# Developing a Reliability Model of CNC System under Limited Sample Data Based on Multisource Information Fusion

Authors: Chong Peng; Yuzhen Cai; Guangpeng Liu; T. W. Liao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The reliability of the computer numerical control (CNC) system affects its processing performance and is a major concern in the manufacturing industry today. However, existing reliability models to assess the reliability of the CNC system often exhibit relatively large errors due to inadequate treatment of small samples. In order to get around the constraint of limited lifetime failure data and take full advantage of existing reliability parameters in traditional reliability models, a multisource information fusion-based reliability model grounded on Bayesian inference is proposed to deal with the small sample size. The prior distributions are derived by using the probability encoding method and conjugate distribution based on the idea of multisource information fusion. Then, using the Jensen–Shannon divergence (JSD) to measure the similarity between prior information and field observation data, a constrained optimization problem is established to determine the respective weight of prior information and field observation data. Finally, by conducting the reliability analysis of repairable CNC systems, the validity of the proposed model and its prior distribution derivation method are verified.


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## Body

## 1. Introduction

Enterprises and end users are increasingly demanding CNC systems with high reliability due to the rapid advancement in technology. The CNC system, as the core part of the CNC machine tool, plays a significant role in the manufacturing industry today, whose reliability directly affects production costs and the performance of the machine tool. A reliability model is the premise and basis of reliability research of product life cycle. How to build a more accurate reliability model based on available reliability data is the key to improve the reliability of electromechanical equipment, especially the CNC system. Traditional reliability modeling [1], based on the law of large numbers and the central limit theorem, combined with statistical principles such as maximum likelihood estimation (MLE) and the least square method, has achieved good results in the research of reliability modeling and assessment with small lifetime failure data. Yalcinkaya and Birgoren [2] proposed using the Bayesian–Weibull method to solve a problem for which confidence interval was difficult to estimate under the condition of small sample. Yang et al. [3] put forward the Weibull neural network model and validated the model using field failure data obtained from 23 sets of CNC machine tools. Peng et al. [4] proposed a bathtub-shaped failure rate model based on the piecewise intensity function and verified the validity of the proposed model as well.However, as the result of increased reliability of the CNC system in modern days, only limited lifetime failure data are available. Applying traditional method to modeling system reliability based on small sample without any modification will inevitably lead to decreasing model accuracy in the obtained model parameters and hence lose the confidence level of the assessment outcome.Information fusion can integrate multiple sources of information to interpret data better and provide a more accurate assessment of the system. Many scholars carried out research studies along this line based on the multisource information fusion method. Mathon et al. [5] used the Dempster–Shafer evidence theory (DST) to fuse collected field data and expert judgement information and obtained a better model applied to uncertainty surrounding permeability. Xu et al. [6] proposed a multisource fuzzy incomplete information fusion method based on the information entropy theory and verified its effectiveness.The methods mentioned above are not appropriate for building the reliability models of the CNC system because of highly complex calculations involved. With the development of the Markov Chain Monte Carlo (MCMC) methods [7–10], the limitations of computational complexity have been overcome. Accordingly, another fusion method, Bayesian inference, has been further studied. Yang et al. [11] obtained field failure time data of machine tools and carried out reliability assessment and modeling based on the Bayesian theory. Wilson and Fronczyk [12] held the belief that developing a prior distribution based on the Bayes theory using similar systems and expert information is useful for future work in those complex systems with limited reliability information. Liu et al. [13] merged degradation data into failure data of a measurement system based on the Bayes theory and combined it with the degenerate trajectory model to obtain a distribution of pseudofailure life to assess the reliability of metering equipment accurately.This paper proposes a new information fusion-based method (as shown in Figure1) for building a reliability model for the CNC system grounded on Bayesian inference, which can solve the problem of inadequate evaluation accuracy caused by lack of reliability data under limited conditions such as time and cost. By obtaining data that can reflect reliability information, combined with expert-provided information, the accuracy of reliability assessment can be greatly improved. Through the information fusion technology, reliability index can be quickly obtained to shorten test time and improve test efficiency, leading to a new approach for reliability modeling of the CNC system.Figure 1
Structure of the information fusion method.The remainder of the paper is organized as follows. Section2 describes the derivation of prior distribution and the optimization of respective weights of different prior distributions. Section 3 studies the parameter estimation of the model. Section 4 gives an exemplary application for verifying the proposed method. Finally, the paper is concluded.


