Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667

Contents lists available at ScienceDirect

Journal of Statistical Planning and Inference

journal homepage: www.e lsev ier .com/ locate / jsp i

Piecewise deterministic Markov processes applied to fatigue crack growthmodelling

Julien Chiqueta,∗, Nikolaos Limniosa, Mohamed Eidb

aUniversité de Technologie de Compiègne, FrancebCommissariat à L'Énergie Atomique, Saclay, France

A R T I C L E I N F O A B S T R A C T

Available online 24 May 2008

Keywords:Piecewise deterministic Markov process;Markov renewal processAveraging principleMaximum likelihood estimationFatigue crack growth

In this paper, we use a particular piecewise deterministic Markov process (PDMP) to modelthe evolution of a degradation mechanism that may arise in various structural components,namely, the fatigue crack growth. We first derive some probability results on the stochasticdynamics with the help of Markov renewal theory: a closed-form solution for the transitionfunction of the PDMP is given. Then, we investigate somemethods to estimate the parametersof the dynamical system, involving Bogolyubov's averaging principle andmaximum likelihoodestimation for the infinitesimal generator of the underlying jump Markov process. Numericalapplications on a real crack data set are given.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Let us consider a structure subject to a main degradation process (Zt , t ∈ R+) which increases randomly on R∗+ = (0,∞) until

it reaches a critical value � >0, meaning the failure of the structure. It is assumed that Zt is an observable process whose samplepaths are obtained from experimental feedback. The reliability of such a system may be expressed by

R(t) = P(Zt <�), (1)

that is the probability that the process Zt does not reach the failure boundary on the whole observation interval [0, t].The dynamical evolution of the continuous-time, real-valued process Zt is modelled by a first order stochastic dynamical

system of the following form

dZtdt

= C(Zt ,Xt), (2)

where C is a function with the appropriate existence and uniqueness properties. The general form of C is known from physicalassumptions on the real phenomenon. The process Zt describes the time evolution of this phenomenonwhosemodelling throughclassical differential equations is not possible due to, for instance, some random environmental effects or some inhomogeneityin material of the structure. The stochastic process Xt with state space E is thus introduced to handle with the randomness of thesystem.

In our study, Xt is assumed to be a jumpMarkov process. The use of jump processes in the framework of stochastic dynamicalsystems modelling seems interesting from a physical point of view. As a matter of fact, it fits well with the mathematicaldescription of a degradation mechanism, i.e., a process observed on a continuous state space, evolving continuously in time,

∗ Corresponding author.E-mail address: julien.chiquet@utc.fr (J. Chiquet).

0378-3758/$ - see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.jspi.2008.05.034

1658 J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667

Fig. 1. The Virkler data set for the fatigue crack growth problem.

but whose growth rate is likely to change through some “chocs”, even very small, occurring at discrete instants of time. Bothintensities and time occurrences of the chocs are random, as described by the jump Markov process.

The coupled process (Zt ,Xt) has the Markov property and no further assumption is required for Zt . The process (Zt ,Xt) belongsto the class of stochastic models sometimes called piecewise deterministic Markov processes (PDMP). PDMP are non-diffusion typestochastic models, combined with the mixture of deterministic and jump motions. Davis (1984, 1993) gave a full descriptionof these stochastic processes as well as the basic associated stochastic calculus. Eq. (2) has also been referred in the literatureas a stochastic version of the transport equations, used to model some particle systems (see, e.g., Papanicolaou, 1975; Lapeyreand Pardoux, 1998), thus (Zt ,Xt) is also known as a transport process. Ezhov and Skorokhod (1969) also used similar processesunder the name of Switching processes. Koroliuk and Limnios (2005) gave some asymptotic results corresponding to a large classof processes including PDMP. Note also that Jacobsen (2006) recently studied non-homogeneous PDMP and built the associatedlikelihood. We want to give in this paper some practical estimation methods for a particular PDMP dedicated to degradationmodelling.

