SHU-DISSERTATION-2016.Pdf (1.898Mb)

Levy,´ Non-Gaussian Ornstein-Uhlenbeck, and Markov Additive

Processes in Reliability Analysis

A Dissertation

Presented to

the Faculty of the Department of Industrial Engineering

University of Houston

In Partial Fulﬁllment

of the Requirements for the Degree

Doctor of Philosophy

in Industrial Engineering

Yin Shu

August 2016 Levy,´ Non-Gaussian Ornstein-Uhlenbeck, and Markov Additive

Processes in Reliability Analysis

Yin Shu

Approved: Chair of the Committee Qianmei Feng, Associate Professor, Department of Industrial Engineering

Co-Chair of the Committee Hao Liu, Associate Professor, Duncan Cancer Center-Biostatistics Baylor College of Medicine

Committee Members:

Gino Lim, Professor, Department of Industrial Engineering

Jiming Peng, Associate Professor, Department of Industrial Engineering

Edward P.C. Kao, Professor, Department of Mathematics

Suresh K. Khator, Associate Dean, Gino Lim, Professor and Chair, Cullen College of Engineering Department of Industrial Engineering Acknowledgments

First and foremost I would like to express my deepest gratitude to my advisor Prof.

Qianmei Feng. The interesting and beautiful topic proposed by her made this dissertation possible. And it is my great honor that Prof. Hao Liu (Duncan Cancer Center-Biostatistics,

Baylor College of Medicine) has accepted to be my co-advisor. Here I appreciate all their advising, support, encouragement and help to make my Ph.D. research and life stimulating.

I am lucky to be one of their students.

I am especially grateful to Prof. Edward P.C. Kao for his excellent courses and exper- tise in stochastic processes. I also would like to thank Prof. David Coit for his professional comments on my research work. Many thanks to Prof. Gino Lim and Prof. Jiming Peng for serving as my committee members, and for their time, interest, and insightful suggestions.

I also would like to thank all the faculties, staff, and friends who have contributed immensely to my personal and professional time at the University of Houston. I acknowl- edge the ﬁnancial support from Texas Norman Hackerman Advanced Research Program under Grant no.003652-0122-2009 and USA National Science Foundation under Grant no.0970140.

Most of all, I thank my parents and love for their understanding and encouragement.

Their support made me devote my time in completing my Ph.D. study. This dissertation is dedicated to them.

iv Levy,´ Non-Gaussian Ornstein-Uhlenbeck, and Markov Additive

Processes in Reliability Analysis

An Abstract

of a

Dissertation

Presented to

the Faculty of the Department of Industrial Engineering

University of Houston

In Partial Fulﬁllment

of the Requirements for the Degree

Doctor of Philosophy

in Industrial Engineering

Yin Shu

August 2016

v Abstract

Unavoidable degradation is one of the major failure mechanisms of many systems

due to internal properties (mechanical, thermal, electrical, or chemical) and external in-

ﬂuences (temperature, humidity, or vibration). Such degradation in critical engineering

systems (e.g., pipelines, wind turbines, power/smart grids, and mechanical devices, etc.)

takes the form of corrosion, erosion, fatigue crack, deterioration or wear that may lead to

the loss of structural integrity and catastrophic failure. Therefore, developing stochastic

degradation models based on appropriate stochastic processes becomes imperative in the

reliability and statistics research communities.

This dissertation aims to develop a new research framework to integrally handle the

complexities in degradation processes (the intrinsic/extrinsic stochastic properties, com-

plex jump mechanisms and dependence) based on general stochastic processes including

Levy,´ non-Gaussian Ornstein-Uhlenbeck (OU), and Markov additive processes; and to

develop a new systematic methodology for reliability analysis that provides compact and

explicit results for reliability function and lifetime characteristics. First, to handle the

intrinsic stochastic properties and complex jumps, we use Levy´ subordinators and their

functional extensions, Levy´ driven non-Gaussian OU processes, to model the cumulative degradation with random jumps. We then integrally handle the complexities of a degradation process including both intrinsic and extrinsic stochastic properties with complex jump mechanisms, by constructing general Markov additive processes. Moreover, the models are extended to multi-dimensional cases for multiple dependent degradation processes under dynamic environments, where the Levy´ copulas are studied to construct

vi Markov-modulated multi-dimensional Levy´ processes. The Fokker-Planck equations for such general stochastic processes are developed, based on which we derive the explicit results for reliability function and lifetime moments, represented by the Levy´ measures,

the inﬁnitesimal generator matrices and the Levy´ copulas. To analyze the degradation data

series from such degradation phenomena of interest, we propose a systematic statistical

estimation method using linear programming estimators and empirical characteristic func-

tions. We also construct bootstrap procedures for the conﬁdence intervals. Simulation

studies for Levy´ measures of gamma, compound Poisson, positive stable and positive

tempered stable processes are performed. The framework can be recognized as a general

approach that can be used to ﬂexibly handle stylized features of widespread classes of

degradation data series such as jumps, linearity/nonlinearity, symmetry/asymmetry, and

light/heavy tails, etc. The results are expected to provide accurate reliability prediction

and estimation that can be used to assist the mitigation of risk and property loss associated

with system failures.

vii Table of Contents

Acknowledgments ...... iv

Abstract ...... vi

Table of Contents ...... viii

List of Figures ...... xii

List of Tables ...... xv

1 Introduction ...... 1

1.1 Background & Motivation ...... 4

1.2 Problem Statement ...... 7

1.3 Objectives & Contributions ...... 10

1.4 Organization ...... 13

2 Literature Review ...... 15

2.1 Wiener Processes ...... 15

2.2 Compound Poisson Processes ...... 18

2.3 Gamma Processes ...... 21

2.4 Levy´ Processes ...... 23

viii 2.5 Ornstein-Uhlenbeck Processes ...... 25

2.6 Markov Additive Processes ...... 28

3 Life Distribution Analysis Based on Levy´ Subordinators for Degradation

with Random Jumps ...... 31

3.1 Introduction ...... 32

3.2 Preliminaries of Levy´ Processes ...... 35

3.2.1 Characteristics ...... 36

3.2.2 Special Cases of Levy´ Processes ...... 39

3.3 Life Distribution Analysis Based on Levy´ Subordinators ...... 41

3.4 Life Distribution Analysis for Temporally Homogeneous Gamma Process

with Random Jumps ...... 45

3.4.1 Reliability Function Using Traditional Convolution Approach . . 46

3.4.2 Reliability Function Using Levy´ Measures ...... 47

3.5 Numerical Examples ...... 51

3.6 Conclusions ...... 54

4 Levy´ Driven Non-Gaussian Ornstein-Uhlenbeck Processes for Degrada-

tion-based Reliability Analysis ...... 56

4.1 Introduction ...... 56

4.2 Preliminaries ...... 61

4.2.1 Levy-It´ oˆ Decomposition ...... 61

4.2.2 Model Construction ...... 62

ix 4.3 Reliability Function and Lifetime Moments ...... 64

4.3.1 Results Based on Levy´ Subordinators ...... 67

4.3.2 Results Based on Non-Gaussian OU Processes ...... 71

4.4 Numerical Examples ...... 75

4.5 Conclusions ...... 79

5 Markov Additive Processes for Degradation with Jumps under Dynamic

Environments ...... 81

5.1 Introduction ...... 82

5.2 Model Construction ...... 85

5.3 Fokker-Planck Equations for Markov Additive Processes ...... 87

5.4 Reliability Function and Lifetime Moments ...... 94

5.5 Numerical Examples ...... 100

5.6 Conclusions ...... 107

6 Markov-modulated Multi-dimensional Levy´ Processes for Multiple Depen-

dent Degradation Processes under Dynamic Environments ...... 108

6.1 Introduction ...... 108

6.2 Preliminaries ...... 111

6.2.1 Multi-dimensional Levy´ Processes ...... 111

6.2.2 Model Construction ...... 114

6.3 Fokker-Planck Equations ...... 118

6.4 Reliability Function and Lifetime Moments ...... 120

x 6.5 Numerical Examples ...... 124

6.6 Conclusions ...... 129

7 Statistical Inference ...... 130

7.1 Introduction ...... 130

7.2 Estimation for Levy´ Degradation Processes ...... 132

7.2.1 Point Estimates ...... 132

7.2.2 Bootstrap Conﬁdence Intervals ...... 134

7.3 Estimation for OU Degradation Processes ...... 136

7.3.1 Point Estimates ...... 136

7.3.2 Bootstrap Conﬁdence Intervals ...... 139

7.4 Simulation Study ...... 140

7.5 Case Study ...... 146

7.6 Conclusions ...... 150

8 Summary and Discussions ...... 151

References ...... 155

Appendices ...... 181

xi List of Figures

Figure 1.1 Complexities in degradation ...... 7

Figure 3.1 Reliability function for gamma degradation with additional gamma

jumps ...... 52

Figure 3.2 Reliability function for gamma degradation with three jump types . 52

Figure 3.3 Pdf of lifetime for gamma degradation with three jump types . . . 53

Figure 3.4 Hazard rate for gamma degradation with three jump types . . . . . 53

Figure 4.1 Sample paths of Levy´ processes ...... 59

Figure 4.2 Sample paths of non-Gaussian OU processes ...... 60

Figure 4.3 Reliability function w.r.t. time t and failure threshold x based on

˜ Xs(t) ...... 77

Figure 4.4 Reliability function w.r.t. time t and failure threshold y based on Y (t) 77

˜ Figure 4.5 Reliability function w.r.t. time t based on Xs(t) and Y (t) . . . . . 78

˜ Figure 4.6 First moments of lifetime w.r.t. failure threshold based on Xs(t)

and Y (t) ...... 78

Figure 5.1 A sample path of Markov additive process with random jumps

when the environment states change ...... 87

xii Figure 5.2 A sample path of Markov additive process with no jump when the

environment states change ...... 93

Figure 5.3 Reliability function w.r.t. time t and failure threshold x for Case 1 . 104

Figure 5.4 Reliability function w.r.t. time t and failure threshold x for Case 2 . 104

Figure 5.5 Reliability functions w.r.t. time t when x = 15 and x = 20 for both

Case 1 and Case 2 ...... 105

Figure 5.6 First moments of lifetime w.r.t. failure threshold for both Case 1

and Case 2 ...... 105

Figure 5.7 Second moments of lifetime w.r.t. failure threshold for both Case 1

and Case 2 ...... 106

Figure 6.1 A sample path of Markov-modulated two-dimensional Levy´ pro-

cess with random jumps when the environment states change ...... 117

Figure 6.2 The mean of the ﬁrst passage time w.r.t. failure threshold for Case 1 128

Figure 6.3 The mean of the ﬁrst passage time w.r.t. failure threshold for Case 2 129

Figure 7.1 Levy´ density of PS(κ); dashed: estimated when n=25; solid: true . 142

Figure 7.2 Increments of the background driving PS(κ); dashed: estimated

when n=50; solid: true ...... 144

Figure 7.3 Marginal Levy´ density of OU driven by PS(κ); dashed: estimated

when n=50; solid: true ...... 145

˜ Figure 7.4 90% conﬁdence intervals of reliability function for Xs(t) . . . . . 145

Figure 7.5 90% conﬁdence intervals of reliability function for Y (t) ...... 146

xiii Figure 7.6 The capacity losing processes of four 18650 Li-ion batteries . . . . 147

Figure 7.7 Estimation of reliability function when failure threshold is 0.9 . . . 148

Figure 7.8 Estimation of ﬁrst moments of lifetime w.r.t. failure threshold . . . 149

Figure 7.9 Estimation of standard deviation of lifetime w.r.t. failure threshold 149

xiv List of Tables

Table 3.1 Parameter values for models in Chapter 3 ...... 51

Table 4.1 Parameter values for models in Chapter 4 ...... 76

Table 5.1 Parameter values for models in Chapter 5 ...... 103

Table 6.1 Parameter values for models in Chapter 6 ...... 128

Table 7.1 Results of κˆ for PS(κ) ...... 142

Table 7.2 Results of αˆ and κˆ for OU driven by PS(κ) ...... 144

xv Chapter 1

Introduction

The research conducted in this dissertation focuses on exploring Levy´ processes, non-Gaussian Ornstein-Uhlenbeck (OU) processes, and Markov additive processes for modeling complex stochastic degradation processes, with the aim of developing systematic procedures for deriving/evaluating reliability and lifetime characteristics. The results are expected to provide precise reliability prediction and estimation that can be used to assist the mitigation of risk and property loss associated with system failures.

With the advancement in technology, new and complex engineering systems (e.g., wind turbines, power/smart grids, subsea pipelines, and mechanical devices, etc.) have increasingly high reliability. However, they usually experience degradation processes by gradually losing their intended functionality over time. The degradation takes forms of wear, fatigue, erosion, corrosion and aging, etc. [3, 75, 181, 206]. A system fails when the accumulated degradation (e.g., fatigue crack) hits a boundary. Degradation data series typically contain more life-related information than the traditional failure time data, especially for highly-reliable systems. Therefore, developing stochastic degradation models based on appropriate stochastic processes becomes increasingly critical and signiﬁcant in the research community.

Levy´ processes are a class of cadl` ag` (right continuous with left limits) Markov processes with independent and stationary increments [17, 168]. The general Levy´ process

1 mathematically consists of three independent parts: the deterministic linear drift part, the

Wiener/Brownian part, and the pure jump part. As stochastic processes with random jump-

s, Levy´ processes have been widely used in modeling stochastic ﬂuctuations for mathe-

matical ﬁnance [55, 170] and risk management [118, 119, 124]. However, they have not

been well studied for degradation modeling (e.g., no explicit results of life characteristics

based on the general Levy´ degradation process). One of the most important advantages of

using Levy´ processes to model degradation is that their jump parts represented by Levy´ measures can model a great deal of jump mechanisms in degradation. We explore Levy´ processes for analyzing life distribution and reliability characteristics in Chapter 3 and a part of Chapter 4.

OU processes, another class of continuous time continuous state stochastic processes, were named after L.S. Ornstein and G. E. Uhlenbeck [194] in a physical modeling context, where the background driving process is a Wiener process, thus called ordinary or Gaussian OU processes [138]. Non-Gaussian OU processes are the generalization of ordinary OU processes by replacing Wiener processes with non-Gaussian Levy´ processes

(e.g., positive tempered stable processes). They have been recently developed and applied in ﬁnancial models by [21–23]. To our best knowledge, non-Gaussian OU processes have not been used for degradation modeling. In fact, it is nontrivial to obtain a closed-form probability distribution function for an OU process driven by a Levy´ process. The most important advantage of using non-Gaussian OU processes in modeling degradation processes stems from their ﬂexibility in modeling stylized features of degradation data series such as jumps, asymmetry, and heavy tails. We explore non-Gaussian OU processes for

2 studying reliability as part of Chapter 4.

Markov additive processes are a class of binary stochastic processes with one component as an additive process (e.g., Levy´ process) that is modulated by the other component, which is a standard Markov process [42, 46]. They form one of the most popular cases of Markov-modulated processes, and have been applied in queueing and storage systems

[151]. However, Markov additive processes have not been well studied for degradation analysis. Since Markov additive processes can represent the dependence of degradation on external factors (e.g., environments), they are suitable for modeling degradation under the dynamic (time-varying) environment. We explore this superiority in Chapter 5.

Multi-dimensional Levy´ processes can be constructed using Levy´ copulas [55, 109].

The dependence among components of a multi-dimensional Levy´ process can be completely characterized by a Levy´ copula, a function that has the similar properties with the ordinary copula but is deﬁned on a different domain. Markov-modulated multi- dimensional Levy´ processes are multi-dimensional cases of Markov additive processes by setting the additive component to be a multi-dimensional Levy´ process. They are appropriate candidates to model multiple dependent degradation processes. We explore this potential in Chapter 6.

Traditional maximum likelihood estimation and Bayesian estimation are not convenient for such general jump processes without closed-form distributions. [55] provided a highly comprehensive and thorough treatment of Levy´ processes in ﬁnance, covering Levy´ models, simulation and estimation. [21] showed that it is not straightforward to implement traditional likelihood-based estimation procedures for the non-Gaussian OU-based model,

3 although various moment-based methods are simple to use. [107] proposed the cumulant M-estimator (CME) to estimate the parameters in Levy´ processes. We explore this approach to estimate/predict the reliability characteristics in Chapter 7 using degradation data series.

1.1 Background & Motivation

Reliability of systems is one of the most critical concerns in many ﬁelds including energy, health, aerospace, and national defense, etc. In investigating reliability, degradation processes have been considered and analyzed for more than half a century. S- ince the 1970s, stochastic processes have been used to represent degradation evolution in order to handle the stochastic properties in degradation processes. Erhan C¸inlar, Mo- hammad Abdel-Hameed, and William Meeker, among others, have made substantial and foundational contributions to the general area of research in degradation-based reliability analysis. [40–53] mainly focused on developing mathematical theories of stochastic processes (e.g., Markov additive processes, Levy´ systems of Markov additive processes). The work provided the prerequisite mathematical support in degradation modeling using such stochastic processes. [3–15] mainly studied life distribution properties (e.g., increasing failure rate) and the optimal maintenance/inspection policies based on stochastic degradation models. [140–147] developed statistical methods for degradation data analysis. Their research demonstrates that constructing appropriate stochastic processes in degradation modeling is critical in precisely evaluating and predicting reliability characteristics of

4 highly-reliable systems. Therefore, modeling the degradation processes and then deriv-

ing/estimating reliability characteristics have attracted great attention among mathemati-

cians and engineers in recent years. However, the complexities in degradation lead to more

challenging research topics that need to be further explored.

Two complex stochastic properties of degradation processes in investigating relia-

bility are: 1) the internally-induced stochastic evolution that has inherent statistical uncertainties stemming from physical, electrical, thermal, or chemical features of systems, such as molecule/atom structure, arrangement and composition, etc. (e.g., the increments of the wear volume during a ﬁxed time interval follow an inverse Gaussian distribution), and 2) the externally-induced stochastic evolution that stems from dynamic (time-varying) environment factors such as temperature, pressure and humidity, etc. These extrinsic uncertainties in dynamic environments can be described as two different aspects: 1) different evolution patterns of degradation under different environment states, e.g., the degradation process evolves as a positive tempered stable process with drift under one environment state, while it evolves as a compound Poisson process without drift under another state, and 2) an instantaneous random jump occurring in degradation at the time the state of the environment process changes.

In addition, there are complex jump mechanisms embedded in the stochastic evolution of degradation processes. As jumps due to the internal operation (e.g., sudden electrical short, chemical reaction, etc.) under the deterministic environment, internally-induced jumps can be an inﬁnite number of small jumps in any small interval of time, or a ﬁnite number of big jumps that occur according to a probability law. As jumps due to the

5 changes in environment states (e.g., temperature extremes, vibration/tension shocks, etc.),

the emergence of externally-induced jumps follows the law of the environment process.

The size of these jumps is a random variable that can be dependent on the switching

environment states. Moreover, multiple degradation processes in a system are naturally

interdependent including internally-induced dependence due to the internal features and

externally-induced dependence due to the exposure to the same external environment con-

ditions.

All the complexities of degradation processes are illustrated in Figure 1.1. The two

stochastic properties are demonstrated by the stochastic evolution of the binary stochas-

tic processes {X(t),E(t)}, where the two-dimensional stochastic degradation process

X(t) = [X1(t),X2(t)] is modulated by the environment process E(t). Under a certain state of E(t), X(t) evolves as a two-dimensional dependent stochastic process. During different states of E(t), X(t) evolves in different patterns with different jump mechanisms.

In addition, instantaneous jumps in all degradation processes induced by the changes of

E(t) are also properly demonstrated.

The motivation for this dissertation is to provide a new methodology in degradation-

based reliability analysis, where the complexities in degradation, including the internally-

/externally-induced uncertainties, dependence and complex jump mechanisms, are inte-

grally modeled in a broad class of general stochastic processes. Based on the mathematical

theories of such stochastic processes that are analytically appealing, we aim to develop

the systematic procedures in deriving/evaluating life characteristics that can provide the

consolidated and compact results.

6 X(t) X1(t)

X2(t)

t i k l m E(t) j

Figure 1.1: Complexities in degradation

1.2 Problem Statement

Depending on the availability of environmental data, two classes of stochastic models have been generally constructed to represent the degradation processes. When it is not possible to identify the factors or it is not feasible/economically convenient to monitor the external environment (in this case, randomness in the environment process and/or its effect on degradation evolution is ignored and the environment is assumed to be deterministic), the observable degradation process was directly represented by a stochastic process modeling the intrinsic uncertainties with temporal variability [3, 75, 152, 206] (e.g., Wiener process for non-monotonic increments, gamma process for monotonic increments, and

7 compound Poisson process for pure jump increments); or the unobservable degradation process was treated as a latent process, measured and tracked by internal stochastic covariates that are observable marker processes [86, 104, 105, 175, 208] (e.g., Linear processes with random effects, Wiener processes, and Poisson processes). These stochastic models just represented internally-induced uncertainties.

The second class of stochastic models is constructed when the operating environment can be monitored and measured. In [71], a stochastic covariate failure model was studied for assessing system reliability, where the external stochastic covariates were modeled by

Wiener-based diffusion processes. The life distribution was assumed to be explicitly related to such stochastic covariates. However, the stochastic models just represent externally- induced uncertainties with temporal variability. Some advanced stochastic degradation models were constructed in [112–115], where the degradation process is modulated by the environment process. A compound Poisson process and a linear process were used to represent the conditional degradation process, and the corresponding parameters are explicit functions of the environment process, which is represented by a Markov chain with a ﬁnite number of states. These models are motivated by the original ideas from [40], in which Markov additive processes were ﬁrst proposed in degradation modeling.

Advancement in technology have had and will continue to impel researchers to make improvement in stochastic models for degradation-based reliability analysis. Recently, advanced technology in measurement and deployment of sensors/smart chips make it convenient to collect plenty of degradation data and environment data, which are accurate, reliable and real-time. However, the existing stochastic models for representing degradation

8 processes have left a number of practical issues untouched that stem from the complexities in degradation. Such practical problems are itemized as follows.

1. The common linear-, Wiener-, gamma-, inverse-Gaussian- and Poisson-based mod-

els are not ﬂexible in general cases, because it may not be appropriate in reality to

assume that the increments follow normal, inverse-Gaussian or gamma distributions,

or to assume that the jumps occur according to the Poisson laws. In practice, the

degradation data series have complex jump mechanisms, and have asymmetric and

heavy-tail properties when there are big jumps occurring randomly.

2. There are no systematic and extendible procedures in analyzing the degradation

evolution and deriving life characteristics (e.g., reliability function and lifetime mo-

ments) from stochastic degradation processes. Existing approaches are typically

based on the convolution formula, under the condition that the distribution functions

of stochastic processes of interest have closed-forms. They often can not produce

compact and explicit results, e.g., when the degradation with random jumps is mod-

eled by the sum of a stable process and a compound Poisson process.

3. The existing stochastic models for degradation processes under the dynamic envi-

ronment are limited: they are simply based on linear processes and Poisson pro-

cesses; and they did not handle the jumps occurring at the time the state of the

environment process is switching, which is a common phenomenon in degradation.

4. In order to develop general degradation models under the dynamic environmen-

t, it is necessary to describe 1) the general stochastic process for degradation, 2)

9 the general stochastic process for environment, and 3) the effects of environment

on degradation. The development of the characteristics of such processes and the

subsequent reliability analysis based on such characteristics is a nontrivial work,

even for simple cases. The difﬁculty of these stems from 1) the stochastic evolution

of degradation has complex mechanisms such as random jumps, 2) the stochastic

nature of environment, and 3) the distributional derivation for the ﬁrst passage time.

5. Stochastic models for multiple dependent degradation processes are not well devel-

oped. The existing models are mainly based on multi-dimensional Wiener processes,

where the dependence is simply represented by the covariance matrix. These models

cannot handle the jumps, the dependence among the jumps, and the dependence

stemming from the environment process.

These issues can be concisely summarized as 1) the lack of general and systemically analyzable stochastic models, 2) the lack of integrated uncertainties, complexities and dependence management methodology, 3) the lack of systematic procedures on derivation, and 4) the lack of explicit and powerful results/formulas.

1.3 Objectives & Contributions

This dissertation aims to develop a new research framework to integrally handle the complexities in degradation processes (the intrinsic and extrinsic stochastic properties, dependence and complex jump mechanisms) based on general stochastic processes including Levy´ processes, non-Gaussian OU processes and Markov additive processes,

10 and to construct a new methodology for reliability analysis, in which all the critical prob-

lems mentioned in the previous section are well addressed. The results provide efﬁcient

and accurate reliability evaluation and prediction. The speciﬁc objectives are itemized as

follows.

Objective 1: we develop a class of general stochastic models that can represent the

internally-induced stochastic properties, dependence and complex jumps in degra-

dation processes. We ﬁrst consider the Levy´ processes for degradation modeling by

relaxing the assumptions on the law of increments and the law of jumps, in which

the jump mechanisms of degradation are well described by the Levy´ measures. We

further extend Levy´ processes to non-Gaussian OU processes that relax the assump-

tion of the linear mean path (Chapters 3, 4). This provides a cornerstone for the next

level work.

Objective 2: we investigate a class of general stochastic models that can simultaneously

represent the internally- and externally-induced stochastic properties, dependence

and complex jumps in degradation processes. We study Markov-modulated pro-

cesses, a class of binary stochastic processes, in which one component is used to

model the degradation evolution and the other is used to model the environment

evolution (Chapter 5). Multi-dimensional setting is further explored using the Levy´

copulas (Chapter 6).

Objective 3: we explore the potentials of the inverse Fourier transform in deriving

11 compact- and closed-form results for reliability characteristics from Levy´ degrada-

tion processes. We extend the idea to be applied in multiple dependent degradation

modeled by multi-dimensional Levy´ processes (Chapter 3).

Objective 4: we propose to use the Fokker-Planck equations of such analytically appeal-

ing stochastic processes (Levy,´ non-Gaussian OU, and Markov-modulated process-

es) in the derivation of reliability characteristics based on the Laplace transforms.

We develop systematic procedures to derive and obtain explicit and compact results

(Chapters 4, 5, 6).

Objective 5: we propose a systematic statistical inference method using linear program-

ing estimators and empirical characteristic functions. The point estimates of re-

liability function and lifetime moments are obtained by deriving their explicit ex-

pressions. Bootstrap procedures are also constructed for the conﬁdence intervals

(Chapters 7).

To achieve these ﬁve objectives, several fundamental research areas need to be ex-

plored as part of the research methodology. The development of the integrated method-

ology enhances the multidisciplinary research by expanding mathematics, probability and

statistics, and computer science into reliability engineering. The work will provide accu-

rate and realistic new models for degradation analysis, by realizing multiple uncertainty

sources of degradation mechanisms. The framework can be recognized as a general ap-

proach that can be used to handle widespread classes of degradation data sets, including

linear data, nonlinear data, ﬂuctuant data, etc.

