Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 9752920 |

Xiaoping Liu, Zhenyu Wu, Dejun Cui, Bin Guo, Lijie Zhang, "A Modeling Method of Stochastic Parameters’ Inverse Gauss Process Considering Measurement Error under Accelerated Degradation Test", Mathematical Problems in Engineering, vol. 2019, Article ID 9752920, 11 pages, 2019.

A Modeling Method of Stochastic Parameters’ Inverse Gauss Process Considering Measurement Error under Accelerated Degradation Test

Academic Editor: Yongheng Yang
Received06 Mar 2019
Accepted30 Apr 2019
Published21 May 2019


To solve the problem that the individual differences and the measurement errors affect the accuracy of life estimation in accelerated degradation test, the inverse Gauss process with stochastic parameters is applied in the accelerated degradation test with the consideration of the influence of individual differences, and the analysis of measurement uncertainty is carried out. An inverse Gauss accelerated degradation model considering both individual differences and measurement errors is established. In the maximum likelihood estimation of parameters, Genetic Algorithm and Monte Carlo integral are used to solve the problems caused by complex integral and the unobservable measurement errors in the calculation process. Finally, the proposed method is verified by the Monte Carlo simulation under the constant accelerated stress and step accelerated stress and the illustrative example of electrical connectors under the constant acceleration stress, respectively. The results show that the modeling tool is useful for improving the accuracy of the life prediction and the reliability evaluation.

1. Introduction

Prognostics Health Management (PHM) is a new management method of complex engineering system, which integrates fault diagnosis, life prediction, and health management. And it has been widely applied in different fields, such as communications, electronics, military, and aviation. As a key component of PHM, an appropriate life prediction method is significant for the product reliability evaluation, and it contributes to formulate reasonable health management schemes. It also improves the safety of industrial production; especially it saves the maintenance cost [13].

With the development of manufacturing technology, massive products with high reliability and long life (e.g., GaAs laser and electrical connector, etc.) have emerged in various fields of industrial production. It is very difficult to obtain the life information which satisfies the practical demands by the traditional life test and the accelerated life test. At present, the reliability evaluation and life prediction methods based on degradation test are widely used; however, under rated conditions, the degradation rate of the products is relatively slow and the degradation information is relatively little. The accelerated degradation test (ADT) nicely solves this problem. Among them, the Constant Stress Accelerated Degradation Test (CSADT) and the Step Stress Accelerated Degradation Test (SSADT) have been favored by many scholars for their advantages of fast acquisition speed and mature test methods in obtaining degradation data.

As the most critical part of the degradation analysis, the degradation models are currently divided into the regression model based on degradation trajectory analysis and the stochastic process model to characterize product performance degradation. Influenced by the changes in the external environment, the manufacturing process and the operation conditions, the degradation process of the products is with some randomness. Therefore, it is reasonable to use the stochastic process to model product degradation, and this is also a widely used modeling method at this stage. Three kinds of commonly used stochastic process models are respectively the Gamma process model, the Wiener process model, and the Inverse Gaussian (IG) process model. However, the degradation trajectories of some products cannot be accurately fitted by the first two models, such as the GaAs laser [4] and the electrical connector [5]. Recently, as a stochastic process with independent increment, the IG process has been increasingly applied in the modeling of degradation. Ye [6] defined the IG process as a composite Poisson process with independent but not necessarily identical distribution in the extreme case, which compensates the physical significance of the IG process. Wang et al. [4] described the degradation characteristics of the products through the IG process and solved the problem of parameter estimation by the maximum likelihood estimation (MLE) method. Peng [7] used the Bayesian method to optimize the parameter estimation in IG process. Then Peng [8] took the individual differences of the similar products into consideration, solved the parameter estimation problem by the EM algorithm, and gave a clear analytical prediction expression of the remaining life. Aiming at handling the impact of catastrophe data on prediction, Xu [9] proposed a self-adaptive IG process which has a great significance on improving the accuracy of prediction.

Many scholars have done a lot of research on the accelerated degradation test using IG process. Wang [10] optimized the generalized IG process under the SSADT. Based on the invariance principle of acceleration coefficient, Wang [11] proposed an accelerated degradation modeling method using IG process with stochastic parameters; Ye [5] represented the individual differences in IG process through the randomization of parameters and optimized the CSADT design; Wu [12] proposed a multiobjective optimization design method and optimized CSADT using IG process. The measurement errors have not been investigated in those studies; however, that affects the fitting degree of the model and the accuracy of life estimation [13]. Therefore, Li [14] integrated the error item into the modeling in SSADT optimization design using IG process; however, the possible influence by the individual differences was neglected in Li’s study. Furthermore, in ADT, the individual differences between similar products and measurement uncertainties are ubiquitous and they considerably affect the accuracy of life prediction [15, 16]. Hence, it is necessary to establish an IG process degradation model with the consideration of both measurement errors and individual differences.