As a standard example of a degradation mechanism, the fatigue crack growth problem is handled in this paper to validatethe models. Fig. 1 represents the extensive set of crack growth data obtained by Virkler et al. (1979) during rigorously controlledlaboratory measurements. Roughly speaking, the data set consists in 68 specimens of the same material containing initially thesame artificially made 9mm-crack, that is Z0 = 9mm. These samples were exposed to the same cyclic loading. The sample pathsrepresent the evolution in time of the crack size for the 68 specimens, whose measures were performed until they reach thecritical value�=49.8mm. One can see the large variability of the crack size despite awell controlled experiment. Thus, stochasticdynamical systems were found well adapted to the modelling of fatigue crack growth (see, e.g., Lin and Yang, 1985; Bogdanoffand Kozin, 1985; Sobczyk, 1993; Tanaka, 1999, for an introduction).

The paper is organized as follows: in the next section, we present the family of stochastic dynamical systems involving jumpMarkov processes of the general form (2) and the settings associated to themodelling of degradationmechanisms.WeuseMarkovrenewal theory to obtain a solution of the transition function of the system, with which an expression of the reliability can begiven. The third section is dedicated to statistical inferences for such a system, more particularly concerning the estimation of thegenerator of the jumpMarkov process Xt , whose paths cannot be directly observed. The only available data concern the process Zt ,whose paths are defined on random length time intervals. Finally, numerical applications for the crack growth problem are given.

2. The model settings

Let (Zt , t ∈ R+) be an increasing stochastic process on R∗+, with initial condition Z0 = z. Looking toward reliability analysis,

we define for the process Zt a set of working states U = [z,�), with 0 < z <�, and a set of down states D = [�,∞). Its evolution iscontinuous in time with values in the real positive axis, thus Zt necessarily passes through the point � while reaching the set ofdown states.

J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667 1659

The evolution of Zt is modelled through the following stochastic dynamical system

dZtdt

= C(Zt ,Xt), Z0 = z, (3)

with the following assumptions:

A. 1. The process (Xt , t ∈ R+) is an irreducible Markov process with a finite state space E, initial law �i = P(X0 = i), for all i ∈ E,stationary distribution � = (�i)i∈E and a matrix generator A = (aij)i,j∈E such as

aij�0 for all i� j,

and

aii = −ai = −∑

k∈E,i� k

aik.

A. 2. The function C : R∗+ ×E −→ R∗

+ is measurable, strictly positive and of the Lipschitz type, that is, there is a function f : E −→ Rsuch that, for x, y ∈ R∗

+ and i ∈ E,

|C(x, i) − C(y, i)|� f (i)|x − y|.

A. 3. We have P(Z0 = z) = 1, with 0 < z <�.

Because Xt is Markov, it can be easily seen that the coupled process (Zt ,Xt) is a Markov homogeneous process on the statespace R∗

+ × E, provided that Z0 and X0 are independent. Obviously, the process Zt evolves in a deterministic way between thejumps of the process Xt , occurring at random discrete instants of time S0 = 0 < S1 < · · · < Sn < · · ·. Hence, between the jumps, theevolution of a path of Zt is fully determined by a classical deterministic differential system. For instance, say that Xt = i for t < S1where S1 stands for the first jump time of Xt . Then the Cauchy problem

dZtdt

= C(Zt , i), Z0 = z,

has a unique solution denoted by �z,i(t), that is (Zt = �z,i(t))0� t<S1 . We can thus build in a piecewise manner a unique solution Ztfor system (3) on the successive jump time intervals [Sn, Sn+1) of Xt .

There is much to say about the Markov process (Zt ,Xt). First, its generatorB (see, e.g., Lapeyre and Pardoux, 1998) is given by

Bf (z, i) = C(z, i)��z

f (z, i) +∑j∈E

aijf (z, j),

with i ∈ E, z ∈ R∗+ and f ameasurable and differentiable function on the first argument. Unfortunately, the infinitesimal generator

cannot always be exploited easily numerically. Thus, it can be more suitable to focus on an other important function of (Zt ,Xt),namely the transition probability function P(t):

Pij(z,B, t) := P(Zt ∈ B,Xt = j|Z0 = z,X0 = i). (4)

It is defined for all z ∈ R∗+, i, j ∈ E and B ∈ B, withB being the Borel �-field of R∗

+. The transition function permits to give a simpleexpression of the probability distribution �(t) of (Zt ,Xt), using a conditional probability development, thus characterizing wellthe PDMP:

�j(B, t) = P(Zt ∈ B,Xt = j)