## 2. Multisource Information Fusion Method Based on Bayesian Inference

The method of multisource information fusion comes from the theory of multisensor fusion, which collects and fuses data from different sources. The related issues in developing a reliability model mainly include the choice of model, quantification of multisource information, and fusion of different prior distributions [14], as shown in Figure 1. Choosing a model depends on the nature of highly reliable and long-life equipment. Weibull distribution, as a traditional reliability modeling method, can be used to analyse the reliability of the CNC system [15]. Accordingly, the failure rate function is(1)λt=βθtθβ−1,θ>0,β>0,t≥0,where β is the shape parameter of Weibull. When taking different β values, the shape of Weibull distribution is greatly affected. The scale parameter θ determines the mean value of a Weibull distribution.The quantization of information mainly uses the method of probability encoding and conjugate distribution. The fusion of different prior distributions uses the JSD to measure the similarity between prior information and field data, and a constrained optimization problem is established to determine the respective weights of prior information and field data. Finally, the reliability model for limited sample data is developed, by combining the probability encoding and the JSD method with Bayesian inference.


### 2.1. Bayesian Model

The Bayesian model is used to integrate different prior distributions [16]. The posterior distribution is(2)πθx=hx,θmx=pxθπθ∫Θpxθπθdθ,where πθ is the prior distribution for the model parameter. Let pθx be the likelihood function that is obtained based on the probability density function of the chosen reliability model.For the reliability analysis of the CNC system, the commonly used reliability model is Weibull distribution. The corresponding Bayesian model [17] is(3)πβ,θt=Ltβ,θπβ,θ∬β>0θ>0Ltβ,θπβ,θdβdθ=Ltβ,θ∑inwiπiβ,θ∬β>0θ>0Ltβ,θ∑inwiπiβ,θdβdθ,where Ltβ,θ is the likelihood function for the collected lifetime failure data. ∑inwiπiβ,θ is the fusion prior distribution that is generated by information fusion using data gathered from multiple sources.


### 2.2. Derivation of Prior Distribution Based on Failure Time Data

Deriving the prior distribution following Weibull distribution is one of the key issues in Bayesian analysis of lifetime data, which requires a great deal of integration and calculation. Therefore, an alternative approach is taken here [18], which involves transforming Weibull distribution into exponential distribution whose conjugate prior distribution determines the prior distribution of Weibull distribution.Based on the failure rate function given in equation (1), the reliability function is(4)Rt=exp−tθβ.The probability density function is(5)ft=βθtθβ−1exp−tθβ.The cumulative probability distribution function is(6)Ft=∫0t1βθtθβ−1exp−tθβdt.Assumingh=t, α=θ, and X=tβ, equation (6) turns into(7)FX=∫0X1αexp−hαdh.The probability density function after transformation is(8)fx=1αexp−xα,where fx is exponential distribution and 1/α is the parameter of exponential distribution. If β is known, then 1/α is only associated with θ; subsequently, the prior distribution of θ can be introduced by the conjugate distribution of exponential distribution. Under the same working environment and the same type of system, the shape parameters β of a particular CNC system are theoretically identical. If the assumed shape parameter deviates from the actual, there will be some errors. Therefore, it is possible to reduce the error to some extent by deriving the shape parameter through historical failure time data. Subsequently, the prior distribution of Weibull distribution [19] can be obtained according to the conjugate prior distribution of exponential distribution, that is,(9)πα=baΓa1αa+1exp−bα.


### 2.3. Derivation of Prior Distribution Based on Expert Information

Information provided by experts can be used to increase sample size and to improve the accuracy of reliability assessment for small samples. Therefore, the probability encoding method as shown in the data mining section of Figure1 [20], specifically the probabilities and values method is applied to obtain statistical information of expert information data in the form of probability density functions obtained by setting parameter type and quantifying expert information data. The information obtained by experts includes values of scale parameters and shape parameters and corresponding probability values, respectively [21]. Since the scale parameter and the shape parameter correspond to failure type and life expectancy of the CNC system, respectively, experts can give parameter values easily based on their experience. Using equation (10), expert information can be transformed into parameter information and the prior distribution of Weibull distribution can be generated from the expert information.The inverse cumulative distribution function (CDF) is(10)Pθ=∫−∞Vθfθ;μθ,σθdθ,where fθ;μθ,σθ is a parametric CDF.The probabilistic encoding method has the following steps:Step 1: determine variables to be collected and collect expert informationStep 2: select an appropriate Bayesian update methodStep 3: convert the expert information into model parameters using the inverse CDF functionStep 4: update model parameters using the Bayesian method