The paper is organized as follows. Section 2 introduces the IG process degradation model considering both measurement errors and individual differences. The acceleration models of the constant stress and the step stress using IG process are introduced in Section 3. In Section 4, a MLE method based on Monte Carlo integral and Genetic Algorithm (GA) is proposed, which solves the problem of complex integrals and unobservable measurement errors effectively in parameter estimation process. In Section 5, the efficiency and reasonability of the proposed approach are validated via Monte Carlo simulation study by two examples: the CSADT and the SSADT. And the proposed approach is validated by an illustrative example of electrical connector in Section 6. The conclusions are finally drawn in Section 7.

2. Degradation Modeling

In this section, we adopt the IG model with stochastic parameters to characterize individual differences, with the consideration of the effect of measurement errors through incorporating error terms, and establish a degradation model for products.

As a newly introduced stochastic process model, the IG model shows a good fitting characteristic for the degradation process of many products [4, 5]. According to the previous description, the measurement errors of the degraded data are inevitable due to the interference of various noises during the degradation process; therefore, it is necessary to be considered in the degradation modeling. Set to be the observed degradation data and the real degradation data of product performance. In this paper, the increment of degradation data is used as the modeling object. And the degradation model can be expressed as

where and denote the increment of the degradation inspections and the true degradation of the product's performance characteristic at time t, respectively. is the increment of measurement errors which follows a normal distribution as . The follows the IG distribution, and it is expressed as . The probability density function (PDF) of and can be defined as

where is the time scale function and denotes the increment of time scale function. Under the condition that the failure threshold is , it is considered that the first hitting time (FHT) of IG process is product life T:

Therefore, the cumulative distribution function (CDF) of the life for the IG process can be obtained [4]

The corresponding PDF is

To reflect the individual differences between the similar products, Wang and Xu [17] proposed a method to incorporate the random effects in the Inverse Gaussian process by letting , with PDF

where denotes a gamma function, and (3) can be considered as the conditional distribution under the condition of being a constant; the expression of can be calculated by , then

Based on the concept of FHT, the CDF of IG process with random effects can be obtained [4]

where is the student t-distribution with degrees of freedom. The corresponding PDF can be obtained as follows:

From the above exposition, in the case of rated stress, the reliability analysis and the life estimation of the product degradation process described by IG process can be carried out through the analytical solutions.

3. Modeling of Accelerated Degradation

The acceleration model can reflect the effect of the stress change upon the degradation process by establishing the relationship between accelerated stress and model parameters. Considering that the mean of the IG process is and the variance is , we assume that is a stress dependent function and ignore the effect of stress on the parameter in order to reflect the effect of stress upon the speed and the volatility of degradation simultaneously. This section takes the temperature stress as an example to illustrate the relationship between the parameters and the accelerated stress by the Arrhenius model

where and are undetermined parameters, stress level . is the accelerated stress after standardization, which is defined as

where denotes the normal working stress, is the maximum accelerated stress, .

The CSADT and the SSADT have been widely used in all kinds of product tests, such as LED [18], electrical connector [17], and SLD [19]. We suppose that there are units participating in ADT from the beginning of the experiment to cut-off time .

In the CSADT, let denote m higher stress levels, denote the degradation path of the unit under , and , .

As for SSADT, assume that there are units in test, where each of them is tested under a total number of stress levels , which can be expressed as

Under the stress level , each unit has been monitored for times, . Suppose that the monitoring intervals for each unit is , and , denotes the complete degradation path under SSADT, the relation between and can be given as (14).

4. Statistical Inference

In this paper, the generalized IG process is adopted. For the convenience of parameter estimation, set time function and the model parameters . Since the measurement errors are integrated into model, the solution of the MLE becomes complicated. On the basis of considering individual difference and measurement errors simultaneously, a MLE method based on combination of Monte Carlo integral and GA for the IG process in ADT is proposed. The SSADT is taken as an example to make statistical inference, and the CSADT can draw corresponding calculation results with reference to it.

Let denote the increment of degradation inspections of the unit at time under the stress and denote the increment of the true degradation, and is the increment of measurement errors which is independent and identically distributed. By (1) we can obtain

For the SSADT, denote the complete degradation path under SSADT of units and denote the total measurement times of a single unit, . And the expression of can be obtained as follows:

contains the observation error, which makes and follow different distributions. By (15), let denote and follow IG distribution with random effects. The expression of the maximum likelihood function can be obtained by the full probability formula

During each measurement, the absolute value of each measurement error should be less than the degradation inspection. Thus, the integral bounds of in (17) is . Considering that the direct integral of (17) is complicated, we adopt a simplified method referring to [14] as follows.