=∑i∈E

∫R∗

+P(Zt ∈ B,Xt = j|Z0 = y,X0 = i)�z(dy)�i

=∑i∈E

�iPij(z,B, t),

where �z is the Dirac distribution at point z.Expression (1) of the reliability can be advantageously defined as a function of the coupled process (Zt ,Xt) rather than as

an expression of Zt only. Concerning (Zt ,Xt), U × E is the set of working states while D × E is the set of down states. Moreover,we consider here that the system is not a reparable one. Indeed, we have

R(t) = P((Zt ,Xt) ∈ U × E) =∑j∈E

�j(U, t) =∑i,j∈E

�iPij(z,U, t). (5)

1660 J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667

We also introduce the hitting time to D for the process Zt , that is the failure time, defined by

= inf{t�0 : Zt ∈ D} = inf{t�0 : (Zt ,Xt) ∈ D × E}. (6)

The distribution function F of is directly linked to the reliability through F(t) = 1 − R(t).A Markov process is a special case of a Markov renewal process, thus we can associate to (Zt ,Xt , t ∈ R+) the extended Markov

renewal process (Zn, Jn, Sn,n ∈ N), where

Zn = ZSn , Jn = XSn , n ∈ N,

with (Sn)n�0 being the jump times of Xt . TheMarkov renewal function(t), which represents at time t themean number of timesa set of R × E is visited by the process (Zt ,Xt), starting from a given point (z, i), is given by

ij(z,B, t) = Ez,i[#transitions in B, j] =∑n�0

Q (n)ij (z,B, t),

where Q is the semi-Markov kernel defined by

Qij(z,B, t) = P(Zn+1 ∈ B, Jn+1 = j, Sn+1 − Sn� t|Zn = z, Jn = i).

The successive n-fold convolution Q (n) for t�0 are defined in a Stieltjes sense, that is

Q (0)ij (z,B, t) = 1{i=j}1B(�z,i(t)), Q (1)

ij (z,B, t) = Qij(z,B, t), t >0,

and, for n�2,

Q (n)ij (z,B, t) =

∑k∈E

∫R+

∫ t

0Qik(z, dy, ds)Q

(n−1)kj (y,B, t − s).

It follows from results of Markov renewal theory (e.g., Limnios and Opri �san, 2001) the following proposition.

Proposition 1. The transition function P(t) is given by the following Markov renewal equation

Pij(z,B, t) = gij(z,B, t) +∑k∈E

∫R+

∫ t

0Qik(z, dy, ds)Pkj(y,B, t − s), (7)

where

Qij(z,B, dt) = aije−ait��z,i(t)(B) dt,

and

gij(z,B, t) = 1{i=j}1B(�z,i(t))e−ait ,

where 1B(x) is the indicator function of the subset B, i.e., 1B(x)=1 if x ∈ B and 0 otherwise. Solution of (7) is obtained through a Markovrenewal argument, and

Pij(z,B, t) =∫

R+

∫ t

0ij(z, dy, ds)gjj(y,B, t − s). (8)

Expression (8) can be computed numerically, thus with (5), the reliability R(t) of the system as well as the cumulativedistribution function of the failure time F can also be computed. An example of a numerical computation of R(t), based uponexpression (8), is detailed in Chiquet and Limnios (in press) for a given system with general form (3).

As statistical inference for the dynamical system (3) is themain concern of this paper, we shall consider K independent samplepaths of Zt obtained from lab experiments. The paths are denoted by (Zkt )k=1,. . .,K

, which are supposed to be the only available

data for estimating the whole dynamical system. The observation procedure is the following: each path Zkt is observed on somerandom time intervals [0, k ∧ Tobs], with the k's being K independent copies of the failure time defined in (6). It means thateach path of Zt is observed until it reaches the failure threshold �, provided that < Tobs. Here, Tobs is a real positive constantrepresenting the maximal time of observation, beyond which the lab experiment become too expensive. If we always can affordto observe the system until it reaches �, then we just put Tobs = ∞, what is the case for the Virkler data set seen on Fig. 1. Wepoint out that Tobs is the same for any path. We also denote by tk the successive realizations of the random variables k ∧ Tobs.Fig. 2 shows a schematic representation of the observation procedure for three paths of Zt and the corresponding paths of Xt ,which are obviously defined on the same random time intervals as the corresponding Zkt 's.