12 1.4 Organization

The dissertation summarizes the results in Shu, et al. [176–180]. Chapter 2 provides

a comprehensive literature review for the related research on degradation modeling, Levy´ processes, non-Gaussian OU processes, and Markov-modulated processes. To handle the intrinsic stochastic properties with complex jumps, Chapter 3 presents the degradation models using Levy´ processes. The complex jump mechanisms including inﬁnite activities

(IA) and ﬁnite activities (FA) in degradation are well described by the Levy´ measures.

Based on the inverse Fourier transform, the new closed-forms of reliability function and probability density function of lifetime are derived. In Chapter 4, we further explore the

Levy´ subordinators and extend the model by using non-Gaussian OU processes, which can model the degradation processes with non-linear mean paths. Based on the Fokker-Planck equations, the new explicit results in Laplace expressions for both reliability function and lifetime moments are derived. To handle both the intrinsic and extrinsic stochastic properties with complex jumps, in Chapter 5, we integrally model the complexities of degradation processes under the dynamic environment by constructing Markov additive processes. The

Fokker-Planck equations are developed for such binary processes. The explicit results for reliability function and lifetime moments are derived. Chapter 6 explores the multi- dimensional settings for the stochastic processes of interest based on Levy´ copulas to handle the dependence among multiple degradation processes. The Markov-modulated multi- dimensional Levy´ processes are constructed to model the multiple dependent degradation

13 processes under dynamic environments. Chapter 7 provides the systematic estimation procedures for our stochastic models. The simulation and case studies are performed. Finally,

Chapter 8 is devoted to make summary and discuss some potential research directions following this dissertation.

14 Chapter 2

Literature Review

In this chapter, we do a thorough literature review for the stochastic processes related to the research in this dissertation. Wiener processes, compound Poisson processes, and gamma processes are widely used in degradation-based reliability analysis, since the required mathematical derivations are relatively straightforward based on the existing closed-form distribution functions of these processes. The fundamental mathematical theories of general Levy´ processes, non-Gaussian OU processes, and Markov additive processes are summarized. Some advanced mathematical theories explored by mathematicians and their applications in ﬁnancial economics are enthusiastically studied and reviewed in this chapter.

2.1 Wiener Processes

Wiener processes were named after Norbert Wiener [209], who has made a great con- tribution in the mathematical description of Brownian motion (named after Robert Brown, who described the random motion of particles suspended in a ﬂuid). Wiener processes are a special case of Levy´ processes with stationary and independent increments following a normal distribution. Based on this property, [171] developed some advanced mathematical theories of Wiener processes. They proved that certain classes of random variables

15 associated with Wiener processes are inﬁnitely divisible. They analyzed the inﬁnitesimal structure and developed the theory of stochastic integrals of Wiener processes. Theoretical research on Wiener processes can be found in [62, 172].

In ﬁnance, the price of an option is directly related to the prices of its underlying stocks. Wiener processes are well used in describing the logarithm of stock prices under the seminal Black-Scholes option pricing model [27, 148]. Since then, many extended models based on Wiener processes have been developed. [97] studied the pricing of a

European call on an asset that has a stochastic volatility. The option price was determined in a series form for the case in which the stock price was instantaneously uncorrelated with the stochastic volatility. [195] first studied the default measures for individual firms and assessed the effect of default risk on the equity returns based on such an option pricing model, where the logarithm of the market value of a firm’s underlying assets was described by a Wiener process, and the default was defined as the failure to service debt obligations.

Similar studies can be found in [58, 85, 89]. These Wiener-based models have had a profound impact on ﬁnancial economics.

A Wiener process is appropriate for modeling degradation that changes nonmono- tonically because it can have non-negative and negative increments alternately. To derive the reliability function from a Wiener degradation process, the failure time can be deﬁned as the ﬁrst passage time to the failure threshold [39, 56, 127]. Using the fact that the

ﬁrst passage time of the Wiener process follows the inverse Gaussian distribution, Wiener- based degradation models have been well developed for reliability analysis. [206–208] proposed some stochastic degradation models where the underlying process is a Wiener

16 process. In [206], the author considered a Wiener process with normal-distributed measurement errors in order to ﬁt the degradation data measured from imperfect instruments, procedures and environments. Inference procedures were constructed based on maximum likelihood estimation and some practical extensions were discussed. In [207], a Wiener process with a time scale transformation was taken into account to model the degradation process. The model and inference methods were illustrated with a case application involv- ing self-regulating heating cables. This model did not consider measurement errors. In

[208], a degradation model with stochastic time-varying covariates was constructed. The degradation is called a latent process that is unobservable, and the stochastic covariate is called a marker process that is observable [108]. These two processes were represented by a bivariate Wiener process, based on which the failure time distribution was derived.

Several candidate markers were discussed for extensions.

Many extensions have been done based on Wiener degradation processes. [69] considered there are many cracks evolving simultaneously in a system, and proposed models to describe the stochastic behavior of fatigue cracks. In the models the growth rates of cracks were described by Wiener processes. The distribution of failure time and its properties were studied, and the parameters were estimated by the maximum likelihood method. [183, 184] presented statistical methods for the remaining useful life estimation based on Wiener processes. A recursive ﬁlter algorithm and Bayesian updating framework were studied. The updating is done at the time that a new piece of degradation data becomes available. Gyros in an inertial navigation were used as a practical case study to demonstrate the superiority of the proposed model. [152, 153] developed degradation

17 models with several accelerating variables for statistical inference based on both observed

failure values and degradation measurements. The underlying degradation process that

they assumed was a geometric Brownian motion (the logarithm of the degradation process

was assumed to be a Wiener process). The inverse Gaussian distribution provided a good

approximation to the failure distribution. Many other Wiener-based degradation analysis

can be found in [64, 188, 192].

2.2 Compound Poisson Processes

Poisson processes were named after Simeon´ Denis Poisson [161]. They are a class of counting processes and their standard deﬁnitions can be found in [167]. Compound

Poisson processes are a generalisation of Poisson processes and are a special case of Levy´ processes with stationary and independent increments evolving by pure jumps. The jumps occurrence is following a Poisson process and their sizes are independent and identically distributed (i.i.d.) random variables. The mathematical theories of compound Poisson processes including Poisson random measures, corresponding Levy´ measures, and charac-

teristic functions are introduced in [168]. Due to these available theories, they are popular

in modeling random events that occur at random discrete times.

In biology, [99] introduced a compound Poisson process to model the rate variation

across lineages on a tree, where the events of substitution rate change were placed onto

a phylogenetic tree according to a Poisson process. The parameters of the model were

estimated based on the Metropolis-Hastings-Green algorithm. [116] proposed a compound

18 Poisson process to describe the neutral changes of gene expression over the evolutionary time, where the occurrence of mutations on the DNA level follows a Poisson process and the effects of a mutation on a gene’s expression are i.i.d. random variables. The observable difference in the expression of a gene between two samples was generated by the difference of two independent compound Poisson processes.

In ﬁnance, compound Poisson processes were ﬁrst used in the classical Cramer-

Lundberg model [57] in risk analysis, where the number of claims follows a Poisson process, and the claim sizes are i.i.d. random variables. The ruin probability can be studied based on the well-known Pollaczek-Khinchine formula. [18, 19] studied the ruin probabilities in compound Poisson models, in which the bounds and asymptotic formulas of the ruin probability were obtained from the n-fold convolution, under speciﬁc conditions on the claim sizes (sub-exponentially distributed). [67, 199] extended the classical compound Poisson risk process to the one perturbed by a Wiener process, and obtained the Lundberg inequality and the asymptotic formula of the ruin probability by the renewal theory. The convolution formula for the probability of ruin was derived when the distributions of the individual claim amounts are subexponential as well as exponential, or a combination of exponential distributions. [213] did a further step by considering a constant force of interest to invest the surplus, under these perturbed compound Poisson risk processes. They investigated the asymptotic behavior and bounds of the ruin probability as the initial reserve goes to inﬁnity. They also obtained a Laplace expression of the ultimate ruin probability. [191] obtained optimal dividend strategies considering the compound

Poisson insurance risk model when strictly positive transaction costs are included. They

19 constructed a numerical procedure to deal with general distributions of claim sizes.

The estimation of the ruin probability relies on the estimation of the underlying compound Poisson processes. This research has been studied by [73, 74]. Several efficient estimation methods for compound Poisson processes have been developed. [34] first introduced the decompounding methods to estimate the parameters of a compound Pois- son process. They studied statistical properties such as the asymptotic normality and the asymptotic validity of bootstrap confidence regions of the estimator. [54, 68] investigated the nonparametric estimation of the jump density of a compound Poisson process from the discrete observation of one trajectory. The estimation procedure was based on the explicit inversion of the operator giving that the law of the increments is a nonlinear transformation of the jump density.

For systems subject to sporadic jump degradation, a compound Poisson process is one of the appropriate candidates to model the cumulative degradation. [75] considered a device subject to a sequence of shocks occurring randomly according to a Poisson process, where a compound Poisson process is proposed to model the accumulated damage. The reliability function is expressed based on the convolution formula, and the increasing hazard rate property is studied. The same model was discussed to describe the fatigue damage of materials in [186]. [204, 205] modeled the degradation process with non-decreasing paths by a generalization of Poisson process, which is a marked point process. The degradation process was assumed to be generated by a position-dependent marking of a doubly stochastic Poisson process, where the jump intensity depends on a non-negative random variable, compared to a compound Poisson process. They discussed the characteristics of

20 such processes and derived reliability functions for some special cases. [4, 91] introduced the life distribution and its properties for systems subject to pure jump damage processes.

2.3 Gamma Processes

Gamma processes are a special case of Levy´ processes with stationary and independent increments following a gamma distribution. They are monotonically increasing because of the positivity of the gamma variable. The Levy´ measure and characteristics of gamma processes were introduced in [168]. Some remarkable mathematical properties such as quasi-invariance of gamma processes were introduced in [193, 212]. They showed that gamma processes are renormalized limits of stable processes and emphasized the deep similarity between the gamma process and the Brownian motion.

In ﬁnance, [111] proposed to use gamma processes to describe the jumps of asset price. Based on the Malliavin calculus for jump processes, they derived the Greeks formulas for derivative securities with both continuous and discontinuous European payoff structures, by making use of a scaling property of gamma processes with respect to the

Girsanov transform. [66] considered to use gamma processes to model aggregate claims in risk theory. They discussed Bayesian estimation for parameters of gamma processes and presented a method for simulating gamma processes. The values of ruin probability including lower and upper bounds were tabulated.

The gamma process is suitable for modeling degradation that progresses in one direction by a sequence of small increments. [3] was the ﬁrst study to use a gamma process

21 for modelling a wear process, in which life distribution properties such as the increasing failure rate were discussed. [125] later presented a gamma model incorporating a random effect for degradation. They used the model to ﬁt the crack-growth data in [102] and presented goodness-of-ﬁt tests (the data was used in [135]). The calculations for failure time distributions were also presented. [20] studied the gamma degradation models with time-dependent covariates. The cases of parameterized and completely unknown mean degradation functions were investigated, and the maximum likelihood estimator of reliability and degradation characteristics were presented based on the degradation data measured without errors.

[152, 153] constructed a new degradation model by incorporating an accelerated test variable based on gamma processes. Both observed failures and degradation measures were used for parameter inference of the life distribution based on the maximum likelihood estimation, which is illustrated by using real data for carbon-ﬁlm resistors and fatigue crack size. [132] proposed a maintenance policy for gamma degrading systems.

[29] performed the parametric inference in a perturbed gamma degradation process. De- tailed discussions were given in [196] that provided an overview and survey for applying gamma processes to model degradation under the maintenance context. The methods for estimation, approximation, and simulation of gamma processes were also reviewed. [70] suggested a non-stationary stochastic process to model the wear and derived the reliability function, where the underlying process is a gamma process.

22 2.4 Levy´ Processes

Levy´ processes were named in honor of the French mathematician Paul Levy.´ They form a rich class of Markov processes with independent and stationary increments. Their basic mathematical theories were established in the 1930s. Since then, a large number of new theories as well as novel applications in such diverse areas as physics, econometrics and mathematical ﬁnance, have been developed.

The mathematical theories of Levy´ processes have been well developed in many books, monograph and papers. [168] analytically elaborated both basic and advanced knowledge of Levy´ processes by connecting with inﬁnitely divisible distributions. The basic part of the monograph covers characteristic functions, special cases and distributional properties; and the advanced part of the monograph covers the Levy-Itoˆ decomposition and Wiener-Hopf factorizations. [17] introduced both the classical theory and stochastic analysis related to Levy´ processes. Basic ideas behind stochastic calculus such as martingales, stopping times, change of measures and stochastic integration were described systemically. [65, 124] considered the ﬂuctuation theory of Levy´ processes. The results focused on the case of Levy´ processes with jumps in only one direction, including subordinators and spectrally negative processes, for which recent theoretical advances have yielded a high degree of mathematical tractability.

There are intense activities focusing on the application of Levy´ processes in ﬁnancial economics, especially on the option pricing. Levy´ models relax the restrictive assumptions of the Black-Scholes-Merton model by allowing jumps in the underlying asset price and

23 have become popular. [36] considered the problem of pricing contingent claims on a stock

whose price process is modelled by a geometric Levy´ process, in exact analogy with the ubiquitous geometric Brownian motion model. For a class of options that allow early exercise (Bermudan options), [81, 82, 134] used the Levy´ process to model the log-return

process of the asset that is underlying the option on a ﬁnite time horizon, and developed

the valuation of the Bermudan option by inverse Fourier and Hilbert transforms. The same

models based on Levy´ processes for pricing options can be referred to [77].

The total revenue of an insurance company is a deterministic increasing process cor-

responding to the accumulation of premiums minus a stochastic process representing the

offset from independent claims sequentially through time. [118, 119] used Levy´ processes

without positive jumps to model insurance risk processes, in which the jumps represent

claim payments, and they studied the properties of the ruin probability, which corresponds

to the distribution of the ﬁrst passage time of the risk process above a speciﬁed bound.

They did not obtain the explicit results for the ﬁrst passage time. [100, 101] studied a

general perturbed risk process, where the cumulative claims were modelled by a subordi-

nator with ﬁnite expectation, and the perturbation was modeled by a spectrally negative

Levy´ process with zero expectation. They derived the survival function for this kind of

risk processes. The calculation of ruin probability for the general case is nontrivial, which

involves the n-fold convolution and trickish integrals.

The application of Levy´ processes to ﬁnance has continued to be a highly active and

fast moving area. [55] provided a monograph that is a highly comprehensive and thorough

treatment of Levy´ processes in ﬁnance, covering Levy´ models, simulation and estimation.

24 They studied and introduced the signiﬁcance of using Levy´ processes in modeling ﬁnancial

time series compared to Brownian motion-based models. A helpful introduction of Levy´ processes in ﬁnance, typically aimed at pricing ﬁnancial derivatives, can be comprehended in [170].

Levy´ processes are a class of cadl` ag` (right continuous with left limits) homogeneous

Markov processes. They are suitable to model degradation processes with random jumps.

A limited number of studies have used general Levy´ processes to model the degradation, and a closed-form life distribution is obtained only when considering the special cases of

Levy´ processes. [5] used Levy´ processes to model the wear and studied its life distribution properties where the threshold is assumed to be random. [211] used special cases of Levy´ processes to model the degradation and jump damages: a gamma process for wear and a compound Poisson process for random jump damages, respectively. They assumed that the threshold is exponentially distributed, which leads to a closed-form lifetime distribution.

[154] derived the distributions of the ﬁrst-passage time and the last-passage time based on Levy´ degradation processes. [128] considered the class of degradation-threshold-shock models where the degradation is modeled by a process with independent and stationary increments.

2.5 Ornstein-Uhlenbeck Processes

OU processes were named after Leonard Salomon Ornstein and George Eugene Uh- lenbeck [194] in a physical modelling context, where the background driving process is

25 a Wiener process, thus called ordinary or Gaussian OU processes [138]. In the ﬁeld of physics, the ordinary OU process is represented by the classic Klein-Kramers dynamics

[123]. In the ﬁeld of ﬁnance, the ordinary OU process is well known as the Vasicek model

[198], in which the interest rate was modeled by such a process. The same interest rate model was studied in [36], where the parameters of ordinary OU process were estimated based on the generalized method of moments. [90] used the ordinary OU process to model the instantaneous net yield of oil in a two-factor model for pricing ﬁnancial and real assets that are contingent on the price of oil. [26] proposed to use the ordinary OU process to model the noise in the ﬁeld of metrology, which is important in evaluating the measurement system capabilities. In survival analysis, [1] considered a model where the individual hazard rate is a squared function of an ordinary OU process and studied the survival distributions. [160] systemically introduced the probability theories of ordinary

OU processes including Markov properties, stochastic integrals and martingales.

Non-Gaussian OU processes are generalisation of ordinary OU processes by replacing the background driving Wiener process with the non-Gaussian Levy´ process (Levy´ process without Gaussian part). [21–23] presented the most important research contributions in the ﬁeld of non-Gaussian OU processes for ﬁnancial models during the recent two decades. In [21], they proposed to use a non-Gaussian OU process to model the stochastic volatility in the stochastic differential equations of the log-price of stocks and the log-exchange rates. The background driving non-Gaussian Levy´ process has no negative increments (e.g., Levy´ subordinators). They showed that it is not straightforward to

26 implement the traditional likelihood-based estimation procedures for the proposed models, although various moment-based methods are simple to use. In [22], they analyzed the probability properties of the realized volatility in the context of the non-Gaussian OU- based model, which provided simple quasi-likelihood results that could be used to perform a computationally simple estimation. In [23], they studied the probability properties of the integrated non-Gaussian OU process, which is an important component in the expression of the log-price of stocks or the log-exchange rates, and described the integrated variance in the proposed stochastic volatility models. They explored the tail behavior of the integrated OU process that reﬂects the tail behavior of the return of underlying stocks.

The efficient statistical inference includes the estimation of non-Gaussian OU processes is significant in applying such related models in financial data analysis. [162] proposed an indirect inference method for a class of stochastic volatility models for financial data based on non-Gaussian OU processes. They combined a quasi-likelihood estimator derived from maximizing an approximative Gaussian quasi-likelihood function with simulations, and then applied a method of moments to obtain the indirect estimator, which is better than the pure quasi-likelihood estimator. [189] developed an efficient and explicit estimation procedure for non-Gaussian OU processes based on their characteristic functions, assuming that the marginal law belongs to a parametric family indexed by a parameter vector. The approach can deal with a general case of processes having both positive and negative jumps. The peculiar form of the characteristic functions of non-

Gaussian OU processes and its relation with the characteristic functions of the underlying

Levy´ process were exploited in [190]. Based on the inversion of characteristic function,

27 they provided fast and reliable simulation procedures for OU processes. Simulation based estimation procedures for non-Gaussian OU processes were discussed in [88, 93, 166] .

The approximate results were often implemented since it is difﬁcult to accurately simulate the jumps in the corresponding Levy´ processes.

2.6 Markov Additive Processes

Markov additive processes are a broad class of Markov-modulated processes. Their general deﬁnitions and fundamental mathematical theories were introduced by [42, 46].

Since then, many advanced probability theories have been developed. The first passage time is defined as the first time a stochastic process hits a bound. [59] studied the law of the first passage time of a Markov additive process, where the additive component is a spectrally negative Levy´ process (all jumps in the Levy´ process are negative). Based on the theory of Jordan chains, they characterized the law by a matrix function. To express such matrix, a matrix equation needs to be solved, which is intractable. [149] studied the

Wiener-Hopf factorization for a Markov additive process with the multi-dimensional additive component and its generating multi-dimensional reflected process (the reflection takes place when the additive component hits the value zero) using the matrix analytic approach, and then derived a closed-form formula for the stationary distribution of this reflected process. [103] analyzed the matrix exponent of a Markov additive process with nonnegative jumps, focusing on the roots of the generalized Cramer-Lundberg´ equation that plays an important role in the fluctuation theory such as the first passage time. The same topic of

28 special cases of Markov additive processes such as Markov-modulated Brownian motion

and Markov-modulated compound Poisson process, can be found in [110, 163].

Markov additive processes are typically implemented in the ﬁelds of risk manage-

ment, ﬁnancial, queueing theory and reliability. [150] considered a Markov additive pro-

cess in the risk and queueing applications and studied the hitting probabilities at upper lev-

els, where the additive process linearly increases or decreases when the background state

is unchanged, and the process may have upward jumps at the transition instants. Asymp-

totic behavior of the ruin probability was studied when the initial reserve goes to inﬁnity,

given that the distributions of claim sizes have light tails. [63] studied the extremes of a

continuous time Markov additive process with one-sided jumps. The Laplace transforms

of extreme distributions were given in terms of two matrices. The results were applied in

determining the steady-state buffer-content distribution of several single-station queueing

systems. However, the underlying work such as solving a nonlinear matrix equation is

intractable.

In degradation analysis, [40] ﬁrst introduced Markov additive processes in modeling

a wear process and shocks under a Markov dynamic environment. In the model, the shock

arrival rate, the shock sizes and the wear out rate were governed by a random environment,

which evolves as a Markov process; and the Levy´ processes were proposed to represent the additive component. There are no explicit results for reliability characteristics due to the complexity of the evolution of Markov additive processes. [80] analyzed the optimal replacement policy by considering a Markov additive process to model the shocks that occur continuously during a time interval.

29 Recently, special cases of Markov additive processes, Markov-modulated linear and compound Poisson processes were studied in degradation analysis. Explicit results of reliability characteristics were derived by [112–115]. Considering the same model, [84] calculated the limiting average availability of a system, assuming that such system is main- tained through inspection and perfect repair when it fails.

30 Chapter 3

Life Distribution Analysis Based on Levy´ Subordinators for Degrada- tion with Random Jumps

In this chapter, for a component or a system subject to stochastic degradation with sporadic jumps that occur at random times and have random sizes, we propose to model the cumulative degradation with random jumps using a single stochastic process based on the characteristics of Levy´ subordinators, the class of non-decreasing Levy´ processes. Based on the inverse Fourier transform, we derive a new closed-form reliability function and probability density function for lifetime, represented by Levy´ measures. The reliability function derived using the traditional convolution approach for common stochastic models such as gamma degradation process with random jumps, is revealed to be a special case of our general model. Numerical experiments are used to demonstrate that our model performs well for different applications, when compared with the traditional convolution method. More importantly, it is a general and useful tool for life distribution analysis of stochastic degradation with random jumps in multi-dimensional cases.

31 3.1 Introduction

Engineering systems usually deteriorate and lose their intended functionality due to wear, fatigue, erosion, corrosion and aging. The continuous deteriorating process commonly experiences sporadic jumps due to discrete damages. Stochastic processes are typically used to represent the inherent statistical uncertainty of a degradation process, e.g., compound Poisson process, gamma process, Wiener process. However, there is a lack of research on using a single stochastic process to describe degradation with random jumps. Degradation with random jumps is a process of stochastically continuous degradation with sporadic jumps that occur at random times and have random sizes. In this chapter, we intend to model the overall change volume of degradation with random jumps using one stochastic process based on the characteristics of Levy´ subordinators, the class of non-decreasing Levy´ processes. Based on the inverse Fourier transform, we derive a new closed-form reliability function and probability density function (pdf) for lifetime of a component or a system subject to a degradation process with random jumps. The reliability function is constructed and represented by a certain Levy´ measure corresponding to a certain Levy´ degradation process.

For systems subject only to sporadic jump damages, a compound Poisson process, a stochastic process with independent and identically distributed (i.i.d.) jumps that occur according to a Poisson process, is one of the appropriate candidates to model the cumulative damages. For degradation due to wear only, a gamma process and a Wiener process are good candidates to model a wear process. The gamma process is suitable for

32 modeling degradation that progresses in one direction due to its property of independent and nonnegative increments. A Wiener process is appropriate for modeling degradation that changes non-monotonically because it can have non-negative and negative increments alternately.

In practice, however, few systems experience a pure sporadic jump damage process or degradation only. Due to random covariates, a degradation process is typically impacted by sporadic jump damages. By considering degradation with random jumps, a typical approach to calculate reliability is using convolution formula [158]. However, when the wear process has different probability laws from the random jump damages, the calculation becomes less straightforward. For example, when we use a Wiener process

(or a gamma process) for wear and a compound Poisson process for sporadic jumps with normally-distributed jump sizes (or gamma-distributed jump sizes), it is straightforward to derive the reliability function by using convolution; however, the calculation becomes more complex when we consider a Wiener degradation process with gamma jumps. [197] used a gamma process to model wear and a Poisson process with jump sizes following a peaks-over-threshold distribution to model random loads, and the computation of reliability is relatively extensive. In addition, the traditional gamma-Poisson-based models may not be suitable enough to ﬁt the general degradation data, especially when there are complex jump mechanisms that cannot be well described by gamma or Poisson laws.

In order to overcome the aforementioned problems, in this chapter, we propose to model a nondecreasing degradation process with random jumps using a single Levy´ subordinator. In our model, we can specify different Levy´ measures to describe different

33 jump mechanisms in degradation, which enables our methods to ﬁt many different types of degradation data sets. By using the inverse Fourier transform, we further derive the closed-form reliability function and pdf of lifetime for a system or a component subjec- t to a degradation process with random jumps, represented by the Levy´ measure. The calculation for reliability is simple enough to be implemented in practice. More importantly, based on mathematical theories in multi-dimensional Levy´ measures, our model in this chapter provides a new framework to analyze multi-degradation processes in multi- component systems.

The organization of the chapter is as follows. Section 3.2 begins with the key no- tions of the general Levy´ process, and then introduces the special cases of Levy´ processes typically used in the literature. In Section 3.3, we derive the reliability function and pdf of lifetime for systems subject to degradation with random jumps described by Levy´ subordinators, based on the Fourier inversion theorem. Section 3.4 studies the reliability of temporally homogenous gamma degradation with different random jumps, a special case of Levy´ subordinators. Numerical examples are illustrated in Section 3.5, and conclusions are given in Section 3.6.

Notation

• Euclidean space: RK ,K ∈ N

K P • Inner product: hx, yi = xiyi i=1

K 1/2 1/2 P 2 • Euclidean norm: |x| = hx, xi = xi i=1

• Borel probability measures: M RK

34 • Borel σ-algebra: B RK

• Levy´ process: X˜ (t)

˜ • Levy´ subordinator: Xs (t)

• Characteristic function: φ

• Levy´ measure: ν

• Levy´ symbol: η

• Standard Brownian motion or Wiener process: B (t)

• Temporally homogeneous gamma process: G (t)

• Compound Poisson process: C (t)

• Indicator function: Ix∈S

• Convolution of ﬁnite measures: µ1 ∗ µ2

3.2 Preliminaries of Levy´ Processes

Levy´ processes are stochastic processes whose increments in nonoverlapping time intervals are independent and stationary in time. Their importance in modelling degradation processes stems from [17, 168]: 1) they are analogues of random walks in continuous time; 2) they form special subclasses of Markov processes, for which the analysis is much simpler and provides a valuable guidance for the general case; 3) they are the simplest

35 examples of random motion whose sample paths are right-continuous and have a number

(at most countable) of random jump discontinuities occurring at random times, on each

ﬁnite time interval; and 4) they include a number of important processes as special cases,

such as Wiener, compound Poisson, negative binomial, gamma, inverse Gaussian, recipro-

cal gamma, positive hyperbolic, positive hyperbola, positive stable, and positive tempered

stable processes, etc. Therefore, Levy´ process can serve as an important tool for the study

of degradation-based reliability theory. In this section, we introduce Levy´ processes along

with their properties and characteristics on Euclidean space, where the increments can be

positive or negative.

