Inspection; structural health monitoring; reliability; Bayesian analysis; updating; decision analysis; value of information

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1 Cite as: Straub D. (2014). Value of inforation analysis with structural reliability ethods. Structural Safety, 49: Value of Inforation Analysis with Structural Reliability Methods Daniel Straub Engineering Risk Analysis Group, Technische Universität München Abstract When designing onitoring systes and planning inspections, engineers ust assess the benefits of the additional inforation that can be obtained and weigh the against the cost of these easures. The Value of Inforation (VoI) concept of the Bayesian statistical decision analysis provides a foral fraework to quantify these benefits. This paper presents the deterination of the VoI when inforation is collected to increase the reliability of engineering systes. It is deonstrated how structural reliability ethods can be used to effectively odel the VoI and an efficient algorith for its coputation is proposed. The theory and the algorith are deonstrated by an illustrative application to onitoring of a structural syste subjected to fatigue deterioration. Keywords Inspection; structural health onitoring; reliability; Bayesian analysis; updating; decision analysis; value of inforation Value of inforation 1/31

2 1 Introduction When it is required to ake decisions under uncertainty and risk, one often has the possibility to gather further inforation prior to aking the decision. Such inforation reduces the uncertainty and thus facilitates iproved decision aking. This explains the success of structural health onitoring (SHM), advanced inspection ethods, reote sensing and other onitoring techniques for civil infrastructures, to which I will refer collectively as onitoring systes. As experienced engineers are well aware, collecting the inforation coes at a price that is not always justified by its benefit. Unfortunately, this is often discovered only after the installation of a onitoring syste. A atheatical fraework exists for quantitatively assessing the benefit of a onitoring syste prior to installing it: the value of inforation (VoI) analysis fro Bayesian statistical decision theory [1-3] that has been considered by civil and structural engineers since the early 1970s [4]. The late Prof. Wilson Tang was one of the first to notice the potential of Bayesian ethods and VoI concepts to optiize engineering decisions [5-7]. In his paper published in 1973 [5], he described Bayesian updating of probabilistic odels of flaws with inspection results, which preceded the optiization of inspections in aircraft and offshore structures subject to fatigue deterioration in the 1970s and 80s [8-12]. These works were aong the first applications of Bayesian decision analysis for optiizing the collection of inforation in an industrial context. Siilar efforts were ade in the field of transportation infrastructure anageent, based on Markovian deterioration odels [13]. In recent years, the optiization of onitoring systes through explicit coputation of the VoI has found increased interest in various fields of civil and infrastructure engineering. Explicit coputation of the VoI for optiizing inspections and structural health onitoring in deteriorating structures was proposed in [14-18]. Optiization of sensor placeent based on VoI has been studied in [19]. In geotechnical engineering, which has always been strongly relying on onitoring, the effect of inforation quality has been investigated [20]; an explicit quantification of the VoI for head onitoring of levees is described in [21]. In the field of natural hazards, the VoI concept has been applied for prioritizing post-earthquake inspections of bridges [22] and for quantifying the value of iproved cliate odels when designing offshore structures against extree wave loads [23]. VoI analysis is and has been applied in any other fields of engineering and science, including oil exploration [24] and environental health risk anageent [25]. Deterining the VoI requires significant odeling and coputational efforts. Coputationally efficient evaluations of the VoI was considered ainly in the field of Value of inforation 2/31

3 achine learning and artificial intelligence [19, 26-28]. In these areas, prediction odels used for the VoI coputations are typically based on known probabilistic dependences aong a potentially large nuber of rando variables. In contrast, in infrastructure and civil engineering, prediction odels are often based on advanced physically-based odels, which describe the onitored phenoena. As an exaple, when planning the onitoring of a bridge, one can ake use of detailed echanical odels of the structure. Furtherore, the onitoring is often installed not to guide the every-day operation of the syste, but for early detection of deterioration or daages that ay ipair the safety of the syste. These applications otivate the cobination of the VoI concept with structural reliability ethods, which were developed to efficiently copute the probability of syste failure via advanced physically-based odels. This paper presents the odeling and coputation of VoI based on structural reliability ethods. A odeling fraework is proposed, which is especially suitable when probabilistic physically-based odels of the onitored systes and processes are available, e.g. in structural engineering applications. On this basis, a coputationally efficient algorith is developed for estiating the VoI. The fraework and the algorith are illustrated through an application to onitoring of a structure subject to fatigue deterioration, which deonstrates the effectiveness and efficiency of the proposed approach. The paper closes with a discussion on the difficulties encountered in deterining the VoI in realistic engineering probles. 2 Value of inforation analysis 2.1 Decision-theoretic fraework As a preise, I assue that all consequences (costs of onitoring, itigation actions as well as failure consequences) can be expressed either in onetary values or in a coon easure of utility U. I adopt the classical expected utility fraework [29] according to which an optial decision under uncertainty is the one axiizing the expected utility E[U]. For siplicity, I further restrict the presentation to situations in which all consequences can be expressed as onetary costs C and in which utility is proportional to C, corresponding to a risk-neutral decision aker. The optial decision is thus the one that iniizes the expected cost E[C]. It is straightforward to adapt the ethods presented in this paper to the case of a risk-averse decision aker or to situations with non-onetary consequences, if preferences of the decision aker can be expressed through utility functions. Value of inforation 3/31