J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667 1661

Sta

tesp

ace

o

f Zt

Time

ZtZt Ztpoint {Δ}

zS

tate

spac

e E

of X

t

Time

Xt

Xt

Xt

t1 = Tobst2 t3

2

31

132

Fig. 2. Observation scheme of the dynamical system.

3. Estimation

In this sectionwedevelop the schemeused for the estimation of the system,wherewe just have someKpaths of the observabledegradation process Zt . The estimation scheme of (3) splits in two parts:

(1) First, we study the asymptotic behavior of the system using the Bogolyubov's averaging principle (see Bogolyubov andMitropol'skii, 1961). With this technique, the random component “disappears” and we obtain a deterministic system thatcorresponds to the mean behavior of (3). A regression analysis is performed on the asymptotic, deterministic system so thatwe estimate the fixed parameters in function C, whose general form is assumed to be known.

(2) Second, we turn back to the remaining estimation of the random part of the system, that is, of the jump Markov process Xt .To this purpose, we estimate the paths of Xt which are not directly observed, as well as its state space. We then build theassociated likelihood function, keeping in mind that these paths are defined on random time intervals. The generator of Xt isestimated by maximizing an approached likelihood function.

3.1. Bogolyubov's averaging principle

An approximation of Zt is obtained by analyzing system (3) in a series scheme (see, e.g., Koroliuk and Limnios, 2005), that isby studying the weak convergence, when � → 0, of

dZ�t

dt= C(Z�

t ,Xt/�), Z�0 = z, (9)

where Xt is assumed to be ergodic. In fact, the change of scale t → t/� is performed for Xt in order to see the behavior of thedynamical system when the random component Xt just adds the information it would add after a very long time of observationof (9), since t/� → ∞ when � → 0. This so-called averaging approximation was first introduced by Bogolyubov (see Bogolyubovand Mitropol'skii, 1961) who showed that (9) converges weakly when � → 0 to the following deterministic system

dztdt

= C(zt), z0 = z, (10)

with zt the limit deterministic process and C a mean function defined by

C(z) = limT→∞

1T

∫ T

0C(z,Xt) dt a.s.

1662 J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667

In the particular case where Xt is an ergodic jump Markov process with a stationary law �, we have

C(z) =∑i∈E

C(z, i)�i.

Through this averaging technique, we have a limit deterministic system (10) associated to stochastic differential system (3). Thefixed parameters appearing in the function C are the same as the ones appearing in C but in (10) the random part was “eliminated”: with the K sample paths Zkt , we can perform a classical regression analysis on (10) to estimate the fixed parameters of C.

By this mean, we obtain the mean deterministic process, that is zt , associated to the dynamical system. The only unknownsthat remain are the initial law �, the generator A and the state space E of the random process Xt . The stationary law � will beobtained from A through

�A = 0,∑i∈E

�i = 1.

3.2. Estimating the paths and the state space of the jump Markov process

Prior to any estimation of A or �, some representations of the paths (Xkt )k=1,. . .,K

are needed. To this purpose, it is assumed

that there exists a function G from expression (3) giving Xt as a function of Zt and its derivative denoted by Zt = dZt/dt. Such afunction does not always exist. Indeed, when the stochastic process Xt is a linear additive or a multiplicative term in the functionC, we may easily find the corresponding function G. This is the case for the so-called Paris–Erdogan model, which will be used todescribe the crack growth mechanism of the Virkler data. Hence, we may obtain a first estimation for the Xk

t 's with the followingrelationship:

Xkt = G(Zkt ,

Zkt ), (11)

where the derivative of Zt can be estimated through

Zkt =Zkt+�t − Zkt

�t, (12)

with �t being the time discretization step of the data set.Due to the way the Xk

t are computed, that is through expression (11), we obtain some noisy paths taking their values in R,which values may be quite nearby. Our model requires a finite state space for the underlying jumpMarkov process Xt with somepiecewise shape paths, thus it is appropriate to “regroup” the values which are very close from each other to an unique state.By these means, we build a state space which is an approximation of the real state space. Two cases may occur: first, if the statespace E of the process Xt is assumed to be known, each point of the Xk

t 's is rounded to the closest element of E, according to agiven distance (e.g., euclidean). Second, if E is not known, which is the most common case, a non-hierarchical clustering methodcan be used to fit as well as possible with the Xk

t 's. In Chiquet et al. (2006), we applied the classical K-means algorithm to estimatethe state space of Xt . The application of one of this clustering methods leads to K approximated paths Xk

t , still defined on therandom time interval [0, k ∧ Tobs]. The Xk

t are then used for further estimations linked to the jump Markov process Xt , namelyfor estimating � and A.

