The th l B-spline basis function of degree q evaluated at z INTRODUCTION In modern industries, high-quality products are expected to perform for years to decades.
Degradation testing is a method developed in the reliability literature to obtain product reliability by using the degradation of performance characteristics, such as the light intensity [1], crack length [2], and material elongation [3]. With the development of highly-quality products, degradation testing under a normal use condition is not efficient to meet industrial requirements due to the limitation of testing times. In practice, accelerating variables such as temperature, voltage, or pressure are incorporated into the degradation testing so that the failures can be reached in less time. In some cases, the physical property being tested may only not survive the measurement procedure; that is, the measurement can only be done through destructive means. This kind of testing is referred to as the Accelerated Destructive Degradation Testing (ADDT).
In the reliability literature for degradation data analysis, Whitmore and Schenkelberg [4], Lu, et al. [5], Nelson [6], Crk [7] conducted early research in degradation modeling. Chen, et al. [8] summarized the methods for degradation data including non-destructive and destructive degradation testing. Nelson [9] used accelerated life testing as a tool to estimate performance degradation. Lu and Meeker [10] used the repeated-measures degradation testing (RMDT) data to estimate the failure time. Meeker, et al. [11] developed a nonlinear mixed-effects model for RMDT data. Gorjian, et al. [12] and Meeker, et al. [13] made an introductory level description of some degradation models. In related literature, Lu and Meeker [10] used RMDT data to estimate the failure time and Meeker, et al. [11] proposed a nonlinear mixed-effects model for analyzing RMDT data. Gorjian, et al. [12] and Meeker, et al. [13] further developed statistical degradation models for reliability analysis. Xu, et al. [14] proposed a nonlinear general path model for analyzing degradation data with dynamic covariates.
In the ADDT literature, several parametric models have been developed to analyze ADDT data.
Shi, et al. [15] designed an ADDT plan to study combinations of different accelerating variables.
Tsai, et al. [16] proposed an optimal design approach for ADDT. , Lim and Yum [17], and Jeng, et al. [18] developed methods for ADDT planning and analysis. In recent years, Xiao and Ye [19] analyzed destructive degradation tests for highly reliable products with random initial degradation values. Hong, et al. [20] proposed models in which the acceleration can be represented by stochastic processes under certain conditions. Tsai, et al. [16] developed parametric models to analyze ADDT data collected from different materials with multiple accelerating levels.
Although the existing parametric ADDT models have been widely used in practice, the common disadvantages of parametric models are their being application specific and the significantly biased conclusions if the model assumption is not satisfied. Instead of a case-by-case parametric modeling approach, Xie, et al. [21] proposed a semiparametric model to analyze ADDT data. In this research, we propose a general ADDT model with a nonparametric part to describe the degradation path and a parametric part to describe the accelerating-variable effect. The parametric part takes both the scale and shape parameters of the response distribution into consideration. The major contribution of this paper is two-fold. First, the proposed model is more general and robust than the existing semiparametric model in the literature. Through a simulation study and real-world case studies, we later show that the performance of the proposed general ADDT model is much more improved over the existing one. Second, the traditional optimization method cannot be directly applied to estimating the model parameters due to the complexity of the objective function. To overcome this challenge, we develop an efficient model parameter estimation method based on self-adaptive differential evolution.
The remainder of this paper is organized as follows. In Section II, we present the notation and propose the general ADDT model. In Section III, we develop the method for model parameter estimation. In Section IV, we conduct a simulation study to investigate the performance of the developed methods. In Section V, we apply the proposed model to analyzing ADDT data of two
Here the lower and upper boundary points are repeated q times, where q is the polynomial degree. In the following, we define the ordered knot sequence as 1 2 2
, , r q d d + + L for simplicity. The B-spline basis functions are defined recursively. For degree 0, the first spline function is
and for degree q , the function is
. With the combination of the nonparametric and parametric portions, the model in ( 1) is represented in its entirety as
Here, l γ is the coefficients of the B-spline basis function , ,
η . The proposed model in (7) falls within the class of shape-scale-acceleration models. For every stress level i , ( ; ) 4). When the SHAF=1, the time acceleration rate / is determined by the SCAF and so does not depend on a specific time point. When the SHAF≠ 1, the time acceleration rate also varies with time. Thus, the introduction of the SHAF allows for more flexibility in modeling the time acceleration rate.
