Uncertainty in Prognostics and Systems Health Management€¦ · Uncertainty in Prognostics and...

Uncertainty in Prognostics and Systems Health ManagementShankar Sankararaman1 and Kai Goebel2

1 SGT Inc., NASA Ames Research Center, Moffett Field, CA [email protected]

2 NASA Ames Research Center, Moffett Field, CA [email protected]

ABSTRACT

This paper presents an overview of various aspects of uncer-tainty quantification and management in prognostics and sys-tems health management. Prognostics deals with predictingpossible future failures in different types of engineering sys-tems. It is almost practically impossible to precisely predictfuture events; therefore, it is necessary to account for the dif-ferent sources of uncertainty that affect prognostics, and de-velop a systematic framework for uncertainty quantificationand management in this context. Researchers have devel-oped computational methods for prognostics, both in the con-text of testing-based health management and condition-basedhealth management. This paper explains that the interpreta-tion of uncertainty for these two different types of situationsis completely different. While both the frequentist (based onthe presence of true variability) and Bayesian (based on sub-jective assessment) approaches are applicable in the contextof testing-based health management, only the Bayesian ap-proach is applicable in the context of condition-based healthmanagement. This paper illustrates that the computation ofthe remaining useful life is more meaningful in the contextof condition-based monitoring and needs to be approachedas an uncertainty propagation problem. Further, uncertaintymanagement issues are discussed and possible solutions areexplored. Numerical examples are presented to illustrate thevarious concepts discussed in the paper.

1. INTRODUCTION

Prognostics deals with predicting the future behavior of en-gineering systems and one has to acknowledge that the fu-ture is invariably clouded with uncertainty. Therefore, it isneither feasible nor meaningful to pretend that predictionsregarding the functioning of such engineering systems willbe always (or ever) precise. Instead, one has to deal with

Shankar Sankararaman et al. This is an open-access article distributed un-der the terms of the Creative Commons Attribution 3.0 United States Li-cense, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original author and source are credited.

the presence of uncertainty by accounting for the differentsources of uncertainty and by rigorously processing them inthe appropriate manner. Methods for quantifying uncertaintyin prognostics can be broadly classified as offline prognosticsand online prognostics. Methods for offline prognostics arebased on thorough testing before and/or after operating an en-gineering system, whereas methods for online prognostics arebased on monitoring the performance of the engineering sys-tem during operation. While online prognostics has not seenreceived much attention yet (although arguably it might bemore important to provide an uncertainty assessment “on thego”, i.e., during operation), offline prognostics has been ex-plored in a number of different domains, such as crack growthanalysis (Sankararaman, Ling, Shantz, & Mahadevan, 2011;Sankararaman, Ling, & Mahadevan, 2011), structural dam-age prognosis (Farrar & Lieven, 2007; Coppe, Haftka, Kim,& Yuan, 2010), electronics (Gu, Barker, & Pecht, 2007), andmechanical bearings (Liao, Zhao, & Guo, 2006), to mentiona few. An important criterion for the implementation of suchoffline-testing methods is the availability of a large number ofrun-to-failure data of engineering components and systems.This restricts this approach to smaller engineering compo-nents when it may be affordable to run several such compo-nents to failure. It may not be practically feasible to extendthis approach to large scale systems since the cost of failingsuch systems is prohibitively high.

In an online health monitoring context, the performance of asystem needs to be assessed, its state of health needs to beestimated, and its remaining useful life needs to be calculatedduring operation. The estimation of remaining useful life ismore meaningful and useful before one reaches the failurepoint, during the operation of the system. There exist severalchallenges in applying uncertainty quantification methods foronline health monitoring purposes. Some methods for un-certainty propagation assume certain distribution types of theRUL prediction (such as Gaussian) and then focus on esti-mating the distribution parameters. It is necessary to ques-tion whether this approach is valid, as further explored later

International Journal of Prognostics and Health Management, ISSN2153-2648, 2015 010 1

INTERNATIONAL JOURNAL OF PROGNOSTICS AND HEALTH MANAGEMENT

in the present paper. Other methods seek to account for un-certainty in prognostics using Bayesian filtering techniqueslike Kalman filtering (Swanson, 2001) and particle filter-ing (Zio & Peloni, 2011). Those approaches may fall shortof accurately representing uncertainty since filtering can beused only to estimate the health state of the system but cannotbe used for future prediction. Therefore, it is necessary to re-sort to other statistical and computational approaches that cancompute the uncertainty in future predictions and remaininguseful life (Sankararaman & Goebel, 2013b).

A primary issue is in understanding the philosophicaldifferences between testing-based health management andcondition-based health management. In this context, Saxenaet al. (Saxena, Sankararaman, & Goebel, 2014) have classi-fied prognostic algorithms into four types of categories, de-pending on how they are implemented and applied; whilethree of these four (Types I, II and IV) are related to testing-based health management, Type-III is directly related tocondition-based health management. The differences be-tween these approaches significantly influence the interpreta-tion of uncertainty (Sankararaman & Goebel, 2013c; Celaya,Saxena, & Goebel, 2012). Such interpretation is key to guidedifferent types of decision-making activities during the oper-ation of engineering systems.

The paper focuses on providing an overview of the state-of-the-art in the topic of uncertainty quantification and manage-ment in prognostics and health monitoring. To begin with,the significance of uncertainty in prognostics is explained indetail in Section 2. Then, the various aspects of uncertainty intesting-based health management and condition-based healthmanagement are discussed in detail in Section 3 and 4, andthe differences between these two approaches are clearly ex-plained. It is also explained that the prediction of remain-ing useful life is more meaningful only in the context ofcondition-based health management, and this topic is dis-cussed in further detail. The importance of statistical tech-niques comprising of uncertainty propagation methods andsensitivity analysis tools in the context of prognostics and re-maining useful life prediction are explained. Numerical ex-amples are presented in Sections 3 and Section 4, to illustratethe various concepts discussed in this paper. Finally, conclu-sions are presented in Section 5.

2. SIGNIFICANCE OF UNCERTAINTY IN PROGNOSTICS

In an ideal scenario, it would be possible to perfectly and pre-cisely predict the behavior of engineering systems and facil-itate decision-making with a significant amount of trust andconfidence. However, this is not possible in practical engi-neering applications. First of all, it is almost impossible to beable to accurately predict the operating conditions and envi-ronmental conditions under which the system operates. Fur-ther, the future loading demands on the system cannot be pre-

cisely known in advance; for example, the future behavior ofa simple electric vehicle depends upon several factors such asthe driving terrain, climatic conditions, desired speed and ac-celeration, characteristics, properties, and parameters of theinternal batteries, remaining charge, etc. While some factorsare internal to the engineering system, other factors are ex-ternal to the system. In order to be able to account for all ofthese factors and perform prognostics, it is necessary to ac-knowledge the presence of uncertainty in all of these factorsand develop a systematic framework in order to account forthese uncertainties in prognostics.