3.3. Maximum likelihood method

Concerning the initial law of the jump Markov process Xt , it is assumed that X0 is independent of the generator and of thecensoring time . We thus have the classical empirical estimator ( see, e.g., Sadek and Limnios, 2005)

�i =1K

K∑k=1

1{Xk0=i}. (13)

It turns out that the estimation problem of the infinitesimal generator A requires further attention. In order to build thelikelihood function for A on the Xk

t 's, we need to give an expression of the density function of a path of Xt defined on a randomtime interval [0, ]. To this purpose, we first remind fundamental results of statistical inference for jump Markov processes,obtained by Billingsley (1961) and Albert (1962), who built the density function on the space of sample paths defined on thesame fixed time interval [0,�] with � a real positive constant. The following notation is required in the sequel:

• (Sn,n ∈ N), the jump times of the process Xt , with S0 = 0;• Wn = Sn+1 − Sn, for n ∈ N, the sojourn times in the successively visited states;• (Jn,n ∈ N), the so-called embedded Markov chain of the Markov process Xt , that is Jn = XSn , for n ∈ N;• N(�), the number of jumps in [0,�], that is the largest integer for which SN(�)��;

J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667 1663

• Nij(�) the number of transitions from state i to state j observed in [0,�], that is,

Nij(�) =N(�)∑k=1

1{Jk−1=i,Jk=j};

• Vi(�) the total length of time spent in state i on [0,�], that is,

Vi(�) =∫ �

01{Xs=i} ds.

Hence, the historyH� of the process Xt on the fixed time interval [0,�] can be represented as an ordered sequence:

H� = ((J0,W0), . . . , (WN(�)−1,WN(�)−1), (JN(�),U�)),

where U� = � − SN(�). It means that the path starts at state J0 at time zero, remains in J0 for W0 units of time, makes a jump to J1,and so on, until it makes the final jump to JN(�) where it stays for U� units of time, that is until the censoring time �.

The probability density function fA associated to a sample path defined on a fixed time interval [0,�], was given by Albert(1962). We put h� =H�( ) a realization of the history of Xt associated to a given sample path. It can be shown that

fA(h�) = �(x0)∏i,j∈Ei� j

anij(�)ij e−aijvi(�).

The lower case lettersnij(�),vi(�) have beenused towrite down the realizations associated to the corresponding randomvariables.These results can be slightly generalized to get an expression of the density for a path defined on a random time interval [0, ],

with the random censoring variable. Let us putH the history of Xt when observed on the random interval [0, ]. A realizationh� =H( )( ) of the history associated to a sample path of Xt censored by has the same structure as in the fixed time case. Yetthe density fA, of a randomly censored path h� is written as a function of the conditional density fA| and of the density functionf of through

fA,(h�) = fA|(h�| = �)f(�). (14)

Let us now build the likelihood function based upon the density fA,, yet under the observation scheme of Fig. 2, that is when thepaths are not necessarily observed until the random failure time , but until ∧ Tobs, that is still random. We have K paths of Xk

testimated from Section 3.2, defined on [0, tk], with tk being a realization of k ∧ Tobs. We associate K ordered sequences hktk to the

estimated paths Xkt 's.

Let us also introduce the following function, associated to the kth path:

Ik(tk) =

{f(tk) if k�Tobs,P( > Tobs) if k > Tobs.