3.2.1 Characteristics

To make our model general, and provide a framework for multi-degradation process-

es, we introduce Levy´ processes on Euclidean space. Deﬁnition 3.1 ([17]). X˜ (t) , t ≥ 0 is a Levy´ process deﬁned on a probability space

(Ω, F,P ) , Ω ∈ RK , F ∈ B RK , P ∈ M RK , if

• X˜ (0) = 0 with probability of 1;

˜ • X (t) has independent and stationary increments: for n ∈ N and 0 ≤ t1 < t2 <

˜ ˜ ··· tn+1 < ∞ , the random variables X (ti+1) − X (ti) , 1 ≤ i ≤ n are indepen-

dent and the distribution of X˜ (s + t) − X˜ (s) does not depend on s;

• X˜ (t) is stochastically continuous: for all ε > 0, s > 0,

lim P |X˜ (t) − X˜ (s) | > ε = 0. t→s

36 Characteristic functions are a primary tool for analysis when the distributions have

no analytic forms, especially for Levy´ processes. On Euclidean space, let φX (u) =

R ihu,xi ihu,Xi RK e PX (dx) = E e denote the characteristic function of a random variable √ K X, where PX is the distribution function of X, i = −1 and u ∈ R . More generally, if

µ ∈ M RK , the set of all Borel probability measures on RK , then

Z ihu,xi φµ (u) = e µ (dx) . RK

Characteristic functions have many useful properties, and readers can refer to [137] for more details. One important property that can be used to analyze the sum of independent variables is described in Deﬁnition 3.2.

K Definition 3.2 ([17]). The convolution µ of two finite measures µ1 and µ2 on R , denoted by µ = µ1 ∗ µ2, is a measure defined by

ZZ K µ (B) = Ix+y∈Bµ1 (dx) µ2 (dy) ,B ∈ B R , RK ×RK where B RK is the Borel σ-algebra on Euclidean space.

If X ∼ µ1,Y ∼ µ2, and X and Y are independent, then X + Y ∼ µ , and

φX+Y (u) = φX (u) φY (u) , which implies that the characteristic function of the sum of independent random variables is the product of the characteristic functions of individual random variables. Characteristic functions of Levy´ processes are characterized by Levy´ measures or Levy´ symbols. Next we give the deﬁnition of Levy´ measure, and a Levy´ symbol can be represented by a Levy´ measure.

37 Deﬁnition 3.3 ([17]). A measure ν on RK is a Levy´ measure if Z min{1, |x|2}ν (dx) < ∞, ν ({0}) = 0. RK Based on the Levy´ Khintchine formula, Levy´ process X˜ (t) has a speciﬁc form for

its characteristic function. More precisely, for all t ≥ 0, x ∈ RK ,

ihu,X˜(t)i tη(u) φX˜(t) (u) = E e = e , (3.1)

where Z 1 ihu,xi η (u) = ihb, ui − hu, aui + e − 1 − I|x|<1ihu, xi ν (dx) 2 RK is the Levy´ symbol, in which ν is the Levy´ measure, b is a constant on RK , and a is a positive deﬁnite symmetric K ×K matrix. Levy´ measure is the most important element of a Levy´ process, in a sense that if we specify a Levy´ measure, we can get the corresponding

Levy´ process and its characteristic function.

Lemma 3.1 ([168]). Let K=1. A Levy´ process is a subordinator if and only if a = 0,

R ν (dx) = 0, R min{1, x}ν (dx) < ∞, and the drift b − R xν (dx) ≥ 0. (−∞,0) R+ 0

Levy´ subordinators form the class of nondecreasing Levy´ processes, taking values

in R+ ≡ (0, ∞). Based on (3.1) and Lemma 3.1, a one-dimensional Levy´ subordinator

˜ Xs (t) has the characteristic function

˜ φ (u) = E eiuXs(t) = etηs(u), X˜s(t)

where Z ∗ iux ηs (u) = ib u + e − 1 ν (dx) , R+

∗ R is the Levy´ symbol, and b = b − 0

38 3.2.2 Special Cases of Levy´ Processes

Levy´ processes are stochastic processes with independent and stationary increments over time. Some special Levy´ processes have been widely used to model degradation processes in the literature, such as Wiener process and gamma process for wear, and compound Poisson process for pure jump damages. The Levy´ measures and Levy´ symbols for these common special cases are introduced in this section.

3.2.2.1 Linear Process

When a = ν = 0, b 6= 0, the Levy´ symbol in (3.1) becomes η (u) = ihb, ui, and the

ithb,ui ˜ characteristic function in (3.1) is φX˜(t) (u) = e , indicating that X (t) = bt, where b is a constant. Therefore, X˜ (t) is a deterministic linear process, which is not suitable for modeling stochastic degradation processes.

3.2.2.2 Brownian Motion/Wiener Process

1 When a 6= 0, b 6= 0, ν = 0, the Levy´ symbol becomes η (u) = ihb, ui − 2 hu, aui

1 t[ihb,ui− 2 hu,aui] and φX˜(t) (u) = e , which is the characteristic function of Brownian motion

with drift b. The case a = 1, b = 0, ν = 0 is usually called standard Brownian motion or

Wiener process (B (t) , t ≥ 0) , which has a Gaussian density

1 |x|2 − 2t ρt (x) = K e . (2πt) 2

Wiener process and Brownian motion with drift are not suitable for modeling monotoni-

cally increasing wear processes, because their increments are not always positive.

39 3.2.2.3 Temporally Homogeneous Gamma Process

When a = 0, b 6= 0, ν 6= 0, if ν is a ﬁnite measure, we have

Z η (u) = ihb∗, ui + eihu,xi − 1ν (dx) , RK

∗ R where b = b − 0<|x|<1 xν (dx). Next, we ﬁnd the special form for ν to obtain the temporally homogeneous gamma process.

For a gamma process on R ≡ R1, if the shape parameter α (t) = αt, t ≥ 0 (i.e., the second condition in Deﬁnition 3.1 is satisﬁed), it is a temporally homogeneous gamma

βαtxαt−1e−βx process G(t). G (t) has a density fG(t) = Ga (x|αt, β) = Γ(αt) , x > 0, t ≥ 0. Then

the characteristic function of G (t) can be expressed as

αt Z ∞ −βx β β iux e φG(t) (u) = = exp αt ln = exp αt e − 1 dx . β − iu β − iu 0 x

Therefore, the temporally homogeneous gamma process is a special case of Levy´ process, and its Levy´ measure is ν(dx) = αx−1e−βxdx, and Levy´ symbol is η (u) =

R ∞ iux e−βx ∗ α 0 (e − 1) x dx, with a = 0, b = 0. The temporally homogeneous gamma pro-

cess is a Levy´ process that is always positive and strictly increasing, and it is suitable for

modeling strictly increasing wear processes with a linear mean path, αt/β.

3.2.2.4 Compound Poisson Process

For a Poisson process with parameter λ, N (t) ∼ Poisson(λt), and P (N(t) = n) =

n e−λt(λt) ∗ n! , for n = 0, 1, 2, ··· . Let (J (n) , n ∈ N(t)) be the jump size described by a

sequence of i.i.d. random variables taking values in R with distribution µJ , which is

40 independent of N(t). The compound Poisson process C (t) is deﬁned as follows:

C (t) = J ∗ (1) + ··· + J ∗ (N (t)) .

Based on Deﬁnition 3.2, we obtain the characteristic function of compound Poisson process:

∞ n ! ∞ n iu P J∗(k) −λt iuC(t) X X e (λt) n φ (u) = E e = P (N (t) = n)E e k=1 = φ (u) C(t) n! J n=0 n=0 Z iux = exp (λt (φJ (u) − 1)) = exp λt e − 1 µJ (dx) . R Therefore, for a compound Poisson process C (t), the Levy´ measure is ν(dx) =

λµJ (dx), and the Levy´ symbol is

Z iux ηC (u) = e − 1 λµJ (dx) . R

The sample paths of C (t) are piecewise constant on ﬁnite intervals with jump dis-

continuities at random times. It is suitable for modeling pure jump damages.

3.3 Life Distribution Analysis Based on Levy´ Subordinators

˜ We use the Levy´ subordinator Xs(t) to represent the monotonically nondecreasing

volume of degradation with random jumps up to time t. A component or a system fails

˜ when Xs(t) exceeds a failure threshold x, assuming that it subjects to one degradation

˜ process that begins with Xs(0) = 0. To simplify the formula, we assume the failure

threshold is a constant, and it is straightforward to extend the model when the failure

threshold is a random variable.

41 The lifetime of the system is deﬁned as the ﬁrst passage time of the stochastic pro-

cess:

˜ Tx = inf{t : Xs(t) > x}.

˜ Since Xs(t) is nondecreasing, we have

˜ {Tx ≥ t} ≡ {Xs(t) ≤ x}.

Then the reliability function can be deﬁned as

R (t) = P (T ≥ t) = P X˜ (t) ≤ x = F (x) . x s X˜s(t) (3.2)

The relationship in (3.2) holds for a broad class of stochastic processes that have monotonically nondecreasing paths. In this section, we present a method based on the inverse Fourier transform to derive the reliability function for systems subject to a degradation process with jumps that can be described by a Levy´ subordinator. Although the pdf of Levy´ process is not readily available, we have the expression of its characteristic function. Since there is a one-to-one correspondence between the cumulative distribution function (cdf) and the characteristic function, we can obtain one of them if the other one is known. Based on the Fourier inversion theorem, Shephard [173] provided the following remarkable theorems describing the cdf as the function of φ(u) for a random variable.

Lemma 3.2 ([173]). If the probability density function f and the characteristic function

φX (u) are integrable in the Lebesgue sense, then under the assumption that the mean for the random variable of interest exists, the following equality holds:

1 1 Z ∞ e−iux FX (x) = − ∆ φX (u) du, 2 2π 0 u iu

42 where ∆ ρ (u) = ρ (u) + ρ (−u). u

The following Lemma 3.3 is the multivariate generalization of Lemma 3.2.

Lemma 3.3 ([173]). If the probability density function f and the characteristic function

φX (u) are integrable in the Lebesgue sense, then under the assumption that the mean for

the multi-dimensional random variable of interest exist, the following equality holds:

K Z ∞ Z ∞ −ihu,xi (−2) e ∗ K ··· ∆ ∆ ··· ∆ φX (u) du = z (x) , (2π) 0 0 u1 u2 uK iu1iu2 ··· iuK

where

∗ K K−1 z (x) = 2 F (x1, ··· xK ) − 2 (F (x2, x3, ··· , xK ) + ··· + F (x1, ··· , xK−2, xK−1))

K−2 K + 2 (F (x3, x4, ··· , xK ) + ··· + F (x1, ··· , xK−3, xK−2)) + ··· + (−1) .

The mean’s existence is a sufﬁcient but not a necessary condition for the results to

hold. It can be removed by using the principal value of the integral [87]. Lemma 3.2 turns

out to be a special case of Lemma 3.3 that deals with multi-dimensional variables. For

an example of two-dimensional variables, if we know the characteristic function φX (u),

∗ we can get the expression of z (x1, x2) = 4F (x1, x2) − 2F (x1) − 2F (x2) + 1 based on

Lemma 3.3. If we know the characteristic function of each variable, φX1 (u1) and φX2 (u2),

we can have F (x1) and F (x2) based on Lemma 3.2. Finally, we can solve for the joint

distribution function F (x1, x2) of X1 and X2. Integration rules for the computation of the

multivariate distribution function are described in [174].

In this chapter, we focus on one-dimensional Levy´ degradation processes. When

˜ K = 1, for all t ≥ 0, u ∈ R, the characteristic function of a Levy´ subordinator Xs(t) is

43 ˜ available. For Xs(t), we derive the reliability function and pdf of lifetime in the following corollaries, respectively.

Corollary 3.1. For systems subject to stochastic degradation with random jumps that

can be described by Levy´ subordinators, assuming the failure threshold value is x, the reliability function represented by the Levy´ measure is

1 1 Z ∞ e−iux Z R (t) = − ∆ exp t ib∗u + eiux − 1ν (dx) du, (3.3) u 2 2π 0 iu R+

∗ R where b = b − 0

Corollary 3.2. For systems subject to stochastic degradation with random jumps that can

be described by Levy´ subordinators, assuming the failure threshold value is x, the pdf of

lifetime represented by the Levy´ measure is

∂R (t) f (t) = − ∂t   −iux ∗ R iux Z ∞ e exp t ib u + (e − 1)ν (dx) (3.4) 1 R+   = ∆ −1 du, 2π u   0 iu ib∗u + R (eiux − 1)ν (dx) R+

∗ R where b = b − 0

For systems subject to degradation with random jumps that can be described by Levy´ subordinators, we can ﬁrst specify a certain Levy´ measure and then calculate the reliability function and pdf using (3.3) and (3.4). [112] gave explicit results for wear processes in

Markovian environment, which requires to use multi-inverse algorithms to calculate. Al- though they are not explicit, our results in (3.3) and (3.4) can be computed comparatively cheap based on [173]. The advantages of our results are twofold: 1) they are general

44 because we can specify different Levy´ measures to ﬁt different types of degradation data

sets, while the models in the literature become special cases of our models, and 2) they

provide a methodology to deal with complex random jumps in degradation processes, e.g.,

when the distributions of jumps size are not additive.

3.4 Life Distribution Analysis for Temporally Homogeneous Gamma

Process with Random Jumps

To demonstrate the advantages of our models, we present the life distribution analysis

for a degradation process represented by a sum of a temporally homogeneous gamma pro-

cess and a compound Poisson process. For this case, we can get a closed-form reliability

function using the traditional convolution approach, which enables the direct comparison

with our proposed method. Nevertheless, we can also choose different Levy´ measures to

illustrate our general model, such as a Levy´ measure for a sum of a positive tempered stable process and a compound Poisson process. In this case, however, no comparison can be directly made, because the traditional convolution approach cannot generate a closed- form reliability function considering that the positive tempered stable process does not have a closed-form distribution.

45 3.4.1 Reliability Function Using Traditional Convolution Approach

We ﬁrst present the reliability function derived from the traditional convolution approach for the temporally homogeneous gamma process with random jumps. If a degradation process with sporadic jumps can be well described by the sum of a temporally homogeneous gamma process and a compound Poisson process that are independent, the traditional convolution approach provides the reliability function:

 N(t)  ˜ X ∗ R (t) = P Xs (t) ≤ x = P G (t) + Ji ≤ x i=0

∞  N(t)  X X ∗ (3.5) = P G (t) + Ji ≤ x |N (t) = n P (N (t) = n) n=0 i=0 ∞ n ! n X X e−λt(λt) = P G (t) + J ∗ ≤ x . i n! n=0 i=0

n ∗ ∗ ∗ P ∗ If the jump size follows a gamma distribution, Ji ∼ Gamma(α , β ), then Ji ∼ i=0 Gamma(nα∗, β∗). If the scale parameter of G (t) is the same as β∗, i.e., β = β∗, then n P ∗ ∗ G (t) + Ji ∼ Gamma(αt + nα , β). The reliability function for this special case is i=0

∞ n X Γ(αt + nα∗, xβ)e−λt(λt) R (t) = 1 − , (3.6) Γ(αt + nα∗) n! n=0

∗ R ∞ αt+nα∗−1 −y ∗ R ∞ αt+nα∗−1 −y where Γ(αt + nα ) = y=0 y e dy , Γ(αt + nα , xβ) = y=xβ y e dy, and x is the threshold value.

We can see that (3.6) is derived based on two assumptions: 1) the jump size follows

∗ a gamma distribution, and 2) the scale parameters of G(t) and Ji are the same. When

β 6= β∗, it is still manageable to derive the reliability function using the convolution of

46 ∗ gamma distributions [106]. However, if Ji follows a different distribution than a gamma distribution (such as inverse Gaussian distribution, Levy´ distribution, Pareto distribution,

or stable distribution), it becomes complex to calculate the reliability function in (3.5).

Our approach in Corollary 3.1 is capable of dealing with these cases by using the Levy´ measure.

3.4.2 Reliability Function Using Levy´ Measures

In this section, we use our new approaches in Corollaries 3.1 and 3.2 to derive the reliability function and pdf for a temporally homogeneous gamma process with random jumps. As given in Section 3.2.2.3, Levy´ measure for a temporally homogeneous gamma

−1 −βx process is ν1(dx) = αx e dx, and Levy´ measure for a compound Poisson process

is ν2(dx) = λµJ (dx). Then for a Levy´ subordinator that is sum of a temporally homo-

geneous gamma process and a compound Poisson process, the characteristic function is

derived to be

Z αe−βx φ (u) = φ (u) = exp t eiux − 1 + λµ0 dx , X˜s(t) G(t)+C(t) J R+ x

0 where µJ is the pdf of the jump size. Therefore, we can model this Levy´ subordinator

−1 −βx by specifying its Levy´ measure to be ν = ν1 + ν2 = αx e dx + λµJ (dx). Based on

Corollary 3.1, the reliability function is

Z ∞ −iux Z −βx 1 e iux αe 0 R (t) = − ∆ exp t e − 1 + λµJ dx du. u 2 0 2πiu R+ x (3.7)

47 Based on Corollary 3.2, the pdf of lifetime is ∂R (t) f (t) = − ∂t   αe−βx ∞ −iux R iux 0 (3.8) Z e exp t R (e − 1) x + λµJ dx =  + du. ∆  −1  0 u 2πiu R (eiux − 1) αe−βx + λµ0 dx R+ x J The results in (3.7) and (3.8) can be applied to the jump size following a general

distribution µJ deﬁned on [0, ∞). In the following, we derive the reliability function and

pdf for three different jump types.

3.4.2.1 Gamma-distributed Jump Sizes

If the jump size follows a gamma distribution,

βα∗ xα∗−1e−β∗x µ0 = Ga (x|α∗, β∗) = , x > 0, J Γ(α∗) then the characteristic function for the compound Poisson process is Z α∗ α∗−1 −β∗x iux β x e φC(t) (u) = exp λt e − 1 ∗ dx R+ Γ(α ) Z α∗ α∗−1 −β∗x Z α∗ α∗−1 −β∗x iux β x e β x e = exp λt e ∗ dx − ∗ dx R+ Γ(α ) R+ Γ(α ) Z α∗ α∗−1 −β∗x iux β x e = exp λt e ∗ dx − 1 R+ Γ(α ) ∗ !! β∗ α = exp λt − 1 . β∗ − iu Then the reliability function in (3.7) is

∗ !!! 1 Z ∞ e−iux β αt β∗ α R (t) = − ∆ exp λt ∗ − 1 du. (3.9) 2 0 u 2πiu β − iu β − iu

(3.9) is a general formula for reliability function of systems subject to degradation

described by the sum of a gamma process and a compound Poisson process with gamma-

distributed jumps, regardless of β = β∗ or not, while (3.6) is only valid for the case of

48 β = β∗. The pdf of lifetime in (3.8) is derived to be ∂R (t) f (t) = − ∂t ∗  ∗ α  αt λt β −1 −iux β β∗−iu Z ∞  e β−iu e  = ∆  du.  α α∗ −1  0 u  β β∗  2πiu ln β−iu + λ β∗−iu − 1

3.4.2.2 Levy-distributed´ Jump Sizes

If the jump size follows a different distribution than a gamma distribution, we can also derive the reliability function and pdf using (3.7) and (3.8). When the jump size follows a Levy´ distribution, the pdf is given as [16]  q  ξ exp(− ξ )  2π 2(x−$)  3 for x > $ > 0 0 2 µJ (x; $, ξ) = (x − $)    0 otherwise. Levy´ distribution is a continuous probability distribution of a non-negative random variable when $ > 0. It has different probability laws from a gamma distribution, leading to complex calculation in the convolution approach. Since the characteristic function of

√ Levy´ distributed variable is eiu$− −2iuξ [92], the reliability function of a gamma process

with additional Levy-distributed´ jumps is derived from (3.7) to be ! 1 Z ∞ e−iux β αt √ R (t) = − ∆ exp λt eiu$− −2iuξ − 1 du. 2 0 u 2πiu β − iu

The pdf of lifetime in (3.8) is ∂R (t) f (t) = − ∂t  αt √  β λt eiu$− −2iuξ−1 ∞ −iux ( ) Z e β−iu e = ∆  du.  α √ −1  0 u β iu$− −2iuξ 2πiu ln β−iu + λ e − 1

49 3.4.2.3 Inverse Gaussian-distributed Jump Sizes

The pdf of the inverse Gaussian distribution is

 r 2  ϑ ϑ(x − ς)  exp{− } for x > 0 0 2πx3 2ς2x µJ (x; ς, ϑ) =   0 otherwise, where ς > 0 is the mean, and ϑ > 0 is the shape parameter. It has different probability laws from a gamma distribution. Since the characteristic function of an inverse Gaussian-

q 2 ϑ 1− 1− 2iuς distributed variable is e ς ϑ , the reliability function of a gamma process with additional inverse Gaussian-distributed jumps in (3.7) is

∞ αt q 2 !!! 1 Z e−iux β ϑ 1− 1− 2iuς R (t) = − ∆ exp λt e ς ϑ − 1 du. 2 0 u 2πiu β − iu

The pdf of lifetime in (3.8) is

∂R (t) f (t) = − ∂t √   2   ϑ 1− 1− 2iuς ς ϑ αt λte −1 β ∞  −iux  Z  e β−iu e    = ∆ −1 du. u  q 2 !!  0  α ϑ 1− 1− 2iuς   β ς ϑ  2πiu ln β−iu + λ e − 1

Beyond Levy´ measures covered in this section, we can specify additional Levy´ measures for model construction in order to ﬁt the corresponding degradation data. Some inter-

−2κ −κ−1 1 2 δγ κx exp(− 2 γ x) esting Levy´ measures have been studied in [25], such as ν(dx) = Γ(κ)Γ(1−κ) dx, x, δ > 0, 0 < κ < 1, γ ≥ 0 for the positive tempered stable process PTS(κ, δ, γ).

50 3.5 Numerical Examples

We consider the crack growth process in a device, which is subject to degradation

due to fatigue and a variety of overloads that can occur in manufacturing, deployment, and

˜ operation phases. We use a Levy´ subordinator Xs(t) to represent the growth of a crack at time t, speciﬁcally, a temporally homogeneous gamma process with random jumps.

−1 −βx Then the Levy´ measure is ν = αx e dx + λµJ (dx). In particular, we consider three different distributions to model the jump size: gamma, Levy´ and inverse Gaussian. The speciﬁc values for the parameters are given in Table 3.1. A device fails when the crack length exceeds the threshold x.

Table 3.1: Parameter values for models in Chapter 3

Parameters Values Parameters Values α 5 $ 1 β 0.8 ξ 0.002 λ 3 ς 1 α∗ 1 or 10 ϑ 1 β∗ 0.8 or 15 x 50

51 1

0.9

0.8

0.7

0.6

0.5

Reliability 0.4

0.3

0.2 Traditional method:beta*=beta=0.8 0.1 New method:beta*=beta=0.8 New method:beta*=15 0 0 1 2 3 4 5 6 7 8 9 10 t

Figure 3.1: Reliability function for gamma degradation with additional gamma jumps

0.9

0.8

0.7

0.6

0.5

Reliability 0.4

0.3

0.2 Gamma-type 0.1 Levy-type Inverse Gaussian-type 0 0 1 2 3 4 5 6 7 8 9 10 t

Figure 3.2: Reliability function for gamma degradation with three jump types

52 0.5 Gamma-type 0.45 Levy-type Inverse Gaussian-type 0.4

0.35

0.3

0.25 Pdf

0.2

0.15

0.1

0.05

0 0 1 2 3 4 5 6 7 8 9 10 t

Figure 3.3: Pdf of lifetime for gamma degradation with three jump types

5 Gamma-type 4.5 Levy-type Inverse Gaussian-type 4

3.5

2.5

Hazard rate 2

1.5

0.5

0 0 1 2 3 4 5 6 7 8 9 10 t

Figure 3.4: Hazard rate for gamma degradation with three jump types

53 Figure 3.1 shows the reliability function over time of devices subject to a gamma

degradation with additional gamma-distributed jumps. When the parameter β∗ = β = 0.8, both the traditional convolution approach in (3.6) and our proposed model in (3.9) can solve the problem, showing the same curve of R(t). When β∗ 6= β, the reliability curve is provided by our model in (3.9), and the convolution approach becomes complex in this case.

Figure 3.2 shows the reliability functions over time of devices subject to a gamma degradation with three different jump distributions. It demonstrates that for Levy´ and inverse Gaussian distributed jump sizes, which have different probability laws from gamma, we can readily calculate the reliability by using our proposed model. Figure 3.3 and

Figure 3.4 illustrate the pdfs and hazard rates for the lifetime of the devices. As expected, we can see that the hazard rates increase over time for all three cases with nondecreasing degradation processes.

3.6 Conclusions

One of the challenging aspects in reliability analysis is how to formulate the reliability function from a degradation process that a system or a component experiences.

In this chapter, we presented a novel model concerning the stochastic mechanism of a complex degradation process that also subjects to random jumps. Based on inverse Fourier transforms, the reliability function and pdf of lifetime were derived. Our model is general because we can specify different Levy´ measures to ﬁt different types of degradation data

54 sets, and the models in the literature become special cases of our model. In addition, by providing a methodology to deal with complex random jumps in degradation processes, our method can solve the problems that the traditional convolution method cannot readily solve, i.e., when the distribution of jumps sizes is not additive. Our new method provides a convenient and general way to evaluate the system reliability. The analysis for reliability is simple enough to be implemented in practice.

More importantly, the model provides a framework for reliability analysis of multi- degradation processes in multi-component systems. To derive the reliability function for multi-Levy´ degradation processes on RK , we need to construct multi-dimensional Levy´ measures. If the multi-degradation processes are dependent, the construction of the multi- dimensional Levy´ measures can refer to Levy´ copula theory [55, 109], which will be explored in Chapter 6. In order to apply the model to degradation data analysis, statistical inference on Levy´ measures is another research direction, which will be developed in

Chapter 7.

Levy´ subordinators studied in this chapter are a class of homogeneous Markov processes with a linear mean path over time t. For non-linear degradation paths, special non- homogeneous subordinators (e.g., non-homogeneous gamma processes) have been studied in the literatures [154, 201], which are essentially non-decreasing additive processes. Our model can be readily extended to the class of non-homogeneous subordinators when the

Levy´ measures are available.

