Immune Clone Maximum Likelihood Estimation of Improved Non-homogeneous Poisson Process Model Parameters

REN Li-na (任丽娜), RUI Zhi-yuan (芮执元), LEI Chun-li (雷春丽)

School of Mechanical and Electronical Engineering, Lanzhou University of Technology, Lanzhou 730050, China

Immune Clone Maximum Likelihood Estimation of Improved Non-homogeneous Poisson Process Model Parameters

REN Li-na (任丽娜), RUI Zhi-yuan (芮执元)*, LEI Chun-li (雷春丽)

SchoolofMechanicalandElectronicalEngineering,LanzhouUniversityofTechnology,Lanzhou730050,China

Aiming at the solving problem of improved non-homogeneous Poisson process(NHPP) model in engineering application, the immune clone maximum likelihood estimation(MLE) method for solving model parameters was proposed. The minimum negative log-likelihood function was used as the objective function to optimize instead of using iterative method to solve complex system of equations, and the problem of parameter estimation of improved NHPP model was solved by immune clone algorithm. And the interval estimation of reliability indices was given by using fisher information matrix method and delta method. An example of failure truncated data from multiple numerical control (NC) machine tools was taken to prove the method. and the results show that the algorithm has a higher convergence rate and computational accuracy, which demonstrates the feasibility of the method.

improvednon-homogeneousPoissonprocess;immuneclonealgorithm;maximumlikelihoodestimation(MLE) ;intervalestimation;multipleNCmachinetools

Introduction

Numerical control (NC) machine tool is a complex repairable electromechanical system. Being affected by the working environment of machine tools, proficiency of operators, maintenance strategy, and many other factors, the times between failures of a repairable system are generally not independent and identically distributed[1]. Therefore, in minimal repair, by taking machine tool failure process as a stochastic point process, using non-homogeneous Poisson process(NHPP) theory to establish the reliability model is an effective method for the reliability analysis of repairable systems such as NC machine tools. The failure intensity of pump and reliability of engine lifetime were analyzed respectively by adopting NHPP theory and verified the validity of the model in Refs.[2-3]. Wangetal.[4]established the reliability assessment model of minimal repair for NC machine tools based on power law process (PLP). It has shown that when the times between failures have obvious trend of rising or falling, NHPP model is more suitable to describe the failure process of NC machine tools. Cuietal.[5]analyzed the operational reliability of aviation onboard repairable products by adopting NHPP theory and obtained a general law of fault statistics for the onboard products. To make the model accord with engineering practice better, Yangetal.[6]put forward an improved NHPP model based on the PLP model, analyzed certain subsystem faults of tank and armored vehicles, and accurately obtained the failure intensity at the initial time. But the solution of model parameters has become another difficulty in engineering application. The traditional method by using iteration method to solve complex system of equation to obtain the maximum likelihood estimation (MLE) of model parameters is no longer applicable. In view of this, this paper presents an immune clone MLE method for solving the parameters of improved NHPP model.

1 Improved NHPP Model

PLP is one of the commonly used NHPP. It is applicable to describe the failure process of repairable mechanical systems. But when the shape parameter of the model is not 1, the failure intensity at initial time may be zero or infinitely great, which is not in consistency with the actual situation. Thus, Yangetal. proposed the improved NHPP model[6]and its failure intensity function is defined as

λ(t)=λ0+λβtβ -1,λ0,λ,β,t>0,

(1)

whent=0,λ(0)=λ0, namely the value reflects the actual failure intensity of NC machine tools at the initial time. Then the expected number of failures (namely the cumulative failure intensity function) up to a generic timetis given by

M(t)=λ0t+λtβ.

(2)

2 Model Parameters and Reliability Indices Estimation

2.1 Point MLE of model parameters

(3)

Then formNC machine tools with failure truncated data, the likelihood function relative to the above data is given by

(4)

Take the logarithms at both sides of Eq. (4) to obtain corresponding log-likelihood function as

(5)

wheretnjis the failure truncated time of thejth NC machine tool (j=1, 2, …,m);njis the number of failures occurred during the observation interval (0,T] of thejth NC machine tool.

Generally, the MLE of model parameters can be obtained by the following method: setting the first partial derivative of Eq. (5) with respect to each parameter and make them equal to zero, then the equations are obtained and the iterative method is used to get the model parameters estimation. But the method needs to set the initial value and iterate over and over again, if the initial value selection is not reasonable, it is likely to be trapped in local extremum. Furthermore, with the increase of model parameters, the calculating process of this method becomes more and more complex with low efficiency. Sometimes it even can’t solve. Therefore, take a negative number for Eq. (5) and minimize it as the objective function. Convert the problem of parameter estimation into an optimal problem with constraints. Using the intelligent optimization algorithms to solve can be regarded as a good choice.

Therefore, setting the first partial derivatives of Eq. (5) with respect toλ0andλas 0 gives

(6)

(7)

(8)

(9)

Substituting Eq. (9) into Eq. (5), a log-likelihood function with two unknown parameters can be obtained as

(10)

Thus, the MLE ofλandβcan be obtained by minimizing -lnL, but it needs to meet the constraint condition

(11)

2.2 Model parameters solution based on immune clone algorithm

According to the derivation stated in Section 2.1, the mathematical model (optimization model) of solving the model parameters can be obtained, which can be described as follows.

Problem: make the negative logarithmic likelihood function minimum, namely min(-lnL).

