Straub (2009): Stochastic modeling of deterioration processes through DBN 1/32
This article appeared in:
Journal of Engineering Mechanics, Trans. ASCE, 2009, 135(10): 1089-1099.
http://dx.doi.org/10.1061/(ASCE)EM.1943-7889.0000024
Stochastic modeling of deterioration processes through dynamic
Bayesian networks
Daniel Straub
1
Abstract
A generic framework for stochastic modeling of deterioration processes is proposed,
based on dynamic Bayesian networks (DBN). The framework facilitates computationally
efficient and robust reliability analysis and, in particular, Bayesian updating of the model
with measurements, monitoring and inspection results. These properties make it ideally
suited for near-real time applications in asset integrity management and deterioration
control. The framework is demonstrated and investigated through two applications to
probabilistic modeling of fatigue crack growth.
Keywords
Bayesian analysis; cracking; deterioration; inspection; Markov chain; monitoring;
stochastic processes.
1
Associate Professor, Engineering Risk Analysis Group, Technische Universität München, Arcisstr. 21,
80290 Munich, Germany, straub@tum.de
Straub (2009): Stochastic modeling of deterioration processes through DBN 2/32
Introduction
The modeling of deterioration is subject to significant uncertainty, which arises from a
simplistic representation of the actual physical processes (typically through empirical or
semi-empirical models) and from limited information on material, environmental and
loading characteristics. This uncertainty has been addressed in stochastic models of
deterioration processes in the past (Committee on Fatigue and Fracture Reliability 1982;
Yang 1994; Melchers 1999). Additionally, observations of the deterioration processes or
influencing factors (e.g., from inspection and monitoring) have been included in the
models through Bayesian updating (Tang 1973; Madsen et al. 1986), in particular in the
context of life-cycle optimization and inspection planning (Thoft-Christensen and
Sørensen 1987; Pedersen et al. 1992; Straub and Faber 2006). In principle, the methods
of structural reliability enable efficient Bayesian updating of any stochastic model with
any kind of information (Madsen et al. 1986). In reality, however, algorithmic difficulties
occur (Sindel and Rackwitz 1998), which can make the computations cumbersome and
hinder the implementation in software that is run by engineers who are not experts in
structural reliability methods.
This paper proposes a novel computational framework for evaluating stochastic
deterioration models. The strength of the framework is its computational efficiency and
robustness when performing Bayesian updating. It is based on Bayesian networks (BN), a
modeling tool that originated in computer science (Pearl 1988; Jensen 2001; Russell and
Norvig 2003), but has recently had a number of applications in engineering risk and
reliability analysis (Faber et al. 2002; Friis-Hansen 2004; Grêt-Regamey and Straub 2006;
Langseth and Portinale 2007). Few researchers have applied BN in the context of
deterioration modeling. Friis-Hansen (2001) studies the application of BN for
deterioration modeling and inspection planning by means of an example considering
fatigue crack growth; Montes-Iturrizaga et al. (2009) use BN for optimizing inspection
efforts for offshore structures subject to multiple deterioration mechanisms; Attoh-Okine
and Bowers (2006) present an empirical model of bridge deterioration using Bayesian
networks. In contrast to these previous publications, which make use of the BN
Straub (2009): Stochastic modeling of deterioration processes through DBN 3/32
capabilities mainly for modeling the system aspects or for optimizing inspection and
maintenance decision, the present paper focuses on the deterioration modeling. The
resulting computational framework enables efficient and robust reliability updating for
realistic deterioration models. By robust it is understood that the reliability updating can
be performed in an automated manner, not requiring the input from an expert in
reliability analysis. This is in contrast to existing efficient computational methods, such
as first/second-order reliability methods (FORM/SORM), importance sampling or subset
simulation, and facilitates the implementation in software that can be used by the lay
engineer for the planning of inspection, repair and maintenance activities, as well as in
automated alarm systems based on monitoring data.
The proposed framework is demonstrated and investigated through two applications
considering fatigue crack growth, which are representative for a large number of
deterioration mechanisms subject to uncertainty.
Relation to Markov process models
Dynamic Bayesian networks (DBN) can be interpreted as a generalization of Markov
process models, which have frequently been applied for the modeling of deterioration
(Bogdanoff and Kozin 1985; Spencer and Tang 1988; Cesare et al. 1992; Ishikawa et al.
