System Reliability Estimation and Confidence Regions from Subsystem and Full System Tests

009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 0-, 009 FrB4. System Reliability Estimation and Confidence Regions from Subsystem and Full System Tests James C. Sall Abstract This aer develos a rigorous and ractical method for estimating the reliability with confidence regions of a comlex system based on a combination of full system and subsystem (and/or comonent or other) tests. It is assumed that the system is comosed of multile rocesses (e.g., the subsystems and/or comonents within subsystems), where the subsystems may be arranged in series, arallel (i.e., redundant), combination series/arallel, or other mode. Maximum likelihood estimation (MLE) is used to estimate the overall system reliability. Interestingly, for a given number of subsystems and/or comonents, the likelihood function does not change with the system configuration; rather, only the otimization constraints change, leading to an aroriate MLE. The MLE aroach is well suited to roviding asymtotic or finite-samle confidence bounds through the use of Fisher information or bootstra Monte Carlo-based samling. Keywords: System identification, system reliability, arameter estimation, otimization, bootstra, maximum likelihood, Fisher information matrix, data fusion.. INTRODUCTION This aer considers the roblem of estimating the reliability for a comlex system based on a combination of information from tests on the subsystems, comonents, or other rocesses within the system, and, if available, tests on the full system. A key motivation for this setting comes from the fact that it is often difficult or infeasible to directly evaluate the reliability of comlex systems through a large number of full system tests alone. Such a difficulty may arise, for examle, when the full system is very costly or dangerous to oerate and/or when each full system test requires the destruction of the system itself. Nevertheless, it is also often the case that there are at least a few tests of the full system available; it is obviously desirable to include such information in the overall reliability assessment. Such The author is with the Johns Hokins University Alied Physics Laboratory, Laurel, Maryland 073-6099 U.S.A. (james.sall@jhual.edu). This work was suorted in art by the JHU/APL IRAD Program and U.S. Navy Contract N0004-03-D- 6606. The author thanks Dr. Coire Maranzano for helful discussions and for verifying some of the calculations. Some details have been eliminated from this aer to meet ACC length constraints. In articular, the roof of convergence for the estimator and a numerical study are available from the author uon request. full systems tests are often critical to hel guard against ossible mismodeling of the relationshi between the subsystems and full system in calculating overall reliability. This aer develos a method based on rinciles of maximum likelihood for estimating the overall system reliability from a combination of full system and subsystem or other tests. Certainly, other aroaches exist for estimating system reliability when the subsystems are indeendent (see, e.g., Hwang et al., 98; and Ramírez-Márquez and Jiang, 006). However, these aroaches do not allow for easy inclusion of limited full system tests when available, and do not generalize to include systems where the subsystems may be statistically deendent. (Note that the inequalitybased reliability method of Hill and Sall, 007, does allow for such deendence in roducing a bound to the full system reliability, but this method requires certain airwise subsystem tests that may not be feasible in ractice.) A key art of the aroach here is the calculation of uncertainty (confidence) bounds on the estimates. We discuss the Fisher information matrix as a basis for asymtotic bounds and also discuss a bootstra-based method for comuting confidence regions when the asymtotic bounds are inaroriate. This bootstra aroach deals with the inadequacies of traditional methods based on the asymtotic normality of the MLE. Several other aroaches have been roosed to deal with the inadequacy of asymtotic normality in the context of using subsystem tests to estimate full system reliability. For examle, Myhre and Saunders (968) use the asymtotic chi-squared distribution of the log-likelihood ratio to deal