4 Following the classical structural reliability odeling fraework [30], the uncertainty associated with the phenoena under consideration is characterized by a vector of rando variables. The relation between and the events of interest is a deterinistic one, e.g. the failure event is described through the liit state function g F () as F = {g F () 0}. In this fraework, odel uncertainties are included through additional rando variables in. In a classical decision analysis under uncertainty, the goal is to identify the actions a that iniize E[C], e.g. the aintenance and repair actions a that ensure an optial balance between the cost of a and the risk associated with failure. Additionally, inforation can be collected prior to aking the action decision a. Therefore, a so-called test decision e is ade on what inforation to collect (e stands for experients). This is, e.g., a decision on the design of a onitoring syste or a decision on the inspection schedule. The extended decision proble is to find the cobination of onitoring decision e and action decision a that iniizes E[C]. This proble is known in the literature as preposterior decision analysis [4]. These probles can be graphically odeled through decision trees and influence diagras, Figure 1. The decision tree explicitly depicts all possible states of rando variables and decisions. In contrast, the influence diagra provides a ore concise representation, which additionally reflects the causal relations between the rando variables and the decisions. Ipleentations of the influence diagra for coputing the VoI can be found in [16, 31]. Figure 1.The basic decision proble when planning onitoring and inspection easures: (a) decision tree and (b) corresponding influence diagra. Here it is assued that the cost of onitoring c e (e) and the cost of the action and syste state c(a, x) are additive. Value of inforation 4/31

5 This paper focuses on the coputation of the value of inforation (VoI) of a given onitoring syste. The optiization of the onitoring syste (the test decision e) is not explicitly considered. However, the VoI is the total expected net benefit of a given onitoring syste and is thus the central part of any preposterior decision analysis. The optial onitoring syste is the one axiizing the VoI inus the cost of onitoring. In the following, the optiization of the decision a is presented prior to considering the onitoring results. This follows the logic that onitoring results enable iproved action decisions and that their benefit can thus only be quantified when explicitly odeling the action decision. 2.2 Prior decision optiization Before applying onitoring, the optiization of the decision a ust be based on the prior knowledge, characterized through the prior probability distribution of. The prior optiization proble is: a opt = arg in E [c(a, )] a = arg in c(a, x)f (x)dx. a (1) c(a, x) is the cost associated with a given set of actions a and realization x, and E denotes the expectation with respect to. Throughout the paper I use the notation dx 1 dx n. dx = In engineering decision probles involving reliability, the consequences typically depend on discrete events describing the syste state, such as failure F or a set of daage levels (e.g., in perforance-based earthquake engineering). In the structural reliability fraework, these events correspond to doains in the outcoe space of. Let E 1, E 2,, E denote the utually exclusive, collectively exhaustive syste states in the general case. (If only failure F is of interest, it is E 1 = F and E 2 = F.) The optiization proble can then be written as a opt = arg in c Ei (a) Pr(E i ). (2) a Here, c Ei (a) is the cost associated with event E i and decision a. Let C prior denote the expected cost associated with this optial decision a opt, i.e. Value of inforation 5/31

6 C prior = in c Ei (a) Pr(E i ) = c Ei (a opt ) Pr(E i ). (3) a The probability of E i is coputed using structural reliability ethods as Pr(E i ) = I(x Ω Ei )f (x)dx = f (x)dx x Ω Ei. (4) I( ) is the indicator function and Ω Ei is the doain in the outcoe space of corresponding to event E i. In structural reliability, Ω Ei is defined in ters of liit state functions g i (x). In case of a single liit state function, it is Ω Ei = {g i (x) 0}. More generally, Ω Ei is defined through unions and intersections of ultiple {g i (x) 0}, [32]. To ake these abstract concepts ore apprehensible, consider the siple exaple of a echanical syste that either functions during its entire service life or fails at soe point in tie. For siplicity, the tie of the failure is considered irrelevant. The failure event is described as F = {g F () 0}, where includes paraeters describing deterioration of the syste. It is possible to perfor aintenance actions during the service life. Let a 0 (do nothing) and a (aintenance) denote the two decision alternatives. If aintenance is perfored, instead of F one needs to consider F, failure of the aintained syste. This event is described through a corresponding liit state function g F ( ) as F = {g F () 0}. Therefore, there are four distinct events (syste states): E 1 = F F, E 2 = F F, E 3 = F F, and E 4 = F F. The cost of aintenance is c. The cost of failure is c F. The expected cost of the two action alternatives are: E[C a 0 ] = E [c(a 0, )] = c(a 0, )f (x)dx = {c F I[g F () 0]}f (x)dx (5) = c F I[g F () 0]f (x)dx = c F Pr(F), Value of inforation 6/31

7 E[C a ] = E [c(a, )] = c(a, )f (x)dx = {c + c F I[g F () 0]} f (x)dx (6) = c + c F I[g F () 0] f (x)dx = c + c F Pr(F ). The integrals in Eqs. (5) and (6) are classical structural reliability probles and can be solved e.g. with FORM/SORM or sapling-based ethods. The results in Eq. (5) and (6) are rather trivial, and the decision optiization a opt = arg in(e[c a 0 ], E[C a ]) is straightforward. Nevertheless, it will later prove beneficial to a explicitly odel the utually exclusive events E 1, E 2, E 3, E 4. The costs associated with these events and decisions a 0 or a are suarized in Table 1. Table 1. Costs associated with the utually exclusive events E 1 = F F, E 2 = F F, E 3 = F F, and E 4 = F F. c E1 (a 0 ) = 0, c E2 (a 0 ) = c F, c E3 (a 0 ) = 0, c E4 (a 0 ) = c F, c E1 (a ) = c, c E2 (a ) = c, c E3 (a ) = c + c F, c E4 (a ) = c + c F. Since it is F = E 2 E 4 and F = E 3 E 4, it should be clear that the optiization following Eq. (2) gives the sae result as the above solution through Eqs. (5) and (6). Note: For any applications, it is initially ore intuitive to odel the effect of a decision through changes of the probability distribution of, i.e. by replacing f (x) with f a (x a) in Eq. (1). In the above exaple, this would signify not aking a distinction between F and F. The effect of the aintenance would instead be odeled through the difference between f a (x a 0 ) and f a (x a ), resulting in Pr(F a 0 ) and Pr(F a ). However, such a odeling approach is less rigorous, as the odel paraeters before and after the aintenance are in fact different rando variables. For this reason, the intuitive approach has several disadvantages over the proposed approach: (a) The rando variables representing the sae odel paraeter before and after an action are often dependent. This cannot be odeled with the intuitive approach, where the paraeter is represented with only one rando variable that is defined through two or ore conditional distributions. With the proposed approach, odeling the dependence is straightforward since the paraeter is represented through ultiple rando Value of inforation 7/31