### 2.4. JSD Method for Determination of Weights

Multisource information fusion aims at integrating information from different dimensions or levels and obtaining more accurate reliability assessment result than using the single information source alone. The ultimate goal of information fusion is to obtain a joint probability distribution. In detail, prior distributions are reasonably assigned proper fusion weights and then integrated into the joint probability distribution. Information entropy measures the total uncertainty in a probability distribution; thus, the similarity between prior information and field test information can be quantified through their respective information entropy. Accordingly, the respective weights of different prior distributions can be determined prior to being used in the integrated prior distribution.The probability density information entropy is(11)Hmx=−Elogmxnx=−∫Ωπxlogmxnxdx,where E⋅ is mathematical expectation of m⋅. In equation (11), nx denotes the noninformation prior density function, and the noninformation means that information of the sample is completely unknown. The noninformation density function of the Weibull position parameter can be taken as nx=1 [22].The Kullback–Leibler divergence (KLD), a.k.a. relative entropy, can be used to calculate the similarity between two probability distributions. For probability distributionsAt and Bt, the KLD is defined as(12)KLDAB=∫Θatlogatbtdx,where at and bt are the probability densities of At and Bt, respectively. Note that KLDAB≠KLDBA. Thus, the KLD is an asymmetric and noncommutative measure.The Jensen–Shannon Divergence [23] is another measure that can be used to measure the similarity between two probability distributions based on the KLD:(13)JSDAB=12KLDAM+KLDBM,where M is the mean probability distribution function of distributions A and B.(14)M=12πiθ+π0θ.The JSD has the following properties:(15)JSDAB≥0,JSDAB=JSDBA,JSDAB=0.when A = B.Compared with the KLD, the JSD can better measure the similarity between the probability density function of prior distribution and the probability density function of field test data. The importance of each prior distribution is determined by the JSD, then weightswi of each distribution are assigned, and the multisource information is fused using the respective weights wi [24].The fusion prior distribution is defined as(16)πθ=∑inwiπiθ.According to the properties of the JSD, the smaller the JSD value obtained by combining the fusion prior distribution,πθ and the prior distribution of field test data π0θ, the greater the weight is(17)JSD∑inwiπiθπ0θ=12KLD∑inwiπiθM+KLDMπ0θ=HM−12H∑inwiπiθ+Hπ0θ.The weights are bounded between zero and one and summed to one. Hence, a constrained optimization problem can be formulated as follows:(18)minHM−12H∑inwiπiθ+Hπ0θ,0≤wi≤1,∑inwi=1.


## 2.1. Bayesian Model

The Bayesian model is used to integrate different prior distributions [16]. The posterior distribution is(2)πθx=hx,θmx=pxθπθ∫Θpxθπθdθ,where πθ is the prior distribution for the model parameter. Let pθx be the likelihood function that is obtained based on the probability density function of the chosen reliability model.For the reliability analysis of the CNC system, the commonly used reliability model is Weibull distribution. The corresponding Bayesian model [17] is(3)πβ,θt=Ltβ,θπβ,θ∬β>0θ>0Ltβ,θπβ,θdβdθ=Ltβ,θ∑inwiπiβ,θ∬β>0θ>0Ltβ,θ∑inwiπiβ,θdβdθ,where Ltβ,θ is the likelihood function for the collected lifetime failure data. ∑inwiπiβ,θ is the fusion prior distribution that is generated by information fusion using data gathered from multiple sources.


## 2.2. Derivation of Prior Distribution Based on Failure Time Data

Deriving the prior distribution following Weibull distribution is one of the key issues in Bayesian analysis of lifetime data, which requires a great deal of integration and calculation. Therefore, an alternative approach is taken here [18], which involves transforming Weibull distribution into exponential distribution whose conjugate prior distribution determines the prior distribution of Weibull distribution.Based on the failure rate function given in equation (1), the reliability function is(4)Rt=exp−tθβ.The probability density function is(5)ft=βθtθβ−1exp−tθβ.The cumulative probability distribution function is(6)Ft=∫0t1βθtθβ−1exp−tθβdt.Assumingh=t, α=θ, and X=tβ, equation (6) turns into(7)FX=∫0X1αexp−hαdh.The probability density function after transformation is(8)fx=1αexp−xα,where fx is exponential distribution and 1/α is the parameter of exponential distribution. If β is known, then 1/α is only associated with θ; subsequently, the prior distribution of θ can be introduced by the conjugate distribution of exponential distribution. Under the same working environment and the same type of system, the shape parameters β of a particular CNC system are theoretically identical. If the assumed shape parameter deviates from the actual, there will be some errors. Therefore, it is possible to reduce the error to some extent by deriving the shape parameter through historical failure time data. Subsequently, the prior distribution of Weibull distribution [19] can be obtained according to the conjugate prior distribution of exponential distribution, that is,(9)πα=baΓa1αa+1exp−bα.