Define a two-variable function

can be simplified as

Formula (19) converts into the expectation of the two-variables function to . Then, an approximate calculation is conducted as follows:

In order to calculate the parameter configuration in the case of minimum , the MLE of unknown parameters is proposed based on Monte Carlo integral and Genetic Algorithm:(1)The upper limit of is calculated by formulation (21) and setting the step length to be , a series of can be obtained:(2)For each , simulate times by normal distribution random number. By substituting into (20), the approximate maximum likelihood function expression can be obtained as follows:(3)In the maximum likelihood function, there are no other unknown parameters except the remaining unknown parameters . In this case, the parameter estimation is regarded as an unconstrained minimum optimization problem. Since the maximum likelihood estimation requires to be the maximum, here is regarded as an objective function and as the variable. By calling the gatool in Matlab, the minimum and the corresponding can be obtained by setting parameters ranges, population sizes, and iteration times in the corresponding item columns. A series of and its corresponding can be obtained by performing the operations above under all values of . According to the definition of maximum likelihood estimation, the minimum can be obtained by comparison. The corresponding and are the optimal parameters for the estimation. The flow chart of parameter estimation is shown in Figure 1.

The approximated value of is given by Monte Carlo method, and it is substituted into the expression of maximum likelihood function, which solves the unknown . The optimal values of other parameters are obtained through GA, and the parameter including is selected by comparing the value of a series of . Finally, the unknown parameters estimation of model is realized. Here are three points for explanation.(i)In step (1), for all the units within two adjacent intervals, the variance of is less than variance of because the fluctuation of includes the random effect, individual difference of and the effects of .(ii)In step (3), as the objective function, the maximum likelihood function is searched, and its numerical value is simplified, so there is a certain deviation from the true value, which is only as the basis for the selection of the optimal parameter . The actual value is calculated by substituting the optimal parameter into (17).(iii)In the searching process, since is carried out under the condition that is determined and the variable is only , is denoted as in the description above. However, since is also a variable in the whole process of parameter estimation, we finally express as . There is no difference in numerical values between and .

5. Simulation Study

In order to verify the correctness of the proposed modeling method and the effectiveness of the parameter estimation method, the simulation and verification of the IG process under CSADT and SSADT are carried out, respectively. For the convenience of following description, M0 is used as the IG model with the consideration of both individual differences and measurement errors; M1 is used as the IG model with the consideration of individual differences proposed by Ye in [5], and M2 as the IG model with the consideration of measurement errors proposed by Ye in [14].

5.1. CSADT

In this section a simulation study for the CSADT is conducted. Assume that the observation times of each unit are equal under each accelerated stress, and the time intervals are also equal. 15 groups of degradation data are simulated under three accelerated stresses of 60°C, 80°C, and 100°C, and the working stress is 40°C. The failure threshold and the individual differences are simulated by parameter randomization. The interval is 100h within 12 measurements. The degradation data and the measurement time are shown in Table 1.





As for the three IG models mentioned above, the estimated parameters and real values are separately given in Table 2. Introducing the Akaike Information Criterion (AIC) as the standard to evaluate model fitting degree, and combining with the maximum value of log likelihood function (Log-LF), the three IG models mentioned above can be evaluated. The calculation formula of AIC is given as



where denotes the number of undetermined parameters in model and the AIC values of the three IG models are shown in Table 2. To reflect the accuracy of the proposed method in life estimation, the mean time to failure (MTTF) of the models is also listed in Table 2. For proving the correctness of the parameter estimation method proposed above, the total mean square errors (TMSE) of parameter estimated values to real values tested in 500 and 1000 times simulation are respectively shown in Table 3, and the calculation formula of TMSE is given as



where denotes the true values of parameters, denotes the estimated values of parameters, and denotes the numbers of simulation.

Table 2 shows that the AIC value of M0 is smaller than the others, which indicates a good fitness. The MTTF is closer to the real value, reflecting its high precision in life estimation. From Table 3, it can be observed that the proposed parameter estimation method has high precision under the results of multiple simulations and can be used to estimate the parameters of proposed model reliably. The reliability curve of each model is shown in Figure 2. It can be observed that the fitting degree of M0 to real model is the highest, and this proves the correctness of proposed method in CSADT.

5.2. SSADT

The simulation study for SSADT is conducted in this section. Assume that units are tested in three accelerated stresses at 60°C, 80°C, and 100°C in sequence and the working stress is 40°C. The failure threshold and the measurement times of each unit are equal at each accelerated stress. The measurement interval is the same, 100h. 8 groups of the degenerate data are simulated in SSADT, and the measurement times are set to be 12 times, 4 times for each stress. The degradation data and the measurement times are shown in Table 4.