8 variables, which are included jointly in and can thus be correlated. (b) The intuitive approach hinders the standardization of the coputations. With the proposed approach, the optiization of the action decision in Eq. (2) is copletely general. It is only necessary to identify the correct liit state functions and the corresponding doains Ω Ei. (c) The proposed structural reliability based odeling facilitates the coputations of the VoI, as presented later. 2.3 Perfect inforation Perfect inforation corresponds to the hypothetical situation in which there is no uncertainty on. In this case, the decision aker is always able to select the best action a. In the outcoe space of, one can identify doains in which one action is optial, Figure 2. These doains are described by one or ultiple liit state functions and are equal to the doains Ω Ei or unions thereof. The case shown in Figure 2 corresponds to the siple exaple provided earlier. The decision a (aintenance) is optial only under E 2 = {g F () 0} {g F () > 0}, i.e. when the coponent fails without aintenance F but does not fail with aintenance F. For all other events, i.e. for E 1 E 3 E 4, a 0 (do nothing) is optial. This follows directly fro Table 1 (given that the cost of aintenance is saller than the cost of failure, c < c F ). Note that perfect inforation eans here that the failure is known to occur (or not) at the tie of aking the decision, i.e. before it actually does occur (or not). Figure 2. The optial action corresponding to each outcoe x = [x 1 ; x 2 ] can be described through liit state functions g(x). The situation shown here corresponds to the basic exaple of section 2.2 with two action alternatives. Action a (aintenance) is optial only in the event E 2 = {g F () 0} {g F () > 0}, else action a 0 (do nothing) is optial. To foralize the concept of decisions under perfect inforation, let a opt (x) denote the optial decision for given x. It is Value of inforation 8/31

9 a opt (x) = arg in c(a, x). a (7) When coparing the cost associated with this optial action a opt (x) to the one of choosing the prior optial action a opt, the so-called conditional value of perfect inforation (CVoPI) is obtained for given syste state = x: CVoPI(x) = c(a opt, x) c(a opt (x), x). (8) Since the costs are uniquely deterined by the events E i, the CVoPI depends only on which event E i occurs. Therefore, the CVoPI can be expressed for given E i : CVoPI Ei = c Ei (a opt ) c Ei (a opt,i ), (9) where the notation a opt,i is introduced for the optial decision under event E i. Fro Eq. (9) it can be observed that the CVoPI is non-zero only if the optial decision under a known syste state E i, a opt,i, differs fro the optial decision that is taken under prior inforation, a opt. In case a opt,i CVoPI cannot be negative. a opt, the forer will lead to lower cost and therefore the A-priori, the true value of and the true event E i are not known. Nevertheless, it is possible to copute the hypothetical value of perfect inforation (VoPI) [1], defined as the expected value of the CVoPI: VoPI = E [CVoPI()] = [c(a opt, x) c(a opt (x), x)]f (x)dx = c(a opt, x)f (x)dx c(a opt (x), x)f (x)dx (10) = in c(a, )f (x)dx a in c(a, x) f (x)dx a = C prior in c(a, x) f (x)dx a, or alternatively as: Value of inforation 9/31

10 VoPI = CVoPI Ei Pr(E i ) = C prior c Ei (a opt,i ) Pr(E i ). (11) This value thus corresponds to the difference in expected utility between the situation a-priori and the expected utility under a situation of perfect inforation. The VoPI is the upper liit of the value any onitoring syste can have, irrespective of its capabilities. Any onitoring syste that is ore expensive than the VoPI will not be efficient. By definition, the VoPI cannot be negative, since in c(a, x)dx a This follows also fro the fact that the CVoPI cannot be negative. in a c(a, x) dx. Note: Viable action alternatives ay exist, which are not optial under any known x and therefore would not appear in Figure 2, but which ay be optial under conditions of uncertainty. 2.4 Iperfect inforation In real applications, onitoring systes provide only iperfect inforation on. Most easureents are subject to rando errors or uncertainty. But even in the absence of these, onitoring systes are iperfect because they do not provide direct inforation on all. As an exaple, for a case where perfect knowledge of aterial paraeters is available, if the future loading reains uncertain, the failure event cannot be predicted with certainty. Additionally, ost easureents are indirect, in particular for onitoring of existing structures and geotechnical applications [33] Bayesian updating Iperfect inforation can be used to learn about and, consequently, about the events E 1,, E. Bayesian updating is the atheatical fraework for learning the probability distribution of and the probabilities of E 1,, E with new iperfect inforation [5, 34, 35]. Following Bayes rule, the conditional distribution of given an observation Z, the posterior distribution, is: f Z (x) = L(x)f (x). (12) L(x)f (x)dx Value of inforation 10/31

