and search globally. And it is an attempt for the parameter estimation study of W-n mixture distribution.

The rest of paper is organized as follows: In Section 2, the W-n mixture distribution for the rainflow matrix is introduced. In Section 3, the proposed method based on GA is illustrated in detail. Comparative analysis are shown in Section 4, the feasibility and effectiveness of the proposed method are proven through the simulation study and case study. Section 5 ends the paper by presenting some conclusions and discussions.

#### 2. W-n Mixture Distribution Model

In general, an arbitrary rainflow matrix can be defined as the form of mixture distribution functions [3].which satisfies the following constraints:where represents a vector of component unknown parameters. Constant stands for a weighting factor, whereas represents a component PDF of two random variables which is the ranges and means points of the rainflow matrix. The corresponding vector parameter contains two parts, which are the Weibull distribution and normal distribution, respectively.

The random variables corresponding to the th component distribution are statistically independent as preconditions for the establishment of the mixture of joint W-n distributions [22]. According to the difference between the 3-parameter Weibull distribution and the 2-parameter Weibull distribution, the load ranges are shifted by threshold . Therefore, the mixture of joint Weibull with 3 parameters and normal distributions is shown as follows:The cumulative distribution function (CDF) can be expressed asThe mixture of joint Weibull with 3 parameters and normal distributions is obtained through inserting (3) into (1).where and represent the scale and shape parameters of the Weibull distribution, is the threshold, and and stand for the mean value and standard deviation of the normal distribution, respectively.

#### 3. Parameters Estimation Based on GA

The Maximum Likelihood estimation fits the parameters of the mixture PDF by maximizing the likelihood function (LF). To facilitate the calculation, (8), the log likelihood function (LLF), is used instead of the LF. For EM algorithm, the Newton-Raphson method is involved to solve the maximized LLF equation [22]. However, unsuitable selection of initial values may cause the fact that the solution does not converge.

GA [23] is a parallel and efficient global search method. The chromosome, population, and generation are involved. The unknown parameter is set as a chromosome representing the individual’s original characteristic. The unknown parameters of (8) form a set of chromosomes called population here. The dimension of the chromosome* D* represents the number of unknown parameters. The number of the initial populations* NP* is typically from 10 to 200. Through the operations of selection, crossover, and mutation of the populations, a new set of individuals is generated. The process of evolving from the predefined individuals to a new set of individuals is named a generation. And the number of generations* NG* is the number of evolutions. The general range of* NG* is 100~1000. GA is an optimization process. Equation (8) is the objective function (OBF). During the operations of selection, crossover, and mutation, the offspring are produced. The offspring meet the requirements of OBF which are selected as the parents of the next generation. The new parents produce new offspring through combination which depends on the crossover rate . The value range of is 0.25~1. According to the mutation rate with the value range of 0.001~0.1, the mutation step is handled. Then the offspring obtained after mutating become the next generation. The above procedures are repeated until the OBF values obtained by the parents and the offspring are nearly the same. Finally, the offspring are the desired parameter estimation results.

In addition, when dealing with the data, the number of components of the mixture distribution should be known. The Akaike information criterion (AIC) [24] is used to estimate the components number . The value of AIC varies with the value of and represents the number of estimated parameters. The best mixture distribution model will be the model with lowest AIC value.

Above all, the process chart of the proposed method is shown in Figure 1.