When the acceleration level is at its highest, i.e., max max 0 x x -= , ( ; , )
indicating that the degradation path no longer relies on α and β . Under this situation, the corresponding baseline degradation path for the scale-shape-acceleration model is
The first derivative of ( ; ) i H t x with respect to , ( ;
According to the assumption of monotone decreasing behavior in the degradation path, Equation (9) should always be negative. As the B-spline basis functions are nonnegative, this indicates that for 2 l p ≤ ≤ , the condition
is sufficient for a monotone decreasing degradation path when the degree of the basis function 1,2 q = . A low degree of spline ( 4 q ≤ ) and a small number of interior knots (1 5 γ ≤ ≤ ) are usually sufficient to provide a good fit to the data. It was proved in the literature [22] that the optimal number of knots is in the order of 1/5 n , where n is the number of observation. Thus, a small number of knots (1 5 γ ≤ ≤ ) is sufficient for the modest size of datasets, such as the datasets in the case study of this research.
To estimate the unknown parameters in the proposed model (7), we develop a likelihood-based optimization method. As shown in Figure 1, the developed method is an iterative process that consists of two parts: (1) knot selection of B-splines, and (2) parameter estimation. The parameter estimation is based on the initial selected knot setting from part 1, and the updated knot selection is based on the profile likelihood value calculated from part 2.
To find the best knot setting, we use Akaike Information Criterion (AIC), which considers both the profile log-likelihood values and the number of model parameters. For a given knot setting, the model parameters can be estimated by maximizing profile log-likelihood values. The objective function of the corresponding optimization problem is nonlinear and complex. As a result, the traditional gradient-based optimization methods cannot be directly applied. To overcome this difficulty, we develop an efficient parameter estimation method by applying the framework of selfadaptive differential evolution (SaDE) (). The SaDE method is a population-based stochastic search strategy which aims to solve global optimization problems like other evolution algorithms.
In the classical differential evolution (DE) method (), the control parameters and learning strategies highly dependents on the problems under consideration so that significant time needs to be spent to try through various strategies and adjust the corresponding parameters. The key improvement of the SaDE over the classical DE method is that the learning strategy and parameter settings are gradually self-adapted during the evaluation process based on the learning experience. Besides that, The authors of [23] also give the numerical analysis of the convergence of the SaDE algorithm which shows that the convergence speed of SaDE performs much better than the original DE algorithm especially in the high-dimension situation.
After obtaining the optimum profile log-likelihood by using the developed optimization method, the AIC value can be calculated and used as the performance index to select optimal knot setting of the B-splines. Thus, the two parts run iteratively until the best knot setting and the optimal model parameters are obtained. The details of the parameter estimation and knot selection stages are discussed in Sections 3.1 and 3.2, respectively. Iteratively updating
, , )) ( ( ; , , )) ( ( ; , , )) ( ( ; , , )) .
( ( ; , , )) ( ( ; , , ))
In equation (10), 1 ( , , )
( , , , , , , ) ,,,,, ) ( ,,,,,, )
To calculate the objective function of the SaDE algorithm which is the profile log-likelihood function of ( , ) α β , the corresponding estimation of
which is a quadratic objective function with linear constraints and can be solved by traditional optimization methods (e.g., the dual method in Goldfarb and Idnani [24] or the Hinge algorithm in Meyer [25]).
After obtaining ( ) ˆm γ , the estimates of
The parameter estimates of ( ) ( )
] .
Upon convergence, the estimates of σ ρ ′ can be obtained by fitting an unconstrained model.
The goal of the SaDE method is to maximize the profile log-likelihood of ( , ) α β , which is calculated as setting and choose the best group of knots with the smallest AIC value. In Equation (17), edf is the effective degrees of freedom, which is calculated as the degrees of freedom in γ plus four degrees of freedom for the parameters ' ( , , , )
α β σ ρ . With 1 p -linear constraints, the effective degrees of freedom in γ has a value from 1 to p, where p corresponds to the release of the constraint of (11).
Letting q denote the degree of the B-spline functions, the knot selection procedure can be summarized as the following two steps.
Step 1. Determine the optimal number of interior knots , opt q r by minimizing (17). The knots are to be evenly distributed in the given interval.