## 2.3. Derivation of Prior Distribution Based on Expert Information

Information provided by experts can be used to increase sample size and to improve the accuracy of reliability assessment for small samples. Therefore, the probability encoding method as shown in the data mining section of Figure1 [20], specifically the probabilities and values method is applied to obtain statistical information of expert information data in the form of probability density functions obtained by setting parameter type and quantifying expert information data. The information obtained by experts includes values of scale parameters and shape parameters and corresponding probability values, respectively [21]. Since the scale parameter and the shape parameter correspond to failure type and life expectancy of the CNC system, respectively, experts can give parameter values easily based on their experience. Using equation (10), expert information can be transformed into parameter information and the prior distribution of Weibull distribution can be generated from the expert information.The inverse cumulative distribution function (CDF) is(10)Pθ=∫−∞Vθfθ;μθ,σθdθ,where fθ;μθ,σθ is a parametric CDF.The probabilistic encoding method has the following steps:Step 1: determine variables to be collected and collect expert informationStep 2: select an appropriate Bayesian update methodStep 3: convert the expert information into model parameters using the inverse CDF functionStep 4: update model parameters using the Bayesian method


## 2.4. JSD Method for Determination of Weights

Multisource information fusion aims at integrating information from different dimensions or levels and obtaining more accurate reliability assessment result than using the single information source alone. The ultimate goal of information fusion is to obtain a joint probability distribution. In detail, prior distributions are reasonably assigned proper fusion weights and then integrated into the joint probability distribution. Information entropy measures the total uncertainty in a probability distribution; thus, the similarity between prior information and field test information can be quantified through their respective information entropy. Accordingly, the respective weights of different prior distributions can be determined prior to being used in the integrated prior distribution.The probability density information entropy is(11)Hmx=−Elogmxnx=−∫Ωπxlogmxnxdx,where E⋅ is mathematical expectation of m⋅. In equation (11), nx denotes the noninformation prior density function, and the noninformation means that information of the sample is completely unknown. The noninformation density function of the Weibull position parameter can be taken as nx=1 [22].The Kullback–Leibler divergence (KLD), a.k.a. relative entropy, can be used to calculate the similarity between two probability distributions. For probability distributionsAt and Bt, the KLD is defined as(12)KLDAB=∫Θatlogatbtdx,where at and bt are the probability densities of At and Bt, respectively. Note that KLDAB≠KLDBA. Thus, the KLD is an asymmetric and noncommutative measure.The Jensen–Shannon Divergence [23] is another measure that can be used to measure the similarity between two probability distributions based on the KLD:(13)JSDAB=12KLDAM+KLDBM,where M is the mean probability distribution function of distributions A and B.(14)M=12πiθ+π0θ.The JSD has the following properties:(15)JSDAB≥0,JSDAB=JSDBA,JSDAB=0.when A = B.Compared with the KLD, the JSD can better measure the similarity between the probability density function of prior distribution and the probability density function of field test data. The importance of each prior distribution is determined by the JSD, then weightswi of each distribution are assigned, and the multisource information is fused using the respective weights wi [24].The fusion prior distribution is defined as(16)πθ=∑inwiπiθ.According to the properties of the JSD, the smaller the JSD value obtained by combining the fusion prior distribution,πθ and the prior distribution of field test data π0θ, the greater the weight is(17)JSD∑inwiπiθπ0θ=12KLD∑inwiπiθM+KLDMπ0θ=HM−12H∑inwiπiθ+Hπ0θ.The weights are bounded between zero and one and summed to one. Hence, a constrained optimization problem can be formulated as follows:(18)minHM−12H∑inwiπiθ+Hπ0θ,0≤wi≤1,∑inwi=1.


## 3. Parameter Estimation Based on MCMC

Solving the integral problem of the posterior distribution of the Bayesian method is difficult and obtaining the result through mathematical integral is also complex, so the Bayesian model after the fusion is used to perform analysis of lifetime failure data by the MCMC method, as shown in the Bayesian inference of Figure1. MCMC [25] is a Markov chain establishing a posteriori stationary distribution. MCMC establishes the Markov chain of posterior smooth distribution, and it can then be used to generate samples to obtain posterior distribution samples for statistical inference. By doing so, it can eliminate the need for numerical calculation of integrand to ensure the effective implementation and stability of the solution process.The posterior distributionπβ,θt as target distribution, according to the experience of selecting a uniform distribution as proposed distribution, generates a Markov chain Θi=βi,θi,i=1,2,…,n with the following specific steps:(1)
Generate an initial valueΘ0 from the proposed distribution(2)
Iterate oni=1,2,…(a)
Produce values ofΘ′ from the proposed distribution(b)
Generate a random valueu from uniform distribution(c)
Calculate the acceptance probability and if(19)u≤πΘ′tgΘiΘ′πΘitgΘ′Θi,then Θi+1=Θ′; Otherwise, Θi+1=Θi, where gΘiΘ is the proposed distribution.