11 The likelihood function L(x) describes the relation between the onitoring outcoe event Z and the uncertain variables. It is defined as [36]: L(x) Pr(Z = x). (13) In the structural reliability context, the onitored quantities are odeled as functions q i of, and the onitoring outcoe event Z can be expressed by eans of q i (). The ost coon case is that of easureents y = [y 1,, y ] of quantities q() = [q 1 (), q 2 (),, q ()]. If easureent errors ε i are additive and statistically independent rando variables, the relation between easureents y i and x is y i = q i (x) + ε i. Thus the event Z is defined as Z = {y = q() + ε}, with ε = [ε 1,, ε ]. It follows that y i q i (x) = ε i, and the likelihood function of the onitoring outcoe y is L(x) = f Y (y x) = f εi [y i q i (x)]. (14) For further details on how to odel observations with likelihood functions, the reader is referred to [35, 37]. In structural reliability analysis, the explicit coputation of f Z (x) can be circuvented and instead Pr(E i Z) can be obtained directly fro the definition of the conditional probability as Pr(E i Z) = Pr(E i Z) Pr(Z). (15) As shown in [37], structural reliability ethods can be used to copute both the nuerator and the denoinator in Eq. (15). Thereby it is relevant to distinguish between two classes of onitoring outcoes: Those that provide inequality inforation and those that provide equality inforation [37, 38]. Monitoring outcoes of the inequality type can be characterized through a function h() as follows: Z = {h() 0}. (16) Exaples include onitoring outcoes such as deforations are larger than a threshold or no defect found. For inequality inforation, reliability updating is straightforward, since h() can be interpreted as a liit state function describing the event Z and any of the available structural reliability ethods can be applied. In this case, it is not necessary to explicitly forulate the likelihood function. Value of inforation 11/31

12 Monitoring outcoes of the equality type are thus called because they can be described by an equality, such as Z = {y = q() + ε} introduced above. In the general case we can write Z = {y = Y}, where Y is the onitoring outcoe as predicted by the odel (for the special case it is Y = q() + ε ). Exaples include easureents of defect sizes, deforations or loads. Most onitoring outcoes are of this for. For equality observations, easureents are best described through the likelihood function L(x), such as Eq. (14). Direct application of Eq. (15) is not straightforward for equality inforation, because it requires the solution of surface integrals to copute the probabilities [39, 40]. An efficient alternative was proposed by the author in [37], which is based on transforing the likelihood function L(x), into equivalent inequality inforation, which can be efficiently and effectively cobined with existing structural reliability ethods to evaluate Eq. (15). A siple solution to updating with equality inforation, which is also a special case of the ethod proposed in [37], is to first update the distribution of following Eq. (12) and then copute the conditional probability Pr(E i Z) by perforing reliability analysis with the posterior PDF f Z (x). Using a Monte Carlo siulation approach, an estiate of Pr(E i Z) is obtained as: Pr(E i Z) = I(x Ω Ei )f Z (x)dx = I(x Ω Ei ) L(x)f (x) dx L(x)f (x)dx = I(x Ω E i )L(x)f (x)dx L(x)f (x)dx n MCS k=1 I(x k Ω Ei )L(x k ) n MCS L(x k ) k=1. (17) where x k, k = 1,, n MCS, are saples fro the prior PDF f (x). The Monte Carlo procedure is generally inefficient as it requires a large nuber of saples n MCS to achieve sufficient accuracy. For ore efficient ethods, the reader is referred to [37] Conditional value of inforation Once an observation Z has been ade and the conditional Pr(E i Z), i = 1,, have been coputed, decision optiization is in analogy to the prior decision optiization of Eq. (2): Value of inforation 12/31

13 a opt Z = arg in a c Ei (a) Pr(E i Z). (18) The only difference to Eq. (2) is the replaceent of Pr(E i ) with the conditional Pr(E i Z). The optiization conditional on Z is called posterior decision analysis and does not represent a coputational issue once Bayesian updating has been perfored. For a given Z, it is furtherore possible to copute the conditional value of inforation: CVoI Z = c Ei (a opt ) Pr(E i Z) c Ei (a opt Z ) Pr(E i Z). (19) Note that the CVoI is zero if the posterior optial decision a opt Z is the sae as the a-priori optial decision a opt, and positive otherwise. The CVoI in itself is uninteresting. Once the observation Z is ade, Pr(E i Z) represents the new state of nature according to which decisions should be ade; it is futile to copare a opt Z to the results of the original prior decision analysis a opt. The true interest is in the value of inforation (VoI) of the onitoring syste before an observation Z is ade Value of inforation (VoI) The VoI is the expected value of the CVoI with respect to all possible easureent outcoes VoI = E[CVoI]. In case of a finite nuber of utually exclusive easureent outcoe events Z 1, Z l, (onitoring outcoes of the inequality type), it is l VoI = CVoI Zj Pr(Z j ) j=1 l = Pr(Z j ) [ c Ei (a opt ) Pr(E i Z j ) c Ei (a opt Zj ) Pr(E i Z j )] j=1 (20) l = C prior [ Pr(Z j ) in c Ei (a) Pr(E i Z j )]. a j=1 The first ter in the last line follows fro the fact that l c Ei (a opt ) Pr(E i Z j ) Pr(Z j ) j=1 l = c Ei (a opt ) Pr(E i Z j ) j=1 (21) Value of inforation 13/31

14 = c Ei (a opt ) Pr(E i ) = C prior. This shows that the expected cost associated with the prior decision does not depend on the onitoring outcoe (as it clearly should not). In case of continuous easureent outcoes Z = {Y = y} (onitoring outcoes of the equality type), it is VoI = CVoI Z f Y (y)dy Y = C prior [ f Y (y) in c Ei (a) Pr(E i Y = y) dy], Y a (22) wherein f Y (y) is the joint PDF of the possible onitoring outcoes Y. Nuerical ethods are necessary to copute the integral in Eq. (22). Thereby, the conditional Pr(E i Y = y) will have to be coputed any ties. A coputationally efficient procedure for coputing Pr(E i Y = y) repetitively is thus crucial for coputing the VoI in this case. Such a procedure is proposed in the following section. 3 Coputationally efficient VoI analysis with structural reliability ethods The coputationally costly part in coputing the VoI is the evaluation of the syste odel as a function of x, which is required to assess the syste perforance through I(x Ω Ei ), and to copute the onitored quantities, h i (x). Efficiency is thus easured by the nuber of odel evaluations. For onitoring systes whose outcoe is of the inequality inforation type, efficient algoriths for evaluating Pr(E i Z) are available through structural reliability ethods, e.g. based on FORM or advanced siulation cobined with Eq. (15) [37, 38, 41]. These ethods require coputing I(x Ω Ei ) and h i (x) only for a sall nuber of values of x, typically in the order of as long as the diension of is liited. Once Pr(E i Z j ), i = 1,,, j = 1,, l, is coputed, the VoI is obtained through Eq. (20). Value of inforation 14/31