In fact, uncertainty plays an important role in a series ofactivities that are related to prognostics and health manage-ment, as indicated in Fig. 1. To begin with, the behaviorof the system under consideration is uncertain; its inputs,states, and parameters may be uncertain at any generic time-instant. The mathematical models (that may be built usingdata or using physics or a hybrid combination of both dataand physics) are not an accurate representation of the systemand this may lead to modeling errors and uncertainties. Theuse of sensors and data processing tools (both pre-processingand post-processing) are essential components of PHM, andadd further uncertainty. In turn, the results of diagnostics,prognostics, and the prediction of remaining useful life arerendered uncertain (Sankararaman, 2015). As a result, it isimportant to evaluate the performance of prognostic algo-rithms (Sankararaman, Saxena, & Goebel, 2014) and developmetrics that directly account for such uncertainty. It would beideal if PHM requirements were to acknowledge the presenceof such uncertainty in order to facilitate robust verification,validation, and certification under uncertainty.

The current state-of-the-art research is still focused on quan-tifying uncertainty in diagnostics and prognostics, and needsto evolve in order to successfully address all of the abovechallenges, particularly those in terms of requirements, ver-ification, validation, and certification. However, prior to ad-dressing these challenges, it is important to understand thesignificance of uncertainty and its impact on prognostics anddecision-making. When information regarding uncertaintyis used for decision-making, it can be useful to quantify theamount of risk involved in different types of decisions. Riskconsists of two important components: the likelihood of oc-currence of adverse events and the cost associated with theoccurrence of adverse events. While the latter can be directlyquantified by analyzing the different types of losses that oc-cur due to such occurrence of adverse events, the former canonly be quantified by rigorously accounting for the differentsources of uncertainty in prognostic and decision-making ac-tivities.

It is a common misconception that the effect of uncertaintycan be included at latter stages of the analysis when the fun-damental deterministic problem has been solved without ac-

2


System

Inputs

States

Properties

Models

Diagnosis

Sensors Data

Processing

Prognosis

Remaining Useful Life Prediction

Performance Evaluation

Evaluation Metrics Requirements

Verification, Validation, and Certification

Figure 1. PHM-related Activities

counting for uncertainty. It is necessary to account for un-certainty right from the initial stages of system-level concep-tion through analysis, design, testing, and operations. Duringthese stages, there are several types of activities that need tobe performed in order to accurately account for the effect ofuncertainty in prognostics.

In the context of prognostics and health management, uncer-tainties have been discussed from representation, quantifica-tion, and management points of view (Hastings, D. and Mc-Manus, H., 2004; Orchard, Kacprzynski, Goebel, Saha, &Vachtsevanos, 2008; Tang, Kacprzynski, Goebel, & Vachtse-vanos, 2009). While these three are different processes, theyare often confused with each other and interchangeably used.In this paper, the various tasks related to uncertainty quantifi-cation and management are classified into four, as explainedbelow. These four tasks need to performed in order to ac-curately estimate the uncertainty in the RUL prediction andinform the decision-maker regarding such uncertainty.

1. Uncertainty Representation and Interpretation: Thefirst step is uncertainty representation and interpretation,which in many practical applications, is guided by thechoice of modeling and simulation frameworks. Thereare several methods for uncertainty representation thatvary in the level of granularity and detail. Some com-mon theories include classical set theory, probability the-ory, fuzzy set theory, fuzzy measure (plausibility and be-

lief) theory, rough set (upper and lower approximations)theory, etc. Amongst these theories, probability theoryhas been widely used in the PHM domain (Celaya etal., 2012); even within the context of probabilistic meth-ods, uncertainty can be interpreted and perceived in twodifferent ways: frequentist (classical) versus subjective(Bayesian). While the former interpretation of uncer-tainty implies that uncertainty exists only when there isnatural randomness across multiple nominally identicalexperiments, the latter facilitates associating uncertaintyeven with events that are not random and such uncer-tainty is simply reflective of the analyst’s belief regardingthe occurrence or non-occurrence of such events.

2. Uncertainty Quantification: The second step is uncer-tainty quantification, that deals with identifying and char-acterizing the various sources of uncertainty that may af-fect prognostics and RUL estimation. It is important thatthese sources of uncertainty are incorporated into modelsand simulations as accurately as possible. The commonsources of uncertainty in a typical PHM application in-clude modeling errors, model parameters, sensor noiseand measurement errors, state estimates (at the time atwhich prediction needs to be performed), future loading,operating and environmental conditions, etc. The goalin this step is to address each of these uncertainties sep-arately and quantify them using probabilistic/statisticalmethods. The Kalman filter is essentially a Bayesian tool

3


for uncertainty quantification, where the uncertainty inthe states is estimated continuously as a function of time,based on data which is also typically available continu-ously as a function of time.

3. Uncertainty Propagation: The third step is uncertaintypropagation and is most relevant to prognostics, sinceit accounts for all the previously quantified uncertain-ties and uses this information to predict (1) future statesand the associated uncertainty; and (2) remaining usefullife and the associated uncertainty. The former is com-puted by propagating the various sources of uncertaintythrough the prediction model. The latter is computed us-ing the estimated uncertainty in the future states alongwith a Boolean threshold function which is used to in-dicate end-of-life. In this step, it is important to under-stand that the future states and remaining useful life pre-dictions are simply dependent upon the various uncer-tainties characterized in the previous step, and therefore,the distribution type and distribution parameters of futurestates and remaining useful life should not be arbitrar-ily chosen. Sometimes, a normal (Gaussian) distributionhas been assigned to the remaining useful life prediction;such an assignment is erroneous and the true probabilitydistribution of RUL needs to be estimated though rig-orous uncertainty propagation of the various sources ofuncertainty through the state space model and the EOLthreshold function, both of which may be non-linear inpractice.

4. Uncertainty Management: The fourth and final step isuncertainty management, and it is unfortunate that, inseveral articles, the term “Uncertainty Management” hasbeen used instead of uncertainty quantification and/orpropagation. As a result, there are few publications thatdirectly address the issue of uncertainty management. Ingeneral, uncertainty management is a term used to referto different activities which aid in managing uncertaintyin condition-based maintenance during real-time opera-tion. There are several aspects of uncertainty manage-ment. One aspect of uncertainty management attemptsto answer the question: “Is it possible to improve theuncertainty estimates?” The answer to this question liesin identifying which sources of uncertainty are signifi-cant contributors to the uncertainty in the RUL predic-tion. For example, if the quality of the sensors can beimproved, then it may be possible to obtain a better stateestimate (with lesser uncertainty) during Kalman filter-ing, which may in turn lead to a less uncertain RUL pre-diction. Another aspect of uncertainty management dealswith how uncertainty-related information can be used inthe decision-making process. Future research needs tosignificantly focus on the different aspects of uncertaintymanagement and develop computational methods for thispurpose.