By adopting the parameter estimation method proposed above, the results of the parameter estimation of simulation data in SSADT are obtained and shown in Table 5, and compared with the other two models. According to the analysis of Log-LF, AIC, and MTTF, it can be observed that the parameter estimation precision of model M0 is higher and the value of AIC is lower, which reflects a better fitness under the consideration of individual difference and measurement error. For M1, since the measurement errors cannot be considered, the parameter estimation precision of it is reduced and the goodness of fit is lower than that of model M0. Since in the model M2 only the measurement errors are considered and it cannot randomize the parameters, its accuracy of life estimation and fitness are reduced. This is the same as the results from the simulation study of CSADT. It can be observed from the MTTF of the three models that the life estimation of M0 is closer to the one from the real model compared to M1 and M2. The same results can be observed from the reliability curves of three models, as shown in Figure 3.



6. Illustrative Example

Stress relaxation is the phenomenon that the strain on a component remains unchanged while the stress gradually decreases. With the increase of working temperature, the contact of electrical connectors often fails due to stress relaxation. Yang [20] verified that the IG process has a good fitting effect in the degradation test of electrical connectors. In this section, the case of stress relaxation test [20] is used for modeling analysis. Set the time function and the failure threshold . The normal working temperature stress is 40°C, and three accelerated temperature stresses are 65°C, 85°C, and 100°C, respectively. The experimental data and the measurement time are shown in Tables 6 and 7. The degradation trend of stress relaxation is shown in Figure 4.

TemperatureIDStress Relaxation/%




TemperatureMeasurement time epochs/h


For validating the superiority of the proposed method in the modeling of stress relaxation of electrical connectors, the three proposed models are evaluated by a combination of Log-LF and AIC, as shown in Table 8. And to validate goodness of fit of IG model for the stress relaxation of electrical connectors, the quantile-quantile plot (Q-Q plot) is shown in Figures 57.



From Table 8, it can be observed that comparing with the models M1 and M2, M0 has a higher Log-LF and a smaller AIC in the three stress level CSADT, which indicates that the model considering measurement errors and individual differences have better fitting performance than the other two under three stress levels. Through the comparison of Q-Q plot, it can also be observed that the quantile of model M0 displays a good linearity to test data, which reflects a good fitness compared with the other two models under three stress levels. Thus, it is meaningful to consider individual differences and measurement errors simultaneously.

Through the reliability curve of three IG models, the advantage of the proposed method in reliability analysis can be seen more intuitively, as shown in Figure 8. It can be observed that the model M1 evaluates reliability conservatively compared with M0, and the maintenance time is estimated early, which leads to lower utilization of product. The model M2 has a high reliability but may lead to the lag of maintenance and increase the risk of early failure of product. Therefore, it is reasonable to evaluate the reliability of products via the model M0.

7. Conclusion

In this paper, considering the influence of individual differences and measurement errors of life estimation in ADT, the errors are incorporated into the IG process with stochastic parameters. The IG degradation model considering individual differences and measurement errors simultaneously is proposed in ADT. Considering the characteristics of the unobservable measurement errors and the complex integral problem in the process of MLE, a MLE method is proposed by combining the Monte Carlo integral with GA, and the method has been verified to be accurate and reliable in the simulation study. The simulations study towards CSADT and SSADT have verified the correctness of the proposed model considering individual differences and measurement errors in the fitting degree of the degradation trajectory and the life estimation. And the illustrative example of electrical connector in CSADT has also demonstrated the efficiency and reliability of the proposed model.

Based on the research in this paper, the following points should be further studied: (1) it is worth investigating the analytical expression of life estimation using IG process considering individual differences and measurement errors simultaneously, based on the FHT concept; more details are given in [21]; (2) it may be of interest to reduce the effect of measurement errors by the filtering method; more details can be found in [13]; (3) it is significant to establish an effective maintenance plan based on proposed model in actual engineering; (4) a further study should focus on accelerated degradation tests considering measurement errors when both the mean and scale parameters are stress dependent; (5) considering that there are still some deviations between the estimated parameters and the real parameters of the model, it may be of interest to consider the interval estimation of parameters.

Data Availability

The simulation study and illustrative example data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare no conflicts of interest.


The authors are grateful for the financial support from the National Natural Science Foundation of China [No. 51875499] and Innovation Funding for Postgraduates in Hebei Province [No. CXZZSS2018035, CXZZBS2017040].


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Copyright © 2019 Xiaoping Liu 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.

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