Estimating the parameters of 3-p Weibull distribution using particle swarm optimization: A comprehensive experimental comparison

doi:10.1016/j.amc.2015.06.043

Applied Mathematics and Computation

Volume 268, 1 October 2015, Pages 201-226

https://doi.org/10.1016/j.amc.2015.06.043 Get rights and content

Highlights

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The Weibull distribution is used lifetime distributions in reliability engineering.
•
The estimation of the parameters of this distribution is essential.
•
Maximum likelihood (ML) estimation is a common method for parameter estimation.
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The working principle of ML estimation based on maximizing the likelihood function.
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We used particle swarm optimization as parameter estimation tool in 3-p Weibull.

Abstract

Weibull distribution plays an important role in failure distribution modeling in reliability studies and the estimation of its parameters is essential in the most real applications. Maximum likelihood (ML) estimation is a common method, which is usually used to elaborate on the parameter estimation. The maximizing likelihood function formed for the parameter estimation of a three-parameter (3-p) Weibull distribution is a quite difficult problem. Hence, the heuristic approaches must be used to discover good solutions. Particle swarm optimization (PSO) is a population based heuristic optimization technique developed from swarm intelligence. The performance of PSO greatly depends on its control parameters such as inertia weight and acceleration coefficients. Slightly different parameter settings may direct to very different performance. This paper gives a comprehensive investigation of different PSO variants (according to inertia weight procedures, acceleration coefficients, particle size, and search space) in the parameter estimation problem of 3-p Weibull distribution. Three explanatory numerical examples are given to show that PSO approach variants exhibit a rapid convergence to the maximum value of the likelihood function in less iteration, provides accurate estimates and PSO method is satisfactory for the parameter estimation of the 3-p Weibull distribution.

Introduction

Forecasting a crucial data such as lifespan of a machine part or the duration of treatment of a serious disease in real life requires real data samples in order to construct a distribution model before forecasting operation. Each real time application can be modeled by a specific statistical distribution such as Normal, Gamma or a Weibull distribution and for an accurate estimation, the data samples should be fitted to projected distribution function. Hence, to obtain the distribution properties, the accurate parameter estimation of the related function becomes inevitable.

Since an inappropriate selection of distribution function leads a wrong estimation, one main problem in real time forecasting is the determination of the statistical distribution and the Weibull distribution [1] is the most widely used distribution in reliability and lifetime studies, because it is extremely flexible in fitting random data and adaptable in different shape distribution, it has been used in many different fields like material science, engineering, physics, chemistry, meteorology, medicine, pharmacy, economics and business, quality control, biology, geology, and geography [2]. It is a known fact that three-parameter Weibull distribution family is extremely flexible and can fit very well an extremely wide range of empirical observations [3].

Successful application of Weibull distribution depends on having acceptable statistical estimates of the unknown parameters. Estimating the parameters of the 3-p Weibull distribution family is intrinsically a very difficult task compared with two parameter cases and in order to estimate these parameters numerous approaches have been conducted in the literature.

Section snippets

Literature survey

The cumulative distribution function (cdf) and the corresponding probability density function (pdf) of the three-parameter Weibull distribution is given by $F_{X} (x) = 1 - e^{- {(\frac{x - γ}{η})}^{β}}$ $f_{X} (x) = \frac{β}{η} {(\frac{x - γ}{η})}^{β - 1} e^{- {(\frac{x - γ}{η})}^{β}}$ where x > γ ≥ 0, η > 0 and β > 0 are location, scale and shape parameter, respectively [3].

The estimation methods conducted in literature may be summarized as follows. Nosal and Nosal [4] used Monte Carlo methods and array processing language to investigate the performance of the gradient random

Particle swarm optimization

Particle swarm optimization (PSO) is biologically inspired technique derived from the collective behavior of bird flocks, first introduced by Kennedy and Eberhart [32] and Eberhart and Kennedy [33]. PSO, known as an optimizer, is a population-based, self-adaptive search optimization technique [41]. PSO consists of a set of solutions (particles) called population. Each solution consists of a set of parameters and represents a point in multidimensional space. All the particles in the swarm act

Implementation of PSO to 3-p Weibull parameter estimation and numerical examples

For ML estimation, in order to estimate the parameters of not only the 3-p Weibull distribution but also any distribution type, the kind of distribution for which the samples come from should be known and for an accurate and sensitive estimation the samples should be enough in size. For performance evaluation of the PSO algorithm, several real values of the parameters of the 3-p Weibull distribution specified in vector $θ = (γ, η, β)$ along with different sample sizes n are used. For parameters, $θ = (2,$

Conclusions

Estimation of the 3-p Weibull distribution occurs in many real-life problems because this distribution is an important model especially for reliability and lifetime studies. It is important to estimate the unknown parameters exactly for modeling. In this paper, different from the existing studies in the literature, we discuss performance of the PSO algorithm in terms of inertia weights, acceleration coefficients, particle size and search space dimension, on the 3-p Weibull parameter estimation.

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View full text

Estimating the parameters of 3-p Weibull distribution using particle swarm optimization: A comprehensive experimental comparison

Highlights

Abstract

Introduction

Section snippets

Literature survey

Particle swarm optimization

Implementation of PSO to 3-p Weibull parameter estimation and numerical examples

Conclusions

Reliab. Eng. Syst. Saf.

Environ. Pollut.

Math. Comput. Simul.

Comput. Stat. Data Anal.

Appl. Math. Comput.

J. Stat. Plann. Inference

Appl. Energy

Comput. Stat. Data Anal.

Appl. Math. Comput.

Expert Syst. Appl.

Appl. Math. Modell.

Stat. Methodol.

Appl. Math. Modell.

Appl. Math. Comput.

Appl. Math. Modell.

Appl. Math. Modell.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Soft Comput.

A statistical theory of the strength of material

Ing. Vetensk. Acad. Handl

Three-parameter Weibull generator for replacing missing observations

Fitting the three-parameter Weibull distribution: review and evaluation of existing and new methods

IEEE Trans. Dielectr. Electr. Insul.

The method of three-parameter Weibull distribution estimation

Acta Commentationes Universitatis Tartuensis Math.

Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start

J. Comput. Appl. Math.