Most of the research in the PHM community pertains to thetopics of uncertainty quantification and propagation; few ar-ticles have directly addressed the topic of uncertainty man-agement. Even within the realm of uncertainty quantificationand propagation, the estimates of uncertainty have sometimesbeen misinterpreted. For example, when statistical principlesare used to estimate a parameter, there is an emphasis on cal-culating the estimate with the minimum variance. When thisprinciple is applied to RUL estimation, it is important notto arbitrarily reduce the variance of RUL itself. Celaya etal. (Celaya et al., 2012) explored this idea and explained thatthe variance of RUL needs to be carefully calculated by ac-counting for the different sources of uncertainty. The calcula-tion of RUL is, arguably, the most important component of aprognostics and health management system, and this topic isdiscussed in detail in the rest of this paper. Though the major-ity of this paper focuses on calculating RUL in the context ofcondition-based monitoring, some fundamental principles oftesting-based health management are discussed, particularlyfrom the perspective of uncertainty quantification, in order toexplain the philosophical differences between these two ap-proaches.

3. TESTING-BASED HEALTH MANAGEMENT

In testing-based prognostics (referred to as “reliability-basedtesting” in some publications), the remaining useful life istypically calculated by testing multiple nominally identicalspecimens of the engineering component/system. It may benoted that the term “remaining” in “remaining useful life”may not be applicable to all types of testing. This is because,testing is typically carried out before the engineering systemis under operation. The term “time-to-failure” is more appro-priate for testing-based health management. It is importantnot to confound “time-to-failure” and “remaining useful life”.The appropriate interpretation of the latter will be clarifiedin the next section, while discussing about condition-basedhealth management.

Assume that a set of run to failure experiments have beenperformed with high level of control, ensuring same usageand operating conditions. The time to failure for all the nsamples (ri; i = 1 to n) are measured. It is important tounderstand that different time-to-failure values are obtaineddue to inherent variability across the n different specimens,thereby confirming the presence of physical probabilities ortrue randomness. The various factors that contribute are:

1. Inherent variability in properties and characteristics ofthe nominally identical specimens

2. Inherent variability across the loading conditions experi-enced by each of the individual specimens

3. Inherent variability in operating and environmental con-ditions for each of the individual specimens

Assume that these random samples belong to an underly-

4


ing probability density function (PDF) fR(r), with expectedvalue E(R) = µ and variance V ar(R) = σ2. The goal ofuncertainty quantification is to characterize this probabilitydensity function based on the available n data. Theoretically,an infinite amount of data is necessary to accurately estimatethis PDF; however, due to the presence of limited data, theestimated PDF is not accurate. Hence, lack of infinite dataadds some additional uncertainty to the aforementioned listof sources of uncertainty. Statistical approaches, both fre-quentist and subjective, express uncertainty regarding the es-timate itself. However, frequentist and subjective analystsquantify and express this uncertainty in completely differentways. The following discussion is based on the assumptionthat the underlying PDF fR(r) is Gaussian, since closed formexpressions for uncertainty are readily available for this case.Whenever appropriate and necessary, remarks are providedfor non-Gaussian distributions.

3.1. Confidence Intervals: Frequentist Approach

Since R is Gaussian, estimating the parameters µ and σ isequivalent to estimating the PDF. In the context of physicalprobabilities (frequentist approach), the “true” underlying pa-rameters µ and σ are referred to as “population mean” and“population standard deviation” respectively. Let x and s de-note the mean and the standard deviation of the available ndata. As stated earlier, due to the presence of limited data, thesample parameters (x and s) will not be equal to the corre-sponding population parameters (µ and σ). The fundamentalassumption in this approach is that, since there are true but un-known population parameters, it is meaningless to talk aboutthe probability distribution of any population parameter. In-stead, the sample parameters are treated as random variables,i.e., if another set of n data were available, then another re-alization of x and s would have been obtained. Using thesample parameters (x and s) and the number of data available(n), frequentists construct confidence intervals on the popu-lation parameters (µ and σ).

Confidence intervals can be constructed for both µ andσ (Haldar & Mahadevan, 2000). Consider multiple nominallyidentical specimens of an engineering component. The term“nominally identical” implies that there is inherent variabilityin the properties and behavior of these specimens. Supposethat these specimens have been subjected to failure analysis,and their time-to-failure times are available. If the true prob-ability distribution of time-to-failure across multiple speci-mens is assumed to be Gaussian, the (1 − α)% confidenceinterval of the mean run-to-failure time can be calculated as:[x− tα

2

s√n

, x+ tα2

s√n

],

where x, s, and n denote the sample mean, sample standarddeviation, and number of samples respectively. If the run-to-failure times are given by {100, 105, 98, 110, 92, 97, 85,

120, 93, 101}, then x = 100.10, s = 9.87, n = 10, and the95% confidence interval on the mean run-to-failure is givenby [93.98, 106.22]. Using the properties of the chi-squaredistribution (χ2), the confidence interval on the variance canbe calculated as:[

(n−1)s2χ21−α

2

, (n−1)s2χ2α2

].

For this numerical example, the corresponding confidence in-terval on the standard deviation is given by [6.79, 18.02].While the above expressions for confidence intervals on meanand standard deviation are applicable only to Gaussian distri-butions, similar confidence intervals can also be constructedfor other types of distributions; in general, it is easier to con-struct confidence intervals for mean than it is for standarddeviation (or, equivalently, variance).

Nevertheless, it is important that these confidence intervalsbe interpreted correctly. To begin with, the above confidenceintervals will decrease as more data are available; therefore,the width of these confidence intervals is simply related to thenumber of data. The actual uncertainty in the run-to-failuretimes is given only by the estimate of the standard deviation,and this uncertainty is the result of variability (in materialproperties, operating conditions, etc.) across all the nomi-nally identical specimens. Further, as stated earlier, the inter-pretation of confidence intervals may be confusing and mis-leading. A 95% confidence interval on µ does not imply that“the probability that µ lies in the interval is equal to 95%”;such a statement is wrong because µ is purely deterministicand physical probabilities cannot be associated with it. Therandom variable here is in fact x, and the confidence inter-val is calculated using x. Therefore, the correct implicationis that “the probability that the estimated confidence intervalcontains the true population mean is equal to 95%”. Thus, itis easy to understand that the width of the confidence inter-vals is indicative of lack of infinite data and the actual valueof the standard deviation is indicative of the uncertainty in R.