with the roblem of having confidence intervals outside the unit interval [0, ]. Easterling (97) treats the system reliability derived from subsystem estimates as an estimate from data having a binomial distribution. Then it is ossible to use standard results on the exact distribution of the binomial estimate to get the confidence interval, yielding an exact solution in the secial case where the system is comosed of one subsystem (i.e., system = subsystem) and an intuitively aealing aroximation when there are two or more To avoid the cumbersome need to reeatedly refer to tests on subsystems, comonents, and other rocesses within the system as the key source of information other than full system tests, we will usually only refer to subsystem tests; subsystem tests in this context should be considered a roxy for all ossible test information short of full system tests. 978--444-454-0/09/$5.00 009 AACC 5067

subsystems. Coit (997) considers the case where there are a large number of subsystems in either a series or arallel configuration; in the series configuration, the logarithm of the system reliability is aroximately normally distributed by central limit theorem effects. Hence, a log-normal distribution is assumed for the system reliability, roviding the basis for the confidence interval. Ramírez-Márquez and Jiang (006) focus on methods for estimating the variance of the reliability estimates, and then use these variance estimates together with normal or binomial aroximations to the distribution of the estimates to form confidence intervals.. THE LIKELIHOOD FUNCTION AND MLE FORMULATION. General Formulation Consider a system comosed of rocesses, tyically subsystems and/or comonents of subsystems. The subsystems may be arranged in series, arallel (i.e., redundant), combination series/arallel, or other form (e.g., standby systems), subject to being able to write down a robabilistic characterization of the system that leads to a likelihood function (Leemis, 995, Cha., includes a thorough discussion of the many tyes of systems frequently encountered in ractice). While we assume that the test data for estimating the reliability are statistically indeendent, we do not, in general, require that the rocesses be statistically indeendent when oerating as art of the full system. (In rincile, it is also ossible to formulate a likelihood function based on test data that are statistically deendent. That is, the testing outcome for rocess j may be statistically deendent on the outcome for rocess i, for some i j. We do not ursue that extension here.) In the discussion below, the term oerationally [in]deendent is used to refer to the case where the rocesses are either statistically indeendent or deendent, as relevant, when oerating as art of the full system. The general MLE formulation involves a arameter vector θ, reresenting the arameters to be estimated, together with an associated log-likelihood criterion L(θ). Let ρ and ρ j reresent the reliabilities (success robabilities) for the full system and for rocess j, resectively, j =,,...,. The vector θ = [ρ, ρ,, ρ ] T ; ρ is not included in θ because it will be uniquely determined, or at least bounded, from the ρ j and ossibly other information via relevant constraints. When ρ is uniquely determined by a function of θ, then the estimate ˆρ, as determined from alying this function to the MLE of θ (say ˆθ ), is the MLE of ρ. This invariance of MLE alies even though the maing from θ to ρ is not generally one-to-one and may not be continuous (see, e.g., Zehna, 966). Ultimately, we are interested in an estimate and confidence region for ρ, as derived from the MLE for θ. The secific definition of θ will deend on the details of the system. Interestingly, for a given definition of θ, the definition of L(θ) will not deend on how the subsystems are arranged in the full system. That is, L(θ) is the same regardless of whether, say, the subsystems are in series or arallel. However, the MLE will change as a function of the system arrangement. This is a consequence of the constraints in the otimization roblem that is solved to roduce the MLE, as illustrated below. It is generally assumed, at a minimum, that success/failure data are available on the rocesses within the full system. As mentioned above, it is also generally assumed that success/failure data are available directly