15 In the following, I focus on the VoI analysis for onitoring systes whose outcoes are of the equality inforation type, since this is the ore coon situation and identifying coputationally efficient solutions is less straightforward. An efficient solution to calculating the VoI in Eq. (22) is proposed. The ain difficulty lies in the need for integrating over Y. For this task, Monte Carlo ethods see appropriate. The siplest solution is offered by crude Monte Carlo siulation (MCS), as proposed in [16]. With the MCS approach, Eq. (22) is approxiated by n SY VoI C prior [ in c Ei (a) j=1 a n MCS I(x k Ω Ei )L(x k y j ) k=1 n MCS k=1 L(x k y j ) ]. (23) Here I have eployed the MCS approxiation of Pr(E i Z) given in Eq. (17). L(x k y j ) denotes the likelihood of x k, where the dependence on the onitoring outcoe y j is ade explicit. The x k, k = 1,, n MCS, are saples fro f (x), and the y j, j = 1,, n SY, are saples fro f Y (y). Following [16], the latter can be obtained by sapling first x k, then deterining the distribution f Y (y x k ), which is equal to the likelihood function Eq. (14), and sapling fro this distribution. This introduces a correlation between the saples of and Y, but this is not critical. The coputationally expensive part is the evaluation of the odel, which is required for deterining I(x k Ω Ei ) and L(x k y j ). In ost probles, one odel evaluation will be required to deterine I(x k Ω Ei ) and L(x k y j ) for every x k. The coputational effort is thus approxiately proportional to n MCS. Note that sapling fro the conditional f Y (y x k ) is inexpensive, and it ay thus be desirable to choose n SY > n MCS. In this case, several saples of easureent outcoes Y are generated based on the sae saple x k. This introduces a correlation aong the saples of Y, which ay becoe relevant when the proble diension n is large. The MCS approach is inefficient for probles involving reliability, where relevant onitoring outcoes (i.e. those which trigger itigation actions) are expected to occur with a sall probability only and where the probabilities of relevant events E i are sall. To obtain accurate solutions with MCS, therefore, a large nuber of odel evaluations would be necessary. For this reason, an iportance sapling (IS) schee is proposed in the following. With IS, n IS weighted saples x k of are generated according to the IS density ψ (x). The IS estiate of Pr (E i Y = y j ) is: Pr(E i Y = y j ) n IS k=1 w (x k )I(x k Ω Ei )L(x k y j ) n IS, (24) w (x k )L(x k y j ) k=1 Value of inforation 15/31

16 with iportance sapling weight w (x) = f (x) ψ (x). (25) For the integration over Y, the entire outcoe space of Y is relevant. Furtherore, for fixed x k, evaluations of the likelihood function are inexpensive. For these reasons, an MCS approach to the integration over Y would be coputationally efficient. Unfortunately, when perforing an IS over, the siple sapling schee for Y, which is applicable in the MCS approach, is not available. For this reason, IS ust also be used for the integration over Y. With an IS approach, the VoI is approxiated by: VoI C prior 1 n ISY C prior 1 n ISY n ISY w Y (y j ) in c Ei (a) Pr(E i Y = y j ) a j=1 n ISY n IS k=1 w Y (y j ) in c Ei (a) w (x k )I(x k Ω Ei )L(x k y j ) n a IS. w (x k )L(x k y j ) j=1 k=1 (26) The iportance sapling weight w Y (y) is w Y (y) = f Y(y) ψ Y (y). (27) where ψ Y (y) is the IS density of Y. The PDF of Y, f Y (y), is unknown, but it can be estiated as n IS f Y (y) 1 w n (x k )f Y (y j x k ). (28) IS k=1 Noting that L(x k y j ) = f Y (y j x k ), it can be seen that this estiator of f Y (y) is equal to the denoinator in Eq. (26) divided by n IS. Cobining Eqs. (26) (28), the final IS estiator of VoI is obatained: n ISY VoI C prior n ISY n IS ψ Y (y j ) in c E a i (a) w (x k )I(x k Ω Ei )L(x k y j ). (29) j=1 n IS k=1 It reains to select efficient IS densities for and Y, which is crucial for the efficiency of the IS approach. The key to identifying an IS density ψ (x) for is to consider the hypothetical Value of inforation 16/31

17 situation of perfect inforation illustrated in Figure 2. Under perfect inforation, the regions in the outcoe space of, in which a particular decision is optial, can be identified. Ideally, the IS density is focused on those parts of these regions with highest probability density. In agreeent with classical IS approaches in structural reliability [e.g., 42], these correspond to the areas around the so-called ost likely failure points (MLFPs), also called design points. When transforing all liit state functions, and hence the doains describing the events E i, into the space of standard noral rando variables U, these are the areas closest to the origin. One possibility, which is eployed in the application exaple described later, is to choose a kernel density for ψ (x), with kernels centered around the MLFPs and the origin. For Y, the original PDF f Y (y) would be an effective sapling density, as discussed earlier. Since f Y (y) is not known, a sapling density ψ Y (y) that is an approxiation to f Y (y) can be obtained fro a few saples of f Y (y x k ), where the x k are the saples used in (29), thus avoiding additional odel runs. 4 Decisions at ultiple points in tie In ost applications, action decisions a can be ade at ultiple points in tie, at which different aounts of inforation fro the onitoring syste are available. A classic exaple is onitoring and inspection of deteriorating structures [43]. Consider two points in tie t 1 and t 2 with t 1 < t 2, at which decisions a(t 1 ) and a(t 2 ) are ade. The onitoring outcoe of t 1 is available only for the decision a(t 1 ), whereas both onitoring outcoes are available for aking the decision a(t 2 ). To copute the VoI, the action decisions ust be optiized sequentially, following the sequence of available inforation. An approxiate solution is obtained as follows for possible action decisions at ties t 1, t 2,, t T. Expanding Eq. (18), one gets: a opt Z (t 1 ) = arg in a(t 1 ) [ a opt Z (t 2 ) = arg in a(t 2 ) [ in c(e i, a(t 1 ),, a(t T )) Pr(E i Z(t 1 ))] (30) a(t 2 ) a(t T ) in c(e i, a opt Z (t 1 ), a(t 2 ),, a(t T )) Pr(E i Z(t 2 ))] (31) a(t 3 ) a(t ) a opt Z (t T ) = arg in c(e i, a opt Z (t 1 ),,, a opt Z (t T 1 ), a(t T )) Pr(E i Z(t T )). (32) a(t T ) Value of inforation 17/31