55 Chapter 4

Levy´ Driven Non-Gaussian Ornstein-Uhlenbeck Processes for Degrada- tion-based Reliability Analysis

In this chapter, we further explore Levy´ subordinators and extend the model using

Levy´ driven non-Gaussian OU processes. To obtain explicit results of reliability function and lifetime moments, we propose to use Fokker-Planck equations for both Levy´ subordinators and their corresponding OU processes. The most important advantage of the models stems from the ﬂexibility of such processes in modeling stylized features of degradation data series such as jumps, linearity/nonlinearity, symmetry/asymmetry, and light/heavy tails. Numerical experiments are used to demonstrate that our general models perform well and are applicable for analyzing a great deal of degradation phenomena.

More importantly, they provide us a new methodology to deal with degradation processes under dynamic environments.

4.1 Introduction

Chapter 3 gave a new closed-form of reliability function for degradation described by Levy´ subordinators, a class of non-decreasing Levy´ processes, which is consistent with some observed physical degradation phenomena. The advantages of using Levy´ subordinators were also demonstrated. With independent and stationary increments, however,

56 all Levy´ processes have linear mean paths, i.e., the mean of Levy´ processes is linear with respect to (w.r.t) time. To overcome the limitation from the linear mean property, Gaussian

OU processes driven by a Wiener process have been developed for survival analysis [1].

However, the assumptions of no jumps and Gaussian distribution (symmetric and light- tailed, i.e., all the positive moments are ﬁnite) are not consistent with many degradation phenomena. In this chapter, to ﬂexibly handle stylized features of degradation data series such as complex jumps, linearity/nonlinearity, symmetry/asymmetry, and light/heavy tails, we propose to model stochastic degradation with independent or dependent increments using Levy´ subordinators or OU processes driven by Levy´ subordinators (i.e., non-Gaussian

OU processes), respectively. To the best of our knowledge, non-Gaussian OU processes have not been used in degradation modeling. In fact, it is nontrivial to obtain a closed- form distribution function for an OU process driven by a Levy´ process. For these general stochastic degradation processes, we construct systematic procedures to derive the explicit expressions for reliability function and lifetime moments using Fokker-Planck equations.

Our proposed new models offer a general approach for modelling stochastic degradation with complex jump mechanisms using a broad class of Levy´ processes and their functional extensions.

Fokker-Planck equations provide us a way to analyze probability laws for stochastic processes, especially for those without closed-form distributions. Fokker-Planck equations represent the fascinated research work from mathematicians in the ﬁeld of stochastic processes. As the partial differential equations (PDE) of the probability density function- s, they describe the time evolution of probability density for stochastic processes, and

57 are thus useful in quantifying random phenomena, such as propagation of uncertainty.

We can ﬁnd the Fokker-Planck equations for Weiner-based processes in many textbooks

[117, 165]. For such processes, it is straightforward to derive the Fokker-Planck equations, because of the absence of jump mechanisms. However, for Levy-based´ processes, explicit results of Fokker-Planck equations cannot be easily derived, due to the difﬁculty in obtaining the expression for the adjoint operators of the inﬁnitesimal generators associated with Levy-based´ processes [187]. Some interesting results of Fokker-Planck equations for Levy-based´ processes are in [61, 169, 187], and [164] gave a numerical algorithm to calculate the mean exit time for Levy´ systems.

In this chapter, we consider a single degradation process with random jumps in a system, i.e., a process of stochastically continuous degradation with sporadic jumps that occur at random times and have random sizes. The system fails when the degradation process hits a boundary. We ﬁrst use Levy´ subordinators, a class of Levy´ processes with non-decreasing sample paths, to model the evolution of degradation with linear mean paths (Figure 4.1: Wiener process (0, 1); gamma process (20, 20); compound Poisson process with jump density 2 and jump size following gamma distribution (1, 10); Levy´ subordinator a: inverse Gaussian process (0.5, 0.1); Levy´ subordinator b: positive stable process (0.9); Levy´ subordinator c: positive stable process (0.92)). We then propose a functional extension of Levy´ subordinators, non-Gaussian OU processes (OU processes driven by Levy´ subordinators), to model degradation processes with nonlinear mean paths

(Figure 4.2: non-Gaussian OU process a: OU process driven by inverse Gaussian process

(0.5, 0.1) and α = 0.2; non-Gaussian OU process b: OU process driven by positive stable

58 process (0.9) and α = 0.1; non-Gaussian OU process c: OU process driven by positive

stable process (0.92) and α = 0.1).

Wiener process Gamma process Compound Poisson process X(t) X(t) X(t) −0.5 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

t t t

Levy subordinator a Levy subordinator b Levy subordinator c X(t) X(t) X(t) 0 5 10 15 0 2 4 6 8 10 12 14 0 2 4 6 8 10

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

t t t

Figure 4.1: Sample paths of Levy´ processes

Figure 4.1 shows sample paths of three commonly-used Levy´ processes (Wiener process, gamma process, and compound Poisson process), and three Levy´ subordinators with different jump mechanisms speciﬁed by different Levy´ measures. Figure 4.2 illustrates sample paths of OU processes driven by Levy´ subordinators, a class of non-Gaussian OU processes, and they are the solutions of stochastic differential equations (SDE) driven by

Levy´ subordinators. The sample data are simulated using R(YUIMA) [33]. In practice, many degradation processes in highly reliable systems have similar paths to those in Figure

4.2: they increase slowly at the early stage, but increase sharply when the degradation is

59 accumulated. In these cases, the linear mean path of a Levy´ subordinator is not appropriate

to represent the degradation.

Non−Gaussian OU a Non−Gaussian OU b Non−Gaussian OU c Y(t) Y(t) Y(t) 0 10 20 30 40 0 5 10 15 20 25 30 35 0 10 20 30

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

t t t

Figure 4.2: Sample paths of non-Gaussian OU processes

For both general Levy´ subordinators and non-Gaussian OU processes, the probability distributions are not analytically available. In addition, the analytical derivation is intractable for non-Gaussian OU processes. In this chapter, we tackle these challenges by using the corresponding Fokker-Planck equations, and then derive explicit expressions for reliability function and lifetime moments in terms of Laplace transform. The results are compact enough to compute and evaluate reliability characteristics conveniently. More importantly, by introducing Fokker-Planck equations to stochastic degradation analysis, our work provides a new methodology for reliability analysis of complex degradation phenomenon, such as multi-degradation processes under dynamic environments.

The organization of this chapter is as follows. Section 4.2 begins with the famous

Levy-It´ oˆ decomposition, and then describes the model construction. In Section 4.3, we derive the explicit expressions of reliability function and lifetime moments for systems subject to degradation described by Levy´ subordinators and non-Gaussian OU processes,

60 respectively, based on Fokker-Planck equations. Numerical examples are illustrated in

Section 4.4, and conclusions are given in Section 4.5.

4.2 Preliminaries

4.2.1 Levy-It´ oˆ Decomposition

We begin with the deﬁnition of Poisson random measure from [168]. A random variable J has a Poisson distribution with a mean 0 if J = 0 almost surely (a.s.); and J has a Poisson distribution with a mean +∞ if J = +∞ a.s..

¯ Deﬁnition 4.1 ([168]). Let (Λ, B, ν) be a σ-ﬁnite measure space. Given Z+ =

¯ {0, 1, 2, ··· , +∞}, a family of Z+-valued random variables {J (A): A ∈ B} is called a Poisson random measure on Λ with an intensity measure ν, if the following conditions hold:

• For every A, J(A) has a Poisson distribution with a mean ν(A);

• If A1,A2, ··· An are disjoint, then J(A1),J(A2), ··· J(An) are independent; and

• For every w, J(·, w) is a measure on Λ.

Lemma 4.1 (The Levy-It´ oˆ Decomposition [17]). If X˜(t) is a Levy´ process, then there exist

K b ∈ R , a Brownian motion Ba with a covariance matrix a, and an independent Poisson random measure J on R+ × RK such that, for each t ≥ 0,

Z Z ˜ X(t) = bt + Ba(t) + y J (t, dy) − ν (t, dy) + yJ (t, dy), |y|<1 |y|≥1

61 where ν (t, dy) is the mean of the Poisson random measure J (t, dy).

The intensity measure ν (t, dy) is often called the Levy´ measure. According to the property of independent and stationary increments of Levy´ process, and from Levy-´

Khintchine formula [168], ν (t, dy) = ν (dy) t.

4.2.2 Model Construction

We assume there is a single degradation path with random jumps occurring in a

˜ system. We use a Levy´ subordinator Xs (t) and a non-Gaussian OU process Y (t) to model the degradation evolution with linear and non-linear mean paths, respectively.

˜ Based on Lemmas 3.1 and 4.1, for Levy´ subordinator Xs (t),

Z Z ˜ Xs(t) = bt + y J (t, dy) − ν (t, dy) + yJ (t, dy), (4.1) 0

R where the continuous degradation is modeled by b − 0

We can specify different forms of Levy´ measures in order to model different complex jump mechanisms. If we specify ν(dx) = αx−1e−βxdx for small jumps in an inﬁnitesimal time interval, then the Levy´ subordinator in (4.1) is a temporally homogeneous gamma process (a gamma process with stationary increments) G (t), which has a density fG(t) =

βαtxαt−1e−βx Ga (x|αt, β) = Γ(αt) , x > 0, t > 0. G (t) has an inﬁnite number of small jumps

in a ﬁnite time interval, thus having inﬁnite activity. The inverse Gaussian process has

the similar feature, but with more very small jumps than G (t), since its Levy´ density

ν0(x) = ν(dx)/dx approaches to inﬁnity faster than the gamma process does as x goes

62 to zero. Another case is the positive stable process whose probability distribution is in general unknown in simple form. For big jumps occurring according to the Poisson law, we can specify ν(dx) = λµJ (dx), and then the Levy´ subordinator is a compound Poisson process C (t) with a jump density λ and a jump size distribution µJ . C (t) has a finite number of jumps over a finite time interval, i.e., finite activity. Another example of Levy´ subordinators with finite activity is the negative binomial process. Compared to C (t), in the negative binomial process, the interarrival times of jumps are not exponentially distributed and the variance of the number of jumps in a time interval is larger than the corresponding mean [25].

˜ A non-Gaussian OU process Y (t) is the solution of a SDE driven by Xs(t):

˜ dY (t) = αY (t) dt + dXs (t) . (4.2)

Proposition 4.1. The non-Gaussian OU process resulted from (4.2) is

Z t αt α(t−ξ) ˜ Y (t) = e Y (0) + e dXs(ξ). 0

Proof. If f(t, y) ∈ C1,2, then based on Taylor series, we have

∂f ∂f 1 ∂2f 1 ∂2f ∂2f df = dt + dy + (dt)2 + (dy)2 + dydt. ∂t ∂y 2 ∂t2 2 ∂y2 ∂y∂t

−αt ∂f −αt ∂f −αt ∂2f Let f(t, y) = ye , then ∂t = −αye , ∂y = e , and ∂y2 = 0. We obtain ∂f ∂f df = dt + dy = −αye−αtdt + e−αtdy = e−αtdx, ∂t ∂y and

Z t αt αt −αξ yt = e y0 + e e dxξ. 0

63 Y (0) represents the initial state of the degradation, and we assume Y (0) = 0 a.s. as many new systems have not accumulated degradation when they are ﬁrstly operated. We assume α > 0, which guarantees that the degradation process is non-decreasing. Y (t) is expressed as

Z t α(t−ξ) ˜ Y (t) = e dXs(ξ) 0 Z t Z Z = eα(t−ξ) bdξ + y J(dξ, dy) − ν(dξ, dy) + yJ (dξ, dy) 0 0

b−R yν(dy) 0

the random jumps are modeled by the Poisson random measure R t eα(t−ξ) R yJ (dξ, dy). 0 R+

As illustrated in Figure 4.2, the mean degradation path of Y (t) is exponential w.r.t t, in-

˜ stead of linear of Xs(t).

4.3 Reliability Function and Lifetime Moments

˜ The system fails when the degradation process Xs(t) or Y (t) exceeds a failure thresh-

˜ old x or y. Based on Xs(t), the lifetime and its moments are deﬁned respectively as

˜ n n Tx = inf{t : Xs(t) > x},M(TX , x) = E(Tx ).

The reliability function can be deﬁned as

R (x, t) = P (T ≥ t) = P X˜ (t) ≤ x = F (x) . X x s X˜s(t)

64 n n Based on Y (t), similar deﬁnitions are Ty = inf{t : Y (t) > y},M(TY , y) = E(Ty ),

and RY (y, t) = P (Ty ≥ t) = P (Y (t) ≤ y) = FY (t) (y). For many new systems that

˜ have not accumulated degradation when they are ﬁrstly operated, we have Xs(0) = 0 a.s.,

and

R (x, 0) = P X˜ (0) ≤ x = F (x) = h(x), X s X˜s(0)

∂F ˜ (x) p(x, 0) = Xs(0) = δ(x), ∂x

where h(x) = I[0,∞)(x) is the unit step function (or the Heaviside step function), and δ(x)

is the Dirac delta function. Similarly, we have

RY (y, 0) = P (Y (0) ≤ y) = FY (0) (y) = h(y), ∂F (y) p(y, 0) = Y (0) = δ(y). ∂y

˜ In addition, RX (0, t) = P (T0 ≥ t) = P Xs (t) ≤ 0 = I(−∞,0](t), and RY (0, t) =

P (T0 ≥ t) = P (Y (t) ≤ 0) = I(−∞,0](t).

To obtain expressions of reliability functions and lifetime moments, we need to study

˜ the probability laws of Xs(t) and Y (t). Since there are no closed-form distribution functions for general Levy´ subordinators, it is a challenge to derive the explicit expressions for reliability functions and lifetime moments. As PDEs of probability density functions,

Fokker-Planck equations [187] provide us a way to overcome the challenge in analyzing probability laws for stochastic processes we are interested in, especially for those without closed-form distributions. The Fokker-Planck equation, also known as the Kolmogorov forward equation, describes the time evolution of probability density for stochastic processes.

65 Let L be an operator, and L∗ be the adjoint operator of L, then

Z Z Lf (x)g (x) dx = f (x) L∗g (x) dx. R R

Let p(x, t) be the probability density function for a stochastic process X(t), and the

Fokker-Planck equation is

∂p(x, t) = L∗p(x, t), ∂t where L∗ is the adjoint operator of the inﬁnitesimal generator L of X(t):

E {f (X ) |X = x} − f (x) Lf(x) = lim t+∆t t . ∆t→0 ∆t

Laplace transform of p(x, t) w.r.t. t is deﬁned to be

Z pL(x, ω) = e−ωtp (x, t) dt, ω > 0. R+

Laplace transform of pL(x, ω) w.r.t. x is

Z pLL(u, ω) = e−uxpL (x, ω)dx, u > 0. R+

Lemma 4.2. Let RLL(u, ω) be the Laplace expression of reliability function R(x, t), then

RLL(u, ω) = u−1pLL(u, ω).

Proof. From the deﬁnition of the reliability function, we have

∂R (x, t) p(x, t) = . ∂x

The Laplace transform of p(x, t) w.r.t. t is

Z Z ∂R (x, t) ∂RL (x, ω) pL(x, ω) = e−ωtp (x, t) dt = e−ωt dt = . R+ R+ ∂x ∂x

66 The Laplace transform of pL(x, ω) w.r.t. x is

Z ∂RL (x, ω) Z pLL(u, ω) = e−ux dx = e−uxdRL (x, ω) R+ ∂x R+ Z −ux L L −ux = e R (x, ω) |R+ − R (x, ω) de R+

= uRLL(u, ω).

4.3.1 Results Based on Levy´ Subordinators

˜ For degradation with random jumps described by a Levy´ subordinator Xs(t), we

n derive the explicit expressions of RX (x, t) and lifetime moments M(TX , x) in terms of

Laplace transform, represented by Levy´ measures. Using the procedure similar to [112], the results are presented in Theorems 4.1 and 4.2.

Theorem 4.1. For degradation with random jumps described by a Levy´ subordinator, the

Laplace expression of reliability function is

Z −1 LL −1 ∗ −uy RX (u, ω) = u ω + b u − (e − 1)ν (dy) , R+

where b∗ ≥ 0, ν is the Levy´ measure.

˜ Proof. Let p(x, t) be the probability density function of a Levy´ subordinator Xs(t). Based

˜ on [187], the Fokker-Planck equation for Xs(t) is

∂p(x, t) ∂p(x, t) Z ∂p(x, t) = −b + p(x − y, t) − p(x, t) + Iy∈(0,1)y ν (dy) . (4.4) ∂t ∂x R+ ∂x

67 For (4.4), we do Laplace transform of p(x, t) w.r.t. t for both sides,

ωpL(x, ω) − p (x, 0)

L Z L (4.5) ∂p (x, ω) L L ∂p (x, ω) = −b + p (x − y, ω) − p (x, ω) + Iy∈(0,1)y ν (dy) . ∂x R+ ∂x For (4.5), we do Laplace transform of pL(x, ω) w.r.t. x for both sides, then

ωpLL(u, ω) − 1 Z LL −uy LL LL LL = −bup (u, ω) + e p (u, ω) − p (u, ω) + Iy∈(0,1)yup (u, ω) ν (dy) . R+

∗ R Let b = b − 0

Based on Lemma 4.2, we obtain

Z −1 LL −1 ∗ −uy RX (u, ω) = u ω + b u − e − 1 ν (dy) . R+

Remark 4.1. For (4.4), we do Laplace transform of p(x, t) w.r.t. x for both sides,

h ˜ i Z E e−uXs(t) = pL(u, t) = e−uxp (x, t) dx, R+ then we have L Z ∂p (u, t) L −uy L L L = −bup (u, t) + e p (u, t) − p (u, t) + Iy∈(0,1)yup (u, t) ν (dy) ∂t R+ Z = −b∗u + e−uy − 1ν (dy) pL(u, t). R+ Solving this ordinary differential equation (ODE), we have

h ˜ i Z E e−uXs(t) = pL(u, t) = pL(u, 0) exp t −b∗u + e−uy − 1 ν (dy) . R+

68 Since pL(u, 0) = 1, this is consistent with the characteristic function of Levy´ subor-

dinators.

Before we use Theorem 4.1 to derive the Laplace expression for the moments of

lifetime Tx as Theorem 4.2, we introduce an important relation in Lemma 4.3.

˜ ∂ R x Lemma 4.3. Denote Q(x, t) = − ∂t R(x, t), where R(x, t) = 0 p(v, t)dv, and

˜LL n ∂nQ˜LL(u,ω) ˜LL ˜ Qn (u, ω) = (−1) ∂ωn , where Q (u, ω) is the Laplace expression of Q(x, t).

Let M L(T n, u) be the Laplace expression of lifetime moments, then

L n ˜LL M (T , u) = Qn (u, 0).

Proof. Since Q˜LL(u, ω) = R e−ωtQ˜L (u, t)dt, we have R+ n ˜LL Z n −ωt ˜LL n ∂ Q (u, ω) n ∂ e ˜L Qn (u, ω) = (−1) n = (−1) n Q (u, t)dt ∂ω R+ ∂ω Z = tne−ωtQ˜L (u, t)dt. R+ And as Z M(T n, x) = tnQ˜ (x, t)dt, R+

we obtain Z L n n ˜L ˜LL M (T , u) = t Q (u, t)dt = Qn (u, 0). R+

Theorem 4.2. For degradation with random jumps described by a Levy´ subordinator, the

Laplace expression of lifetime moments is

Z −n L n −1 ∗ −uy M (TX , u) = n!u b u − e − 1 ν (dy) , R+

69 where b∗ ≥ 0, ν is the Levy´ measure.

Proof. The Laplace transform of Q˜(x, t) w.r.t t is Z ˜L −ωt ∂ L Q (x, ω) = − e RX (x, t) dt = h(x) − ωRX (x, ω) . (4.6) R+ ∂t

For (4.6), we do Laplace transform w.r.t. x on both sides, then

˜LL LL −1 Q (u, ω) = −ωRX (u, ω) + u .

From Theorem 4.1, we have Z −1 Q˜LL(u, ω) = −ωu−1 ω + b∗u − e−uy − 1ν (dy) + u−1. R+

From Lemma 4.3, " # ∂nQ˜LL(u, ω) M L(T n , u) = Q˜LL(u, 0) = (−1)n , X n ∂ωn ω=0 where  −1  nn ∗ R −uy o " n LL # ∂ ω + b u − (e − 1)ν (dy) ∂ Q˜ (u, ω) R+ = −u−1 ω  ∂ωn  ∂ωn  ω=0 ω=0  −1  n−1n ∗ R −uy o ∂ ω + b u − R (e − 1)ν (dy) − u−1 n +  ,  ∂ωn−1  ω=0 and n o−1 ∂n−1 ω + b∗u − R (e−uy − 1)ν (dy) R+ ∂ωn−1 Z −n = (−1)n−1 (n − 1)! ω + b∗u − e−uy − 1ν (dy) . R+ Therefore, we have Z −n L n −1 ∗ −uy M (TX , u) = n!u b u − e − 1 ν (dy) . R+

70 4.3.2 Results Based on Non-Gaussian OU Processes

For degradation with random jumps described by the non-Gaussian OU process Y (t),

n we derive the explicit expressions of RY (y, t) and lifetime moments M(TY , y) in terms of

Laplace transform, represented by Levy´ measures. The results are presented in Theorem

4.3 and 4.4.

Theorem 4.3. For degradation with random jumps described by a non-Gaussian OU process Y (t), the Laplace expression of reliability function is

Z ∞ LL −1 F (v,u,ω) RY (u, ω) = −u e g(v)dv, u where F (v, u, ω) = R u f(v0, ω)dv0, f(v, ω) = ω + b∗v − R (e−vz − 1)ν (dz) /αv, v R+ and g(v) = −1/αv. In addition, b∗ ≥ 0, ν is the Levy´ measure.

Proof. Let p(y, t) be the probability density function of Y (t). Based on [187], the Fokker-

Planck equation for Y (t) is

∂p(y, t) ∂yp(y, t) ∂p(y, t) = −α − b ∂t ∂y ∂y (4.7) Z ∂p(y, t) + p(y − z, t) − p(y, t) + Iz∈(0,1)z ν (dz) . R+ ∂y

For (4.7), we do Laplace transform of p(y, t) w.r.t. t for both sides,

∂ypL(y, ω) ∂pL(y, ω) ωpL(y, ω) − p (y, 0) = −α − b ∂y ∂y Z L L L ∂p (y, ω) + p (y − z, ω) − p (y, ω) + Iz∈(0,1)z ν (dz) . R+ ∂y (4.8)

71 For (4.8), we do Laplace transform of pL(y, ω) w.r.t. y for both sides, then

∂pLL(u, ω) ωpLL(u, ω) − 1 = αu − bupLL(u, ω) ∂u Z −uz LL LL LL + e p (u, ω) − p (u, ω) + Iz∈(0,1)zup (u, ω) ν (dz) . R+

∗ R Let b = b − 0

∂pLL(u, ω) Z αu = ω + b∗u − e−uz − 1ν (dz) pLL(u, ω) − 1. ∂u R+ Let f(u, ω) = ω + b∗u − R (e−uz − 1)ν (dz) /αu, and g(u) = −1/αu. We R+ have

∂pLL(u, ω) = f(u, ω)pLL(u, ω) + g(u), ∂u with pLL(∞, ω) = 0. By solving this ODE, we have

Z ∞ Z ∞ LL − R ∞ f(v0,ω)dv0 R ∞ f(v0,ω)dv0 F (v,u,ω) p (u, ω) = −e u e v g(v)dv = − e g(v)dv, u u

R u 0 0 where F (v, u, ω) = v f(v , ω)dv .

Then based on Lemma 4.2, we have

Z ∞ LL −1 F (v,u,ω) RY (u, ω) = −u e g(v)dv. u

Remark 4.2. For (4.7), we do Laplace transform of p(y, t) w.r.t. y for both sides,

Z E e−uY (t) = pL(u, t) = e−uyp (y, t) dy, R+

72 then we have

∂pL(u, t) ∂pL(u, t) = αu − bupL(u, t) ∂t ∂u Z −uz L L L + e p (u, t) − p (u, t) + Iz∈(0,1)zup (u, t) ν (dz) R+ ∂pL(u, t) Z = αu + −b∗u + e−uz − 1ν (dz) pL(u, t). ∂u R+ By using the method of characteristics to solve this ﬁrst order PDE, we have

Z t Z E e−uY (t) = pL(ueαt, 0) exp −b∗ueαr + e−ueαry − 1 ν (dy) dr . 0 R+

Since pL(ueαt, 0) = 1, we have

Z t Z E e−uY (t) = exp −b∗ueαr + e−ueαry − 1 ν (dy) dr . 0 R+

We use Theorem 4.3 to derive the transform expression for the moments of lifetime

Ty in Theorem 4.4.

Theorem 4.4. For degradation with random jumps described by a non-Gaussian OU process Y (t), the Laplace expression of lifetime moments is

Z ∞ L n n −1 1−n n−1 F (v,u) M (TY , u) = (−1) u nα (lnu − lnv) e g(v)dv, u where F (v, u) = R u f(v0)dv0, f(v) = b∗v − R (e−vz − 1)ν (dz) /αv, and g(v) = v R+

−1/αv. In addition, b∗ ≥ 0, ν is the Levy´ measure.

Proof. The Laplace transform of Q˜(y, t) w.r.t t is

Z ˜L −ωt ∂ L Q (y, ω) = − e RY (y, t) dt = h(y) − ωRY (y, ω) . (4.9) R+ ∂t

73 For (4.9), we do Laplace transform w.r.t. y on both sides, then

˜LL LL −1 Q (u, ω) = −ωRY (u, ω) + u .

From Theorem 4.3, we have

Z ∞ Q˜LL(u, ω) = u−1ω eF (v,u,ω)g(v)dv + u−1. u

From Lemma 4.3, we have

" # ∂nQ˜LL(u, ω) M L(T n, u) = Q˜LL(u, 0) = (−1)n , Y n ∂ωn ω=0 where

 n R ∞ F (v,u,ω)  " n LL # ∂ Q˜ (u, ω) ∂ ω u e g(v)dv = u−1 ∂ωn  ∂ωn  ω=0 ω=0  n R ∞ F (v,u,ω)  ∂ u e g(v)dv = u−1 ω  ∂ωn  ω=0  n−1 R ∞ F (v,u,ω)  ∂ u e g(v)dv + u−1n ,  ∂ωn−1  ω=0 and

 n−1 R ∞ F (v,u,ω)  ∂ u e g(v)dv  ∂ωn−1  ω=0 Z ∞ ∂n−1eF (v,u,ω) = n−1 g(v)dv u ∂ω ω=0 "Z ∞ Z u n−1 # 1 0 F (v,u,ω) = 0 dv e g(v)dv u v αv ω=0 Z ∞ = α1−n (lnu − lnv)n−1 eF (v,u)g(v)dv. u

74 Therefore, we have

Z ∞ L n n −1 1−n n−1 F (v,u) M (TY , u) = (−1) u nα (lnu − lnv) e g(v)dv. u

4.4 Numerical Examples

To illustrate our models, we use an interesting Levy´ measure

κ 1 ν(dx) = dx, Γ(1 − κ) xκ+1 where x > 0, 0 < κ < 1, which represents a positive stable process PS(κ), a Levy´ subordinator, whose distribution is in general unknown in closed-form [25]. Notice that if

κ is close to 0, the process propagates with big jumps; and if κ is close to 1, the process evolves with small jumps. The distribution of this variable is asymmetric and heavy-tailed, i.e., it does not have moments of order κ and above.