A practical challenge is that, in many applications, it may notbe possible to know what type of probability distribution (forexample, Gaussian distribution had been “assumed” in theabove discussion) needs to be assumed to in order to calcu-late the above confidence intervals; obviously, the procedurefor calculation of confidence intervals depends on the choiceof distribution type (Gaussian, Weibull, lognormal, etc.), andthe presence of such distribution type uncertainty further addsto the confusion regarding the interpretation of confidenceintervals. As the sample size increases, the confidence in-tervals for the mean and standard deviation may get nar-rower. This may be misleading since the confidence intervalsshould be interpreted only based on the underlying assump-tion of distribution type (which might have been wrong tobegin with). Computational methods are being developed todeal with distribution type uncertainty (Sankararaman & Ma-

5


hadevan, 2013a), however they have not been implemented inprognostics and health management applications.

3.2. Probability Distribution: Bayesian Approach

Alternatively, it is also possible to address the problem ofcomputing fR(r) purely from a subjective (Bayesian) point ofview. One important difference now is that the Bayesian ap-proach does not clearly differentiate between “sample param-eters” and “population parameters”. The probability distribu-tion of µ is directly computed using the available data (recallthat this was impossible in the frequentist approach since µis the underlying mean that is precise but unknown), and thisuncertainty is referred to as the analyst’s degree of belief forthe underlying true parameter µ. Similarly, the probabilitydistribution of σ can also be computed using Bayes’ theorem.

Consider a set of time-to-failure times, given by ri (i = 1to n). In order to compute the probability distribution ofµ and σ, the first step is construct their joint likelihoodas (Sankararaman & Mahadevan, 2011):

L(µ, σ) ∝m∏i=1

fR(ri|µ, σ) (1)

The maximum likelihood estimate of the parameters P canbe calculated by maximizing the above expression. Instead ofmaximizing the likelihood, the entire likelihood function canbe used to construct the PDF of the distribution parameters.Further, sometimes time-to-failure data may also be availablein terms of intervals. For example, intermittent inspectionsmay be performed to check whether failure has occurred ina specimen; if failure is found to have occurred between 10minutes and 11 minutes, the resultant time to failure is ac-tually an interval. The above likelihood-based approach canalso be extended to account for interval data, in order to com-pute the uncertainty in the distribution parameters.

This approach is generally applicable for any type of para-metric probability distribution, where the probability densityfunction (PDF) can be expressed as fR(r|P ). If R is Gaus-sian, then P represents the vector of mean and standard de-viation. Let f(P ) denote the joint PDF of the distributionparameters P . It is easy to apply Bayes theorem, choose uni-form prior density (f ′(P ) = h), and calculate the joint PDFas:

f(P ) =hL(P )∫hL(P )dP

=L(P )∫L(P )dP

(2)

Note that the uniform prior density function can be definedover the entire admissible range of the parameters P . For ex-ample, the mean of a normal distribution can vary in (−∞,∞) while the standard deviation can vary in (0,∞) because thestandard deviation is always greater than zero. Both theseprior distributions are improper prior distributions becausethey do not have finite bounds.

For the above numerical example, i.e., if the run-to-failuretimes are given by {100, 105, 98, 110, 92, 97, 85, 120, 93,101}, the probability distribution of µ and σ can be calculatedas shown in Figs. 2 and 3.

85 90 95 100 105 110 115 1200

0.02

0.04

0.06

0.08

0.1

µ

f µ(µ

)

Figure 2. PDF of µ

0 10 20 30 400

0.05

0.1

0.15

0.2

σ

f σ (

σ )

Figure 3. PDF of σ

Recall that one realization of the parameters (µ and σ)uniquely define the PDF fR(r). However, since the param-eters are themselves uncertain, R is now represented by afamily of distributions (Sankararaman & Mahadevan, 2011,2013b). This family of distributions will “shrink” to the trueunderlying PDF (denoted by fTR (r)) as the number of avail-able data increases, and asymptotic PDF (as the number ofdata increases) is simply reflective of the variability (in mate-rial properties, operating conditions, etc.) across all the nom-inally identical specimens. Alternative to the family of PDFsapproach, a single unconditional PDF of R, which includesboth the variability in X and the uncertainty in the distribu-tion parameters P , as:

fR(r) =

∫fR(r|P )f(P )dP (3)

Note that the right hand side of Eq. 3 is not conditioned onP anymore. Some researchers refer to this PDF fR(r) as thepredictive PDF (Kiureghian, 1989) of R. The predictive PDFfor the above numerical example is shown in Fig. 4.

Note that the predictive PDF fR(r) will indicate the presence

6


50 100 1500

0.01

0.02

0.03

0.04

R

f R (r )

Figure 4. Predictive PDF of R

of larger uncertainty in R than the original PDF fTR (r), be-cause the former accounts for the lack of infinite data. Asthe number of data increases, fR(r) will tend towards fTR (r).Of course, this is true only when the correct distribution typewas assumed for R; in many cases, the choice of distributiontype (referred to as “statistical model” by some researchers)is a challenge by itself, and contributes to additional uncer-tainty (Sankararaman & Mahadevan, 2013a).

3.3. Summary

To summarize, the treatment of uncertainty in testing-basedprognostics relies on classical reliability methods and prob-ability concepts. These concepts have been used in the so-called Type-I and Type-II prediction methods in the contextof prognostics and health management (Saxena et al., 2014).Further, methods that use predictive analytics concepts forhealth prediction collect health monitoring data from mul-tiple components and systems, and therefore, the results ofthese methods also need to be interpreted similar to testing-based prognostics approaches. On the other hand, condition-based prognostics methods are significantly different fromthose methods discussed earlier in this section, as explainedin the following section.

4. CONDITION-BASED HEALTH MANAGEMENT

Most of the discussion pertaining to testing-based prognosticsis not applicable to condition-based monitoring and prognos-tics. The distinctive feature of condition-based monitoringis that each component/subsystem/system is considered byitself, and therefore, “variability across specimens” is non-existent. Any such “variability” is spurious and must not beconsidered. At any generic time instant tP at which prognos-tics needs to be performed, the component/subsystem/systemis at a specific state. The actual state of the system is purelydeterministic, i.e., the true value of each state is completelyprecise, however unknown. Therefore, if a probability distri-bution is assigned for this state, then this distribution is sim-ply reflective of the analyst’s knowledge regarding this state

and cannot be interpreted from a frequentist point of view.Thus, by virtue of definition of condition-based monitoring,physical probabilities are not present here, and a subjective(Bayesian) approach is only suitable for uncertainty quantifi-cation.