It is not hard to see that the optimization process mentioned above includes multiple constraint conditions. Therefore, in the process of solution using intelligent optimization algorithms, the design of operator will be relatively complicated. It needs to seek a simple and efficient operator to achieve optimization. Artificial immune clone algorithm, as a modern heuristic intelligent optimization algorithm[7], is suitable for solving nonlinear numerical optimization problems with constraints since it only covers cloning variation operator, and it is simple and practicable with relatively strong robustness. Therefore, the problem of parameter estimation of improved NHPP model is solved by using the immune clone algorithm. The main steps of the algorithm[8]are as follows.

(1) Initialization: set the population size asNand the clone scale asM. Set the clone variation rate asPm.

(2) Generation of initial antibody group: randomly generate initial antibody group and encode antibodies under the circumstance which satisfies the requirements of completeness, integrity, and non redundancy.

(3) Calculation of the fitness: calculate the fitness between antibody and antigen, namely calculate the affinity (make the negative log likelihood function (seen in Eq. (10)) as the objective function, and the affinity is the matching degree between the objective function and candidate solutions).

(4) Clonal expansion, selection, and variation[8]: selectnantibodies of the highest affinity with the antigen and perform cloning expansion, selection, and variation on the selected antibodies. Substitute themantibodies of the lowest affinity withmantibodies generated randomly among the group to form a new generation of antibody group.

(5) Termination conditions determination: when the termination conditions are satisfied, exit and output the results. Otherwise, return to step (3) and continue to calculate.

2.3 Point MLE of reliability indices

Clearly, once we have the MLE of the model parameters, point MLE of reliability indices, such as the failure intensity functionλ(t) and thes-expected number of failures up to a given timet, sayM(t), the instantaneous mean time between failurestIMTBF, and the cumulative mean time between failurestCMTBFcan be easily obtained by

(12)

(13)

(14)

(15)

2.4s-confidence intervals

Since negative confidence lower limits may be generated during the solution ofs-confidence intervals by using the asymptotic normal distribution characteristics of parameters’ MLE, the characteristic of asymptotic logarithmic normal distribution is used to solves-confidence intervals in this paper. Then the 1-μapproximate confidence interval for model parameters or reliability indices is given by

(16)

(17)

From Eq. (5), the second derivatives and the second partial derivatives of the model parameters are given by

According to delta-method[10], the variance of the cumulative failure intensity is:

(18)

SubstitutingM(t) withtIMTBFor other reliability indices, the variance of corresponding reliability indices can be obtained. And thes-confidence intervals of reliability indices can be calculated according to the variance and covariance of model parameters.

3 Example of Application

The data, given in Table 1, contain 37 failure times of two NC machine tools in three and a half years, with failure truncation (in cumulative operating hours) taken from a machine tool plant, and the NC machine tools are in reliability deterioration phase.

Table 1 NC machine tools failure data

Select the parameter individual search space as 30 coding spaces, the population size as 200,the clone scale as 20, the variation rate as 0.005, and the evolving algebra as 500. After 20 times of calculation, the optimized result is selected. The calculation results are shown in Table 2.

Table 2 Statistical results after 20 times of isolated operation of immune clone algorithm

From Table 2, we can see that the immune clone algorithm has a relatively higher convergence rate, requiring about 19.28 s to finish the calculation. It can preliminarily state that it is possible to use the method proposed in this paper to solve model parameters. But whether the accuracy of the solution is good or not, it can be verified by model fitting effects. Define the evaluation index for goodness-of-fit[11]as:

(19)

Fig.1 The curve of failure intensity

Fromtheshapeparameterβ=1.526,wecanconcludethatthemachinetoolisexperiencingdegradationphenomenaandthefailureintensityisonanincreasingtrend.ThecurveoffailureintensityisshowninFig.1.Thefailureintensityattheinitialtimeisλ0=0.00977,indicatingthatthemachinetoolshasthepossibilityoffailingattheinitialtimeofdatacollection,ifitisn’tconsideredinthemodel,theassessmentresultmaybepossiblenotaccurateandprovideainaccuratemaintenancestrategy.SubstitutethecalculationresultsofmodelparametersintoEqs. (12)-(15)respectivelytogetthepointestimationsofreliabilityindices,andthenthes-confidenceintervalscanbecalculatedaccordingtoEqs. (16)-(18).ThecalculationresultsareshowninTable3.ReliabilityindicescurvesareshowninFigs.2-4.

Table 3 Point and interval estimations of reliability indices

Fig.2 The curve of cumulative failure intensity

Fig.3 The curve of cumulative mean time between failures

Fig.4 The curve of instantaneous mean time between failures

From Table 3, we can see that, when the machine tool runs to 7511.75 h, the cumulative mean time between failures is 375.2 h, while the instantaneous mean time between failures is only 277.78 h. It is suggested to take measures to arrange an overhaul as soon as possible to prevent sudden shutdown during machine tools’ operation which will cause negative impacts on the production efficiency and benefits of the enterprise.

4 Conclusions

In evalution of the reliability of NC machine tools, to make the model conform to the engineering practice better, some influence factors are generally taken into the model through increasing model parameters. However, at the same time of model optimization, the difficulty of model parameters solution also increases. The iteration method used to solve the MLE of model parameters is no longer applicable. Therefore, translate the problem of parameter estimation into an optimal problem, and use the immune clone algorithm to estimate parameters of improved NHPP model, an example of failure truncated data from multiple NC machine tools is taken to prove the proposed method. The calculation results show that the immune clone algorithm has stronger robustness, higher convergence rate, and computational accuracy, thus the computational problem of the improved NHPP model in engineering application is solved.

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National CNC Special Project, China (No. 2010ZX04001-032); the Youth Science and Technology Foundation of Gansu Province, China (No. 145RJYA307)

1672-5220(2014)06-0801-04

Received date: 2014-08-08

* Correspondence should be addressed to RUI Zhi-yuan, E-mail: zhiy_rui@163.com

CLC number: TG659 Document code: A