When the degradation evolution can be described by this positive stable process, the

Laplace expression of reliability function based on Theorem 4.1 is

LL −1 κ −1 RX (u, ω) = u {ω + u } .

Based on Theorem 4.2, the Laplace expression of lifetime moments is

L n −nκ−1 M (TX , u) = n!u .

When the evolution of the degradation can be described by the non-Gaussian OU process driven by PS(κ), the Laplace expression of reliability function based on Theorem

75 4.3, is

∞ −1 −1 1 κ Z −1 −1 1 κ LL −1 α ω−1 α κ u −(α ω+1) −α κ v RY (u, ω) = α u e v e dv. u

Based on Theorem 4.4, the Laplace expression of lifetime moments is

Z ∞ n−1 −1 1 κ κ L n n −1 1−n α κ (u −v ) −1 −1 M (TY , u) = (−1) u nα (lnu − lnv) e (−α v )dv u n−1 Z ∞ X −1 1 κ −1 1 κ −1 −n i i i α κ u n−1−i −1 −α κ v = u nα Cn−1 (−1) (lnu) e (lnv) v e dv. i=0 u ˜ The speciﬁc values for the parameters are given in Table 4.1. Sample paths of Xs(t) and Y (t) are showed in Figure 4.1 (Levy´ subordinator b) and Figure 4.2 (non-Gaussian

OU b), respectively. The system fails when the degradation exceeds the respective failure threshold. The inversion algorithms for Laplace transform [2, 31] were implemented to invert Laplace expressions in order to compute the values of reliability and lifetime moments.

Table 4.1: Parameter values for models in Chapter 4

Parameters Value Parameters Value x; y [0,30] κ 0.9 α 0.1

76 1

0.8

0.6

Reliability 0.4

0.2

0 15 30 10 20 5 t 10 0 0 x

˜ Figure 4.3: Reliability function w.r.t. time t and failure threshold x based on Xs(t)

0.8

0.6

Reliability 0.4

0.2

0 15 30 10 20 5 t 10 0 0 y

Figure 4.4: Reliability function w.r.t. time t and failure threshold y based on Y (t)

77 1 OU y=15 0.9 Levy x=15 OU y=20 0.8 Levy x=20

0.7

0.6

0.5

Reliability 0.4

0.3

0.2

0.1

0 0 2 4 6 8 10 12 14 16 18 20 t ˜ Figure 4.5: Reliability function w.r.t. time t based on Xs(t) and Y (t)

25 Levy OU

15 M(T)

0 0 5 10 15 20 25 30 x(y) ˜ Figure 4.6: First moments of lifetime w.r.t. failure threshold based on Xs(t) and Y (t)

78 Figure 4.3 and Figure 4.4 show the reliability with respect to time t and the failure

˜ threshold for Xs(t) and Y (t), respectively. The reliability decreases as the time increases,

and it increases as the threshold increases. Figure 4.5 shows the reliability with respect

to time t when the failure thresholds are 15 and 20, respectively. The reliability based on

˜ Y (t) decreases faster than that based on Xs(t). Figure 4.6 illustrates the ﬁrst moment of lifetime with respect to failure threshold. The mean failure time based on Y (t) is less than

˜ that based on Xs(t) for the same threshold. These observations correspond to the evolution

˜ of Y (t) and Xs(t): the mean path of Y (t) is exponential with respect to time t, while the

˜ mean path of Xs(t) is linear with respect to time t. Besides the Levy´ measure used in this example, we can specify different Levy´ measures to ﬁt the corresponding degradation data, in order to construct models and analyze reliability and lifetime.

4.5 Conclusions

In this chapter, we presented models concerning the stochastic mechanism of a complex degradation process that also subjects to random jumps. Based on the Fokker-Planck equation, we derived explicit results for reliability function and lifetime moments in terms of Laplace transform. The Laplace expressions of reliability function and lifetime moments are represented by Levy´ measures. Our model is general because we can specify many different Levy´ measures to handle many different kinds of degradation data sets.

The models in the literature become special cases of our models.

79 Our method provides a convenient and general way to evaluate the system reliability. When the degradation data is available, our results are explicit and compact enough for effective and efﬁcient statistical inference on lifetime characteristics. The reliability estimation is expected to be more accurate, because our models integrally consider all the stylized features of degradation data series including temporal uncertainty, jumps, independence/dependence, linearity/nonlinearity, symmetry/asymmetry, and light/heavy tails. Based on the precise reliability estimation and prediction, a proper and valuable maintenance policy can be proposed and implemented.

One of the challenging aspects in reliability analysis is how to formulate reliability functions for degradation processes under dynamic environments. Our model based on

Fokker-Planck equations provides a new methodology to overcome this challenge. We will focus on deriving Fokker-Planck equations for degradation processes under dynamic environment in Chapter 5.

80 Chapter 5

Markov Additive Processes for Degradation with Jumps under Dy-

namic Environments

We use general Markov additive processes (Markov-modulated Levy´ processes) to

integrally handle the complexities of degradation including internally- and externally-

induced stochastic properties with complex jump mechanisms. The background com-

ponent of the Markov additive process is a Markov chain deﬁned on a ﬁnite state space;

the additive component evolves as a Levy´ subordinator under a certain background state, and may have instantaneous nonnegative jumps occurring at the time the background state switches. We derive the Fokker-Planck equations for such Markov-modulated processes, based on which we derive Laplace expressions for reliability function and lifetime moments, represented by the inﬁnitesimal generator matrices of Markov chain and the Levy´ measure of Levy´ subordinator. Our models are ﬂexible in modeling degradation data with jumps under dynamic environments. Numerical experiments are used to demonstrate that our general models perform well.

81 5.1 Introduction

During the life of many critical systems (e.g., wind turbines, power/smart grids, and mechanical devices, etc.), there are some external time-varying variables/factors that continuously govern the progress of the stochastic degradation of the systems. Such variables are called stochastic covariates (e.g., dynamic environments such as temperature, humidity, or vibration). Incorporating this externally-induced uncertainty together with internally-induced uncertainty in modeling degradation is a challenging research work, especially when there are many complex jumps stemming from both internal features (mechanical, thermal, electrical, or chemical) of the system and instantaneous state changes of external variables/factors.

To integrally handle the complexities of degradation including both internally- and externally-induced stochastic properties with complex jump mechanisms, we propose to develop degradation models under dynamic environments using a broad class of general

Markov additive processes (Markov-modulated Levy´ processes), where the background component is a Markov chain with finite states, the additive component evolves as a Levy´ subordinator under a certain background state, and may have instantaneous nonnegative jumps occurring at the time the background state switches. We develop the Fokker-Planck equations of such analytically appealing stochastic processes in order to derive reliability characteristics. We also develop systematic procedures for deriving and obtaining the explicit and compact results, represented by infinitesimal generator matrices and Levy´ measures. Using Markov-modulated Levy´ processes, our general models are flexible in

82 modeling stylized features of degradation data series under dynamic environments. Our new framework provides a new methodology in reliability analysis.

Without considering external factors, stochastic processes such as Wiener processes, gamma processes and compound Poisson processes are directly used to represent degradation processes when the degradation is observable [75, 125, 184, 188, 192]. When the degradation is unobservable, it is treated as a latent process, measured and tracked by internal stochastic covariates that are observable marker processes [104, 126, 175, 182, 208].

These markers (e.g., diagnostic factors such as mileage traveled of an auto) provide information about the progress of degradation processes that can be used to infer the reliability function or the hazard function. To conduct reliability/survival analysis, [126, 208] used a bivariate Wiener process to describe the correlation of the degradation process and the marker process, and then formulated the reliability function based on the ﬁrst passage time of the Wiener process. Some models directly deﬁned the hazard function as an explicit function of the marker process [104, 175, 182].

In biostatistics, the marker processes are stochastic processes representing time- varying covariates that track the health of a system under study in the language of [108].

[104, 105] considered the marker processes as associated variables that continuously measure the progress of an individual towards the ﬁnal expression of the disease (failure).

Assuming a simple additive model for the relationship between the marker process and the hazard function, the survival distribution of time to failure was expressed, where the Poisson process was used to represent the marker process. [210] reviewed models in survival analysis under the framework that the hazard function explicitly represents

83 the effects of markers. Typically they discussed the model where the marker processes are Wiener-based diffusion processes and the relationship between the hazard function and markers is quadratic. [86] constructed the model using a nonparametric frame to describe the dependency of the hazard on marker variables. Regarding the efﬁcient use of marker information, [139] proposed a heuristic approach in estimating parameters of survival functions. [175] studied the distributions of the residual time in acquired immune deﬁciency syndrome diagnosis based on markers that carry valuable information about disease progression. They derived the residual time distribution for several combinations of marker processes and marker-dependent hazard functions. However, all these stochastic models just represent internally-induced uncertainty with temporal variability.

Markov additive processes are a class of binary stochastic processes with one component as an additive process (e.g., Levy´ process) that is modulated by the other component, which is a standard Markov process [42, 46]. They can integrally handle the complexities of degradation processes under dynamic environments. Special Markov additive processes, including Markov-modulated linear processes and Markov-modulated compound

Poisson processes, have been used to represent the linear deterministic degradation with

Poisson-type jumps under discrete and ﬁnite state Markov environments [112–115]. The explicit results were readily derived due to the simple nature of the Poisson process. These models are motivated by the original ideas from [40], in which general Markov additive processes were ﬁrst proposed in degradation modeling, but have not been well developed, e.g., no explicit results of reliability characteristics.

The organization of this chapter is as follows. In Section 5.2, we describe the model

84 construction. In Section 5.3, we derive the Fokker-Planck equations of general Markov

additive processes. In Section 5.4, we derive the explicit expressions of reliability function

and lifetime moments for systems subject to degradation under the dynamic environment.

Numerical examples are illustrated in Section 5.5, and conclusions are given in Section

5.6.

5.2 Model Construction

We consider a system subject to degradation with random jumps, which is a process

of stochastically continuous degradation with sporadic jumps that occur at random times

and have random sizes. In addition, the degradation process is modulated by the environ-

ment process. To model the evolution of this type of degradation process, we propose to

use a general Markov additive process {X (t) ,E (t)}, in which the cumulative degrada-

tion by time t is represented by a nondecreasing continuous time cadl` ag` (right continuous

with left limits) stochastic process X(t), and the environment process is represented by a temporally homogeneous continuous time cadl` ag` Markov process E(t) with ﬁnite state

P space Ξ = {0, 1, ··· , n}. Let G = (rij), rii = − rij, i, j ∈ Ξ denote the transition rate j6=i matrix (inﬁnitesimal generator matrix) of E(t).

Moreover, the evolution of X(t) depends on the states of E(t). X(t) evolves as a non-decreasing Levy´ process (i.e. a Levy´ subordinator) when the state of E(t) is unchanged. When E(t) = i ∈ Ξ, b(E(t)) = b(i), and ν(E(t), dy) = ν(i, dy). In practice, the changes of environment states, such as instantaneous temperature increase or decrease, can

85 induce certain damages to the system, modeled by the jumps in the degradation process.

Therefore, we assume there is an additional random nonnegative jump in X(t) when the state of E(t) changes. When E(t) changes from state i to state j, the distribution of the jump is denoted as Dij(z), deﬁned on R+. For i = j, Dij(dz) = δz(0), which is a Dirac delta function. {X (t) ,E (t)} is a class of Markov additive process (see C¸inlar [42]). To simplify the formulae, we assume the initial state X(0) = 0,E(0) = 0 a.s., but it is easy to generalize the formulae with X(0) = c, E(0) = k, c ∈ R+, k ∈ Ξ.

To integrally handle internally- and externally-induced stochastic properties with complex jump mechanisms, X(t) can be expressed as

Z t Z t Z X(t) = b(E(ξ−))dξ + y J (E(ξ−), dξ, dy) − ν (E(ξ−), dy) dξ 0 0 0

86 a compound Poisson process (CP); when E(t) = k, X(t) evolves as an inverse Gaussian process (IG); when E(t) = l, X(t) evolves as a gamma process (G); and when E(t) = m,

X(t) evolves as a stable process (S).

X(t) LP CP IG G S

i k l m j E(t)

Figure 5.1: A sample path of Markov additive process with random jumps when the environment states change

5.3 Fokker-Planck Equations for Markov Additive Processes

Without an analytical expression of the probability law for {X (t) ,E (t)}, the development of the characteristics for such processes and the subsequent reliability function is a nontrivial work, even for simple cases. The difﬁculty of these stems from 1) the stochastic

87 evolution of degradation has complex mechanisms such as random jumps, 2) the stochastic

nature of environment, and 3) the distributional derivation for the ﬁrst passage time. We

overcome this challenge by deriving the Fokker-Planck equation of {X (t) ,E (t)}.

Let p(x, i, t) be the probability density function for the Markov additive process

{X(t),E(t)}, and its Fokker-Planck equation is

∂p(x, i, t) = L∗p(x, i, t), ∂t

where L∗ is the adjoint operator of the inﬁnitesimal generator of {X(t),E(t)}. Then it is

important to derive the adjoint operator, and our main result is given in this section. The

Fokker-Planck equation is derived and presented in Theorem 5.1.

Theorem 5.1. For the Markov additive process {X(t),E(t)} described in Section 5.2, the

Fokker-Planck equation is

∂p(x, i, t) ∂ X Z = −b(i) p(x, i, t) + r p (x − z, j, t)D (dz) ∂t ∂x ji ji j∈Ξ R+ Z ∂ + p(x − y, i, t) − p(x, i, t) + Iy∈(0,1)y p(x, i, t) ν (i, dy). R+ ∂x

∞ 2 Proof. Step 1: For each f ∈ C0 (R ) (f is a smooth function and compactly supported),

and for each t > 0, we aim to derive f(X(t + ∆t),E(t + ∆t)) − f(X(t),E(t)).

Both X(t) and E(t) are cadl` ag` processes. We deﬁne X(ξ−) and E(ξ−) as the left

limits at the time point ξ, S = [t, t + ∆t], S1 = {ξ ∈ S : E(ξ) − E(ξ−) = 0}, and

88 S2 = {ξ ∈ S : E(ξ) − E(ξ−) 6= 0}. Then we have

X f(X(t + ∆t),E(t + ∆t)) − f(X(t),E(t)) = f(X(ξ),E(ξ)) − f(X(ξ−),E(ξ−)) ξ∈S X X = f(X(ξ),E(ξ)) − f(X(ξ−),E(ξ)) + f(X(ξ),E(ξ)) − f(X(ξ−),E(ξ−)).

ξ∈S1 ξ∈S2 (5.1)

During a continuous time interval s ⊆ S1, if E(ξ) = e, ξ ∈ s, e ∈ Ξ, dX(ξ) R has a constant part dXC (ξ) = b(e) − 0

th time of the m jump, τ0 = inf{ξ : ξ ∈ s}, τm = inf{ξ : ξ > τm−1 & ∆XJ (ξ) > 0}, where ∆XJ (ξ) = XJ (ξ) − XJ (ξ−), and τ = sup{ξ : ξ ∈ s}. Then we have

X f(X(ξ),E(ξ)) − f(X(ξ−),E(ξ)) ξ∈s M X = f (X (max{τM , τ−}), e) − f (X (τM ), e) + f (X (τm), e) − f (X (τm−1), e) m=1

= f (X (max{τM , τ−}), e) − f (X (τM ), e)

M X + f (X (τm−) + ∆XJ (τm) , e) − f (X (τm−), e) m=1 M X + f (X (τm−), e) − f (X (τm−1), e) . m=1

89 Based on the stochastic integration (see Chapter 4 in [17]), we have

X f(X(ξ),E(ξ)) − f(X(ξ−),E(ξ))

ξ∈S1 Z ∂f = b(E (ξ)) (X (ξ−) ,E (ξ))dξ ξ∈S1 ∂x Z Z ∂f − y (X (ξ−) ,E (ξ))ν (E (ξ) , dy) dξ ξ∈S1 0

f(X(t + ∆t),E(t + ∆t)) − f(X(t),E(t)) Z ∂f = b(E (ξ)) (X (ξ−) ,E (ξ))dξ ξ∈S1 ∂x Z Z ∂f − y (X (ξ−) ,E (ξ))ν (E (ξ) , dy) dξ ξ∈S1 0

ξ∈S2

Notice that for ξ ∈ S1, both E(ξ) and X(ξ−) are predictable. Our calculus is in the

Itoˆ form.

∞ 2 Step 2: For each f ∈ C0 (R ), we aim to derive the inﬁnitesimal generator L of

{X(t),E(t)}:

E (f(X(t + ∆t),E(t + ∆t))|X(t) = x, E(t) = i) − f (x, i) Lf(x, i) = lim . (5.2) ∆t→0 ∆t

As E(t) has the transition rate matrix (inﬁnitesimal generator matrix) G =

90 P (rij), rii = − rij, deﬁning P (E(t + ∆t) = j|E(t) = i) = Pij(∆t), we have j6=i ! P E f(X(ξ) + ME(ξ−),E(ξ),E (ξ)) − f (X(ξ−),E (ξ−)) |X(t) = x, E(t) = i ξ∈S2 lim ∆t→0 ∆t P R (f (x + z, j) − f (x, i))Dij(dz)Pij(∆t) R+ j6=i = lim ∆t→0 ∆t X Z = rij (f (x + z, j) − f (x, i)) Dij(dz) j6=i R+ X Z = rij f (x + z, j)Dij(dz). j∈Ξ R+

Since the Poisson random measure J (dt, dy) has a Poisson distribution with mean

ν (dy) dt, we have

∂ Z ∂ Lf(x, i) = b(i) f(x, i) − y f(x, i)ν (i, dy) ∂x 0

Step 3: We aim to derive L∗, the adjoint operator corresponding to the inﬁnitesimal

generator L in (5.3):

X Z X Z Lf (x, i)p (x, i, t) dx = f (x, i) L∗p (x, i, t) dx. (5.4) i∈Ξ R+ i∈Ξ R+

Using integration by parts, as p(0, i, t) = 0, and p(∞, i, t) = 0, we have

X Z ∂ X Z b(i) f(x, i)p (x, i, t) dx = b(i)p (x, i, t) df(x, i) ∂x i∈Ξ R+ i∈Ξ R+ X X Z = b(i)p (x, i, t) f(x, i)|R+ − b(i)f (x, i) dp(x, i, t) (5.5) i∈Ξ i∈Ξ R+ X Z ∂ = −b(i) p(x, i, t)f (x, i) dx, ∂x i∈Ξ R+

91 and X Z Z (f(x + y, i) − f(x, i))ν (i, dy)p (x, i, t) dx i∈Ξ R+ R+ X Z Z ∂ − Iy∈(0,1)y f(x, i)ν (i, dy)p (x, i, t) dx R+ R+ ∂x i∈Ξ (5.6) X Z Z = (p(x − y, i, t) − p(x, i, t))ν (i, dy)f (x, i) dx i∈Ξ R+ R+ X Z Z ∂ + I y p(x, i, t)ν (i, dy)f (x, i) dx. y∈(0,1) ∂x i∈Ξ R+ R+ By swapping i and j, we have

X Z X Z rij f (x + z, j)Dij(dz)p (x, i, t) dx i∈Ξ R+ j∈Ξ R+ (5.7) X Z X Z = rji p (x − z, j, t)Dji(dz)f (x, i) dx. i∈Ξ R+ j∈Ξ R+ Then from (5.5), (5.6) and (5.7), we have ∂ X Z L∗p (x, i, t) = −b(i) p(x, i, t) + r p (x − z, j, t)D (dz) ∂x ji ji j∈Ξ R+ Z ∂ + p(x − y, i, t) − p(x, i, t) + Iy∈(0,1)y p(x, i, t) ν (i, dy). R+ ∂x ∞ 2 For each f ∈ C0 (R ), we denote u(t) = E (f(X(t),E(t))). Then based on (5.2),

∂ we have Lu(t) = ∂t u(t) (see [187]), and

X Z X Z ∂ Lf(x, i)p (x, i, t) dx = (f(x, i)p(x, i, t))dx. ∂t i∈Ξ R+ i∈Ξ R+ From (5.4), we have

X Z X Z ∂ f(x, i)L∗p (x, i, t) dx = (f(x, i)p(x, i, t))dx, ∂t i∈Ξ R+ i∈Ξ R+ then

∂p(x, i, t) = L∗p (x, i, t) . ∂t

92 Thus we obtain the Fokker-Planck equation

∂p(x, i, t) ∂ X Z = −b(i) p(x, i, t) + r p (x − z, j, t)D (dz) ∂t ∂x ji ji j∈Ξ R+ Z ∂ + p(x − y, i, t) − p(x, i, t) + Iy∈(0,1)y p(x, i, t) ν (i, dy). R+ ∂x

To make comparison, Corollary 5.1 provides the Fokker-Planck equation for the case that there is no jump in X(t) when the state of E(t) changes (Figure 5.2).

X(t) LP CP IG G S

i k l m j E(t)

Figure 5.2: A sample path of Markov additive process with no jump when the environment states change

Corollary 5.1. For the Markov additive process {X(t),E(t)}, assuming there is no jump

93 in X(t) when the state of E(t) changes, the Fokker-Planck equation is ∂p(x, i, t) ∂ X = −b(i) p(x, i, t) + r p (x, j, t) ∂t ∂x ji j∈Ξ Z ∂ + p(x − y, i, t) − p(x, i, t) + Iy∈(0,1)y p(x, i, t) ν (i, dy). R+ ∂x

Proof. If there is no jump in X(t) when the state of E(t) changes, Dij(dz) = δz(0), i, j ∈

Ξ, then in Theorem 5.1, Z p(x − z, j, t)Dji(dz) = p(x, j, t). R+

5.4 Reliability Function and Lifetime Moments

A system fails when the degradation process X(t) exceeds a failure threshold x. The lifetime of the system and its moments are deﬁned respectively as

n n Tx = inf{t : X(t) > x},M(T , x) = E(Tx ).

Since X(t) is nondecreasing, we have {Tx ≥ t} ≡ {X(t) ≤ x}, then the reliability function can be deﬁned as R (x, t) = P (Tx ≥ t) = P (X (t) ≤ x) = FX(t) (x). In this section, for a degradation process under the dynamic environment described by the

Markov additive process {X(t),E(t)}, we derive the explicit expressions of R (x, t) and lifetime moments M(T n, x) in terms of Laplace transform, represented by the inﬁnitesi- mal generator matrix and the Levy´ measure.

Laplace transform of p(x, i, t) w.r.t. t is deﬁned to be Z pL(x, i, ω) = e−ωtp (x, i, t) dt, ω > 0. R+

94 Laplace transform of pL(x, i, ω) w.r.t. x is

Z pLL(u, i, ω) = e−uxpL (x, i, ω)dx, u > 0. R+

The results are presented in Theorems 5.2 and 5.3.

Theorem 5.2. For a degradation process under the dynamic environment that is described by the Markov additive process {X(t),E(t)} in Section 5.2, the Laplace expression of reliability function is

RLL(u, ω) = u−1[1, 0, ··· , 0][A − B]−1[1, 1, ··· , 1]T , where A is a diagonal matrix with diagonal entries ω + b∗(i)u − R (e−uy − 1)ν (i, dy), R+

L ∗ and B = [rijdij], i, j ∈ Ξ. In addition, b (i) ≥ 0, ν is the Levy´ measure, rij, i, j ∈

L R −uz Ξ are entries of the inﬁnitesimal generator matrix of E(t), d (u) = e Dji(dz), ji R+

[1, 0, ··· , 0] is a vector of size n + 1, where the ﬁrst element is 1 and all others are 0, and

[1, 1, ··· , 1] is a vector of size n + 1, where all the elements are 1.

Proof. Based on Theorem 5.1, the Fokker-Planck equation for {X(t),E(t)} is

∂p(x, i, t) ∂ X Z = −b(i) p(x, i, t) + rji p (x − z, j, t)Dji(dz) ∂t ∂x R+ j∈Ξ (5.8) Z ∂ + p(x − y, i, t) − p(x, i, t) + Iy∈(0,1)y p(x, i, t) ν (i, dy). R+ ∂x For (5.8), we do Laplace transform of p(x, i, t) w.r.t. t for both sides,

ωpL(x, i, ω) − p (x, i, 0)

∂pL(x, i, ω) X Z = −b(i) + r pL (x − z, j, ω)D (dz) ∂x ji ji (5.9) j∈Ξ R+ Z L L L ∂p (x, i, ω) + p (x − y, i, ω) − p (x, i, ω) + Iy∈(0,1)y ν (i, dy) . R+ ∂x

95 For (5.9), we do Laplace transform of pL(x, i, ω) w.r.t. x for both sides, then

LL ωp (u, i, ω) − Ii=0 Z LL X −uz LL = −b(i)up (u, i, ω) + rji e p (u, j, ω)Dji(dz) j∈Ξ R+ Z −uy LL LL LL + e p (u, i, ω) − p (u, i, ω) + Iy∈(0,1)yup (u, i, ω) ν (i, dy) . R+

∗ R Let b (i) = b(i) − 0

L R −uz Let d (u) = e Dji(dz), and then the matrix form is ji R+

pLL(u, ω)[A − B] = [1, 0, ··· , 0], where pLL(u, ω) = [pLL(u, 0, ω), pLL(u, 1, ω), ··· , pLL(u, n, ω)], A is a diagonal matrix

∗ R −uy L with diagonal entries ω + b (i)u − (e − 1)ν (i, dy), and B = [rijd ], i, j ∈ Ξ. R+ ij

We have

RLL(u, ω) = u−1[1, 0, ··· , 0][A − B]−1[1, 1, ··· , 1]T .

Remark 5.1. For (5.8), we do Laplace transform of p(x, i, t) w.r.t. x for both sides,

Z pL(u, i, t) = e−uxp (x, i, t) dx, u > 0, R+

96 then

∂pL(u, i, t) X Z = −b(i)upL(u, i, t) + r e−uzpL (u, j, t)D (dz) ∂t ji ji j∈Ξ R+ Z −uy L L L + e p (u, i, t) − p (u, i, t) + Iy∈(0,1)yup (u, i, t) ν (i, dy) R+ Z ∗ −uy L X L L = −b (i)u + e − 1 ν (i, dy) p (u, i, t) + rjidjip (u, j, t). R+ j∈Ξ Solving this ordinary differential equation, we have the solution in the matrix form:

L p (u, t) = [1, 0, ··· , 0] exp {t [B − A0]} ,

∗ R −uy where A0 is a diagonal matrix with diagonal entries b (i)u − (e − 1)ν (i, dy), i ∈ R+

Ξ.

We use Theorem 5.2 to derive the Laplace expression for the moments of lifetime Tx as Theorem 5.3.