By expression (14) of fA, and using the functionI, it turns out that the corresponding log-likelihoodLA for the generator A is

LA(K) =K∑

k=1

log fA|(hktk |k = tk) +K∑

k=1

logIk(tk). (15)

The second term in the right hand of (15) can be written

K∑k=1

logIk(tk) =

∑k:tk<Tobs

log f(tk) +∑

k:tk=Tobs

logP( > tk),

and if we denote by L the number of paths censored by Tobs, with L�K, we have for these L paths P( > tk) = 1 − F(Tobs). Hence,the log-likelihood (15) becomes

LA(K) =K∑

k=1

log fA|(hktk |k = tk) +∑

k:tk<Tobs

log f(tk) + L log(1 − F(Tobs)). (16)

The above expression (16) is quite general. Let us here treat the case where the failure time and the process Xt are independent,which is obviously not true when dealing with the dynamical system (3), but can be considered as a first approximation. If so,

1664 J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667

the structure of the likelihood is almost the same as in Albert (1962). The conditional density fA| turns to fA and the log-likelihoodL0

A to be maximized is just

L0A(K) =

K∑k=1

log fA(hktk ) =K∑

k=1

log∏i,j∈Ei� j

ankij(tk)

ij e−aijvki (tk),

where nkij and vki are, respectively, the number of transitions from i to j and the time spent in i for the kth observed history hk. Ifwe denote nij(K) and vi(K) the corresponding variables for all of the K paths, that is

nij(K) =K∑

k=1

nkij(tk), vi(K) =K∑

k=1

vki (tk),

then we have

L0A(K) =

∑i,j∈Ei� j

nij(K) log aij − aijvi(K).

For an element aij with i� j, we get

aij(K) = nij(K)vi(K)

. (17)

This is a straightforward generalization of the maximum likelihood estimator obtained in Billingsley (1961) and Albert (1962),computed here on several sample paths with random time lengths. In Chiquet and Limnios (2006), estimator (17) was provedto be consistent with an asymptotic normality property, even when defined on randomly censored sample paths, in theindependent case.

In the following numerical application, we use the estimator for the generator of the underlying jump Markov process of thedynamical system, in the framework of fatigue crack growth analysis. This is obviously an approximation because we considerthe failure time and the process Xt independent, which is not the case as seen in expression (6) of the failure time. Although goodnumerical results are obtained with this approximation, it could be significantly improved by introducing the dependency in theestimation procedure.

4. Application to crack growth modelling

To handle with the fatigue crack growth problem, some empirical laws have been suggested in fracture mechanics todescribe the mean behavior of a crack. One of the most versatile one is the law given by Paris and Erdogan (1963), describ-ing deterministically the evolution of the crack size function zt by

dztdt

= C(�K(zt))n, (18)

where C and n are some material constants and �K(z) = Y(z)�S√

�z is the so-called stress intensity factor range. �S is the stressrange of the load applied on the specimen, while Y(z) is a structure geometry function that is approximately equal to 1 whenconsidering large structure. With this assumption, Eq. (18) turns to the much simpler form

dztdt

= a(zt)b, (19)

where a = C(�S)n�n/2 and b = n/2. We suggest to randomized (19) with a right-hand side multiplicative factor given by a jumpMarkov process Xt , thus we model the phenomenon by the following stochastic dynamical system

dZtdt

= a(Zt)b × Xt , Z0 = z, (20)

with z being the initial crack size.Applying the Bogolyubov's averaging principle described in Section 3.1, we have

dZ�t

dt= a(Z�

t )b × Xt/�, Z�

0 = z,

where � >0, a series parameter. The asymptotic behavior, when � → 0, gives

dztdt

= a0(zt)b, z0 = z, (21)

J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667 1665

where a0 = a∑

i∈E�ii. The following stochastic dynamical system is equivalent to (20)

dZtdt

= a0(Zt)b × v(Xt), Z0 = z, (22)

where v(Xt) is a jump Markov process with the same generator as Xt , but a state space E′ that is linked to the state space E of Xt

through the linear mapping v defined by v(x)= x/∑

i∈E�ii, x ∈ E. We consider system (22) rather than (20) so that we “normalize”the process Xt to v(Xt) by dividing by the state mean

∑i�ii. The term a0 absorbs the compensatory term for the normalization.

Hence, practically, the stationary law � does not need to be computed. Moreover, the system is more suitable to regressionanalysis. As a matter of fact, the parameters a0 and b can be estimated by a least squares method on the data set, by taking thelogarithm on both sides of (21). Hence,

ln Zt = ln a0 + b ln Zt + lnv(Xt).

The available data, that is the Zkt 's, can be represented as a N-sample composed with all of the data points from the crack growthcurves:

{(ti, Zti ) : i = 1, . . . ,N}.