18 Z(t i ) refers to all onitoring inforation collected up to tie t i. The coputation of the VoI is then perfored following the second line of Eq. (20), where a opt Z is replaced by the set [a opt Z (t 1 ),, a opt Z (t T )]. The IS solution of Eq. (29) is equally applicable to this case. The above procedure is only an approxiation, as it does not take into account the possibility that it can be beneficial to delay an action because the decision on the appropriate action will be iproved when ore inforation is available later. The procedure will thus underestiate the overall VoI. Nevertheless, for any applications the approxiation will be reasonable. Unfortunately, in the general case the exact coputation of the VoI is associated with an exponential increase in coputation cost with increasing nuber of decision ties. In riskbased inspection planning, this has otivated the identification of decisions based on siple decision rules, e.g. perforing a repair when the easured size of an identified defect exceeds a threshold value [43, 44]. Through such decision rules, the decision at each tie step is readily identified, and the VoI coputation can again follow the second line of Eq. (20). For the special case that the relevant phenoena can be odeled as discrete Markov processes, algoriths for solving partially observable Markov decision processes [13, 45] can be eployed to copute the VoI. These algoriths have a coputation cost that increases only linearly with nuber of tie steps; their disadvantage is the liitation to discrete rando variables. For any proble with discrete rando variables that is odeled through graphical odels, also the liit eory decision diagras (LIMID) ay be viable alternative [22, 46]. 5 Illustrative application to onitoring of a structure subject to fatigue deterioration An illustrative application is presented of the theory and the proposed solution strategy to onitoring of a structural coponent subject to fatigue. For the sake of a clear presentation, I siplify the application, yet it includes ost features of a real application. 5.1 Life-cycle odel The structure has a life-tie of 20 years and is inspected and aintained every 5 years. During these capaigns, the considered coponent ay be inspected and/or replaced. I consider the following action alternatives: Value of inforation 18/31

19 a 0, a 5, a 10, a 15 : the coponent is replaced in year 0, 5, 10 or 15, respectively; a n : the coponent is not replaced. The VoI is coputed for two exeplary options: (1) perfor a easureent in year 5; (2) perfor easureents in year 0 and 5. In case of option (1), alternative a 0 is not relevant, as will be seen fro the prior decision analysis. In case of option (2), two action decisions are considered, one in year 0 and one in year 5, following Section Deterioration odel Fatigue deterioration of the coponent is described by a classical siplified odel taken fro the literature [38]. The crack growth due to the stress ranges ΔS is described by Paris law as dl(n) dn = C [ΔS πl(n)] (33) Here, l is the crack depth, n is the nuber of stress cycles, ΔS is the stress range per cycle (constant stress aplitudes are assued) and C and are epirically deterined odel paraeters. In this forulation of Paris law, the geoetry correction factor is one, which in theory corresponds to the case of a crack in a plate with infinite size. With the boundary condition l(n = 0) = l 0, this differential equation can be solved for the crack depth as a function of tie t [38]: 1 l(x, t) = [(1 2 ) CΔS π (1 2 2 νt + l ) 1 0 ] 2. (34) t is the tie in years and ν is the annual cycle rate, so that n = νt. The event of failure is described by the liit state function g F as a function of l(x, t) and the critical crack depth l c : g F (x, t) = l c l(x, t) (35) Here, the rando variables of the proble are = [l 0 ; ΔS]. The odel paraeters are suarized in Table 2. Value of inforation 19/31

20 Table 2. Paraeters of the crack growth exaple. Variable Distribution Mean c.o.v. l 0 [] exponential 1 1 l c [] deterinistic 50 - ΔS [N 2 ] lognoral [ ] deterinistic C [N ] deterinistic exp ( 33) - ν [yr 1 ] deterinistic Based on the liit state function for failure, Eq. (35), the relevant events, which deterine the consequences and hence the optial action decision, are defined as: E 1 = F(5yr) = {g F (, 5yr) 0}; E 2 = F(5yr) F(10yr) = {gf (, 5yr) > 0} {g F (, 10yr) 0}; E 3 = F(10yr) F(15yr) = {gf (, 10yr) > 0} {g F (, 15yr) 0}; E 4 = F(15yr) F(20yr) = {gf (, 15yr) > 0} {g F (, 20yr) 0}; E 5 = F(20yr) = {gf (, 20yr) > 0}. 5.3 Measureent odel In the inspection capaign at tie t i, the crack depth can be easured. The easureent y ti has independent, additive easureent error ε,ti with PDF f ϵ ( ). The likelihood function describing the easureent is L(x) = f ϵ (y ti l(x, t i )). (36) with f ϵ being a zero-ean noral PDF with standard deviation σ ϵ. Unless otherwise stated, it is σ ϵ = 1. For the case of two easureents y 0 and y 5 in years 0 and 5, the likelihood function is (see also Eq. (14)): L(x) = f ϵ (y 0 l(x, 0yr))f ϵ (y 5 l(x, 5yr)). (37) 5.4 Repair and cost odel For illustrative purposes, I consider two cases: Value of inforation 20/31

