The geometric structure on a degradation model with application to optimal design under a cost constraint
Introduction
Information geometry, introduced by Chentsov [1] and Amari [2], and further studied by Amari et al. [3], Amari [4] and Ay et al. [5], poses new applications and trends in topological and geometrical methods. In information theory, the family of probability density functions can be regarded as a manifold with the parameters play the role of the coordinate system and the Fisher information metric to be considered as a Riemannian metric. This kind of manifold is referred as the statistical manifold. The methods in information geometry provide a way to study the statistical manifold from the geometry point of view and provide solutions to various problems arising in neural networks [6], [7], [8], information theory [9], [10], [11], statistical mechanics [12], [13] and machine learning [14], [15].
In reliability analysis, information geometry is used to discuss the geometry on the statistical manifold induced by different life testing procedures. For instance, [16] studied the Amari–Chentsov structure on the statistical manifold with accelerated life tests data. In Bayesian analysis, [17] proposed a generalized Bayesian prediction rule by using -divergence as the loss function and discussed the asymptotic expansion by using the information geometry method. Zhao et al. [18] proposed a generative model for robust tensor factorization in the presence of both missing data and outliers and developed an efficient inference under a fully Bayesian treatment. All these existing work are based on time-to-failure data, however, for highly reliable products, no failure data or very few failure data can be obtained in a life testing experiment within a reasonable time period under normal operating conditions. Thus, [19] investigated the geometry on a statistical manifold induced by the linear and nonlinear degradation model. Collecting degradation measurements related to the presumed failure over time provides a way to assess the reliability behavior of the product via suitable statistical analysis. Degradation analysis is an important topic in reliability and maintainability engineering. For more information on degradation data analysis, one can refer to [20] and [21].
One of the approaches for degradation analysis is the use of stochastic processes such as the Markov process, the gamma process, and the Wiener process, to characterize the failure mechanisms and uncertainty over time in the degradation processes [22]. Among those stochastic processes, the Wiener process is frequently employed to model degradation data due to its physical interpretation and mathematical properties [23], [24]. Liu et al. [25] proposed a maintenance policy and investigated an optimal preventive replacement policy for the system with the Wiener degradation process. Hu et al. [26] discussed the Wiener degradation model with an accelerated degradation test. Tsai et al. [27] and Guo et al. [28] studied the Wiener process with linear drift. Wang [29] discussed the maximum likelihood inference on the Wiener degradation process with random drift and diffusion parameters. Consider that not all the degradation processes can be properly illustrated by the linear model, [30] suggested a nonlinear Wiener process while the variance is still linear over time. Li et al. [23] further proposed a generalized Wiener process degradation model with two different kinds of transformed time scales, the distribution of the failure time was obtained in closed-form. Zhang et al. [31] suggested a parameter estimation method of Wiener degradation models by minimizing the -divergence.
In information geometry, the Amari–Chentsov structure is a triplet involving the statistical manifold and its Fisher information metric and Amari–Chentsov tensor. Note that both the Fisher information metric and Amari–Chentsov tensor are invariant tensors under sufficient statistics, this leads to the Amari–Chentsov structure is an invariant structure. The structure plays an important role in information geometry and information theory. Thus, it deserves to investigate the Amari–Chentsov structure on the statistical manifold induced by the degradation model. Since its invariance, as applications, the Amari–Chentsov structure is employed to study the optimal design of degradation experiments with the cost constraint. To the best of our knowledge, there are only a few studies done on information geometry of the degradation model.
Motivated by the aforementioned literature, this paper focuses on the information geometry of statistical manifold induced by the nonlinear Wiener process degradation model. The organization of this paper is listed as follows. In Section 2, we describe the generalized Wiener degradation model and discuss the failure time distribution of the generalized Wiener process. In Section 3, the statistical manifold based on the Wiener degradation model is established. Then, in Section 4, we present the geometry on the statistical manifold. The Fisher information metric, Amari–Chentsov tensor and -connection are investigated. Section 5 discusses the parameter estimation and optimal design for the degradation experiment. The results of a Monte Carlo simulation study and a numerical study that illustrate the proposed methodologies are presented in Section 6. Some concluding remarks are presented in Section 7.
Section snippets
Wiener degradation model
Let denote the degradation measurement at time , we consider the Wiener process [23], [32] where and are continuously nonlinear functions of time , is a non-standard Brownian motion that describes the temporal uncertainty of the degradation process, is the standard Brownian motion. If , then the degradation model becomes a stochastic process derived by the standard Brownian motion [28], [33]. For simplicity, we
Statistical manifold induced by the Wiener degradation process
The statistical manifold can be considered as a Riemannian manifold with the probability density function as the element and the Fisher information metric as the Riemannian metric. In this section, we construct the statistical manifold induced by the degradation model . Let be the prefixed inspection time sequence and , , then from the property of the standard Wiener process, the stochastic process has the following properties:
- •
The degradation increments
Geometry on the statistical manifold
Take the logarithm on the both sides of Eq. (3.1), we have the log-likelihood function The -representation of the tangent space is given by which is a linear space of random variables spanned by . In order to obtain the partial derivatives , we use the following notations for the matrix :
- •
,
- •
Parameter estimation
Let and , then the model reduces to the degradation process with drift , which is a nonlinear function with time and parameter . For simplicity, we assume that , that is the degradation measurements are taken at the same time sequences , then the MLEs of the parameters and can be obtained by solving Eqs. (4.1), (4.2) as where
Monte Carlo simulation study and numerical study
In this section, a Monte Carlo simulation study is used to evaluate the performance of the parameter estimation procedures and a numerical study is used to demonstrate the statistical analysis and the determination of the optimal designs proposed in the previous sections.
Concluding remarks
In this paper, we investigated the geometry on a statistical manifold induced by the degradation model with the Wiener process. The degradation path is equipped with nonlinear drift and diffusion coefficients, which subsumes many current degradation processes as special cases. The drift parameter is assumed to be a random variable that describes the diversity from item to item. The PDF, hazard function, cumulative hazard function and the MTTF of the failure time are obtained. The statistical
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Nos. , , , and ), the Fundamental Research Funds for the Central Universities, China (Nos. JBK, JBK and JBK) of China.
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