Optimal design of step-stress accelerated degradation test oriented by nonlinear and distributed degradation process
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Keywords:
Test plan
Robustness
Degradation
Optimal design
Design of experiments
In mixture experiments, the factors under study are proportions of the ingredients of a mixture. The special nature of the factors necessitates specific types of regression models, and specific types of experimental designs. Although mixture experiments usually are intended to predict the response(s) for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients, little research has been done concerning their I-optimal design. This is surprising given that I-optimal designs minimize the average variance of prediction and, therefore, seem more appropriate for mixture experiments than the commonly used D-optimal designs, which focus on a precise model estimation rather than precise predictions. In this article, we provide the first detailed overview of the literature on the I-optimal design of mixture experiments and identify several contradictions. For the second-order and the special cubic model, we present continuous I-optimal designs and contrast them with the published results. We also study exact I-optimal designs, and compare them in detail to continuous I-optimal designs and to D-optimal designs. One striking result of our work is that the performance of D-optimal designs in terms of the I-optimality criterion very strongly depends on which of the D-optimal designs is considered. Supplemental materials for this article are available online.
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In mixture experiments, the factors under study are proportions of the ingredients of a mixture. The special nature of the factors in a mixture experiment necessitates specific types of regression models, and specific types of experimental designs. Although mixture experiments usually are intended to predict the response(s) for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients, little research has been done concerning their I-optimal design. This is surprising given that I-optimal designs minimize the average variance of prediction and, therefore, seem more appropriate for mixture experiments than the commonly used D-optimal designs, which focus on a precise model estimation rather than precise predictions. In this paper, we provide the first detailed overview of the literature on the I-optimal design of mixture experiments and identify several contradictions. For the second-order, special cubic and the degree models, we present I-optimal continuous designs and contrast them with the published results. We also study exact I-optimal designs, and compare them in detail to continuous I-optimal designs and to D-optimal designs. One striking result of our work is that the performance of D-optimal designs in terms of the I-optimality criterion very strongly depends on which of the D-optimal design points are replicated.
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Deteriorate process of products with the effect of various degradation mechanism or complex components often shows a multiphase degradation phenomenon. Misspecification of single-phase degradation model will lead to poor performance. To make full use of degradation signals, a two-stage degradation model considering random effects on deteriorate rate and change time point is established to enhance the accuracy of degradation description and reliability assessment. Online parameter updating and reliability estimation are applied to reflect the equipment status and capability. We also analyze the reliability estimation error that a product with two-stage degradation signals may introduce when change point time has not shown off. Simulation study is conducted to verify the proposed method.
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Today, many products are of high reliability. It takes a long time before its failure. Accelerated degradation testing (ADT) is a common way to assess the reliability of such products. However, sufficient samples are required in constant-stress ADT for lifetime inference. It is difficult to perform the usual constant-stress ADT under the constrain of really limiting samples, like some very expensive products. In this circumstance, usually only a few test samples are available. The step-stress accelerated degradation testing (SSADT) can be used to overcome this difficulty. Due to the differences between samples, the degradation levels of different products may have random initial values. This will bring additional uncertainty for the lifetime inference of the product. Wiener process is adopted in this study to model the step-stress degradation data, considering the inconsistency of the initial degradation performance of the product. The method of parameter inference under this model is given. The effectiveness of the method is verified on specific data.
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Design of experiments (DOE) is a statistical technique for quickly optimizing performance of systems with known input variables. It starts with a screening experimental design test plan involving all of the known factors that are suspected to affect the system's performance (or output). When the number of input variables or test factors is large, the primary experimental objective is to pare this number down into a manageable few. This is usually followed by another designed experiment design or test plan with the objective of optimizing the system's performance. The most common initial and final optimization designs of experiment are called the screening design and the response surface method (RSM). This paper will present some examples in the use of these designs.
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For DRO,the storage testing has following problems——small sample size,long life,and high storage test cost.Aiming at above problem,a step-stress accelerated degradation testing based on Wiener process was put forward and the testing plan was designed.According to failure mechanism analysis and storage micro-environment analysis,the accelerated model was determined.Combined with the result from preliminary testing,the performance degradation of DRO was modeling as Wiener process.The unbiasedness and the asymptotic variance of model parameters were analyzed,and the testing plan was optimized to improve the statistical precision.The validity of the method was verified with simulation method.
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Accelerated life testing
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Degradation tests have been used for the purpose of assessing reliability information. In this paper, we consider a degradation test design problem under a statistical model mis-specification scenario. The Wiener and gamma processes are considered in this research. A gamma process is suitable for describing degradation paths that exhibit monotone behavior. However, if one fits a Wiener process model to the data, the resulting statistical inferences may be affected. Tsai, Tseng, and Balakrishnan studied the effect on the estimated mean time to failure. However, the experimental design problem was not discussed. A lack of the explicit functional form of the estimation variances makes it difficult to find degradation plans that can improve test efficiency. In this paper, functional forms of optimal degradation test plans are proposed under model mis-specification. Furthermore, a weighted ratio objective function considering the prior probability of the true model is used to find robust test plans for practical use. The results from a numerical example show that, by using an appropriate degradation test plan, the estimation variance can be reduced, and the test efficiency can be improved. A simulation study is conducted to investigate the performance of the parameter estimates when the sample size is small.
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The purpose of this article is to persuade experimenters to choose A-optimal designs rather than D-optimal designs for screening experiments. The primary reason for this advice is that the A-optimality criterion is more consistent with the screening objective than the D-optimality criterion. The goal of screening experiments is to identify an active subset of the factors. An A-optimal design minimizes the average variance of the parameter estimates, which is directly related to that goal. While there are many cases where A- and D-optimal designs coincide, the A-optimal designs tend to have better statistical properties when the A- and D-optimal designs differ. In such cases, A-optimal designs generally have more uncorrelated columns in their model matrices than D-optimal designs. Also, even though A-optimal designs minimize the average variance of the parameter estimates, various cases exist where they outperform D-optimal designs in terms of the variances of all individual parameter estimates. Finally, A-optimal designs can also substantially reduce the worst prediction variance compared with D-optimal designs.
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This study investigates and compares Optimal designed Experiment with classical design which is non-Optimal using statistical tools. The two designs were evaluated on the basis of six parameters viz, Information matrix, Dispersion matrix, Prediction variance, A-Efficiency, D-Efficiency and G-efficiency. These six parameters help to determine the better design of the two experiment. Thus, it helps to establish efficient experiment suitable for better estimation of parameters of Linear Regression Model. The result obtained in this research work showed that the D-Optimal design increased the A-Efficiency, D-Efficiency and the G-efficiency of the Initial non optimal design. Furthermore the D-Optimal design maximized the determinant of the Information Matrix, Minimized the determinant of the Dispersion matrix and minimized the trace of the Dispersion matrix. It was therefore established in this research work that the D-Optimal Design Experiment has higher statistical efficiency than the initial non-optimal design. Moreso statistical analysis of the model parameters for both designs established the D-Optimal design experiment produce better models when used for estimating the parameters of Linear Regression models. It is therefore suggested that D-Optimal approach is suitable for fixing a poorly designed experiment. It is therefore recommended for use in estimating the parameters of Linear Regression Models.
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