on the full system. In cases involving deendent subsystems, it may be desirable that the information from the rocesses include some data other than direct subsystem success/failure data in order to obtain the information needed for characterizing the nature of the deendence (we say desirable because it may be ossible to estimate bounds to the system reliability in the absence of such information). For examle, in the deendent-subsystem case discussed in Section 3, obtaining data on one critical comonent aearing within multile subsystems allows for an MLE of ρ; the absence of such data allows the analyst to estimate a lower bound to ρ. We now resent the general MLE otimization roblem. Let Θ reresent the feasible region for the elements of θ. To ensure that relevant logarithms are defined and that the aroriate derivatives exist, it is assumed, at a minimum, that the feasible region Θ includes the restriction that 0 < ρ j < for all j (other restrictions may be included as aroriate). The general MLE formulation is: θ= ˆ arg max L( θ) θ Θ subject to f ( θ, ρ) 0, (.) where f ( ) is some function reflecting the constraints associated with the oeration of the full system. In some common cases (e.g., fully series and fully arallel cases, with the rocesses corresonding to the subsystems as in Subsection.), the inequality in the constraint can be relaced with an equality. Let X reresent the number of successes in n indeendent, identically distributed (i.i.d.) exeriments with the full system and X j reresent the number of successes in n j i.i.d. exeriments with rocess j, j =,,...,. Note that in the discussion below there is no notational distinction between a random variable (vector) and its realization, with the exectation that the distinction should be clear from the context. Let Y reresent the full set of data {X, X, X,, X }. From the assumtion of indeendence of all test data, the robability mass function, say (Y θ, ρ), is: 5068

n ( Y θ, ρ ) = X( )( n X) ρ ρ X system n X n ( ( ) ) X n ( ) n X ( ) X ρ ρ ρ ρ, X X rocesses (.) leading to the log-likelihood function L(θ) log (Y θ, ρ): L( θ) = X logρ + ( n X) log( ρ) + X jlog ρ j + ( nj X j) log( ρ j) + constant, j= (.3) where the constant is not deendent on θ. Note that the generic forms for the above likelihood and log-likelihood aly regardless of the secific layout of the subsystems (series, arallel, combination series/arallel, or oerationally deendent). However, the relationshi between ρ and the ρ j differs according to the layout of the rocesses. In finding the MLE of θ, say ˆθ, this relationshi manifests itself as constraints in an otimization roblem. Tyically, the MLE is determined via finding a root of the score equation ( θ) = 0, or a normalized form of this equation, in the asymtotic samle size case, where the score vector is ρ ρ =. (.4) ρ The elements of the score vector deend on the constraints. A common secial case is where the constraints lead to a unique differentiable function h( ) that relates θ to ρ: ρ = h(θ). In that case, (.3) leads to the following form of the score vector in (.4): X n X ρ ρ X h n X h = +. (.5) ρ ρ X n X ρ ρ We will see several illustrations of the form in (.5) in the examles and theoretical results below.. Fully Series and Fully Parallel Cases To illustrate the general formulation of Subsection., we consider here the two most common extreme cases of fully series systems and fully arallel systems with the rocesses corresonding to oerationally indeendent subsystems. The results below can extend readily to common cases of mixed (oerationally indeendent) series/arallel systems (see, e.g., Leemis, 995, Sect.., for a descrition of such general systems). The results here illustrate the setting mentioned in the context of (.5) above, where there exists a differentiable function ρ = h(θ). From (.), the MLE in the series-subsystem case is found according to θ ˆ = arg max L ( θ) θ Θ subject to ρ= ρj, j= while the MLE in the arallel-subsystem case is found according to θ ˆ = arg max L ( θ) θ Θ subject to ρ= ( ρ j ). j= In the above series and arallel cases, it is straightforward to determine the score vector using (.5). Making the substitution ρ = j= ρ j in eqn. (.5), the j =,,..., elements of the score vector in (.4) (or (.5)) for the series case are: X + X j ( n X) ρ ( nj X j) =. (.6) ρ ρ ( ρ) ρ ( ρ ) j j j j Likewise, making the substitution ρ = ( ρ j ) in j= eqn. (.5), the elements of the score vector in (.4) (or (.5)) for the arallel case are: X ( n n X X ). (.7) ρ ρ ρ ρ j X + j j = + ( ρi ) j j i= ( j) i j Excet for the degenerate settings of X = n in the series case and X = 0 in the arallel case, the solution to ( θ) = 0 must generally be found by numerical search methods. (The two degenerate cases yield ρ ˆ j = ( n+ X j) ( n+ nj) and ρ ˆ j = X j ( n+ nj), resectively, both of which are the natural intuitively obvious solutions.) 