This N-sample can be transformed in {(ln Zti , lnZti )}, with Zti being obtained through (12). Introducing the notation xi = ln Zti , yi =

lnZti and �i = lnv(Xti ), we have the following classical regression problem:

yi = ln a0 + bxi + �i, i = 1, . . . ,N. (23)

When � → 0, that is in the series scheme of the Bogolyubov's principle described in Section 3.1, system (22) tends to (21). It alsomeans that, on average, v(Xt) is 1. The term “average” must be understood here in the series scheme sense of the Bogolyubov'sprinciple. Thus, the lnv(Xti )'s are “close to 0” in the regression equation (23) and correspond to the so-called residuals �i.Minimizing

∑�2i , we get the estimators associated to the well-known least squares method:

a0 = exp(y − bx), b =∑

xiyi − Nxy∑x2i − Nx2

,

with

x = 1N

∑xi, y = 1

N

∑yi.

Some paths (˜v(Xt))k ,k=1,. . .,K, are then obtained explicitly by “reversing” the dynamical system, that is through the function G

given by (11). In the case at hand, we have

(˜v(Xt))k= 1

a0(Zkt )

−b×Zkt .We then apply the K-means clustering method, initially suggested by MacQueen (1967) ( see, e.g. Jain and Dubes, 1988, forimplementation) to reduce the state space of the (˜v(Xt))

k, thus obtaining some (v(Xt))k's suitable for any further estimation of the

jump Markov process. For the Virkler data, the histogram of a 18-state space obtained with the K-means method is representedon Fig. 3. The values of the estimated state space of E′ are also put in the figure. One can see that it has a kind of discretelog-normal dispersion profile, with values distributed around one, what fits well the physical point of view: when v(Xt) = 1,we are in the mean behavior of the asymptotic Paris–Erdogan law (21), corresponding to the assumption made during the leastsquares regression.

Finally, the initial law � and the infinitesimal generator A are estimated with (13) and (17), adapted to the jump Markovprocess v(Xt). Some crack growth curves can thus be simulated with the dynamical system (22) as long as all of the modelparameters have been estimated. With Monte Carlo techniques, some important functions can then be computed. For instance,the cumulative distribution function of Zt is interesting for industrial care, especially for optimizing a maintenance strategy. Itrepresents the probability distribution of the crack size in time. We estimate it for a fixed instant of time t through

Ft(x) = P(Zt �x) = 1M

M∑m=1

1{�mt � x}, x ∈ [z,�], (24)

with (�mt )m=1,. . .,Mbeing M sample paths simulated with the stochastic dynamical system (22). Estimator (24) can be compared

with the empirical cumulative distribution function, that is, computed on the Virkler data set from Fig. 1. Results are representedat Fig. 4 for various values of t.

1666 J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667

00

5

10

Freq

uenc

y (%

)

Clusters

state frequency

0.5398055 1.5110.6496718 3.9730.7204433 5.7920.7795669 7.1620.8322706 7.8010.8875104 9.4240.9437743 9.9411.0017041 11.061.0017041 10.191.1380179 9.1091.2220338 7.8721.3250410 6.3151.4614355 4.3341.6445818 2.6761.9201478 1.3962.4161315 0.3063.9463620 0.06314.270229 0.011

4.54.03.53.02.52.01.51.00.5

Fig. 3. 18- State space estimated for˜v(Xt ) through the K-means method.

50

0.2

0.4

0.6

0.8

1.0

C.D

.F F

t (x)

of

Zt a

t var

ious

inst

ants

t

Cracksize x

datathe model (M = 2000)

t = 102500 t = 152000 t = 201750

45352515

Fig. 4. Cumulative distribution function of Zt at various instants of time.

In the same way, we may estimate the reliability defined here by (1) through

R(t) = 1M

M∑m=1

1{�mt <�}. (25)

Results are compared with the data points on Fig. 5.

5. Concluding remarks and perspectives

Concerning the generator A of the jump Markov process Xt , the estimator used in the previous numerical application isan approximation, since we have used the maximum likelihood estimator for K sample paths censored by a random variableindependent of the process. We pointed out that it was not the case in the framework of dynamical system developed in thepresent paper, as long as the random censoring is also the failure time of the system. In Section 3, we gave the general form ofthe log-likelihood to be maximized. One can see that the density function of appears within (16), and an important issue is togive an formal expression of f if we want to include it in the maximization process. This could be achieved using Proposition 1and consecutive results, what the authors are working on.