The goal in condition-based prognostics is, at any generictime instant tP , to predict the remaining useful life of thecomponent/subsystem/system as condition-based estimate ofthe usage time left until failure. Such computation needs tobe, ideally, performed in real-time. In other words, the per-formance of the system during its operation needs to be ana-lyzed, possible failure modes and future degradation needs tobe predicted, and the remaining useful life needs to be com-puted while the system is under operation. These calculationshelp in operational decision-making activities such as pathplanning, mission routing, etc.

The following prognostics architecture can be used to achievethese goals. First, measurements until time tP are used toestimate the state at time tP . Then, using a degradation-prediction model (that may be model-based or data-driven),future state values (corresponding to time instants greaterthan tP ) are computed, and the first time instant at which afailure threshold is true is calculated; this information is thenused to calculate the remaining useful life. In order to forecastfuture state values, it is also necessary to assume future load-ing conditions (and operating conditions), and this is a majorchallenge in condition-based prognostics. Typically, the an-alyst subjectively assumes statistics for future loading condi-tions based on past experience and existing knowledge; thus,the subjective interpretation of uncertainty is clearly consis-tent across the entire condition-based monitoring procedure,and therefore, inferences made out of condition-based moni-toring also need to be interpreted subjectively. The predictionof degradation (forecasting of future state values) is stoppedwhen failure is reached, as indicated by a boolean thresholdfunction that checks whether failure has occurred or not. Thisindicates the end-of-life (EOL) and the EOL can be directlyused to compute the remaining useful life (RUL) prediction.Note that it is important to interpret the uncertainty in EOLand RUL subjectively.

4.1. Illustrative Example

Consider a generic engineering component whose health stateat any time instant is given by x(t). Consider a simple degra-dation model, where the rate of degradation of the health state(that decreases with time, due to the presence of damage) isproportional to the current health state. This simple modelcan be used for prognostics of different types of engineeringcomponents such as mechanical bearings, valves, capacitors,etc. (Goebel, Saha, & Saxena, 2008; Saxena, Goebel, Simon,& Eklund, 2008; Daigle & Goebel, 2013; Kulkarni, Celaya,Goebel, & Biswas, 2013); sometimes this model can be used

7


to directly represent the state equation, while in some othersituations, this model can represent the change in certain pa-rameters that govern the state equation.

Without loss of generality, this degradation model can bemathematically expressed as:

x(t) ∝ x(t), (4)

If a decrease x(t) corresponds to decrease in the componenthealth, then the constant of proportionality is a negative num-ber, and vice-versa. Since differential equations are usuallysolved by considering discrete time instants, the above equa-tion can be rewritten as:

x(k + 1) = a.x(k) + b, (5)

where k represents the discretized time-index. The conditionthat “the constant of proportionality in Eq. 4 is negative” isequivalent to the condition that “a < 1 in Eq. 5”. For the sakeof illustration, let a denote the loading on the system, b denotethe model of the degradation model above, and let a and b beconstant and time-invariant. In practical examples, more thanone variable may be necessary to represent the loading con-ditions and there may be multiple model parameters and statevariables; further, the loading variables and model parametersmay also be time-varying, just like the state x.

In order to compute the remaining useful life, it is necessaryto chose a threshold function that defines the occurrence offailure. Since x(k) is a decreasing function, the thresholdfunction will indicate that failure occurs when the state valuex becomes smaller than a critical lower bound (l). The firsttime instant at which this event occurs indicates the end oflife, and this time instant can be used to calculate the RUL.Therefore, the remaining useful life (r, an instance of the ran-dom variableR) is equal to the smallest n such that x(n) < l.Therefore RUL can be calculated as

r = inf{n : x(n) < l}, (6)

For a given value of x(0) (or x(tP ), where tP denotes thetime at which prediction needs to be performed), a, b, it possi-ble to calculate the end-of-life and remaining useful life usingthe above set of equations. However, in practical conditions,all of these are uncertain. However, note that the uncertaintyin x(0), a, b are related only to the knowledge regarding thisparticular unit and not an ensemble of units; recall that an en-semble of nominally identical units was considered earlier inSection 3. The presence of these uncertainties leads to un-certainty in the RUL prediction. This leads to the obviousquestion: How to compute the uncertainty in RUL? Prior toanswering this question, the next subsection lists the differentsources of uncertainty in generic condition-based prognosticapplications.

4.2. Sources of Uncertainty

Typically, researchers have classified the different sources ofuncertainty into different categories in order to facilitate un-certainty quantification and management. While it has beencustomary to classify the different sources of uncertainty intoaleatory (arising due to physical variability) and epistemic(arising due to lack of knowledge), such a classification maynot be suitable for prognostics in the context of condition-based monitoring and RUL prediction because, as mentionedearlier, “true variability”’ is not present in condition-basedmonitoring. A completely different approach for classifica-tion, particularly applicable to condition-based monitoring,is proposed in this paper. These sources of uncertainty aregraphically shown in Fig. 5, and enumerated below:

1. Present uncertainty: Prior to prognosis, it is importantto be able to precisely estimate the condition/state of thecomponent/system at the time at which RUL needs to bepredicted. Typically, damage (or faults) are expressedin terms of states, and therefore, estimating the state isequivalent to estimating the extent of damage (or fault).This is related to state estimation and is commonly ad-dressed using filtering. Output data (usually collectedthrough sensors) is used to estimate the state and manyfiltering approaches (Kalman filtering, particle filtering,etc.) are able to provide an estimate of the uncertainty inthe state. In the illustrative example, the state uncertaintyis equal to the uncertainty associated with x(0). Practi-cally, it is possible to improve the estimate of the statesand thereby reduce this uncertainty, by using better sen-sors and improved filtering approaches. It is important tounderstand that the system is at particular state at anytime instant, and the aforementioned uncertainty sim-ply describes the lack of knowledge regarding the “true”state of the system.

2. Future uncertainty: The most important source of un-certainty in the context of prognostics is due to the factthat the future is unknown, i.e. the loading, operating,environmental, and usage conditions are not known pre-cisely, and it is important to assess this uncertainty be-fore performing prognosis. In the illustrative example,the future uncertainty is equal to the uncertainty regard-ing the loading value, i.e., a, from the time of predictionuntil the time of failure. If there is no uncertainty re-garding the future, then there would be no uncertaintyregarding the true remaining useful life of the engineer-ing component/system. However, this true RUL needs tobe estimated using a model; the usage of a model impartsadditional uncertainty as explained below.