3. DEPENDENT SUBSYSTEMS There are obviously innumerable ways in which subsystems can interact while oerating as art of a full system. The reliability analysis for each such system must be handled searately based on the information available. While the reliability analysis with oerationally deendent 5069

subsystems is usually more difficult than with oerationally indeendent subsystems, the MLE may still be available if the roblem can be cast in the form of (.) with an aroriate constraint. For examle, in systems that may be reresented as a series of m generally deendent subsystems, the following exression relates the full system reliability to conditional subsystem reliabilities: ρ= P( S = ) P( S = S = ) P S = { S = } { S = } P m ( Sm = { S } j= j = ) ( ) 3, (3.) where S j = 0 or is the indicator of whether subsystem j is, resectively, a failure or success. Hence, the analyst may have the information needed to imlement (.) with an equality constraint that uniquely defines ρ if data and/or rior information are available to characterize the conditional robabilities on the right-hand side of (3.). Further information on the relatively common setting where deendence gets introduced through shared comonents is available from the author (not included here due to ACC 009 sace constraints). 4. THEORETICAL PROPERTIES This section summarizes the convergence roerties associated with the MLE formulation above. Note that standard i.i.d. MLE theory (e.g., Serfling, 980, Sect. 4.) does not aly because of the different success/failure robabilities associated with the different subsystems. Nevertheless, the structure associated with (.) and (.3) allows us to show that the MLE for ρ will converge to the true full system reliability under reasonable conditions. First, however, we resent a result giving conditions under which there is a unique function h(θ) relating θ to ρ. Lemma. Suose that the constraint in roblem statement (.) can be reresented as an equality f (θ, ρ) = 0 with f being a continuously differentiable function in both θ Θ and 0 < ρ <. For a fixed θ Θ, suose f ( θ, ρ) ρ 0 almost surely (a.s.) at ρ such that f ( θ, ρ) = 0. Then there exists an oen neighborhood of θ and a unique continuously differentiable function h such that for all θ in this neighborhood, ρ = h(θ) and h( θ) f( θ, ρ) f( θ, ρ) = ρ (4.) Proof. The result is a consequence of the imlicit function theorem (e.g., Aostol, 974, Sect. 3.4). Q.E.D. We now resent the following strong (a.s.) convergence result for the MLE of ρ. Let ρ be the true value of the full system reliability and θ = [ ρ, ρ,, ρ ] T be the corresonding true value of θ. Further, let A N = diag var( ρ), var( ρ),, var( ρ ) L L L be a diagonal matrix used to normalize for the variability of the elements in, where N = {n, n, n,..., n } is the collective samle size. Proosition below establishes conditions for ˆρ = ρ + o() being a root of the normalized score equation, A N = 0, where o() is a term going to zero as N gets large in the sense described below. As usual when working with score vectors, however, the Proosition does not guarantee that this solution is unique and/or a global maximum of L(θ). Proosition. For constants 0 < C C + <, suose the feasible region Θ is such that Θ = {θ: C ρ j C + for all j} and that C ρ C +. Further, suose that the constraint in roblem statement (.) can be reresented as an equality f (θ, ρ) = 0 with f being a continuously differentiable function in both θ Θ and C ρ C +. For some strictly ositive constants C and C and all θ Θ and C ρ C +, suose C < f ( θ, ρ) ρ C and C f ( θ, ρ) ρ j C for all j. Then, for the roblem described in (.) and (.3), ˆρ = ρ + o() is an a.s. solution to A N = 0 as n + n, n + n,..., n + n, where, for each j, one of the following three ossibilities holds: (i) n j n = o(), (ii) nn j = o(), or (iii) nj n = O() and