## 4. Case Study

### 4.1. Data Source and Determination of Prior Distribution

The development of the CNC machine tool is strategically important to the equipment manufacturing industry in the globally competitive market. The CNC system is the core of CNC machine tools, and its importance is hence self-evident. Because of its long life and high reliability, it is difficult to collect lifetime failure data, and data size is often too small for reliability modeling. Therefore, in order to test the reliability of the CNC system, a long-term multisample test under the same environment was carried out in the test laboratory as shown in Figure2.Figure 2
The environment of CNC system reliability test laboratory.
(a)
(b)Figures2(a) and 2(b) show the laboratory environment specially built for collecting lifetime failure data, which correspond to the laboratory environment of field data and historical data, respectively. Among them, field data refer to data collected from March 2016 to June 2018 of 30 sets of new CNC systems of the same type running in laboratory, as shown in Figure 2(a). On the other hand, historical data refer to data collected between March 2013 and June 2016, as shown in Figure 2(b).This test has been taken to record the failures of CNC systems, and a small sample of lifetime failure data is obtained, as shown in Tables1 and 2. Table 1 is field failure time data and Table 2 is historical failure time data. Expert judgment information is obtained from two experts, as given in Table 3.Table 1
Field failure time data.
Failure instance
Failure time (h)
1
1162.69
2
156.94
3
200.56
4
49.56
5
395.54
6
231.95
7
318.76
8
125.11
9
231.78
10
126.35
11
174.50
12
91.50Table 2
Historical failure time data.
Failure instance
Failure time (h)
1
829.92
2
366.55
3
94.08
4
147.99
5
450.25
6
73.15
7
152.51
8
23.50
9
280.28
10
17.02
11
24.47
12
9.31
13
342.61
14
203.93
15
269.73
16
113.85Table 3
Expert judgment information and PV analysis result.
Judgment information
V
θ
P
θ
μ
θ
V
β
P
β
μ
β
Expert 1
207
0.17
226
0.62
0.29
0.6753
215
0.42
219
0.71
0.57
0.6924
221
0.86
199
0.84
0.65
0.7015
227
0.89
202
0.97
0.83
0.8764
235
0.96
200
0.99
0.92
0.8495
258
0.97
220
1.02
0.98
0.8164
274
0.99
227
1.13
0.99
0.8974
Expert 2
200
0.10
226
0.64
0.15
0.7436
212
0.32
221
0.69
0.29
0.7453
219
0.65
211
0.73
0.32
0.7768
224
0.85
203
0.84
0.65
0.8015
231
0.91
204
0.87
0.83
0.7764
246
0.93
216
0.95
0.95
0.7855
259
0.98
218
0.99
0.98
0.7864Then, the proposed model is applied to perform reliability analysis of a repairable CNC system.Based on the inverse CDF given in equation (10) and expert provided information in Table 3, combining the Bayesian updating theory with the probability encoding method, the distribution function of the expert information is obtained as follows:(20)π1β,θ=12πσθ2σβ2exp−θ−θ¯22σθ2−β−β¯22σβ2=12π20×0.1exp−θ−214.7922×20−β−0.7822×0.1.In the last section, the process of solving the prior distribution using failure time data was described. According to the collected historical and field failure time data, the mean and variance of the sample data are computed as 85.188 and 5168.521, respectively. On this basis, according to equations (7) and (8), the hyperparameters in equation (9) are calculated as a2 = 2.576 and b2  = 243.181, and the prior distribution of historical failure time data is obtained as(21)π2β,θ=πα=204.7923.404Γ3.4041α4.404exp−204.792α.Similarly,a0=4.03andb0=1100.25. The prior distribution of failure time data can be obtained as(22)π0β,θ=πα=394.022.8Γ2.81α3.8exp−394.02α.The probability density curve of each prior distribution is plotted in Figure3. As shown in Figure 3, the weight ratio of each prior distribution to fuse prior distribution can be predicted by its shape. For example, the expert judgement information is subjective and differs greatly from the field failure data source, so its weight ratio should be relatively small. Recall that the specific weights need to be obtained by calculating the JSD optimization model given in equation (18), to be further described next.Figure 3
Prior distribution probability density curve.