3. Modeling uncertainty: It is necessary to use a func-tional degradation model in order to predict future statebehavior, i.e. model the response of the system to an-ticipated loading, environmental, operational, and usage

8


Sources of Uncertainty

Present State Uncertainty

Sensor noise, gain and bias

Data/Signal Processing tools

Filtering and/or estimation techniques

Future Uncertainty

Loading Conditions

Environmental Conditions

Operating Conditions

Modeling Uncertainty

Input-Output Response Model

End-of-Life Threshold Definition

Model Form

Model Parameters

Prediction Method

Uncertainty

Sampling Errors

Model Approximations

Figure 5. Sources of Uncertainty

conditions. Further, the end-of-life is also defined us-ing a Boolean threshold functional model, that is used toindicate whether failure has occurred or not. These twomodels are jointly used to predict the RUL, and they mayeither be physics-based or data-driven. It may be practi-cally impossible develop models that accurately predictthe underlying reality. Modeling uncertainty representsthe difference between the predicted response and thetrue response (that can neither be known nor measuredaccurately), and comprises of several parts: model pa-rameters, model form, and process noise. While it maybe possible to quantify these terms until the time of pre-diction, it is challenging to know their values at futuretime instants. In the illustrative example, Eq. 5 repre-sents the degradation model, x(n) < l represents theBoolean threshold function that indicates failure, b is amodel parameter, and the uncertainty in b correspondsto one aspect of modeling uncertainty. Another aspectis the choice of the “linear” form of the model in Eq. 5;the underlying physical phenomena may differ from thisassumption.

4. Prediction method uncertainty: Even if all the abovesources of uncertainty can be quantified accurately, it isnecessary to quantify their combined effect on the RULprediction, and thereby, quantify the overall uncertaintyin the RUL prediction. It may not be possible to do this

accurately and this leads to additional uncertainty. Forexample, when sampling-based approaches are used forprediction, the use of limited number of samples causesuncertainty regarding the estimated probability distribu-tion.

4.3. Computing Uncertainty in RUL

The goal in condition-based prognostics is to meaningfullyintegrate the degradation equation along with the failurethreshold equation, and account for the different sources ofuncertainty in x(0), a, and b, and thereby, estimate the uncer-tainty in the remaining useful life. For any given realizationof x0, a, and b, it is possible to compute the first time instant(indicates the end-of-life) at which the failure threshold cri-teria will be valid, i.e., calculate the smallest value of n atwhich x(n) < l. The challenge is to compute the combinedeffect of uncertainty in x(0), a, and b on RUL, and estimatethe probability distribution of RUL.

It can be easily demonstrated that the state value at any futuretime instant can be expressed as a function of the initial statex(0), as:

x(n) = an.x(0) +

j=n−1∑j=0

ajb (7)

Note that that x(n) is decreasing and failure happens when

9


x < l. Therefore, the remaining useful life (r, an instanceof the random variable R) is equal to the smallest n such thatx(n) < l. Therefore RUL can be calculated as

r = inf{n : an.x(0) +

j=n−1∑j=0

ajb < l}, (8)

Assuming that the chosen time-discretization level is in-finitesimally small, it is possible to directly estimate the RULby solving the equation:

ar.x(0) +

j=r−1∑j=0

aj .b = l. (9)

The above equation calculates the RUL (r) as a function ofthe initial state x(0), a and b. Even if the only consideredsource of uncertainty is the state estimate x(0) (that is, a andb are constants), RUL R follows a Gaussian distribution ifand only if it is linearly dependent on x(0). In other words,R follows a Gaussian distribution if and only if Eq. 9 can berewritten as:

α.r + β.x(0) + γ = 0 (10)

for some arbitrary values of α, β, and γ. If it were possible toestimate such values for α, β, and γ, the distribution of RULcan be obtained analytically.

In order to examine if this is possible, rewrite Eq. 9 as:

x(0) =1

ar(l −

j=r−1∑j=0

aj .b) (11)

While x(0) is completely on the left hand side of this equa-tion, r appears not only as an exponent in the denominatorbut is also indicative of the number of terms in the summa-tion on the right hand side of the above equation. Therefore,it is clear that the relationship between r and x(0) is not lin-ear. Therefore, even if the initial state (x(0), a realization ofX(0)) follows a Gaussian distribution, the RUL (r, a real-ization of R) does not follow a Gaussian distribution. Fur-thermore, it is not even possible to analytically estimate thedistribution of RUL. Thus, it is clear that even for a simpleproblem consisting of linear state models, an extremely sim-ple threshold function, and only one uncertain variable that isGaussian, the calculation of the probability distribution of Ris neither trivial nor straightforward.

Practical problems in the prognostics and health managementdomain may consist of:

1. Several non-Gaussian random variables that affect theRUL prediction,

2. A non-linear multi-dimensional state space model,

3. Uncertain future loading conditions

4. A complicated threshold function that may be defined inmulti-dimensional space.

The fact that the distribution of RUL simply depends onquantities such as degradation model and model parameters,threshold function, state estimate, future loading conditions,etc., implies that it is technically inaccurate to artificially as-sign the probability distribution type (or any statistic such asthe mean or variance) to RUL. It is important to understandthat RUL is a dependent quantity and that the probability dis-tribution of RUL needs to be accurately estimated using com-putational approaches. Thus, the RUL (R) needs to be ex-pressed as a function of the different sources of uncertainty;let X denote the vector of all sources of uncertainty, and theaforementioned function be expressed as:

R = G(X) (12)

Now, the problem of computing the uncertainty in the RULprediction can be posed as an uncertainty propagation prob-lem (Sankararaman & Goebel, 2013b), and therefore, it maybe helpful to investigate statistical uncertainty propagationtechniques in order to accomplish this goal.

4.4. Uncertainty Propagation Methods

The most commonly used uncertainty propagation techniqueis Monte Carlo sampling (Caflisch, 1998), which is based ondrawing random samples of independent quantities, and com-puting corresponding realizations of the dependent quantity(in this case, the RUL).

For instance, in the conceptual example, if x(0) followsa Gaussian distribution (with mean and standard deviationequal to 975 and 50 respectively), a follows a uniform distri-bution (with lower and upper bounds of 0.990 and 0.993), andb follows a uniform distribution (with lower and upper boundsof -0.005 and 0 respectively), then the RUL (defined by Eq. 6,where l follows a Gaussian distribution with mean and stan-dard deviation equal to 50 and 5 respectively, thereby reflect-ing the presence of uncertainty in the end-of-life thresholddefinition) can calculated as a probability distribution, usingMonte Carlo sampling (5000 random samples were used forthis numerical illustration). Using unit discretization (i.e., thetime interval between the kth and (k + 1)th instants is equalto one second) for solution, the resultant probability densityfunction (PDF) is shown in Fig. 6. It is clear that this distribu-tion is not a typical parametric distribution (such as normal,lognormal, etc.) and that is why rigorous uncertainty propa-gation methods are necessary to accurately estimate this PDF.