Step 2. Calculate the change of the AIC value that is caused by the deletion of each interior knots in sequence. Remove the knot that leads to the greatest increase of the AIC value. Repeat until no more existing knots can be removed.
Stepwise knot selection methods have been used in the literature including [26][27][28][29]. We start from a set of uniform knots (in percentile ranks) to keep the amount of computation as small as possible. The convergence of the iterative framework is proved by the research work [22],
which is based on the asymptotic analysis and the existing work [30,31].
The detailed steps of the estimation method are listed in the following algorithm 1.
Update the constraint of ( 11) based on the result of knot selection;
Mutation with strategy ( 16) and Crossover to decide whether to accept the mutation result under a given probability;
While (the deletion of any one of the knots would lead to the reduction of the AIC value)
Remove the interior knot whose deletion leads to the greatest reduction of the AIC; End For
At first, the failure threshold f D can be derived by the following equation.
where ˆf µ is the mean time to failure (MTTF) at use condition f
x . Thus, the failure time distribution can be derived based on the general ADDT model as follows.
exp( )
where the failure time T is equivalent to the degradation measurement at time t . The function
Φ ⋅ is the cdf of the standard normal distribution. The quantile function can then be calculated as the inverse of the which the α is 1 ( )
IV.
We conduct a simulation study to investigate the performance of the developed parameter estimation method. In the simulation study, temperature is used as the acceleration factor. We use two types of temperature settings: (1) 3 M = temperature levels at 50°C, 65°C, and 80°C and (2) 6 M = temperature levels at 30°C, 40°C, 50°C, 60°C, 70°C, and 80°C. There are 5 N = measuring times set as 8, 25, 75, 130, and 170 days.
The data are simulated from the following parametric model: 1 and the fitted degradation results as shown Figure 3.
Table 2 shows the simulation results of the two semiparametric models when the data are simulated using parametric model. 3. It can be seen in Table 3 that the general ADDT model proposed in this paper outperforms the existing semiparametric method. Based on the existing simulated results, we further our experiments. For each simulated dataset, the MTTF at 30°C (considered as the normal use condition) for each simulated dataset by using the true parametric model, the semiparametric model [21], and the proposed general ADDT model. The mean, bias, standard derivation (SD), and root MSE of the MTTF for all the three different models are summarized in the following Table 1. The results in Table 1 show that the estimated mean of MTTF using the general ADDT model is closer to the true value, and general ADDT method has a smaller bias, SD and RMSE of the MTTF estimation than the semiparametric model [21]. So the proposed general ADDT model outperforms the existing semiparametric model [21]. performance with the parametric model and an existing semi-parametric model in the literature.
An ADDT of the adhesive formulation K is conducted to estimate the lifetime of a certain kind of adhesive that has a new formulation with a new additive compound to help enhance the performance. The adhesive's viscidity would decrease over time till failure. During the ADDT process, the temperature is the acceleration factor and is set at three levels: 40°C, 50°C and 60°C.
First, ten samples were tested at nominal temperature as baseline units. Next, for each temperature, 30 samples are placed and measured at different times to obtain the degradation paths of the material. The strength of the adhesive is measured in Newtons. Figure 6 shows the raw data of the Adhesive Formulation K.
).
Here, ( ) In this article, we developed a new general model for analyzing ADDT data. The degradation path is modeled using the B-spline and includes both location and scale parameters to describe the acceleration factor effect. We developed methods to select the optimal number and location of interior knots of B-splines based on AIC values. We also developed an MLE method based on self-adaptive differential evolution to estimate the model parameters.
We implement a simulation study and a case study to assess the performance of the developed methods. The results show that the proposed model performs well in fitting degradation paths. We also compare the proposed model to the existing parametric and semiparametric ADDT model.
Both the simulation results and the real-world case studies show that the proposed model outperforms the existing models in terms of model-fitting. Compared to the existing models, the newly proposed model's is more general and flexible to describe complex relationships between the response and the acceleration factors.
The newly proposed model can be applied to a wide range of applications. An interesting future research topic would be using the model as a basis for optimal test planning. Also, we can assess the performance of the model using other types of nonparametric methods, such as cubic splines or wavelets, which are also applied widely in the engineering and science research fields. In addition, this research only focuses on ADDT data; a similar model could be developed for RMDT data or other similar data types.