Theorem 5.3. For a degradation process under the dynamic environment that is described by the Markov additive process {X(t),E(t)} in Section 5.2, the Laplace expression of lifetime moments is

L n −1 −n T M (T , u) = n!u [1, 0, ··· , 0][A0 − B] [1, 1, ··· , 1] ,

∗ R −uy where A0 is a diagonal matrix with diagonal entries b (i)u − (e − 1)ν (i, dy), and R+

L ∗ B = [rijdij], i, j ∈ Ξ. In addition, b (i) ≥ 0, ν is the Levy´ measure, rij, i, j ∈ Ξ are entries

L R −uz of the inﬁnitesimal generator matrix of E(t), d (u) = e Dji(dz), [1, 0, ··· , 0] is a ji R+

vector of size n + 1, where the ﬁrst element is 1 and all others are 0, and [1, 1, ··· , 1] is a

vector of size n + 1, where all the elements are 1.

97 ˜ ˜ Proof. We have P (Tx < t) ≡ P (x, t) = 1 − R(x, t). Then P (dt, x) = −R(x, dt). The

Laplace transform of P˜(dt, x) w.r.t t is

p˜L(x, ω) = −ωRL(x, ω) + h(x), (5.10) where h(x) is the unit step function. For (5.10), we do Laplace transform w.r.t. x for both sides, then

p˜LL(u, ω) = −ωRLL(u, ω) + u−1.

From Theorem 5.2, we have

p˜LL(u, ω) = −ωu−1[1, 0, ··· , 0][A − B]−1[1, 1, ··· , 1]T + u−1.

We denote

∂np˜LL(u, ω) p˜LL(u, ω) = (−1)n , n ∂ωn

and then the Laplace expression of lifetime moments is

n LL L n LL n ∂ p˜ (u, ω) M (T , u) =p ˜n (u, 0) = (−1) n , ∂ω ω=0

where ∂np˜LL(u, ω) n ∂ω ω=0 " # ∂n−1[1, 0, ··· , 0][A − B]−1[1, 1, ··· , 1]T = −u−1n , ∂ωn−1 ω=0 and " # ∂n−1[1, 0, ··· , 0][A − B]−1[1, 1, ··· , 1]T ∂ωn−1 ω=0

n−1 −n T = (−1) (n − 1)![1, 0, ··· , 0][A0 − B] [1, 1, ··· , 1] .

98 Therefore, we have

L n −1 −n T M (T , u) = n!u [1, 0, ··· , 0][A0 − B] [1, 1, ··· , 1] .

To make comparison, Corollaries 5.2 and 5.3 provide the Laplace expressions of

reliability function and lifetime moments, respectively, assuming there is no jump in X(t)

at the time the state of E(t) changes.

Corollary 5.2. For a degradation process under the dynamic environment that is described

by the Markov additive process {X(t),E(t)}, assuming there is no jump in X(t) when the state of E(t) changes, the Laplace expression of reliability function is

RLL(u, ω) = u−1[1, 0, ··· , 0][A − G]−1[1, 1, ··· , 1]T , where A is a diagonal matrix with diagonal entries ω+b∗(i)u − R (e−uy − 1)ν (i, dy), i ∈ R+

Ξ, and G is the inﬁnitesimal generator matrix of E(t).

Corollary 5.3. For a degradation process under the dynamic environment that is described

by the Markov additive process {X(t),E(t)}, assuming there is no jump in X(t) when the state of E(t) changes, the Laplace expression of lifetime moments is

L n −1 −n T M (T , u) = n!u [1, 0, ··· , 0][A0 − G] [1, 1, ··· , 1] ,

∗ R −uy where A0 is a diagonal matrix with diagonal entries b (i)u − (e − 1)ν (i, dy), i ∈ R+

Ξ, and G is the inﬁnitesimal generator matrix of E(t).

99 5.5 Numerical Examples

To illustrate our models, we consider two cases of {X (t) ,E (t)}:

Case 1: There are no jumps in degradation X(t) when the states of environment E(t)

change;

Case 2: There are random jumps in degradation X(t) when the states of environment

E(t) change.

We use a Markov process with two states {0, 1} to model the environment, and its inﬁnitesimal generator matrix is   r r  00 01  G =   .   r10 r11

In Case 2, we use a Levy´ distribution to model the jumps when the environment switches from state 0 to state 1:  q  ξ exp(− ξ )  2π 2(z−$)  3 for z > $ > 0 2 D01(dz) = (z − $)    0 otherwise,

√ L −u$− 2uξ and then d01(u) = e . A gamma distribution is used to model the jumps when the

environment switches from state 1 to state 0:

βαzα−1e−βz D (dz) = , z > 0, 10 Γ(α)

L β α and then d10(u) = ( β+u ) .

100 We use a Levy´ measure to model the Levy´ degradation under the environment state

δγ−2κ1 κ y−κ1−1 exp(− 1 γ2y) ν(0, dy) = 1 2 dy, Γ(κ1)Γ(1 − κ1)

where y, δ > 0, 0 < κ1 < 1, γ ≥ 0, which represents a positive tempered stable process

PTS(κ1, δ, γ) [25].

We use another Levy´ measure to model the Levy´ degradation under the environment state 1:

κ2 1 ν(1, dy) = κ +1 dy, Γ(1 − κ2) y 2

where y > 0, 0 < κ2 < 1, which represents a positive stable process PS(κ2) [25]. Notice

that if κ2 are close to 0, the corresponding stable processes propagate with big jumps; if

κ2 are close to 1, the stable processes evolve with small jumps.

For case 1, the Laplace expression of reliability function based on Corollary 5.2 is

RLL(u, ω) = u−1[1, 0][A − G]−1[1, 1]T  −1 ˜ ˜ V00 V01 −1   T = u [1, 0]   [1, 1]  ˜ ˜  V10 V11 V˜ − V˜ = u−1 11 01 , ˜ ˜ ˜ ˜ V00V11 − V01V10

1 κ1 ˜ ∗ κ ˜ ˜ where V00 = ω + b (0)u − δγ + δ(γ 1 + 2u) − r00, V01 = −r01, V10 = −r10, and

˜ ∗ κ2 V11 = ω + b (1)u + u − r11.

101 The Laplace expression of lifetime moments based on Corollary 5.3 is

L n −1 −n T M (T , u) = n!u [1, 0][A0 − G] [1, 1]  −n V00 V01 −1   T = n!u [1, 0]   [1, 1] .   V10 V11 The ﬁrst and second moments of lifetime for Case 1 are

V − V M L(T 1, u) = u−1 11 01 , V00V11 − V01V10

2 L 2 −1 V11 + V01V10 − V01V11 − V01V00 M (T , u) = 2u 2 , [V00V11 − V01V10]

1 κ1 ∗ κ where V00 = b (0)u − δγ + δ(γ 1 + 2u) − r00,V01 = −r01,V10 = −r10, and V11 =

∗ κ2 b (1)u + u − r11.

For case 2, the Laplace expression of reliability function based on Theorem 5.2 is

RLL(u, ω) = u−1[1, 0][A − B]−1[1, 1]T  −1 V˜ ∗ V˜ ∗ −1  00 01  T = u [1, 0]   [1, 1]  ˜ ∗ ˜ ∗  V10 V11 V˜ ∗ − V˜ ∗ = u−1 11 01 , ˜ ∗ ˜ ∗ ˜ ∗ ˜ ∗ V00V11 − V01V10 1 κ1 ˜ ∗ ∗ κ L ˜ ∗ L ˜ ∗ L where V00 = ω + b (0)u − δγ + δ(γ 1 + 2u) − r00d00, V01 = −r01d01, V10 = −r10d10,

˜ ∗ ∗ κ2 L and V11 = ω + b (1)u + u − r11d11.

The Laplace expression of lifetime moments based on Theorem 5.3 is

L n −1 −n T M (T , u) = n!u [1, 0][A0 − B] [1, 1]  −n V ∗ V ∗ −1  00 01  T = n!u [1, 0]   [1, 1] .  ∗ ∗  V10 V11

102 The ﬁrst and second moments of lifetime for Case 2 are

∗ ∗ L 1 −1 V11 − V01 M (T , u) = u ∗ ∗ ∗ ∗ , V00V11 − V01V10

∗2 ∗ ∗ ∗ ∗ ∗ ∗ L 2 −1 V11 + V01V10 − V01V11 − V01V00 M (T , u) = 2u ∗ ∗ ∗ ∗ 2 , [V00V11 − V01V10]

1 κ1 ∗ ∗ κ L ∗ L ∗ L where V00 = b (0)u − δγ + δ(γ 1 + 2u) − r00d00,V01 = −r01d01,V10 = −r10d10, and

∗ ∗ κ2 L V11 = b (1)u + u − r11d11.

The values for the parameters are given in Table 5.1. The system fails when X(t) exceeds the threshold x. The inversion algorithms for Laplace transform were implemented to invert the Laplace expressions in Theorems 5.2, 5.3 and Corollaries 5.2, 5.3 in order to compute the values of reliability and lifetime moments.

Table 5.1: Parameter values for models in Chapter 5

x [0,30] α 0.2 κ2 0.9 ∗ r00 = −r01 -10 β 50 b (0) 0.02 ∗ r10 = −r11 15 δ 0.6 b (1) 0.01 ξ 0.0001 κ1 0.8 $ 0.01 γ 0.9

103 1

0.8

0.6

Reliability 0.4

0.2

0 20 15 30 10 20 t 5 10 0 0 x

Figure 5.3: Reliability function w.r.t. time t and failure threshold x for Case 1

0.8

0.6

Reliability 0.4

0.2

0 20 15 30 10 20 t 5 10 0 0 x

Figure 5.4: Reliability function w.r.t. time t and failure threshold x for Case 2

104 1

0.9

0.8

0.7

0.6

0.5

Reliability 0.4

0.3

0.2 Case1 x=15 Case2 x=15 0.1 Case1 x=20 Case2 x=20 0 0 2 4 6 8 10 12 14 16 18 20 t

Figure 5.5: Reliability functions w.r.t. time t when x = 15 and x = 20 for both Case 1 and Case 2

30 Case1 Case2 25

15 First moments 10

0 0 5 10 15 20 25 30 x

Figure 5.6: First moments of lifetime w.r.t. failure threshold for both Case 1 and Case 2

105 800 Case1 700 Case2

600

500

400

300 Second moments

200

100

0 0 5 10 15 20 25 30 x

Figure 5.7: Second moments of lifetime w.r.t. failure threshold for both Case 1 and Case 2

Figure 5.3 and 5.4 show the reliability w.r.t. time t and failure threshold x based on general Markov additive processes. For both cases, the reliability decreases as the time increases, and it increases as the threshold increases. Figure 5.5 shows the reliability w.r.t. time t when x = 15 and x = 20 for both cases. The reliability in Case 2 decreases faster than that in Case 1 at the same threshold. Figure 5.6 and Figure 5.7 illustrate the ﬁrst moments and the second moments of lifetime with respect to failure threshold x for both cases. Both the ﬁrst and the second moments of lifetime in Case 2 are less than that in

Case 1 at the same threshold. Besides the Levy´ measures used in this section, we can specify different Levy´ measures to ﬁt the corresponding degradation data, and evaluate their reliability function and lifetime moments.

106 5.6 Conclusions

In this chapter, we developed the systematic procedures to derive compact results for reliability analysis based on the degradation process under the dynamic environment:

Step 1: Derive the inﬁnitesimal generator of the stochastic process of interests;

Step 2: Derive the adjoint operator corresponding to the inﬁnitesimal generator, based on

which the Fokker-Planck equation of such stochastic process is developed;

Step 3: Derive the reliability characteristics of the system in terms of Laplace transform.

Our work in this chapter is summarized as: 1) we model the degradation process under the dynamic environment using the Markov additive process, while most models in the literature were constructed for the deterministic environment; 2) we use the Levy´ subordinator to model the degradation under a certain environment state, and the corresponding Levy´ measure can represent different complex jump mechanisms including infinite activities and finite activities in degradation; 3) our models are general to fit more types of degradation data than those based on gamma/Poisson processes; and 4) we derive the Fokker-Planck equation for a class of general Markov additive processes, and obtain the explicit expressions for reliability function and lifetime moments, which provide a new methodology to deal with multiple dependent degradation processes under dynamic environments.

107 Chapter 6

Markov-modulated Multi-dimensional Levy´ Processes for Multiple

Dependent Degradation Processes under Dynamic Environments

The analysis of multiple dependent degradation processes is a challenging research

work in the reliability ﬁeld, especially for complex degradation with jumps under dy-

namic environments. To integrally handle the jump uncertainties in degradation and the

jump dependence among degradation processes, we construct Markov-modulated multi-

dimensional Levy´ processes to model multiple dependent degradation processes under dynamic environments. The dynamic environment is modeled by a Markov process. Under a certain environment state, the evolution of multiple degradation processes is modeled by a multi-dimensional Levy´ subordinator constructed by the Levy´ copula. All the degradation processes are simultaneously modulated by the dynamic environment. When the environment state switches, there are instantaneous random jumps in all degradation processes.

We develop Fokker-Planck equations and study the ﬁrst passage time for such processes.

Numerical examples are used to illustrate our models in lifetime analysis.

6.1 Introduction

In practice, it is common to observe multiple degradation processes in a component

(e.g., multiple crack growth on a metal surface) or in a multi-component system, where

108 each component is subjected to degradation. All the degradation processes of the system should be considered to describe the health state of the system. These degradation processes are naturally dependent due to the internal features (e.g., mechanical, thermal, electrical, or chemical) and the exposure to the same external environment conditions (e.g., temperature, pressure, humidity, or vibration). These internal and external covariates can be instantaneous, and their presence and intensities can change over time in an unpredictable manner, resulting in randomly changing states of the degradation. Multiple dependent degradation processes can be described using multi-dimensional stochastic processes that have interesting characteristics corresponding to the physical degradation processes. In this chapter, we construct Markov-modulated multi-dimensional Levy´ processes for modeling multiple dependent degradation processes, develop the Fokker-Planck equations, and study the ﬁrst passage time for such processes.

Chapters 3 and 4 developed general models using Levy´ processes and Levy´ driven non-Gaussian OU processes, in order to handle internally-induced stochastic uncertainties with complex jumps and to ﬂexibly ﬁt degradation data series. Incorporating the stochastic uncertainties from dynamic environments, Chapter 5 developed models by constructing general Markov additive processes. All these models focus on a single degradation process without concerning the dependence. [133] used a multi-dimensional Wiener process to model the degradation, in which the dependence among degradation was described by a covariance matrix. However, the Wiener-based models cannot handle the jumps in degradation. As the covariance can only describe the linear dependence, copulas that capture the whole dependence structure (various non-linear and linear forms of dependence among

109 random variables) are good tools to use. The concept of copulas was introduced by [185].

Copulas allow to separate the dependence structure of a random vector from its univariate margins. They provide a complete characterization of the possible dependence structures of a random vector with fixed margins. They can be used to construct multi-dimensional distributions with specified dependence and arbitrary marginal laws. [202] studied a bivariate non-stationary gamma degradation process and characterized the dependence by a copula function. [203] modeled the dependent competing risks with multiple degradation processes and random Poisson shocks using time-varying copulas. However, these gamma- and Poisson-based models are not flexible in general, and the randomness in the environment process and its effects on degradation are not considered.

In this chapter, we use Markov-modulated multi-dimensional Levy´ processes to construct the stochastic models for multiple dependent degradation processes. Under a certain environment state, the degradation evolution evolves as a multi-dimensional Levy´ subordinator constructed by a Levy´ copula, which captures the internally-induced stochastic uncertainties, jumps and dependence. The multi-dimensional Levy´ subordinator is modulated by a Markov process that is used to describe the dynamic environment. When the environment state switches, there are instantaneous random jumps in all degradation processes. This Markov-modulated process can handle the externally-induced stochastic uncertainties, jumps and dependence. The system fails when any of the degradation processes hits a certain failure threshold. The lifetime of the system can be defined as the first passage time of the Markov-modulated multi-dimensional Levy´ process of interest. We develop Fokker-Planck equations and study the first passage time for such processes.

110 The organization of this chapter is as follows. Section 6.2 describes the model

construction. In Section 6.3, the Fokker-Planck equations of general Markov-modulated

multi-dimensional Levy´ processes are derived. Section 6.4 studies the distribution of the

ﬁrst passage time of general Markov-modulated multi-dimensional Levy´ processes, and

derives the explicit expressions of reliability function and lifetime moments. Numerical

examples are illustrated in Section 6.5, and conclusions are given in Section 6.6.

6.2 Preliminaries

In this section, we introduce some fundamentals related to multi-dimensional Levy´

processes and Levy´ copulas. Then we describe model construction based on Markov-

modulated multi-dimensional Levy´ processes, capturing the complexities of a real degra-

dation phenomenon under dynamic environments.

6.2.1 Multi-dimensional Levy´ Processes

˜ Based on the Levy-It´ oˆ decomposition, a K-dimensional Levy´ subordinator Xs(t) =

˜ ˜ ˜ [Xs1(t), Xs2(t), ··· , XsK (t)] can be expressed as

Z Z ˜ Xs(t) = bt + y J(t, dy) − ν(t, dy) + yJ (t, dy), |y|<1 |y|≥1

R where b = [b1, b2, ··· , bK ] ≥ 0, y = [y1, y2, ··· , yK ] ≥ 0, bt − |y|<1 yν (t, dy) ≥ 0, J is

K a Poisson random measure on R+ × R+ , and ν (t, dy) = ν (dy) t is the mean of J (t, dy) R satisfying K min{1, |y|}ν (dy) < ∞. ν (dy) is a K-dimensional Levy´ measure that can R+ be constructed by Levy´ copulas.

111 The range of a K-dimensional Levy´ measure is on [0, ∞]K , while the range of a K- dimensional probability measure is on [0, 1]K . Thus it is inconvenient to model the dependence using the copulas of probability distributions. To overcome this, the Levy´ copulas were studied by [55, 109]. The dependence among components of a multi-dimensional

Levy´ subordinator can be completely characterized by a Levy´ copula, a function that has the similar properties to the ordinary copula but is deﬁned on a different domain.

¯ ¯ ¯ ¯ Deﬁnition 6.1 ([55]). Let S1, S2, ··· , SK be nonempty subsets of R = R∪{∞}∪{−∞} ,

¯ ¯ ¯ and F be a real function of K variables such that Dom F = S1 × S2 × · · · × SK . For a ≤ b (ak ≤ bk, k = 1, 2, ··· ,K), let B = [a, b] be a K-box whose vertices are in Dom F .

Then the F -volume of B is deﬁned by

X VF (B) = sgn(c)F (c), where the sum is taken over all vertices c of B, and sgn(c) is    1 if ck = ak for an even number of vertices, sgn(c) =   −1 if ck = ak for an odd number of vertices.

Deﬁnition 6.2 ([55]). 1) A real function F of K variables is called K-increasing if

VF (B) ≥ 0 for all K-boxes B whose vertices lie in Dom F . 2) Suppose that the domain

¯ ¯ ¯ ¯ of F is S1 × S2 × · · · × SK where each Sk has a smallest element ak. F is said to be grounded if F (t) = 0 for all t in Dom F such that tk = ak for at least one k. 3) If each

¯ Sk is nonempty and has a greatest element bk, then (one-dimensional) margins of F are

¯ functions Fk with Dom Fk = Sk deﬁned by Fk (x) = F (b1, ··· , bk−1, x, bk+1, ··· , bK ) for

¯ all x in Sk.

112 Let F be a K-dimensional distribution function F (x1, x2, ··· , xK ) = P (X1 ≤

x1,X2 ≤ x2, ··· ,XK ≤ xK ), with margins F1,F2, ··· ,FK , then there exists an K-

dimensional copula C such that F (x1, x2, ··· , xK ) = C(F1(x1),F2(x2), ··· ,FK (xK )).

For the Levy´ copula, the role of distribution function is played by the tail integral. We

consider the Levy´ copulas for Levy´ subordinators with Levy´ measures concentrated on

[0, ∞]K .

Deﬁnition 6.3 ([55]). A K-dimensional tail integral is a function U : [0, ∞]K → [0, ∞]

such that

1. (−1)K U is a K-increasing function;

2. U is equal to zero if one of its arguments is equal to ∞;

3. U is ﬁnite everywhere except at zero and U (0, ··· , 0) = ∞.

The tail integral of a Levy´ measure is

K U (x1, x2, ··· , xK ) = ν ([x1, ∞] × [x2, ∞] × · · · × [xK , ∞]) , x1, x2, ··· xK ∈ [0, ∞] .

The margins of a tail integral is

Uk (xk) = U (0, ··· , 0, xk, 0, ··· , 0) .

The Levy´ copulas link multi-dimensional tail integrals to their margins in the same way as the copulas link the distribution functions to their margins.

Deﬁnition 6.4 ([55]). A K-dimensional positive Levy´ copula is a K-increasing grounded

K function C : [0, ∞] → [0, ∞] with margins Ck, k = 1, 2, ··· ,K, which satisfy Ck (u) = u for all u in [0, ∞].

113 Lemma 6.1 ([55]). Let U be the tail integral of a K-dimensional Levy´ process with pos-

itive jumps, and let U1,U2, ··· ,UK be the tail integrals of its components. Then there exists a K-dimensional positive Levy copula C, such that for all vectors [x1, x2, ··· , xK ]

K in R+ ,

U (x1, x2, ··· , xK ) = C (U1 (x1) ,U2 (x1) , ··· ,UK (xK )) .

If U1,U2, ··· ,UK are all continuous then C is unique; otherwise, C is uniquely deter-

mined on Ran U1 × · · · × Ran UK . Conversely, if C is a K-dimensional positive Levy´

copula and U1,U2, ··· ,UK are tail integrals of Levy´ measure on [0, ∞], then the function

C deﬁned above is the tail integral of a K-dimensional Levy´ process with positive jumps

having marginal tail integrals U1,U2, ··· ,UK .

6.2.2 Model Construction

We consider a system consisting of multiple subsystems or components. Any subsys-

tem/component can be subjected to one or more degradation processes with random jumps.

All the degradation processes on different subsystems/components are modulated by the

same environment process. To model the evolution of the multiple degradation processes,

we propose to use a Markov-modulated multi-dimensional Levy´ process {X (t) ,E (t)}, in which the cumulative degradation by time t is represented by a nondecreasing continuous time cadl` ag` (right continuous with left limits) multi-dimensional stochastic process

X(t) = [X1(t),X2(t), ··· ,XK (t)], and the environment process is represented by a temporally homogeneous continuous time cadl` ag` Markov process E(t) with ﬁnite state space

114 P Ξ = {0, 1, ··· , n}. Let G = (rij), rii = − rij, i, j ∈ Ξ denote the transition rate matrix j6=i (inﬁnitesimal generator matrix) of E(t).

The evolution of X(t) depends on the states of E(t). Xk(t), k = 1, 2, ··· ,K e- volves as a nondecreasing Levy´ process (i.e., Levy´ subordinator) when the state of E(t) is unchanged. When E(t) = i ∈ Ξ, we have bk(E(t)) = bk(i), and νk(E(t), dyk) =

νk(i, dyk). All the degradation processes are interdependent, which can be modeled by a K-dimensional Levy´ measure ν(i, dy), y = [y1, y2, ··· , yK ], constructed by a Levy´ copula and the marginal Levy´ measures νk(i, dyk). In practice, the changes of environment states, such as an instantaneous temperature increase or decrease, can induce certain damages to the system, modeled by the jumps in the degradation process. Therefore, we assume there is a random nonnegative jump in each Xk(t), k = 1, 2, ··· ,K when the state of E(t) changes. When E(t) changes from state i to state j, if the jumps in different degradation processes are independent, the distribution of the jump in Xk(t) is denoted as Dijk(zk), k = 1, 2, ··· ,K, deﬁned on R+; if the jumps in different degradation processes are dependent, the distribution of the jump is a K-dimensional distri-

K bution Dij(z), z = [z1, z2, ··· , zK ], deﬁned on R+ , which can be constructed by a cop-

ula and Dijk(zk), k = 1, 2, ··· ,K. For i = j, Dijk(dzk) = δzk (0), k = 1, 2, ··· ,K, and Dij(dz) = δz(0), where δ is a Dirac delta function. We assume the initial state

X(0) = 0,E(0) = 0 a.s., but it is easy to generalize the formulae with X(0) = c, E(0) =

K l, c ∈ R+ , l ∈ Ξ.

115 The degradation can be expressed as

Z t Z t Z X(t) = b(E(ξ−))dξ + y J (E(ξ−), dξ, dy) − ν (E(ξ−), dy) dξ 0 0 |y|<1 (6.1) Z t Z X + yJ (E(ξ−), dξ, dy) + ME(ξ−),E(ξ), 0 |y|≥1 ξ∈[0,t] where b(E(ξ−)) = [b1(E(ξ−)), b2(E(ξ−)), ··· , bK (E(ξ−))] ≥ 0, y = [y1, y2, ··· , yK ] ≥

0, and ME(ξ−),E(ξ) is a K-dimensional random variable following the K-dimensional distribution DE(ξ−),E(ξ)(z), which is independent of E(ξ), for all ξ ∈ [0, t].

Remark 6.1. Xk(t), k = 1, 2, ··· ,K can be expressed as

Z t Z t Z Xk(t) = bk(E(ξ−))dξ + yk Jk (E(ξ−), dξ, dyk) − νk (E(ξ−), dyk) dξ 0 0 0

Each component of the K-dimensional Levy´ subordinator is represented by a one- dimensional Levy´ measure νk. The K-dimensional Levy´ measure ν in (6.1) can be constructed by a Levy´ copula combined with νk, k = 1, 2, ··· ,K. The DE(ξ−),E(ξ)(z) can also be constructed by a copula of the distribution function.

116 X(t) Li Lj Lk Ll Lm X1(t)

X2(t)

t i k l m E(t) j

Figure 6.1: A sample path of Markov-modulated two-dimensional Levy´ process with random jumps when the environment states change

The internally- and externally-induced stochastic properties, dependence and complex jump mechanisms are integrally handled by the general model in (6.1). For internally-induced complexities under a certain state of E(t), X(t) is modeled by a multi- dimensional Levy´ process constructed by a Levy´ copula. For externally-induced complexities during different states of E(t), X(t) evolves in different patterns with different jump mechanisms. In addition, instantaneous nonnegative jumps induced by the change in E(t) are also properly modeled (see Figure 6.1). As illustrated in Figure 6.1, when

E(t) = i, j, k, l or m, X(t) evolves as a two-dimensional Levy´ process Li,Lj,Lk,Ll or

117 Lm, respectively.

6.3 Fokker-Planck Equations

Let p(x, i, t) be the probability density function for the Markov-modulated K-

dimensional Levy´ process {X(t),E(t)}, and its Fokker-Planck equation is

∂p(x, i, t) = L∗p(x, i, t), ∂t

∗ where x = [x1, x2, ··· , xK ], and L is the adjoint operator of the inﬁnitesimal generator

of {X(t),E(t)}. The Fokker-Planck equation is derived and presented in Theorem 6.1.