21 1. Perfect repair: Following a repair, the coponent will not fail. 2. Iperfect repair: Following a repair, the coponent is characterized by a new initial crack depth l 0. The repaired coponent is again subject to fatigue deterioration. The new crack depth l 0 has the sae probability distribution as l 0, but is independent of the latter. The stress range ΔS is the sae before and after repair. The nuber of rando variables thus depends on the odeling of the repair action. In case 1, the set of rando variables is = [l 0 ; ΔS]; in case 2, it is = [l 0 ; l 0 ; ΔS]. In sections 5.5 to 5.8, the coputations and results for case 1 are presented. Because this case includes only two rando variables, it facilitates graphical representation. Extension to case 2 is considered in section 5.9. The following cost odel is applied, which considers discounting of costs: Cost of repair in years 0, 5, 10, 15, respectively: c R0 = , c R5 = , c R10 = , c R15 = Cost of failure in the periods 0-5, 5-10, 10-15, years, respectively: c F5 = , c F10 = 10 6, c F15 = , c F20 = Decision analysis under prior and perfect inforation The doains of the optial actions under perfect inforation are shown in Figure 3. On the left-hand side, the doains are shown in the outcoe space of ; on the right-hand side, the doains are shown in the outcoe space of standard noral rando variables U. The latter are coputed by a transforation x = T(u). In the general case, T can be any of the classical transforations applied in structural reliability analysis, e.g. the Rosenblatt transforation [47] or the Nataf transforation [48]. For the considered exaple, due to the statistical independence of l 0 and ΔS, these reduce to the arginal transforations l 0 = F l0 1 [Φ(u 1 )] and ΔS = F 1 ΔS [Φ(u 2 )]. Φ( ) is the standard noral cuulative distribution function; F 1 l0 ( ) and F 1 ΔS ( ) are the inverse CDFs of l 0 and ΔS. Value of inforation 21/31

22 Figure 3. Optial actions under perfect inforation. (a) In the outcoe space of ; (b) transfored to standard noral space. Circles indicate the ost likely failure points (MLFPs) in standard noral space. The corresponding a-priori probabilities as obtained with FORM are suarized in Table 3. As evident fro the alost linear behavior of the liit state surfaces around the MLFPs visible in Figure 3b, FORM provides accurate results. Table 3. A-priori probabilities (FORM results). Event E 1 : Failure in period 0-5 years E 2 : Failure in period 5-10 years E 3 : Failure in period years E 4 : Failure in period years Probability E 5 : No failure before year Prior decision analysis A-priori, the optial decision a opt is found using Eq. (2). With the cost odel and the probabilities reported in Table 3, the optial decision is a n : do nothing. The corresponding expected cost is calculated with Eq. (3) as Value of inforation 22/31

23 C prior = = Value of perfect inforation VoPI The optial actions under different evidence are: a 0 in case E 1, i.e. the coponent should be replaced in year 0 if it were to fail prior to year 5; a 5 in case E 2, i.e. the coponent should be replaced in year 5 if it were to fail in the period fro year 5 to year 10; and so on. If the coponent does not fail during the service life (event E 5 ), the optial action is a n : do nothing. Given E 5, the conditional value of perfect inforation (CVoPI) is zero, since the optial action in this case is equal to the one found with the prior decision analysis. Given any of the other events, the CVoPI is positive. The value of perfect inforation is deterined with Eq. (11) as VoPI = ( ) = This value is the upper liit of the benefit that can be achieved with any onitoring syste. 5.6 Iportance sapling Based on the identified design points (MLFPs) shown in Figure 3b, an IS density ψ is is selected. The ψ is specified in the space of standard noral rando variables, i.e. ψ U is specified and the saples of are obtained by sapling u k fro U and transforing the saples, x k = T(u k ), according to section 5.5. Following Section 3, a kernel density is selected for ψ U (and consequently for ψ ), with five standard noral distributions as kernels. The first four kernels are centered around the 4 MLFPs shown in Figure 3b, the fifth is centered around the origin. (The origin is the MLFP of the no-failure event E 5.) The resulting ψ U is shown in Figure 4, together with rando saples generated fro ψ U. Value of inforation 23/31

24 Figure 4. Iportance sapling density, shown together with 100 rando saples. Dashed lines indicate the borders of the doains corresponding to E 1,, E 5 (see Figure 3). The IS distribution for the easureent results ψ Y is constructed by evaluating the odel for the saples fro ψ U, sapling corresponding easureent outcoes using f Y and then fitting a joint noral distribution to these saples. 5.7 Bayesian updating Likelihood functions for exeplarily onitoring outcoes are shown in Figure 5. On the lefthand side, likelihood functions are shown for a onitoring outcoe y 5 = 6 in year 5. On the right-hand side, likelihood functions are shown for a onitoring outcoe y 0 = 3 in year 0 and y 5 = 6 in year 5. The effect of the easureent error on the likelihood function is clearly visible. For the case of two easureents and a onitoring syste with sall easureent error σ ϵ = 0.3, one is able to deterine the uncertain paraeters with good accuracy, i.e. the posterior variance becoes sall in this case. With only one easureent, the resulting posterior variance reains significant even in the case of sall easureent error, since different cobinations of paraeters l 0 and ΔS lead to the sae crack size. Value of inforation 24/31

25 Figure 5. Likelihood functions shown for easureent results y 0 = 3 and y 5 = 6 and different easureent accuracy σ ε. The likelihood functions on the left-hand-side are for a easureent at year 5 only, those on the right hand side represent the cobined easureent in year 0 and 5. Dashed lines indicate the borders of the doains corresponding to E 1,, E 5 (Figure 3). 5.8 Value of inforation The value of inforation (VoI) is coputed for the case of one easureent in year 5 and for the case of two easureents in year 0 and 5. In the latter case, the subsequent decision aking outlined in Section 4 is applied to identify the optial decision after the first easureent. To assess the efficiency and accuracy of the proposed IS approach, coputations are perfored with different nubers of saples and are copared to solutions obtained with MCS. The results are suarized in Table 4. The VoI of two easureents is only slightly larger than the VoI of one easureent only. This indicates that with σ ϵ = 1.0 the first easureent in year 0 does not provide uch useful additional inforation. Value of inforation 25/31