J. Chiquet et al. / Journal of Statistical Planning and Inference 139 (2009) 1657 -- 1667 1667

20000

0.2

0.4

0.6

0.8

1.0 + + + + + + + ++ + ++

++

+

+

+ +

++

++

+ + + + + + +

Rel

iabi

lity

R (

t)

Time t × 10−2

+ datathe model (M = 2000)

275025002250

Fig. 5. Reliability estimator compared with the data points.

References

Albert, A., 1962. Estimating the infinitesimal generator of a continuous time, finite state Markov process. Ann. Math. Statist. 38, 727–753.Billingsley, P., 1961. Statistical Inference for Markov Processes. The University of Chicago, Chicago Press.Bogdanoff, J., Kozin, F., 1985. Probabilistic Models of Cumulative Damage. John Wiley, New York.Bogolyubov, N., Mitropol'skii, Y., 1961. Asymptotics Methods in the Theory of Nonlinear Oscillations. Gordon and Breach Science Publishers, London.Chiquet, J., Limnios, N., 2006. Estimating stochastic dynamical systems driven by a continuous-time jump Markov process. Methodol. Comput. Appl. Probab. 8,

431–447.Chiquet, J., Limnios, N., Eid, M., 2006. Modeling and estimating the reliability of stochastic dynamical systems with Markovian switching. In: Proceedings of

European Safety and Reliability Conference, pp. 133–139.Chiquet, J., Limnios, N., A method to compute the transition function of a piecewise eterministic Markov process. Stat. Prob. Let., in press, doi:10.1016/

j.spl.2007.12.016.Davis, M., 1984. Piecewise- deterministic Markov processes: a general class of non-diffusion stochastic models. J. Roy. Statist. Soc. 46 (3), 353–388.Davis, M., 1993. Markov Models and Optimization. Monographs On statistics and Applied Probability, vol. 49. Chapman & Hall, London.Ezhov, I., Skorokhod, A., 1969. Markov processes homogeneous with respect to the second component I, II. Theory Probab. Appl. 14 (3–14, II), 679–692.Jacobsen, M., 2006. Point Process Theory and Applications. Birkhauser, Basel.Jain, A.K., Dubes, R.C., 1988. Algorithms for Clustering Data. Prentice-Hall, Englewood Cliffs, NJ. URL: 〈http://www.cse.msu.edu/∼jain/Clustering_Jain_Dubes.pdf〉.Koroliuk, V., Limnios, N., 2005. Stochastic Systems in Merging Phase Space. World Scientific, Singapore.Lapeyre, B., Pardoux, E., 1998. Méthodes de Monte Carlo pour les Équations de Transport et de Diffusion. Springer, Berlin.Limnios, N., Opri �san, G., 2001. Semi-Markov Processes and Reliability. Birkhauser, Basel.Lin, Y.K., Yang, J.N., 1985. A stochastic theory of fatigue crack propagation. Amer. Inst. Aeronaut. Astronaut. J. 23, 117–124.MacQueen, J.B., 1967. Some methods for classification and analysis of multivariate observations. In: Proceedings of Fifth Berkeley Symposium onMathematical

Statistics and Probability, vol. 1. University of California Press, Berkeley, pp. 281–297.Papanicolaou, G., 1975. Asymptotic analysis of transport processes. Bull. Amer. Math. Soc. 81 (2), 330–392.Paris, P., Erdogan, F., 1963. A critical analysis of crack propagation laws. J. Basic Eng. 85, 528–534.Sadek, A., Limnios, N., 2005. Nonparametric estimation of reliability and survival function for continuous time finiteMarkov processes. J. Statist. Plann. Inference

133, 1–21.1993. In: Sobczyk, K. (Ed.), Stochastic Approach to Fatigue. Springer, Berlin, Wien, New York.Tanaka, H., 1999. Importance sampling simulation for a stochastic fatigue crack growth model. In: Melchers, R., Stewart, M. (Eds.), Proceeding of International

Conference on Applications of Statistics and Probability, vol. 2. pp. 907–914.Virkler, D., Hillberry, B., Goel, P., 1979. The statistical nature of fatigue crack propagation. J. Eng. Mater. Technol. 101, 148–153.