Theorem 6.1. For the Markov-modulated multi-dimensional Levy´ process {X(t),E(t)} described in Section 6.2, the Fokker-Planck equation is

∂p(x, i, t) ∂p(x, i, t) X Z = −hb(i), i + rji p (x − z, j, t)Dji(dz) ∂t ∂x K j∈Ξ R+ Z ∂p(x, i, t) + p(x − y, i, t) − p(x, i, t) + I|y|<1hy, i ν (i, dy), K ∂x R+ where p(x, i, t) is the probability density function of {X(t),E(t)}, x = [x1, x2, ··· , xK ],

y = [y1, y2, ··· , yK ], z = [z1, z2, ··· , zK ], x − y = [x1 − y1, x2 − y2, ··· , xK − yK ],

∂p(x,i,t) ∂p(x,i,t) ∂p(x,i,t) ∂p(x,i,t) x − z = [x1 − z1, x2 − z2, ··· , xK − zK ], = [ , , ··· , ], b(i) = ∂x ∂x1 ∂x2 ∂xK

K K [b1(i), b2(i), ··· , bK (i)] is a constant on R+ , and ν is a Levy´ measure on R+ .

118 For the two-dimensional case:

∂p(x1, x2, i, t) ∂t

∂p(x1, x2, i, t) ∂p(x1, x2, i, t) = −b1(i) − b2(i) ∂x1 ∂x2 X Z + rji p (x1 − z1, x2 − z2, j, t)Dji(dz1dz2) 2 j∈Ξ R+ Z + (p(x1 − y1, x2 − y2, i, t) − p(x1, x2, i, t))ν (i, dy1dy2) 2 R+ Z ∂p(x , x , i, t) ∂p(x , x , i, t) + I√ y 1 2 + I√ y 1 2 ν (i, dy dy ). y2+y2<1 1 y2+y2<1 2 1 2 2 1 2 ∂x 1 2 ∂x R+ 1 2 Proof. The proof is similar to that of Theorem 5.1 in Chapter 5, with all the operations on

K R+ .

Consider a situation where the dependence is only due to the common environment,

i.e., all the degradation processes Xk, 1 ≤ k ≤ K are independent under a certain envi-

ronment state. In this case, the Fokker-Planck equation for {X(t),E(t)} is provided in

Corollary 6.1, without constructing the multi-dimensional Levy´ measure.

Corollary 6.1. For the Markov-modulated multi-dimensional Levy´ process {X(t),E(t)}, when Xk, k = 1, 2, ··· ,K are independent when E(t) is unchanged, the Fokker-Planck

equation is

∂p(x, i, t) ∂p(x, i, t) X Z = −hb(i), i + rji p (x − z, j, t)Dji(dz) ∂t ∂x K j∈Ξ R+ X Z ∂p(x, i, t) + p(x − y˜k, i, t) − p(x, i, t) + I0

y˜k = [0, ··· , 0, yk, 0, ··· , 0], x − y˜k = [x1, x2, ··· , xk − yk, ··· , xK ], x − z =

119 ∂p(x,i,t) ∂p(x,i,t) ∂p(x,i,t) ∂p(x,i,t) [x1 − z1, x2 − z2, ··· , xK − zK ], = [ , , ··· , ], b(i) = ∂x ∂x1 ∂x2 ∂xK

K [b1(i), b2(i), ··· , bK (i)] is a constant on R+ , and νk is a Levy´ measure on R+.

For the two-dimensional case:

∂p(x1, x2, i, t) ∂t

∂p(x1, x2, i, t) ∂p(x1, x2, i, t) = −b1(i) − b2(i) ∂x1 ∂x2 X Z + rji p (x1 − z1, x2 − z2, j, t)Dji(dz1dz2) 2 j∈Ξ R+ Z ∂p(x1, x2, i, t) + p(x1 − y1, x2, i, t) − p(x1, x2, i, t) + I0

6.4 Reliability Function and Lifetime Moments

A system fails when any degradation process Xk(t) exceeds a failure threshold

xk, k = 1, 2, ··· ,K. The lifetime of the system and its moments are deﬁned respectively

n n as Tx = inf{t : X1(t) > x1,X2(t) > x2, ··· ,XK (t) > xK },M(T , x) = E(Tx ). The

reliability function can be deﬁned as

R (x, t) = P (Tx ≥ t) = P (X1 (t) ≤ x1,X2 (t) ≤ x2, ··· ,XK (t) ≤ xK ) = FX(t) (x) .

In this section, for multiple dependent degradation processes under the dynamic environment described by the Markov-modulated multi-dimensional Levy´ process

{X(t),E(t)}, we derive the explicit expressions of R (x, t) and lifetime moments

M(T n, x) in terms of Laplace transform, represented by the inﬁnitesimal generator matrix and the Levy´ measure. The results are presented in Theorems 6.2 and 6.3.

120 Theorem 6.2. For multiple dependent degradation processes under the dynamic environment that is described by the Markov-modulated multi-dimensional Levy´ process

{X(t),E(t)} in Section 6.2, the Laplace expression of reliability function is

RLL(u, ω) = u−1[1, 0, ··· , 0][A˘ − B˘]−1[1, 1, ··· , 1]T ,

˘ ∗ R −hu,yi where A is a diagonal matrix with ω + hb (i), ui − K e − 1 ν (i, dy) as diagonal R+ ˘ L −1 −1 ∗ entries, and B = [rijdij], i, j ∈ Ξ. In addition, u = (u1u2 ··· uK ) , b (i) ≥ 0, ν is

K the Levy´ measure on R+ , rij, i, j ∈ Ξ are entries of the inﬁnitesimal generator matrix of

L R −hu,zi E(t), dji(u) = K e Dji(dz), [1, 0, ··· , 0] is a vector of size n + 1, where the ﬁrst R+ element is 1 and all others are 0, and [1, 1, ··· , 1] is a vector of size n + 1, where all the elements are 1.

Proof. For the Fokker-Planck equation of {X(t),E(t)} in Theorem 6.1, we do Laplace transform of p(x, i, t) w.r.t. t and x for both sides, then

LL ωp (u, i, ω) − Ii=0 Z LL X −hu,zi LL = −hb(i), uip (u, i, ω) + rji e p (u, j, ω)Dji(dz) K j∈Ξ R+ Z −hu,yi LL LL LL + e p (u, i, ω) − p (u, i, ω) + I|y|<1hy, uip (u, i, ω) ν (i, dy) . K R+

∗ R Let b (i) = b(i) − |y|<1 yν (i, dy),

Z ! LL ∗ −hu,yi LL ωp (u, i, ω) − Ii=0 = −hb (i), ui + e − 1 ν (i, dy) p (u, i, ω) K R+ Z X −hu,zi LL + rji e Dji(dz)p (u, j, ω). K j∈Ξ R+

121 L R −hu,zi Let dji(u) = K e Dji(dz), then the matrix form is R+

pLL(u, ω)[A˘ − B˘] = [1, 0, ··· , 0],

where pLL(u, ω) = [pLL(u, 0, ω), pLL(u, 1, ω), ··· , pLL(u, n, ω)], A˘ is a diagonal matrix

∗ R −hu,yi ˘ L with diagonal entries ω+hb (i), ui − K e − 1 ν (i, dy), and B = [rijdij], i, j ∈ Ξ. R+ We obtain

RLL(u, ω) = u−1[1, 0, ··· , 0][A˘ − B˘]−1[1, 1, ··· , 1]T .

We use Theorem 6.2 to derive the Laplace expression for the moments of lifetime Tx

as Theorem 6.3.

Theorem 6.3. For multiple dependent degradation processes under the dynamic en-

vironment that is described by the Markov-modulated multi-dimensional Levy´ process

{X(t),E(t)} in Section 6.2, the Laplace expression of lifetime moments is

L n −1 −n T M (T , u) = n!u [1, 0, ··· , 0][A˘ 0 − B˘] [1, 1, ··· , 1] ,

˘ ∗ R −hu,yi where A0 is a diagonal matrix with hb (i), ui − K e − 1 ν (i, dy) as diagonal R+ ˘ L −1 −1 ∗ entries, and B = [rijdij], i, j ∈ Ξ. In addition, u = (u1u2 ··· uK ) , b (i) ≥ 0, ν is the Levy´ measure, rij, i, j ∈ Ξ are entries of the inﬁnitesimal generator matrix of E(t),

L R −hu,zi dji(u) = K e Dji(dz), [1, 0, ··· , 0] is a vector of size n + 1, where the ﬁrst element R+ is 1 and all others are 0, and [1, 1, ··· , 1] is a vector of size n + 1, where all the elements are 1.

122 Proof. The proof is similar to that of Theorem 5.3 in Chapter 5.

For the special case, assuming Xk, k = 1, 2, ··· ,K are independent when E(t) is unchanged, Corollaries 6.2 and 6.3 give the Laplace expressions of reliability function and lifetime moments, respectively.

Corollary 6.2. For multiple dependent degradation processes under the dynamic environment that is described by the Markov-modulated multi-dimensional Levy´ process

{X(t),E(t)}, when Xk, k = 1, 2, ··· ,K are independent when E(t) is unchanged, the

Laplace expression of reliability function is

RLL(u, ω) = u−1[1, 0, ··· , 0][A´ − B˘]−1[1, 1, ··· , 1]T ,

∗ P R −ukyk where A´ is a diagonal matrix with diagonal entries ω+hb (i), ui− (e − 1)νk(i, R+ k dyk).

Corollary 6.3. For multiple dependent degradation processes under the dynamic environment that is described by the Markov-modulated multi-dimensional Levy´ process

{X(t),E(t)}, when Xk, k = 1, 2, ··· ,K are independent when E(t) is unchanged, the

Laplace expression of lifetime moments is

L n −1 −n T M (T , u) = n!u [1, 0, ··· , 0][A´ 0 − B˘] [1, 1, ··· , 1] ,

∗ P R −ukyk where A´ 0 is a diagonal matrix with diagonal entries hb (i), ui − (e − 1)νk(i, R+ k

dyk).

123 6.5 Numerical Examples

To illustrate our models, we consider two cases of a Markov-modulated two-

dimensional Levy´ process {X1 (t) ,X2 (t) ,E (t)}:

Case 1: X1(t) and X2(t) are independent when E(t) is unchanged;

Case 2: X1(t) and X2(t) are dependent when E(t) is unchanged.

We use a Markov process with two states {0, 1} to model the environment, and its

inﬁnitesimal generator matrix is   r r  00 01  G =   .   r10 r11

We assume the jump sizes in X1(t) and X2(t) induced by the environment state change are independent. For both X1(t) and X2(t), we use a Levy´ distribution to model the jumps when the environment switches from state 0 to state 1:  q  ξ exp(− ξ )  2π 2(z−$)  3 for z > $ > 0 2 D01(dz) = (z − $)    0 otherwise,

√ L −u$− 2uξ and then d01(u) = e . We use a gamma distribution to model the jumps when the

environment switches from state 1 to state 0:

βαzα−1e−βz D (dz) = , z > 0, 10 Γ(α)

L β α and then d10(u) = ( β+u ) .

124 ˜ For Case 1, we use the Levy´ measure to model Xk˜(t), k = 1, 2 under the environment

state i = 0, 1:

κki˜ 1 ν˜(i, dy˜) = dy˜, k k κki˜ +1 k Γ(1 − κki˜ ) y k˜

where y > 0, 0 < κki˜ < 1, which represents a positive stable process PS(κki˜ ).

For Case 2, we construct a two-dimensinal Levy´ process based on its marginal Levy´

−ϑ process PS(κki˜ ) and a parametric family of Levy´ copula under generator g (z) = z , ϑ >

0. Since g(z) is a strictly decreasing convex function from [0, ∞] to [0, ∞], and g (0) =

−1 −1 −ϑ −ϑ ϑ ∞, g (∞) = 0,Cϑ (x, y) = g (g (x) + g (y)) = x + y is a family of Levy´ copula and is sufﬁciently smooth [55]. The two-dimensional Levy´ density is

2 0 ∂ Cϑ (y1, y2) 0 0 ν (x1, x2) = |y1=U1(x1),y2=U2(x2)ν1 (x1) ν2 (x2) ∂y1∂y2 −1 2 −ϑ −ϑ ϑ ∂ y1 + y2 0 0 = |y1=U1(x1),y2=U2(x2)ν1 (x1) ν2 (x2) ∂y1∂y2 − 1 −2 −ϑ−1 −ϑ−1h −ϑ −ϑi ϑ 0 0 = (1 + ϑ)[U1 (x1)] [U2 (x2)] [U1 (x1)] + [U2 (x2)] ν1 (x1) ν2 (x2) Z ∞ −ϑ−1Z ∞ −ϑ−1 0 0 = (1 + ϑ) ν1 (x1) dx1 ν2 (x2) dx2 x1 x2 − 1 −2 "Z ∞ −ϑ Z ∞ −ϑ# ϑ 0 0 0 0 × ν1 (x1) dx1 + ν2 (x2) dx2 ν1 (x1) ν2 (x2) . x1 x2

For PS(κki˜ ):

Z ∞ Z ∞ 0 κki˜ 1 1 1 ν˜ (i, y˜) dy˜ = dy˜ = κ˜ , k k k κki˜ +1 k ki y y Γ(1 − κki˜ ) y Γ(1 − κki˜ ) y k˜ k˜ k˜ k˜

125 then we have

0 1 1 −ϑ−1 1 1 −ϑ−1 ν (i, y1, y2) = (1 + ϑ)[ κ1i ] [ κ2i ] Γ(1 − κ1i) y1 Γ(1 − κ2i) y2 1 − ϑ −2 1 1 −ϑ 1 1 −ϑ × [ κ1i ] + [ κ2i ] Γ(1 − κ1i) y1 Γ(1 − κ2i) y2 κ 1 κ 1 × 1i 2i (6.2) κ1i+1 κ2i+1 Γ(1 − κ1i) y1 Γ(1 − κ2i) y2

κ1iκ2i κ1i ϑ κ2i ϑ = (1 + ϑ) [Γ(1 − κ1i)y1 ] [Γ(1 − κ2i)y2 ] y1y2

− 1 −2 κ1i ϑ κ2i ϑ ϑ × [Γ(1 − κ1i)y1 ] + [Γ(1 − κ2i)y2 ] .

When the degradation evolution under the dynamic environment can be described by the Markov-modulated two-dimensional Levy´ process in Case 1, the Laplace expression of lifetime moments based on Corollary 6.3 is  −n ˜ ˜ W00 W01 L n −1   T M (T , u1, u2) = n!(u1u2) [1, 0]   [1, 1] ,  ˜ ˜  W10 W11 where

˜ ∗ ∗ κ10 κ20 L L W00 = b1(0)u1 + b2(0)u2 + u1 + u2 − r00d00(u1)d00(u2),

˜ L L W01 = −r01d01(u1)d01(u2),

˜ L L W10 = −r10d10(u1)d10(u2),

˜ ∗ ∗ κ11 κ21 L L W11 = b1(1)u1 + b2(1)u2 + u1 + u2 − r11d11(u1)d11(u2).

When the degradation evolution under the dynamic environment can be described by the Markov-modulated two-dimensional Levy´ process in Case 2, the Laplace expression

126 of lifetime moments based on Theorem 6.3 is,  −n W00 W01 L n −1   T M (T , u1, u2) =n!(u1u2) [1, 0]   [1, 1] ,   W10 W11

where

Z ∗ ∗ −u1y1−u2y2 L L W00 = b1(0)u1 + b2(0)u2 − e − 1 ν (0, dy1dy2) − r00d00(u1)d00(u2), 2 R+

L L W01 = −r01d01(u1)d01(u2),

L L W10 = −r10d10(u1)d10(u2),

Z ∗ ∗ −u1y1−u2y2 L L W11 = b1(1)u1 + b2(1)u2 − e − 1 ν (1, dy1dy2) − r11d11(u1)d11(u2). 2 R+

The speciﬁc values for the parameters are given in Table 6.1. Based on (6.2), we have Z −u1y1−u2y2 e − 1 ν (i, dy1dy2) 2 R+ Z κ κ −u1y1−u2y2 1i 2i κ1i ϑ κ2i ϑ = e − 1 (1 + ϑ) [Γ(1 − κ1i)y1 ] [Γ(1 − κ2i)y2 ] 2 y y R+ 1 2

− 1 −2 κ1i ϑ κ2i ϑ ϑ × [Γ(1 − κ1i)y1 ] + [Γ(1 − κ2i)y2 ] dy1dy2 √ Z −u1y1−u2y2 −5/2 = (3/4 π) e − 1 (y1 + y2) dy1dy2 2 R+ u3/2 − u3/2 u1/2u1/2 = − 1 2 = 1 2 − u1/2 − u1/2. u − u 1/2 1/2 1 2 1 2 u1 + u2

The system fails when X1(t) exceeds the threshold x1 or X2(t) exceeds the threshold x2. The inversion algorithms for Laplace transform were implemented to invert the

Laplace expressions in order to compute the values of the mean of the ﬁrst passage time.

127 Figure 6.2 and Figure 6.3 illustrate the mean of the ﬁrst passage time w.r.t. the failure threshold for Case 1 and Case 2, respectively. The mean of the ﬁrst passage time in Case 2 is larger than that in Case 1 at the same threshold using a parametric family of Levy´ copula under generator g (z) = z−ϑ, ϑ > 0.

Table 6.1: Parameter values for models in Chapter 6

∗ x1, x2 [0,30] β 50 b2(0) 0.01 ∗ r00 = −r01 -10 κ10 0.5 b1(1) 0.05 ∗ r10 = −r11 15 κ11 0.5 b2(1) 0.06 ξ 0.0001 κ20 0.5 ϑ 2 $ 0.01 κ21 0.5 ∗ α 0.2 b1(0) 0.02

3.5

2.5

2 Mean 1.5

0.5

0 30 30 20 25 20 10 15 10 5 0 0 x2 x1

Figure 6.2: The mean of the ﬁrst passage time w.r.t. failure threshold for Case 1

128 5

Mean 2

0 30 30 20 25 20 10 15 10 5 0 0 x2 x1

Figure 6.3: The mean of the ﬁrst passage time w.r.t. failure threshold for Case 2

6.6 Conclusions

In this chapter, we construct the Markov-modulated multi-dimensional Levy´ process to model the multiple dependent degradation processes with random jumps under dynamic environments. The internally-induced dependence is modeled by the Levy´ copula. The externally-induced dependence is handled by 1) setting the Levy´ measures as a function of the environment process, and 2) simultaneous jumps in all degradation when the environment states change. The law of the ﬁrst passage time is studied by developing the

Fokker-Planck equation for such the process. The distribution and moments are explicitly expressed by the Levy´ measure, the inﬁnitesimal generator matrix and the parameter in the Levy´ copula.

129 Chapter 7

Statistical Inference

Degradation-with-jump measures are time series data sets containing the information

of both continuous and randomly jumping degradation evolution of a system. For this type

of data, we have constructed new and general degradation models. Traditional maximum

likelihood estimation and Bayesian estimation are not convenient for such general jump

processes without closed-form distributions. We propose a systematic statistical method

using linear programming estimators and empirical characteristic functions. The point

estimates of reliability function and lifetime moments are obtained by their explicit ex-

pressions. We also construct bootstrap procedures for the conﬁdence intervals. Simulation

studies for a stable process and a stable driven OU process are performed. In the case study,

we use a general Levy´ process to ﬁt the battery data and estimate the reliability and lifetime moments of the battery. By integrally analyzing degradation data series embedded jump measures, our work provides efﬁcient and precise results for life characteristics estimation.

7.1 Introduction

With advanced measurement tools such as sensors, degradation data can be measured

and collected effectively and economically, e.g., the Li-ion battery capacity data [28], the

integrated circuit propagation delay data [30], the metal fatigue-crack-growth data [135],

130 and the transistor gain data [206]. The degradation data series over the life cycle reﬂect the

evolution of the system’s health state that contain more information than the sparse failure

time data for reliable systems. Recently, the reliability estimation/prediction based on

degradation measures has gained popularity and become an effective approach, especially

when it is time-consuming and costly to test and collect the failure time data for highly-

reliable systems with advanced and evolving technologies.

In practice, the continuous degradation process commonly experiences complex

jumps due to random damages caused by internal changes and external inﬂuences.

The complex jumps can be consecutive or sporadic, small or large, or their mixture.

Degradation-with-jump measures are time series data sets that are observed from such

degradation phenomena, containing the information of both continuous and randomly

jumping degradation evolution of a system.

For the degradation models constructed using Levy´ or non-Gaussian OU processes, we need to estimate the parameters of the underlying stochastic process using degradation data series, in order to further estimate/predict the reliability characteristics. The certain distribution of independent increments in existing models (Gaussian, Poisson, gamma, or inverse Gaussian) makes the statistical inference straightforward by using the likelihood function or Bayesian approach. For general Levy´ and non-Gaussian OU processes, how-

ever, the traditional maximum likelihood estimation and Bayesian estimation are not con-

venient as the closed-form distributions are not available for such general jump processes.

In this chapter, we develop a systematic approach to estimate reliability characteristics

using degradation-with-jump measures.

131 The organization of this paper is as follows. In Section 7.2, we present the estimation method for Levy´ subordinators. Section 7.3 provides the estimation procedure for non-

Gaussian OU processes. In Section 7.4, we perform simulation studies. Case study for battery degradation data is illustrated in Section 7.5, and conclusions are given in Section

7.6.

7.2 Estimation for Levy´ Degradation Processes

Based on the characteristic function of Levy´ subordinator, we propose to use the cumulant M-estimator (CME) [107] to estimate the parameters in a Levy´ degradation process. The point estimators of life characteristics can be obtained by their explicit expressions. We also construct bootstrap procedures to obtain the conﬁdence intervals for life characteristics.

7.2.1 Point Estimates

Due to the property of independent and identically distributed increments, the CME can achieve a consistent estimator using a single degradation path with enough data points.

When multiple degradation paths are available from the same population, all the paths can be used to do estimation for more accurate results.

A degradation path can be discretized as ~xn = (x1∆, x2∆, . . . , xn∆), where ∆ is the step of the discretely measured data series. We denote Θ as the parameter vector in

η (u) η (u) ≡ η (u; Θ) φ (u) ≡ φ (u; Θ) η (u) = s , i.e., s s , X˜s(1) X˜s(1) . The Levy´ symbol s

132 logφ (u) X˜s(1) , is also called as the cumulant function.

φˆ (u; ~x ) We choose a preliminary estimator X˜s(∆) n , either almost surely

φˆ (u; ~x ) −−→a.s. φ (u; Θ) , X˜s(∆) n X˜s(∆) or in probability

φˆ (u; ~x ) −→P φ (u; Θ) , X˜s(∆) n X˜s(∆) as n → ∞.

Deﬁne wˇ() as an integrable weight function with compact support. wˇ() is symmetric around the origin and is strictly positive on a neighbourhood of the origin. For examples,

−u2 wˇ(u) = Iu∈[−l,l], l > 0, or wˇ(u) = e .

The space of square integrable functions w.r.t wˇ is

Z 2 n 2 o z (w ˇ) = f : R → C|f is measurable and |f(u)| wˇ(u)du < ∞ ,

where C is the complex space.

2 The semi-inner product (f1, f2)wˇ on z (w ˇ) is deﬁned as

Z (f1, f2)wˇ = < f1(u)f2(u)w ˇ(u)du,

where f2(u) is the complex conjugate of f2(u), and

1 2 2 on z (w ˇ) is deﬁned as ||f||wˇ = (f, f)wˇ.

The CME is

ˆ Θn = argmin D(Θ; ~xn), Θ

133 where D(Θ; ~xn) is the weighted difference between cumulants:

D(Θ; ~x ) = ||logφˆ (u; ~x ) − logφ (u; Θ) ||2 n X˜s(∆) n X˜s(∆) wˇ Z = |logφˆ (u; ~x ) − logφ (u; Θ) |2wˇ(u)du X˜s(∆) n X˜s(∆) Z 2 2 = ∆ |ηˆs (u; ~xn) − ηs (u; Θ) | wˇ(u)du.

ˆ With Θn, the point estimators of reliability function and lifetime moments in The-

ˆ ˆ m ˆLL ˆ orems 4.1 and 4.2, RX (x, t) and M(TX , x), can be obtained by inverting RX (u, ω; Θn)

ˆ L m ˆ and M (TX , u; Θn), respectively.

When we have multipe sample paths ~xj = (xj , xj , . . . , xj ), j ∈ {1, 2, ··· ,M}, nj 1∆ 2∆ nj ∆

the CME is

1 2 M Θˆ M = argmin D(Θ; ~x , ~x , ··· , ~x ), P n1 n2 nM nj Θ j=1

D(Θ; ~x1 , ~x2 , ··· , ~xM ) = ||logφˆ u; ~x1 , ~x2 , ··· , ~xM −logφ (u; Θ) ||2 . where n1 n2 nM X˜s(∆) n1 n2 nM X˜s(∆) wˇ

7.2.2 Bootstrap Conﬁdence Intervals

In this section, the sample data are simulated using R(YUIMA) [33]. We construct

m conﬁdence intervals for RX (x, t) and M(TX , x) based on bootstrap simulation with the following steps:

ˆ ˆ 1. Obtain Θn or Θ M by using the CME for one or M sample paths. The sample P nj j=1 ˜ paths can be the real data in practice or the simulated data from Xs(t; Θ) by setting

an initial value for Θ.

134 ˜ ˆ 2. Generate one sample path ~x¨n = (¨x1∆, x¨2∆,..., x¨n∆) from Xs(t; Θn), or M sample

~ j j j j ˜ ˆ paths x¨ = (¨x , x¨ ,..., x¨ ), j ∈ {1, 2, ··· ,M} from Xs(t; Θ M ). nj 1∆ 2∆ nj ∆ P nj j=1

¨ˆ ¨ˆ ~ 3. Get the bootstrap estimates Θn or Θ M using the CME, based on x¨n = (¨x1∆, x¨2∆, P nj j=1 j ..., x¨ ) or ~x¨ = (¨xj , x¨j ,..., x¨j ), j ∈ {1, 2, ··· ,M}. n∆ nj 1∆ 2∆ nj ∆

LL L ˆ¨ ¨ˆ ˆ¨ m ¨ˆ 4. Obtain the bootstrap estimates R (u, ω; Θn) and M (T , u; Θn), or    X  X LL L ˆ¨ ¨ˆ ˆ¨ m ¨ˆ R u, ω; Θ M and M T , u; Θ M . X  P   X P  nj nj j=1 j=1

LL L ˆ¨ ¨ˆ ˆ¨ m ¨ˆ 5. Repeat Steps 2-4 K times to obtain RX;k(u, ω; Θn) and M k (TX , u; Θn), or     LL L ˆ¨ ¨ˆ ˆ¨ m ¨ˆ R u, ω; Θ M and M T , u; Θ M , 1 ≤ k ≤ K. X;k  P  k  X P  nj nj j=1 j=1

LL L ˆ¨ ˆ¨ 6. Implement the inversion algorithm for Laplace transform to invert RX;k and M k ,

ˆ¨ ˆ¨ m obtaining RX;k(x, t) and M k(TX , x), 1 ≤ k ≤ K.