### 4.2. Prior Distribution Fusion Based on JSD

In the previous section, we obtained prior distribution functions based on expert information and historical fault data. Because of the different importance of expert information and historical information, we use the JSD to obtain the optimal weights for quantitatively analysing their respective importance. Subsequently, the fusion prior distribution is computed as(23)πβ,θ=w1×π1β,θ+w2×π2β,θ.Using the numerical solution method provided in the MATLAB optimization toolbox, based on the constrained optimization problem given in equation (18), the weights are obtained as w1 = 0.175 and w2 = 0.825, respectively.


### 4.3. Bayesian Reliability Assessment Based on Information Fusion

The probability distribution function of posterior probability can be obtained by using the likelihood function to combine the field failure data, the acquired prior distribution, and the weights of different prior distributions.The likelihood function of Weibull distribution is(24)Ltβ,θ=∏i=1nβθtiθβ−1exp−tiθβ=βnθnβ∏i=1ntiβ−1exp−∑i=1ntiθβ.The Bayesian posterior distribution is(25)πβ,θt=Ltβ,θπβ,θ∬β>0θ>0Ltβ,θπβ,θdβdθ=Ltβ,θ∑inwiπiβ,θ∬β>0θ>0Ltβ,θ∑inwiπiβ,θdβdθ=βn/θnβ∏i=1ntiβ−1exp−∑i=1nti/θβπβ,θ∬β>0θ>0βn/θnβ∏i=1ntiβ−1exp−∑i=1nti/θβπβ,θdβdθ.In order to get around the numerical calculation difficulty brought by high calculus, the MCMC method is used to estimate the parameters. Figures4(a) and 4(b) show the convergence of MCMC in estimating prior distribution parameters in the multisource information fusion process. The corresponding autocorrelation functions shown in Figures 4(c) and 4(d) can be used to diagnose the convergence of the Markov chain.Figure 4
Parameter iteration analysis process and autocorrelation function diagram for multisource information fusion. (a) The relationship between the number of iterationst and hyperparameter β. (b) The relationship between the number of iterations t and hyperparameter θ. (c) Sample autocorrelation function graph for β. (d) Sample autocorrelation function graph for θ.
(a)
(b)
(c)
(d)Likely, Figures5(a) and 5(b) show the convergence of MCMC in estimating noninformation prior distribution parameters. The corresponding autocorrelation functions shown in Figures 5(c) and 5(d) can be used to diagnose the convergence of the Markov chain. The change of the value indicates the convergence of the Markov chain. Figures 4(c) and 5(d) have higher correlation values and slower convergence indicated by a slight downward trend. Figures 4(d) and 5(c) have an obvious downward trend, indicating a good convergence of MCMC.Figure 5
Parameter iteration analysis process and autocorrelation function diagram for noninformation prior distribution. (a) The relationship between the number of iterationst and hyperparameter β. (b) The relationship between the number of iterations t and hyperparameter θ. (c) Sample autocorrelation function graph for β. (d) Sample autocorrelation function graph for θ.
(a)
(b)
(c)
(d)Using the above-described parameter estimation by MCMC and MLE, the results of final estimated parameters are obtained as shown in Table4, and then, the reliability index, i.e., mean time between failure (MTBF) is calculated.Table 4
Comparison of parameter estimation results.
Parameter
Multisource fusion prior information
MLE
Noninformation priori distribution
Mean
Confidence interval
Mean
Confidence interval
Mean
Confidence interval
β
1.162
1.11591
1.1639
1.0
0.64
1.36
1.158
1.152
1.164
Confidence interval
0.005
0.72
0.012
MTBF
287.487
250
256.672For comparison of different parameter estimation processes, the MCMC method based on the fusion prior distribution, the MCMC method without prior information, and the MLE method are, respectively, used to estimate parameters. As can be seen from the table, the results obtained based on the MCMC method of fusion prior distribution and confidence intervals of MCMC without prior information are both less than those of the traditional MLE method (0.005 < 0.012 < 0.72), indicating that the MCMC method has better accuracy. The confidence interval of the MCMC with prior distribution is smaller than that of MCMC without prior information. Based on the above test results, the validity of the proposed model is verified.