nn j = O(). Comment. The multile limits n + n, n + n,..., n + n are true if and only if one of the following three (mutually exclusive) ossibilities occur: (a) n and n j < for all j, (b) n < and n j for all j, or (c) n and n j for at least one j. The roof considers these three cases in turn subject to the additional constraints (i) (iii) in the Proosition statement. Proof. The conditions of the Proosition are stronger than those of Lemma. Hence, there exists a differentiable function h such that for all θ in an oen neighborhood of ˆθ, ρ = h(θ) with derivative given by (4.). Thus, the score vector is given by (.5). The remainder of the roof is available uon request (not included here due to sace constraints). 5. THE FISHER INFORMATION MATRIX The Fisher information matrix is helful in at least two resects in the reliability estimation roblem: (i) It can be used to determine when the estimation roblem in Section is well osed (i.e., when θ is identifiable) through an evaluation of the conditions ensuring that the information matrix is ositive definite (e.g., Goodwin and Payne, 977,. 04 and 39) and (ii) the inverse mean information matrix is the covariance matrix aearing in the asymtotic distribution of the aroriately normalized MLE. Hence, when combined with the asymtotic normal distribution, the information matrix may be used in constructing confidence regions for the MLE when the samle size is sufficiently large. More generally, the information matrix rovides a summary of the amount of 5070

information in the data relative to θ (e.g., Sall, 003, Sect. 3.3). We restrict our attention below to the fully series and fully arallel subsystems cases (Subsection.), but the analysis can be modified in a straightforward manner for certain other cases, including hybrid series-arallel subsystems cases. The Fisher information matrix F(θ) for a twicedifferentiable log-likelihood function is given by ( ) E L L = T E L F θ = T, (5.) where L T aearing after the last equality above corresonds to the Hessian matrix of the log-likelihood function. Let us first comute F(θ) for the series case using the Hessian-based form aearing after the second equality in (5.). Because Θ = {0 < ρ j < for all j}, it is known that the Hessian matrix is continuous and, consequently, symmetric. From (.6), the elements of the negative Hessian for the series case of interest are: X + X j ( n X) ρ ( nj X j) + + when j = k, ( ) ( ) L ρj ρ ρj ρj = ρ j ρk ( n X) ρ when j k. ( ρ) ρjρk Then, the corresonding elements of the information matrix F(θ) = F jk ( θ ) are: nρ+ n jρj nρ nj + + when j = k, ( ) ρj ρ ρj ( ρj) Fjk ( θ ) = nρ when j k. ( ρ) ρjρk (5.) Likewise, the Hessian can be used to comute F(θ) in the arallel subsystem case. From (.7), the elements of the negative Hessian in the arallel case are: X j ( n+ nj X X j) X ( ρ) + + when j = k, ( ) ( ) L ρ j ρ j ρ j ρ = ρ ( )( j ρk ρ ρ Xρ+ X) when j k, ( ρj)( ρk) ρ leading to the following elements of the information matrix: F jk nj n( ρ ) + njρ( ρj) + when j = k, ρ j ( ρ j ) ρ ( θ ) = (5.3) ( ρ)( ρ nρ+ n) when j k. ( ρj)( ρk) ρ Then, the exression in (5.) or (5.3) can be used to determine if the information matrix is ositive definite, thereby characterizing the identifiability of θ. It is clear that both n and the n j can contribute to the ositive definiteness of F(θ). For examle, if the n j dominate n, then increasing all n j at the same rate (in the sense that nj n k = O() and nk n j = O() for all j, k) is sufficient to achieve the ositive definiteness for sufficiently large samle sizes. It is also ossible to have n subject to the n j growing sufficiently raidly as well. In a ractical alication, it will be necessary to assume a value for θ rior to carrying out the estimation in order to evaluate F(θ). This value for θ may be chosen conservatively or at a tyical level in determining identifiability. The other main interest for alication of the information matrix is determining aroximate confidence regions. However, one of the comlications in using the standard asymtotic normality results is the multile samles sizes, n, n,..., n. Fortunately, the form for F(θ) rovides clarification with resect to the mix of samle sizes. Recall that the standard generic form for the asymtotic distribution of MLEs is, dist samle size( MLE true value ) N( 0, F ) (5.4) dist where denotes convergence in distribution and F is the limit of the mean information matrix (i.e., the limit of the information matrix averaged over the samle size) (e.g., Hoadley, 97; Rao, 973,. 