ˆ¨ 7. Sort RX;k(x, t), 1 ≤ k ≤ K in ascending order for each x and t, obtaining

ˆ¨ ˆ¨ m RX;[k](x, t), 1 ≤ k ≤ K. Sort M k(TX , x), 1 ≤ k ≤ K in ascending order for

ˆ¨ m each x, obtaining M [k](TX , x), 1 ≤ k ≤ K.

ˆ¨ ˆ¨ 8. Compute the 100(1−δ)% conﬁdence intervals for RX (x, t): [RX;[l˙](x, t), RX;[u ˙ ](x, t)]

m ˆ¨ m ˆ¨ m ˙ −1 and for M(TX , x): [M [l˙](TX , x), M [u ˙ ](TX , x)], where l = Φ(2Φ (χ) +

−1 −1 −1 Φ (δ/2))K, u˙ = Φ(2Φ (χ) + Φ (1 − δ/2))K, for RX (x, t) P I ˆ¨ ˆ RX;k(x,t)≤RX (x,t) χ = k , K

135 m for M(TX , x), P I ˆ¨ m ˆ m M k(TX ,x)≤M(TX ,x) χ = k , K

ˆ¨ and Φ is the standard normal distribution function used in [135]. RX;[l˙](x, t) and ˆ¨ RX;[u ˙ ](x, t) are approximate pointwise lower and upper one-sided 100(1 − δ/2)%

ˆ¨ m ˆ¨ m biased-corrected conﬁdence bounds for RX (x, t); and M [l˙](TX , x) and M [u ˙ ](TX , x)

are approximate pointwise lower and upper one-sided 100(1 − δ/2)% biased-

m corrected conﬁdence bounds for M(TX , x) [72].

7.3 Estimation for OU Degradation Processes

7.3.1 Point Estimates

An OU degradation path can be discretized as ~yn = (y1∆, y2∆, . . . , yn∆), where

∆ is the step of the discretely measured data series. Since the OU process driven by a

Levy´ subordinator has dependent increments, the CME cannot be directly used to estimate the parameters (α, Θ) in Y (t). Three steps are proposed to obtain the point estimates of

(α, Θ): (1) estimate α, (2) estimate the increments of the background driving process-

Levy´ subordinator using the estimator of α [32], and (3) the CME is activated to estimate

Θ using the estimated increments. When multipe OU degradation paths are available from the same population, all the paths can be used to do estimation using the same procedure.

The discrete OU process can be expressed as

Z i∆ α∆ α(i∆−ξ) ˜ Yi∆ = e Y(i−1)∆ + e dXs(ξ), 1 ≤ i ≤ n, (i−1)∆

136 which is an analogue of the discrete-time ﬁrst-order autoregression processes (AR(1)) with

nonnegative innovations:

Yi∆ = ρY(i−1)∆ + Zi∆, 1 ≤ i ≤ n,

α∆ R i∆ α(i∆−ξ) ˜ ˜ by setting ρ = e > 1, and Zi∆ = (i−1)∆ e dXs(ξ). Since Xs(t) is a Levy´ subordinator, Zi∆ is nonnegatively independent and identically distributed. Taking the advantage of the nonnegativity of the increments of the background driving process, Levy´ subordinator, we choose the following linear programming estimator for ρ:

ρˆn = min yi∆/y(i−1)∆. 1≤i≤n

The estimator for α is αˆn = logρˆn/∆. Assuming ρ > 0 and the distribution function

F of Zi∆ is regularly varying at zero with exponent ϑ1, (i.e., there exists ϑ1 > 0 such that

a.s. F (ax) ϑ1 lim = x , x > 0), [60] showed that ρˆn −−→ ρ and developed the asymptotic distri- a→0 F (x) butions for both stationary and nonstationary cases. This estimator has been further studied for the stationary case of autoregressive processes (when 0 < ρ < 1) [32]. In another way,

ρˆn can be viewed as the solution to the linear programming problem of maximizing the objective function g(ρ) = ρ subject to n linear constraints yi∆ − ρy(i−1)∆ ≥ 0, 1 ≤ i ≤ n,

and therefore, it is called the linear programming estimator [78, 79]. ρˆn is equal to the

maximum likelihood estimator conditioned on Y0 if Zi∆ is exponentially distributed. [159]

showed that ρˆn is strongly consistent for a broad range of F , including both light-tailed

and heavy-tailed distributions.

137 Based on (4.2), the increment of the Levy´ subordinator is

Z i∆ Xi∆ − X(i−1)∆ = Yi∆ − Y(i−1)∆ − α Y (ξ) dξ. (i−1)∆

The estimated increments by the trapezoidal approximation for the integral can be valued as

y + y x − x = y − y − αˆ ∆ i∆ (i−1)∆ , 1 ≤ i ≤ n. i∆ (i−1)∆ i∆ (i−1)∆ n 2

˜ Then we can estimate Θ in Xs(t) by the CME in Section 7.2 using xi∆ −x(i−1)∆, 1 ≤

ˆ i ≤ n. With αˆn and Θn, the point estimators of reliability function and lifetime moments

ˆ ˆ m ˆLL ˆ in Theorems 4.3 and 4.4 are RY (y, t) and M(TY , y) by inverting RY (u, ω;α ˆn, Θn) and

ˆ L m ˆ M (TY , u;α ˆn, Θn), respectively.

When we have multiple sample paths ~yj = (yj , yj , . . . , yj ), j ∈ {1, 2, ··· ,M}, nj 1∆ 2∆ nj ∆

we have:   j j log  min yi∆/y  1≤j≤M (i−1)∆ 1≤i≤nj αˆ M = , P nj ∆ j=1

yj + yj j j j j i∆ (i−1)∆ x − x = y − y − αˆ M ∆ , 1 ≤ j ≤ M, 1 ≤ i ≤ nj, i∆ (i−1)∆ i∆ (i−1)∆ P nj 2 j=1 and

ˆ j j Θ M = argmin D Θ; x − x , 1 ≤ j ≤ M, 1 ≤ i ≤ nj , P i∆ (i−1)∆ nj Θ j=1 where

j j D Θ; xi∆ − x(i−1)∆, 1 ≤ j ≤ M, 1 ≤ i ≤ nj = ||logφˆ u; xj − xj , 1 ≤ j ≤ M, 1 ≤ i ≤ n − logφ (u; Θ) ||2 . X˜s(∆) i∆ (i−1)∆ j X˜s(∆) wˇ

138 7.3.2 Bootstrap Conﬁdence Intervals

m We construct conﬁdence intervals for RY (y, t) and M(TY , y) based on bootstrap simulation. The steps are:

1. Obtain αˆn or αˆ M using the linear programming estimator for one or M sample P nj j=1 paths. The sample paths can be the real data in practice or the simulated data from

Y (t; α, Θ) by setting a value for (α, Θ).

ˆ 2. Estimate the increments of the background driving Levy´ subordinator, obtaining Θn

ˆ or Θ M using the CME. P nj j=1 ~ ˆ 3. Generate one sample path y¨n = (¨y1∆, y¨2∆,..., y¨n∆) from Y (t;α ˆn, Θn) or M sam-

j j j j ˆ ple paths~y¨ = (ÿ , y¨ ,..., y¨ ), j ∈ {1, 2, ··· ,M} from Y (t;α ˆ M , Θ M ). nj 1∆ 2∆ nj ∆ P P nj nj j=1 j=1   ˆ ¨ˆ ˆ ¨ˆ 4. Obtain the bootstrap estimates (α¨n, Θn) or α¨ M , Θ M using the CME, based  P P  nj nj j=1 j=1 j on ~y¨ = (ÿ , y¨ ,..., y¨ ) or ~y¨ = (ÿj , y¨j ,..., y¨j ), j ∈ {1, 2, ··· ,M}. n 1∆ 2∆ n∆ nj 1∆ 2∆ nj ∆

LL L ˆ¨ ¨ˆ ˆ¨ m ¨ˆ 5. Get the bootstrap estimates R (u, ω; αˆ¨n, Θn) and M (T , u; αˆ¨n, Θn), or   Y  Y  LL L ˆ¨ ¨ˆ ˆ¨ m ¨ˆ R u, ω; αˆ¨ M , Θ M and M T , u; αˆ¨ M , Θ M . Y  P P   Y P P  nj nj nj nj j=1 j=1 j=1 j=1

LL L ˆ¨ ˆ ¨ˆ ˆ¨ m ˆ ¨ˆ 6. Repeat Steps 3-5 K times to obtain RY ;k(u, ω; α¨n, Θn) and M k (TY , u; α¨n, Θn), or     LL L ˆ¨ ¨ˆ ˆ¨ m ¨ˆ R u, ω; αˆ¨ M , Θ M and M T , u; αˆ¨ M , Θ M , 1 ≤ k ≤ K. Y ;k  P P  k  Y P P  nj nj nj nj j=1 j=1 j=1 j=1

LL L ˆ¨ ˆ¨ 7. Implement the inversion algorithm for Laplace transform to invert RY ;k and M k ,

ˆ¨ ˆ¨ m obtaining RY ;k(y, t) and M k(TY , y), 1 ≤ k ≤ K.

139 ˆ¨ 8. Sort RY ;k(y, t), 1 ≤ k ≤ K in ascending order for each y and t, obtaining

ˆ¨ ˆ¨ m RY ;[k](y, t), 1 ≤ k ≤ K. Sort M k(TY , y), 1 ≤ k ≤ K in ascending order for

ˆ¨ m each y, obtaining M [k](TY , y), 1 ≤ k ≤ K.

ˆ¨ ˆ¨ 9. Compute the 100(1−δ)% conﬁdence intervals for RY (y, t): [RY ;[l˙](y, t), RY ;[u ˙ ](y, t)]

m ˆ¨ m ˆ¨ m ˙ −1 −1 and for M(TY , y): [M [l˙](TY , y), M [u ˙ ](TY , y)], where l = Φ(2Φ (χ)+Φ (δ/2))K,

−1 −1 u˙ = Φ(2Φ (χ) + Φ (1 − δ/2))K, for RY (y, t) P I ˆ¨ ˆ RY ;k(y,t)≤RY (y,t) χ = k , K

m for M(TY , y), P I ˆ¨ m ˆ m M k(TY ,y)≤M(TY ,y) χ = k . K

ˆ¨ ˆ¨ RY ;[l˙](y, t) and RY ;[u ˙ ](y, t) are approximate pointwise lower and upper one-sided

ˆ¨ m 100(1 − δ/2)% biased-corrected conﬁdence bounds for RY (y, t); and M [l˙](TY , y)

ˆ¨ m and M [u ˙ ](TY , y) are approximate pointwise lower and upper one-sided 100(1 −

m δ/2)% biased-corrected conﬁdence bounds for M(TY , y).

7.4 Simulation Study

To illustrate our models, we use the Levy´ measure

κ 1 ν(dx) = dx, Γ(1 − κ) xκ+1 where x > 0, 0 < κ < 1, which represents a positive stable process PS(κ).

140 Proposition 7.1. The characteristic function of PS(κ) is

πκ πκ φ (u) = exp −tcos |u|κ 1 − itan sgn(u) , X 2 2 √ where i = −1 and sgn(u) is the sign function.

Proof. Z Z iux κ 1 κ −κ −1 −1 iux ηs (u) = e − 1 κ+1 dx = x dx −x + x e R+ Γ(1 − κ) x Γ(1 − κ) R+ κ Z Z 0 κ Z 0 Z = x−κdx e−yxdy = dy x−κe−yxdx Γ(1 − κ) R+ −iu Γ(1 − κ) −iu R+   κ iπκ Z 0  −(−u) e 2 u < 0 = κ yκ−1dy = −(−iu)κ = −iu  κ −iπκ  −u e 2 u > 0   κ πκ πκ  −(−u) cos + isin u < 0 = 2 2  κ πκ πκ  −u cos − isin u > 0 2 2 πκ πκ = −cos |u|κ 1 − itan sgn(u) . 2 2

Remark 7.1. PS(κ) is a class of general stable processes S(κ, β, γ, δ), where κ ∈

(0, 2] is the index parameter, β ∈ [−1, 1] is the skewness parameter, γ > 0 is the scale parameter, and δ is the shift parameter. The Levy´ symbol of S(κ, β, γ, δ) is

κ κ πκ η (u) = −γ |u| 1 − iβtan 2 sgn(u) + iuδ. Thus by setting 0 < κ < 1, β = 1,

κ πκ πκ −1 γ = cos 2 = |1 − itan 2 | , and δ = 0, we get PS(κ).

In the simulation study, we use one degradation path to illustrate the proposed proce-

dure that is also applied for multiple degradation paths. We choose ∆ = 1 without losing

141 n ˆ 1 P iu(xj −xj−1) the generality. The empirical characteristic function φXs(1) (u; ~xn) = n e is j=1 used as the preliminary estimator. The CME of κ is

1 n − 1 − itan πκ sgnu X iu(xj −xj−1) 2 κ 2 κˆ = argmin ||log( e ) − πκ |u| ||wˇ, κ n |1 − itan | j=1 2 √ where we use wˇ(u) = Iu∈[−l,l] for l > 0 and i = −1.

Table 7.1: Results of κˆ for PS(κ)

n mean mean squared error 20 0.8912641 0.003706321 50 0.9029252 0.001141654 100 0.8987184 0.0007026813 Levy density Levy 0.00 0.05 0.10 0.15 0.20

0 5 10 15 20 25

x Figure 7.1: Levy´ density of PS(κ); dashed: estimated when n=25; solid: true

We simulate one path with n data points from PS(κ) by setting κ = 0.9, and perform estimation to obtain κˆ. We repeat the estimation 1000 times to calculate the mean and the mean squared error (MSE) of κˆ. The optimization problem is solved numerically using a

142 quasi-Newton method. Table 7.1 shows the results of κˆ for PS(κ) based on the simulated

data. The mean is closed to 0.9 and the MSE is small. Figure 7.1 shows the estimated

Levy´ density comparing with the true Levy´ density of PS(κ).

Proposition 7.2. The characteristic function of the OU process driven by PS(κ) is

n πκ πκ o φ (u) = exp −(ακ)−1(eακt − 1)cos |u|κ 1 − itan sgn(u) , Y 2 2 √ where i = −1 and sgn(u) is the sign function.

Proof. For the OU process driven by a general Levy´ subordinator, the characteristic func-

tion can be expressed as

Z t Z E eiuY (t) = exp ib∗ueαr + eiueαry − 1 ν (dy) dr . 0 R+

Then for the OU process driven by PS(κ),

We perform the simulation study using the proposed procedures. We simulate one

path with n data points from the OU driven by PS(κ) by setting α = 0.1, κ = 0.9, and perform estimation to obtain α,ˆ κˆ. We repeat the estimation 1000 times to calculate the mean and the mean squared error of αˆ and κˆ, respectively. Table 7.2 shows the results of

143 Table 7.2: Results of αˆ and κˆ for OU driven by PS(κ)

αˆ κˆ n mean mean squared error mean mean squared error 20 0.1101515 0.0001279552 0.8915346 0.006616674 50 0.1087836 9.227509e-05 0.9078137 0.001910659 100 0.1081567 7.684354e-05 0.9060823 0.001126384

αˆ and κˆ for the OU driven by PS(κ) based on the simulated data. Figure 7.2 shows the estimated increments comparing with the true increments of PS(κ). Figure 7.3 shows the estimated marginal Levy´ density comparing with the true marginal Levy´ density of OU driven by PS(κ) (Proposition 7.2). increments 2 4 6 8

0 10 20 30 40 50

Figure 7.2: Increments of the background driving PS(κ); dashed: estimated when n=50; solid: true

144 Levy density Levy 0.000 0.002 0.004 0.006 0.008 0 10 20 30 40 50 60 70

Figure 7.3: Marginal Levy´ density of OU driven by PS(κ); dashed: estimated when n=50; solid: true

0.9

0.8

0.7

0.6

0.5 Reliability

0.4

0.3

0.2

0.1 0 2 4 6 8 10 12 14 16 18 20 t ˜ Figure 7.4: 90% conﬁdence intervals of reliability function for Xs(t)

145 1

0.9

0.8

0.7

0.6

0.5

Reliability 0.4

0.3

0.2

0.1

0 0 2 4 6 8 10 12 14 16 18 20 t Figure 7.5: 90% conﬁdence intervals of reliability function for Y (t)

Figure 7.4 and Figure 7.5 show the conﬁdence intervals for reliability function based on bootstrap simulation. Besides the Levy´ measure used in this example, we can specify different Levy´ measures to ﬁt the corresponding degradation data, in order to construct models and analyze reliability and lifetime characteristics.

7.5 Case Study

In this section, we apply our model to analyze the degradation data of lithium-ion batteries. The data series are from the randomized battery usage data set [28]. We choose the capacity data of four 18650 lithium-ion batteries, which were tested under the room temperature with random charging and discharging current and time (Figure 7.6). We perform statistical test for the presence of jumps using the bipower variation [24]. The p-value

146 is close to zero, which indicates that there are many jumps in the capacity loss processes

of such batteries. In other words, a jump process is suitable to ﬁt these degradation data

series.

Figure 7.6: The capacity losing processes of four 18650 Li-ion batteries

From Figure 7.6, the capacity loss paths have the linear trend. We use a general Levy´ process to ﬁt the data. The corresponding Levy´ measure is

δγ−2κκy−κ−1 exp(− 1 γ2y) ν(dy) = 2 dy, Γ(κ)Γ(1 − κ)

which can cover: 1) the positive tempered stable process, when y, δ > 0, 0 < κ < 1, γ ≥

0; 2) the positive stable process, when y, δ > 0, 0 < κ < 1, γ = 0; 3) the inverse

Gaussian process, when y, δ > 0, 0 < κ < 1, γ = 0.5; and 4) the gamma process, when

y, δ > 0, κ → 0, γ > 0. Using CME, we obtain δˆ = 2.9884776, γˆ = 2.0335391 and

147 κˆ = 0.1511678. The Laplace expression of reliability estimator based on Theorem 4.1 is

−1 LL −1n 1 κˆo Rˆ (u, ω) = u ω − δˆγˆ + δˆ(ˆγ κˆ + 2u) .

Based on Theorem 4.2, the Laplace expression of lifetime moments’ estimator is

−n L n −1n 1 κˆo Mˆ (T , u) = n!u −δˆγˆ + δˆ(ˆγ κˆ + 2u) .

0.9

0.8

0.7

0.6

0.5

Reliability 0.4

0.3

0.2

0.1

0 0 10 20 30 40 50 60 x 1500 Periods

Figure 7.7: Estimation of reliability function when failure threshold is 0.9

148 90

60 Estimated:(0.9,53.9719) (1,60.0680) 50 Real:(0.9,53.5) (1,62.75)

Mean failure time 30

0 0 0.5 1 1.5 Failure threshold

Figure 7.8: Estimation of ﬁrst moments of lifetime w.r.t. failure threshold

Standard deviation 3

0 0 0.5 1 1.5 Failure threshold

Figure 7.9: Estimation of standard deviation of lifetime w.r.t. failure threshold

149 Figure 7.7 shows the estimation of reliability function for the Li-ion batteries, assuming the failure threshold is 0.9. In Figure 7.8, the solid line is the estimated mean failure time w.r.t. the failure threshold; when the failure threshold is 0.9, the estimated mean failure time is 53.9719, while the real mean failure time is 53.5; when the failure threshold is 1, the estimated mean failure time is 60.068, while the real mean failure time is 62.75.

Figure 7.9 illustrates the estimation of standard deviation of failure time w.r.t. the failure threshold.

7.6 Conclusions

In this chapter, we constructed general stochastic models to integrally handle uncertainties and jumps using Levy´ and non-Gaussian OU processes. Our model can ﬁt a great deal of degradation data with jumps (e.g., linear/nonlinear, light/heavy-tailed). We developed systematic procedures for estimating reliability characteristics based on the linear programming estimator, CME and bootstrap simulation. Our new framework provides explicit results for precise reliability analysis.

150 Chapter 8

Summary and Discussions

Reliability of systems is one of the most critical concerns in many ﬁelds including

energy, automotive industry, health, aerospace, and national defense, etc. Degradation-

based reliability analysis becomes popular and efﬁcient for system evaluation during all the

stages of the systems’ life cycle from design and development to ﬁeld deployment. In this

dissertation, we developed a new research framework to integrally handle the complexities

in degradation processes including internally-induced/externally-induced stochastic prop-

erties, complex jump mechanisms and dependence, based on general stochastic processes

including Levy,´ non-Gaussian OU, and Markov additive processes; and developed a new systematic methodology for reliability analysis that provides compact and explicit results for reliability function and lifetime characteristics.

Chapter 3 proposed to model the cumulative degradation with random jumps using a single Levy´ process. Based on the inverse Fourier transform, new closed-forms of reliability function and probability density function for lifetime are provided. The reliability characteristics derived using the traditional convolution approach for common stochastic models such as gamma degradation process with random jumps, is revealed to be a special case of our general model. In our model, we can specify different Levy´ measures to

describe different jump mechanisms in degradation. The reliability function is constructed

and represented by a certain Levy´ measure corresponding to a certain Levy´ degradation

151 process. The calculation for reliability is simple enough to be implemented in practice.

Chapter 4 extended the model using Levy´ driven non-Gaussian OU processes. To obtain explicit results of reliability function and lifetime moments, we proposed to use

Fokker-Planck equations for both Levy´ subordinators and their corresponding OU pro-

cesses. The models are ﬂexible in modeling stylized features of degradation data series

such as jumps, linearity/nonlinearity, symmetry/asymmetry, and light/heavy tails. The

models can handle the intrinsic stochastic properties and complex jumps. The results are

compact enough to compute and evaluate reliability characteristics conveniently. More

importantly, by introducing Fokker-Planck equations to stochastic degradation analysis,

the work provided a new methodology for reliability analysis of complex degradation

phenomenon.

Chapter 5 integrally handled the complexities of degradation including internally-

and externally-induced stochastic properties with complex jump mechanisms by con-

structing general Markov additive processes (Markov-modulated Levy´ processes). The

background component of the Markov additive process is a Markov chain deﬁned on a

ﬁnite state space; the additive component evolves as a Levy´ subordinator under a cer-

tain background state, and may have instantaneous nonnegative jumps occurring at the

time the background state switches. Fokker-Planck equations for such Markov-modulated

processes were derived, based on which explicit results of reliability function and lifetime

moments were provided. The models are ﬂexible in modeling degradation data with jumps

under dynamic environments.

Chapter 6 extended the models to multi-dimensional cases for multiple dependent

152 degradation processes under dynamic environments, where the Levy´ copulas were studied to construct Markov-modulated multi-dimensional Levy´ processes. The dynamic environment was modeled by a Markov process. Under a certain environment state, the evolution of multiple degradation processes was modeled by a multi-dimensional Levy´ subordinator constructed by the Levy´ copula. All the degradation processes are simultaneously modulated by the dynamic environment. When the environment state switches, there are instantaneous random jumps in all degradation processes. We developed Fokker-Planck equations and studied the ﬁrst passage time for such processes. The models handle the jump uncertainties in degradation and the jump dependence among degradation processes.

Chapter 7 proposed a systematic statistical estimation method for our models using linear programming estimators and empirical characteristic functions. The point estimates of reliability function and lifetime moments were obtained by their explicit expressions.

We also constructed bootstrap procedures for the conﬁdence intervals. Simulation studies for a stable process and a stable driven OU process were performed. By integrally analyzing degradation data series embedded jump measures, the results are expected to provide accurate reliability prediction and estimation that can be used to assist the mitigation of risk and property loss associated with system failures.

Levy´ processes with non-monotonic paths are suitable to model degradation processes in systems that have self-healing properties and/or undergo random maintenance actions. These Levy´ processes may contain Gaussian part and/or negative jumps with Levy´ measures deﬁned on the whole R domain, due to which the ﬁrst passage time of such processes is analytically intractable (see jump diffusion processes [120–122]). Subsequently,

153 the reliability analysis based on such non-monotonic Levy´ processes, OU processes and

Markov additive processes is interesting and challenging to be explored. The asymptotic behaviour of ﬁrst passage time distributions is another research potential.

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180 Appendices

Outcomes (As of August 2016)

Journal Publications

• Yin Shu, Qianmei Feng, and David W. Coit, Life Distribution Analysis Based on

Levy´ Subordinators for Degradation with Random Jumps, Naval Research Logistics

2015, 62(6), 483-492.

• Yin Shu, Qianmei Feng, Edward P.C. Kao, and Hao Liu, Levy´ Driven Non-

Gaussian Ornstein-Uhlenbeck Processes for Degradation-based Reliability Analy-

sis, IIE Transactions 2016, DOI: 10.1080/0740817X.2016.1172743.

• Yin Shu, Qianmei Feng, Edward P.C. Kao, David W. Coit, and Hao Liu, Markov

Additive Processes for Degradation with Jumps under Dynamic Environments, In-

vited to submit to Journal of Quality Technology, (INFORMS-QSR Refereed Best

Paper Finalist, 2015).

• Yin Shu, Qianmei Feng, Edward P.C. Kao, and Hao Liu, Using Degradation-with-

jump Measures to Estimate Life Characteristics, to be submitted, 2016.

• Yin Shu, Qianmei Feng, Edward P.C. Kao, and Hao Liu, First Passage Times of

Markov-modulated Multi-dimensional Levy´ Processes, working paper, 2016.

181 Conference Proceedings

• Yin Shu, Qianmei Feng, and Hao Liu, Laplace Expressions of Reliability Charac-

teristics for Levy´ Subordinators-Based Degradation, Proceedings of the IIE Annual

Research Conference, Nashville, TN, May 30-June 2, 2015, (ISERC-QCRE Best

Student Paper Runner-Up).

Conference Presentations

• Laplace Expressions of Reliability Characteristics for Levy´ Subordinators-Based

Degradation, IIE Annual Conference, Nashville, TN, May 30-June 2, 2015 (NSF

Travel Award, QCRE Best Student Paper Finalist).

• A Physics-of-Failure based Statistical Modeling Framework for Lithium Ion Battery

Degradation, IIE Annual Conference, Nashville, TN, May 30-June 2, 2015.

• Life Distribution Analysis Based on Levy´ Subordinators for Degradation with Ran-

dom Jumps, The 59th Annual Fall Technical Conference, Statistics and Quality:

Solving Problems Today and Tomorrow, Houston, TX, October 8-9, 2015.

• Markov Additive Processes for Degradation with Jumps under Dynamic Environ-

ments, The INFORMS Annual Conference, Philadelphia, PA, November 1-4, 2015

(Invited and QSR Best Refereed Paper Finalist).

• Non-Gaussian Ornstein-Uhlenbeck Processes in Degradation-based Reliability

Analysis, The INFORMS Annual Conference, Philadelphia, PA, November 1-4,

2015.

182 • Using Degradation-with-jump Measures to Estimate Life Characteristics, IIE Annu-

al Conference, Anaheim, CA, May 21-24, 2016 (NSF Travel Award).

• Multi-dimensional Markov-modulated Levy´ Processes for Multi-dependent Degra-

dation under Dynamic Environments, IIE Annual Conference, Anaheim, CA, May

21-24, 2016 (University of Houston-Cullen Fellowship Travel Grant).

183