## 4.1. Data Source and Determination of Prior Distribution

The development of the CNC machine tool is strategically important to the equipment manufacturing industry in the globally competitive market. The CNC system is the core of CNC machine tools, and its importance is hence self-evident. Because of its long life and high reliability, it is difficult to collect lifetime failure data, and data size is often too small for reliability modeling. Therefore, in order to test the reliability of the CNC system, a long-term multisample test under the same environment was carried out in the test laboratory as shown in Figure2.Figure 2
The environment of CNC system reliability test laboratory.
(a)
(b)Figures2(a) and 2(b) show the laboratory environment specially built for collecting lifetime failure data, which correspond to the laboratory environment of field data and historical data, respectively. Among them, field data refer to data collected from March 2016 to June 2018 of 30 sets of new CNC systems of the same type running in laboratory, as shown in Figure 2(a). On the other hand, historical data refer to data collected between March 2013 and June 2016, as shown in Figure 2(b).This test has been taken to record the failures of CNC systems, and a small sample of lifetime failure data is obtained, as shown in Tables1 and 2. Table 1 is field failure time data and Table 2 is historical failure time data. Expert judgment information is obtained from two experts, as given in Table 3.Table 1
Field failure time data.
Failure instance
Failure time (h)
1
1162.69
2
156.94
3
200.56
4
49.56
5
395.54
6
231.95
7
318.76
8
125.11
9
231.78
10
126.35
11
174.50
12
91.50Table 2
Historical failure time data.
Failure instance
Failure time (h)
1
829.92
2
366.55
3
94.08
4
147.99
5
450.25
6
73.15
7
152.51
8
23.50
9
280.28
10
17.02
11
24.47
12
9.31
13
342.61
14
203.93
15
269.73
16
113.85Table 3
Expert judgment information and PV analysis result.
Judgment information
V
θ
P
θ
μ
θ
V
β
P
β
μ
β
Expert 1
207
0.17
226
0.62
0.29
0.6753
215
0.42
219
0.71
0.57
0.6924
221
0.86
199
0.84
0.65
0.7015
227
0.89
202
0.97
0.83
0.8764
235
0.96
200
0.99
0.92
0.8495
258
0.97
220
1.02
0.98
0.8164
274
0.99
227
1.13
0.99
0.8974
Expert 2
200
0.10
226
0.64
0.15
0.7436
212
0.32
221
0.69
0.29
0.7453
219
0.65
211
0.73
0.32
0.7768
224
0.85
203
0.84
0.65
0.8015
231
0.91
204
0.87
0.83
0.7764
246
0.93
216
0.95
0.95
0.7855
259
0.98
218
0.99
0.98
0.7864Then, the proposed model is applied to perform reliability analysis of a repairable CNC system.Based on the inverse CDF given in equation (10) and expert provided information in Table 3, combining the Bayesian updating theory with the probability encoding method, the distribution function of the expert information is obtained as follows:(20)π1β,θ=12πσθ2σβ2exp−θ−θ¯22σθ2−β−β¯22σβ2=12π20×0.1exp−θ−214.7922×20−β−0.7822×0.1.In the last section, the process of solving the prior distribution using failure time data was described. According to the collected historical and field failure time data, the mean and variance of the sample data are computed as 85.188 and 5168.521, respectively. On this basis, according to equations (7) and (8), the hyperparameters in equation (9) are calculated as a2 = 2.576 and b2  = 243.181, and the prior distribution of historical failure time data is obtained as(21)π2β,θ=πα=204.7923.404Γ3.4041α4.404exp−204.792α.Similarly,a0=4.03andb0=1100.25. The prior distribution of failure time data can be obtained as(22)π0β,θ=πα=394.022.8Γ2.81α3.8exp−394.02α.The probability density curve of each prior distribution is plotted in Figure3. As shown in Figure 3, the weight ratio of each prior distribution to fuse prior distribution can be predicted by its shape. For example, the expert judgement information is subjective and differs greatly from the field failure data source, so its weight ratio should be relatively small. Recall that the specific weights need to be obtained by calculating the JSD optimization model given in equation (18), to be further described next.Figure 3
Prior distribution probability density curve.


## 4.2. Prior Distribution Fusion Based on JSD

In the previous section, we obtained prior distribution functions based on expert information and historical fault data. Because of the different importance of expert information and historical information, we use the JSD to obtain the optimal weights for quantitatively analysing their respective importance. Subsequently, the fusion prior distribution is computed as(23)πβ,θ=w1×π1β,θ+w2×π2β,θ.Using the numerical solution method provided in the MATLAB optimization toolbox, based on the constrained optimization problem given in equation (18), the weights are obtained as w1 = 0.175 and w2 = 0.825, respectively.