While Monte Carlo sampling can be accurate, it is compu-tationally expensive and time-consuming, and therefore, re-searchers have focused on developing advanced methods thatare computationally cheaper. These approaches include Latinhypercube sampling (Loh, 1996), adaptive sampling (Bucher,

10


200 250 300 350 400 450 5000

0.002

0.004

0.006

0.008

0.01

Value of R

ProbabilityDensityFunction

Figure 6. RUL: Conceptual Example

1988), importance sampling (Glynn & Iglehart, 1989), un-scented transform sampling (Van Zandt, 2001), etc. Alter-natively, there are analytical methods such as the first-ordersecond moment method (Dolinski, 1983), first-order reliabil-ity method (Hohenbichler & Rackwitz, 1983; Sankararaman& Goebel, 2013a), second-order reliability method (Der Ki-ureghian, Lin, & Hwang, 1987), etc. In addition, there arealso methods such as the efficient global reliability analy-sis (Bichon, Eldred, Swiler, Mahadevan, & McFarland, 2008)method which involve both sampling and the use of analyti-cal techniques. All of these methods empirically calculate theprobability distribution of RUL; while some of these meth-ods calculate the PDF (fR(r)) of RUL, some other methodscalculate the CDF (FR(r)), and some other methods directlygenerate samples from the desired probability density func-tion (fR(r)). Due to some limitations of each of these meth-ods, it may not be possible to accurately calculate the actualprobability distribution of R. Accurate calculation is possi-ble only by using infinite samples for Monte Carlo sampling.Any other method (for example, the use of a limited, finitenumber of samples) will lead to uncertainty in the estimatedprobability distribution, and this additional uncertainty is re-ferred to as prediction-method uncertainty. It is possible todecrease (and maybe eventually eliminate) this type of un-certainty either by using advanced probability techniques orpowerful computing power.

It is necessary to further investigate the aforementioned un-certainty propagation methods, and identify whether they canbe applied to prognostics and health monitoring applications.A few recent publications (Sankararaman, Daigle, & Goebel,2014; Sankararaman, 2015) have investigated the use of cer-tain methods such as Monte Carlo sampling, unscented trans-form sampling, first-order reliability methods, etc. in this re-gard, and promising results have been obtained. Nevertheless,it is necessary to continue future research in this direction anddevelop methods for quantifying uncertainty in the context ofonline prognostics and health management.

4.5. Uncertainty Management in RUL Prediction

Having calculated the uncertainty in the RUL prediction,it is necessary to facilitate uncertainty management from adecision-making point of view in order to facilitate risk miti-gation activities. In this context, some common types of ques-tions are enumerated below:

1. If the variance of RUL is too large, how to control theuncertainty in input conditions in order achieve a desiredamount of reduction in the uncertainty in RUL?

2. If there is a very high probability that the RUL is smallerthan a critical lower limit, then how to decrease thechance of early system failure?

3. Sometimes, when the RUL follows a multi-modal prob-ability (Fig. 7 represents a practical scenario of the re-manning useful life of the power system of an unmannedaerial vehicle subjected to realistic random loading con-ditions, studied earlier by (Sankararaman, 2015)), thenhow to eliminate the mode corresponding to early fail-ure?

0 200 400 600 800 1000 1200

0

1

2

3

4

5

6x 10

−3

Remaining Useful Life (seconds)

Pro

bablity

Density

Function

Figure 7. Multi-modal RUL: UAV Power System

While it is still necessary to develop computational meth-ods to answer above questions, it appears that the method ofglobal sensitivity analysis (Saltelli et al., 2008) shows con-siderable promise in this direction. Using this methodology,it is possible to identify the extent of contribution of the dif-ferent sources of uncertainty to the overall uncertainty in theremaining useful life prediction. Consider Eq. 12, and us-ing global sensitivity analysis, it is possible to calculate thecontribution of each Xi towards the uncertainty in Y . Thisis facilitated through the calculation of the so-called first-order effects (Si1) and total effects indices (SiT ), as indicatedin Eq. 13 and 14 respectively.

Si1 =VXi(EX−i(R|Xi))

V (R)(13)

11


Table 1. Sensitivity Indices: Sensitivity of RUL

Quantity First-order Index Total Effects Indexx(0) 2.5× 10−2 2.5× 10−2

a 9.2× 10−1 9.3× 10−1

b 9.6× 10−5 1.5× 10−4

l 1.5× 10−2 9.6× 10−2

SiT = 1− VX−i(EXi(R|X−i))V (R)

(14)

While the first-order effects index calculates the contributionofXi by itself to Y , the total effects index calculates the con-tribution of Xi to Y by accounting for the interaction of Xi

with all other variables (denoted by X−i). If the first-ordereffects index of a variable is high, then this variable is con-sidered to be important. On the other hand, if the total effectsindex of a variable is low, then this variable is considered tobe less important.

For the numerical example discussed earlier in this section,the first-order effects index and total-effects index are tabu-lated in Table 1.

As it can be seen from Table 1, the single most importantquantity is a, which is representative of the rate of degrada-tion. In this paper, the above indices have been calculatedusing double loop Monte Carlo simulation (with 1000 sam-ples in each loop) to evaluate Eq. 13 and Eq.14. Single loopsampling approaches have also been discussed in the litera-ture (Saltelli et al., 2008).

Thus, the method of global sensitivity analysis can be usedto identify important sources of uncertainty, and this infor-mation can be used to aid uncertainty reduction and man-agement. Nevertheless, further research needs to address thisissue in detail, and develop computational methods that canmake use of statistical techniques such as global sensitivityanalysis for uncertainty management in the context of prog-nostics.

5. CONCLUSION

This paper presented an overview of uncertainty quantifica-tion in prognostics and health management in engineeringsystems. First, the significance of the uncertainty in prognos-tics was explained, and the need for a systematic approachto account for uncertainty in prognostics was discussed. Itwas explained that four different activities — uncertainty rep-resentation and interpretation, uncertainty quantification, un-certainty propagation, and uncertainty management — needto be performed in order to rigorously include the effects ofuncertainty in prognostics and provide useful information fordecision-making under uncertainty. Researchers have pur-sued two different approaches for prognostics, and these twoapproaches are based on testing and condition-based assess-ment. The philosophical differences between these two ap-

proaches were explained and it was demonstrated that theconcept of remaining useful life is more meaningful in thecontext of condition-based assessment since the engineeringsystem is under operation. Further, these differences are usedto analyze the interpretation of uncertainty in prognostics.

Probability and uncertainty can be interpreted in two ways.The frequentist interpretation of uncertainty is applicable inthe presence of true randomness, as is the case in testing-based health management. The Bayesian (subjective) in-terpretation of uncertainty is applicable even while talkingabout events that may not be random, and therefore, this in-terpretation is applicable for both testing-based health man-agement and condition-based health management. In fact,only the Bayesian interpretation of uncertainty is applicablein condition-based health management. Techniques such asKalman filtering, particle filtering, etc. that are commonlyused in condition-based prognostics are collectively knownas Bayesian tracking algorithms, not only because they useBayes’ theorem but also because they are based on the sub-jective interpretation of probability.