45 47). In well-osed roblems, F is a finite-magnitude ositive definite matrix. Hence, to within slower growth terms, the magnitude of the unaveraged Fisher information matrix F(θ) must grow linearly with the increase in the relevant samle size. In the context of the tyical forms for F(θ) above, it is clear that both n and the n j can contribute to the growth in magnitude for F(θ). For examle, as above, if the n j dominate n, then increasing all n j at the same rate (in the sense that nj n k = O() and nk n j = O() for all j, k) is sufficient to achieve the necessary growth in the magnitude of F(θ). It is also ossible to have n subject to the n j growing sufficiently raidly as well. The author is currently ursuing the issue of asymtotic normality in greater detail. 6. BOOTSTRAP CONFIDENCE INTERVALS It is well known that the traditional asymtotic normality-based methods are often inadequate in constructing confidence intervals for reliability estimates. Two factors contribute to this inadequacy: (i) samle sizes that are too small to justify the asymtotic normality and (ii) confidence intervals from the asymtotic normality that fall outside of the interval [0, ] as a consequence of the need to aroximate the true asymmetric distribution with the symmetric normal distribution. The latter factor is 507

exacerbated by the fact that ractical reliability estimates are often very near unity. We now resent a bootstrabased method for constructing confidence intervals for the full system reliability estimate ˆρ under the assumtion that ˆρ is uniquely determined from ˆθ. Lemma resented sufficient conditions for such a function via the imlicit function theorem. Bootstra methods are well-known Monte Carlo rocedures for creating imortant statistical quantities of interest when analytical methods are infeasible (e.g., Efron and Tibshirani, 986; Ljung, 999,. 304 and 334; and Aronsson et al., 006). The stes below describe a arametric bootstra aroach to constructing confidence intervals for ˆρ. Parametric bootstra methods samle from a secified distribution based on using the estimated arameter values; a standard bootstra method samles from the raw data. Ste 0: Treat the MLE ˆθ, and associated ˆρ, as the true value of θ and ρ. Ste : Generate (by Monte Carlo) a set of bootstra data of the same collective samle size N = {n, n, n,..., n } as the real data Y using the assumed robability mass function in (.) and the value of θ and ρ from Ste 0. Ste : Calculate the MLE of θ, say ˆθ boot, from the bootstra data Y in Ste, and then calculate the corresonding full system reliability MLE, ˆρ boot. Ste 3: Reeat Stes and a large number of times (erhas 000) and rank order the resulting ˆρ boot values; one- or two-sided confidence intervals are available by determining the aroriate quantiles from the ranked samle of ˆρ boot values. 7. CONCLUDING REMARKS We have described above an MLE-based aroach for estimating the reliability of a comlex system by combining data from full system reliability tests and subsystem or other tests. The method alies in general systems, where the subsystems may be arranged in series, arallel, combination series/arallel, or other mode. This MLE aroach rovides a means of estimating the reliability of systems with relatively few (or even no) full system tests through the knowledge obtained via subsystem tests. By aroriately formulating constraints in an otimization roblem, the aroach accommodates general relationshis between the subsystems and full system, including statistical deendencies among subsystems oerating within the full system. Interestingly, the likelihood function has the same general form across all settings; only the constraints in the otimization roblem change. The method includes asymtotic (Fisher information-based) and finite-samle (bootstra) methods for characterizing the uncertainty via confidence regions. REFERENCES [] Aostol, T. M. (974), Mathematical Analysis (nd ed.), Addison-Wesley, Reading, MA. [] Aronsson, M., Arvastson, L., Holst, J., Lindoff, B., and Svensson, A. (006), Bootstra Control, IEEE Transactions on Automatic Control, vol. 5(),. 8 37. [3] Coit, D. W. 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