## 4.3. Bayesian Reliability Assessment Based on Information Fusion

The probability distribution function of posterior probability can be obtained by using the likelihood function to combine the field failure data, the acquired prior distribution, and the weights of different prior distributions.The likelihood function of Weibull distribution is(24)Ltβ,θ=∏i=1nβθtiθβ−1exp−tiθβ=βnθnβ∏i=1ntiβ−1exp−∑i=1ntiθβ.The Bayesian posterior distribution is(25)πβ,θt=Ltβ,θπβ,θ∬β>0θ>0Ltβ,θπβ,θdβdθ=Ltβ,θ∑inwiπiβ,θ∬β>0θ>0Ltβ,θ∑inwiπiβ,θdβdθ=βn/θnβ∏i=1ntiβ−1exp−∑i=1nti/θβπβ,θ∬β>0θ>0βn/θnβ∏i=1ntiβ−1exp−∑i=1nti/θβπβ,θdβdθ.In order to get around the numerical calculation difficulty brought by high calculus, the MCMC method is used to estimate the parameters. Figures4(a) and 4(b) show the convergence of MCMC in estimating prior distribution parameters in the multisource information fusion process. The corresponding autocorrelation functions shown in Figures 4(c) and 4(d) can be used to diagnose the convergence of the Markov chain.Figure 4
Parameter iteration analysis process and autocorrelation function diagram for multisource information fusion. (a) The relationship between the number of iterationst and hyperparameter β. (b) The relationship between the number of iterations t and hyperparameter θ. (c) Sample autocorrelation function graph for β. (d) Sample autocorrelation function graph for θ.
(a)
(b)
(c)
(d)Likely, Figures5(a) and 5(b) show the convergence of MCMC in estimating noninformation prior distribution parameters. The corresponding autocorrelation functions shown in Figures 5(c) and 5(d) can be used to diagnose the convergence of the Markov chain. The change of the value indicates the convergence of the Markov chain. Figures 4(c) and 5(d) have higher correlation values and slower convergence indicated by a slight downward trend. Figures 4(d) and 5(c) have an obvious downward trend, indicating a good convergence of MCMC.Figure 5
Parameter iteration analysis process and autocorrelation function diagram for noninformation prior distribution. (a) The relationship between the number of iterationst and hyperparameter β. (b) The relationship between the number of iterations t and hyperparameter θ. (c) Sample autocorrelation function graph for β. (d) Sample autocorrelation function graph for θ.
(a)
(b)
(c)
(d)Using the above-described parameter estimation by MCMC and MLE, the results of final estimated parameters are obtained as shown in Table4, and then, the reliability index, i.e., mean time between failure (MTBF) is calculated.Table 4
Comparison of parameter estimation results.
Parameter
Multisource fusion prior information
MLE
Noninformation priori distribution
Mean
Confidence interval
Mean
Confidence interval
Mean
Confidence interval
β
1.162
1.11591
1.1639
1.0
0.64
1.36
1.158
1.152
1.164
Confidence interval
0.005
0.72
0.012
MTBF
287.487
250
256.672For comparison of different parameter estimation processes, the MCMC method based on the fusion prior distribution, the MCMC method without prior information, and the MLE method are, respectively, used to estimate parameters. As can be seen from the table, the results obtained based on the MCMC method of fusion prior distribution and confidence intervals of MCMC without prior information are both less than those of the traditional MLE method (0.005 < 0.012 < 0.72), indicating that the MCMC method has better accuracy. The confidence interval of the MCMC with prior distribution is smaller than that of MCMC without prior information. Based on the above test results, the validity of the proposed model is verified.


## 5. Conclusions

This paper has presented a new method for developing a reliability model based on multisource information fusion to deal with small sample size. The following achievements are made:(1)
In order to solve the problems of small lifetime failure data, low accuracy of reliability assessment, and incomplete use of reliability parameters, a new method for developing a reliability assessment model based on multisource information fusion is proposed. The prior distribution of combining information from different sources is obtained by using the conjugate distribution and probability encoding method.(2)
A JSD-based method for optimizing weights of multisource information is developed, which is designed to obtain more reasonable weight information through optimization. Moreover, with respect to the difficulty of parameter estimation methods in solving integral equations, MCMC is used to estimate relevant model parameters and guarantee the feasibility of the model in a selected application.(3)
The practical use of the proposed model is demonstrated by using a set of lifetime failure data of repairable CNC systems. For comparison, the MCMC method based on fusion prior distribution, the MCMC method without prior information, and the method of MLE are, respectively, used to estimate parameters. The corresponding confidence intervals are obtained, and the validity of the proposed model and its parameter estimation method are verified. The study is expected to be useful for reliability analysis of repairable systems, especially for small samples, and provides important insights for guiding the development of the preventive maintenance strategy.



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*Source: 3645858-2020-01-16.xml*