26 Table 4. VoI coputed with iportance sapling (IS) and Monte Carlo siulation (MCS). Mean value and standard deviations were evaluated by repeating the coputations 20 ties. 1 easureent 2 easureents Coputation Mean St. dev. Mean St. dev. MCS (10 4 saples) 1447 (264) 1624 (246) IS (10 3 saples) 1443 (124) 1636 (49) IS (10 2 saples) 1586 (385) 1721 (404) The results in Table 4 indicate the effectiveness of the proposed IS solution. Copared to MCS, with IS the nuber of odel evaluations can be reduced by approxiately a factor of 50 to achieve the sae accuracy VoI as a function of easureent accuracy To obtain further insights into the VoI, the resulting VoI is coputed for varying easureent accuracy, i.e. for different values of σ ϵ. The results are suarized in Figure 6. For coparison, also the value of perfect inforation VoPI is shown. For larger easureent uncertainty ( σ ϵ 1 ), two easureents do not provide significantly ore inforation than one easureent. In this case, the second easureent in year 5 is doinating the posterior distribution and the updated probabilities Pr(E i Z). This can be observed graphically by coparing the likelihood functions in Figure 5. For larger easureent uncertainty (upper figures), the differences between one and two easureents are less distinct than for saller easureent uncertainty (lower figures). With a single easureent, the VoI is far fro the VoPI even for σ ϵ = 0. The reason is that knowing the crack size only at one point in tie does not allow one to deterine the odel paraeters = [l 0 ; ΔS] uniquely, as should be evident fro the likelihood function for the case of one easureent and σ ϵ = 0.3 shown in Figure 5. This deonstrates that perfect inforation requires not only zero easureent uncertainty, but also easuring enough relevant quantities. For the case of two easureents shown in Figure 6, the VoI would eventually reach the VoPI as σ ϵ = 0. In this idealized exaple, two exact easureents at two different points in tie are sufficient to exactly deterine the two odel paraeters l 0 and ΔS, thus eliinating all uncertainty. Value of inforation 26/31

27 Figure 6. VoI as a function of easureent accuracy. Coputations are perfored with IS using 10 5 saples. 5.9 Iperfect repair In case of iperfect repair, the proble has three rando variables as discussed in section 5.4. After the repair, the perforance of the coponent is described by the sae echanical odel, but with new initial crack depth l 0 and different starting tie. The liit state function describing a failure τ years after a repair is g FR (x, τ) = l r (x, τ) l c. (38) The function l r is equal to l defined in Eq. (34), wherein l 0 is replaced with l 0. Since a repair is possible already in year 0, the tie τ can be up to 20 years. g FR (x, τ) ust thus be evaluated for τ = 5,10,15,20 years. A total of 25 events E 1,, E 25 ust now be considered, representing cobinations of possible failure ties of the original coponent and of the repaired coponent. For each cobination of events and action a, the corresponding costs can be assigned. These are not reported here for brevity. Exeplarily, event E 10 is equal to failure of the original structure in the period of 5 10 years and no failure of the repaired structure, i.e. E 10 = {g F (, 5yr) > 0 g F (, 10yr) 0 g FR (, 20yr) > 0}. The costs associated with event E 10 are c R0 in case of action a 0, c R5 in case of a 5 and c F10 otherwise. The VoI coputations are then again perfored following Eq. (29). It is found that the consideration of the possible failure following a repair has little ipact on the result. The changes in the results copared to the case with perfect repair are within the range of scatter of the IS results; the results reported in Table 4 are thus also valid for the case Value of inforation 27/31

28 of iperfect repair. The reason for these sall differences in the results? The probability of perforing a repair is only in the order of Therefore, the additional expected cost due to a potential second failure is only inor. This deonstrates that in this application there is no need to explicitly odel the behavior of the repaired coponent when coputing the VoI. 6 Discussion The ai of this paper is to propose a rigorous odeling fraework and efficient coputational algoriths for evaluating the value of inforation VoI of onitoring systes based on structural reliability ethods. An iportance sapling solution was developed and its efficiency was deonstrated through the presented exaple application. To facilitate a graphical interpretation of the results, I studied a proble with only two rando variables. For probles with any rando variables, iportance sapling solutions becoe inefficient. However, in ost applications it will be possible to reduce to nuber of relevant rando variables to less than ten by a-priori identifying the ost influential variables, thus facilitating the use of the proposed solution strategy. Arguably the ost difficult part in realistic applications is the need to explicitly odel the relevant decision processes over the entire service life period of the onitoring syste. As pointed out in [49], contrary to the exaples of Bayesian decision analysis provided in text books, real situations in which the VoI should be estiated are not siple. A ultitude of potential action alternatives a exist, and engineers are not trained to prescribe quantitative rules for which decision to take in which circustances, except for siple cases. The typical behavior of engineers under situations of uncertainty is to collect soe inforation, and to go fro there. Hence the popularity of onitoring systes. Nevertheless, the act of systeatically describing the potential onitoring outcoes and the action alternatives is a highly useful process, even when it is not possible to provide exact and coplete answers. In any instances, the decision odel can be kept quite siple and only ain events and actions ust be included explicitly in the odel. As an exaple, it is often not necessary to explicitly odel the perforance of a syste after repair or other itigation actions [43], as was also found in the exaple application in this paper. It is also iportant to point out that an exact assignent of costs or utility values is often not necessary. For decision aking purposes it is typically sufficient to provide approxiate estiates of costs when coputing the VoI. Value of inforation 28/31