1993; Rocha and Schuëller 1996; Mishalani and Madanat 2002). Markov deterioration
processes are characterized by the fact that for a given condition at time
1
t , the condition
at any future time
21
tt is statistically independent of the condition at any past time
01
tt . It is noted that the Markovian assumption does not hold in engineering practice,
where epistemic uncertainties are prevalent (Yang 1994; Melchers 1999; Mishalani and
Madanat 2002). Epistemic uncertainties are often time-invariant (e.g., uncertainties due to
simplistic parametrical models, due to limited statistical data for empirical models, or due
to incomplete knowledge of influencing parameters), thus invalidating the Markovian
assumption. To overcome this shortcoming, here a deterioration model is formulated that
corresponds to a Markov process model conditional on time-invariant random variables.
The DBN technique enables the efficient computation of such models.
Among Markov process models, it can be distinguished between two fundamentally
different approaches: models that are based on an underlying parametric model (Ishikawa
Straub (2009): Stochastic modeling of deterioration processes through DBN 4/32
et al. 1993) and models that are purely empirical (Cesare et al. 1992; Mishalani and
Madanat 2002). The latter are typically finite space Markov chain models. Although the
DBN approach can be applied to both models, this paper focuses on parametric models,
which are preferable in that they facilitate learning and transferability.
Dynamic Bayesian networks
The textbooks by Pearl (1988), Jensen (2001) and Russell and Norvig (2003) provide an
introduction to BN. In the following, a concise introduction to BN is given, limited to the
case of discrete random variables, i.e., random variables that are defined in a finite space.
BN are probabilistic models based on directed acyclic graphs that represent
()
p
x
, the
joint probability mass function (PMF) of a set of random variables
X . The space of X ,
i.e., the number of outcome states of
X for which
()
p
x
must be computed, increases
exponentially with the number of variables in
X , but BN enable an efficient modeling by
factoring the joint probability distribution into conditional (local) distributions for each
variable.
X
1
X
2
X
3
Figure 1. A simple Bayesian network.
A simple BN is illustrated in Figure 1. It consists of three discrete random variables
123
,,
X
XX.
1
X
is a parent of
2
X
and
3
X
, which are children of the former. The PMF of
each variable is defined conditional on its parents and the joint PMF of this network is
given as a product of these conditional probabilities:
123 1 21 31
,,pxx x px pxx pxx
(1)
Wherein
(|)
ij
p
xx
is the conditional PMF of
i
X
given
jj
Xx
. More generally, the joint
probability mass function for any BN having discrete variables is given as
Straub (2009): Stochastic modeling of deterioration processes through DBN 5/32
1
1
,,
N
Nii
i
ppxx px
xpa
(2)
where
i
pa is the set of realizations of the parents of
i
X . The basic supposition of BNs is
that each variable
i
X is independent of all other variables for given values of the
variables in its Markov blanket, which includes the parents of
i
X , the children of
i
X and
the parents of the children of
i
X
.
The BN allows entering evidence: probabilities in the network are updated when new
information becomes available. For example, when the state of
2
X in the network in
Figure 1 is observed to be
e
, this information propagates through the network and the
joint PMF of
1
X and
3
X change according to Bayes’ rule to
1
1131
13
13
11
,,
,
X
px pex pxx
pxex
pxxe
pe
px pex
(3)
Consequently the marginal posterior probabilities of
1
X and
3
X are also updated. Note
that the common influencing variable
1
X introduces dependence between
2
X and
3
X ,
but evidence can change the dependence among variables in the network; in the above
example, if
1
X is known,
2
X and
3
X become independent. For given sets of evidence, it
is possible to infer the independence assumptions encoded in the graphical structure using
the rules of
d-separation (Pearl 1988).
Dynamic Bayesian networks (DBN) are a special class of Bayesian networks, which
represent stochastic processes. They consist of a sequence of slices, each of which
consists of one or more BN nodes. The slices are connected by directed links from nodes
in slice
i to nodes in slice i+1. Figure 2 shows an example of a DBN. If the model
structure and the conditional probability tables are identical for all slices except the first,
then the DBN is homogenous. As for any BN, the joint PMF of the variables in the DBN
is defined through Equation (2), but a number of inference algorithms are available that
are developed especially for the DBN structure (Murphy 2002).
X
0
X
1
X
2
X
T
Y
T
Y
2
Y
1
Y
0