This paper also explained methods for the computation of re-maining useful life, in the context of condition-based prog-nostics. It was illustrated that it is not possible to analyti-cally calculate the uncertainty in the remaining useful life pre-diction even for certain simple problems involving Gaussianrandom variables and linear state-prediction models. There-fore, it is necessary to resort to computational methodologiesfor such uncertainty quantification and compute the probabil-ity distribution of remaining useful life prediction. Finally,the importance of uncertainty management was explained,and the relevance of global sensitivity analysis methods inthis context was explored. While different types of statisticalmethods for uncertainty quantification and management werediscussed, there are still several challenges that exist in thisregard (Sankararaman & Goebel, 2014), and further researchis necessary to investigate the applicability of these methodsto prognostics and health monitoring applications.

ACKNOWLEDGMENT

The work reported herein was in part funded by the NASASystem-wide Safety Assurance Technologies (SSAT) projectunder the Aviation Safety (AvSafe) Program of the Aero-nautics Research Mission Directorate (ARMD), and by theNASA Automated Cryogenic Loading Operations (ACLO)project under the Office of the Chief Technologist (OCT) ofAdvanced Exploration Systems (AES).

REFERENCES

Bichon, B., Eldred, M., Swiler, L., Mahadevan, S., & McFar-land, J. (2008). Efficient global reliability analysis fornonlinear implicit performance functions. AIAA jour-

12


nal, 46(10), 2459–2468.Bucher, C. G. (1988). Adaptive samplingan iterative fast

monte carlo procedure. Structural Safety, 5(2), 119–126.

Caflisch, R. E. (1998). Monte carlo and quasi-monte carlomethods. Acta numerica, 1998, 1–49.

Celaya, J., Saxena, A., & Goebel, K. (2012). Uncertaintyrepresentation and interpretation in model-based prog-nostics algorithms based on kalman filter estimation.In Proceedings of the Annual Conference of the PHMSociety (pp. 23–27).

Coppe, A., Haftka, R. T., Kim, N. H., & Yuan, F.-G. (2010).Uncertainty reduction of damage growth properties us-ing structural health monitoring. Journal of Aircraft,47(6), 2030–2038.

Daigle, M. J., & Goebel, K. (2013). Model-based prognosticswith concurrent damage progression processes. Sys-tems, Man, and Cybernetics: Systems, IEEE Transac-tions on, 43(3), 535–546.

Der Kiureghian, A., Lin, H.-Z., & Hwang, S.-J. (1987).Second-order reliability approximations. Journal ofEngineering Mechanics, 113(8), 1208–1225.

Dolinski, K. (1983). First-order second-moment approxima-tion in reliability of structural systems: critical reviewand alternative approach. Structural Safety, 1(3), 211–231.

Farrar, C. R., & Lieven, N. A. (2007). Damage prognosis:the future of structural health monitoring. Philosophi-cal Transactions of the Royal Society A: Mathematical,Physical and Engineering Sciences, 365(1851), 623–632.

Glynn, P. W., & Iglehart, D. L. (1989). Importance sam-pling for stochastic simulations. Management Science,35(11), 1367–1392.

Goebel, K., Saha, B., & Saxena, A. (2008). A comparison ofthree data-driven techniques for prognostics. In 62ndmeeting of the society for machinery failure preventiontechnology (mfpt) (pp. 119–131).

Gu, J., Barker, D., & Pecht, M. (2007). Uncertainty assess-ment of prognostics of electronics subject to randomvibration. In AAAI fall symposium on artificial intelli-gence for prognostics (pp. 50–57).

Haldar, A., & Mahadevan, S. (2000). Probability, reliabil-ity, and statistical methods in engineering design. JohnWiley & Sons, Incorporated.

Hastings, D. and McManus, H. (2004). A framework for un-derstanding uncertainty and its mitigation and exploita-tion in complex systems. In Engineering Systems Sym-posium MIT (p. 19). Cambridge MA..

Hohenbichler, M., & Rackwitz, R. (1983). First-order con-cepts in system reliability. Structural safety, 1(3), 177–188.

Kiureghian, A. D. (1989). Measures of structural safety underimperfect states of knowledge. Journal of Structural

Engineering, 115(5), 1119–1140.Kulkarni, C. S., Celaya, J. R., Goebel, K., & Biswas, G.

(2013). Physics based electrolytic capacitor degrada-tion models for prognostic studies under thermal over-stress. International Journal of Prognostics and HealthManagement, 4(5).

Liao, H., Zhao, W., & Guo, H. (2006). Predicting remaininguseful life of an individual unit using proportional haz-ards model and logistic regression model. In Reliabilityand Maintainability Symposium, 2006. RAMS’06. An-nual (pp. 127–132).

Loh, W.-L. (1996). On latin hypercube sampling. The annalsof statistics, 24(5), 2058–2080.

Orchard, M., Kacprzynski, G., Goebel, K., Saha, B., & Vacht-sevanos, G. (2008, oct.). Advances in uncertainty rep-resentation and management for particle filtering ap-plied to prognostics. In Prognostics and Health Man-agement, 2008. PHM 2008. International Conferenceon (p. 1 -6). doi: 10.1109/PHM.2008.4711433

Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni,J., Gatelli, D., . . . Tarantola, S. (2008). Global sensi-tivity analysis: the primer. John Wiley & Sons.

Sankararaman, S. (2015). Significance, interpretation, andquantification of uncertainty in prognostics and re-maining useful life prediction. Mechanical Systemsand Signal Processing, 52, 228–247.

Sankararaman, S., Daigle, M., & Goebel, K. (2014, June).Uncertainty quantification in remaining useful life pre-diction using first-order reliability methods. Relia-bility, IEEE Transactions on, 63(2), 603-619. doi:10.1109/TR.2014.2313801

Sankararaman, S., & Goebel, K. (2013a). A novel compu-tational methodology for uncertainty quantification inprognostics using the most probable point concept. InAnnual conference of the prognostics and health man-agement society.

Sankararaman, S., & Goebel, K. (2013b). Remaining use-ful life estimation in prognosis: An uncertainty prop-agation problem. In 2013 aiaa infotech@ aerospaceconference.

Sankararaman, S., & Goebel, K. (2013c). Why is the re-maining useful life prediction uncertain? In Annualconference of the prognostics and health managementsociety.

Sankararaman, S., & Goebel, K. (2014). Uncertainty in prog-nostics: Computational methods and practical chal-lenges. In Aerospace Conference, 2014 IEEE (pp. 1–9).

Sankararaman, S., Ling, Y., & Mahadevan, S. (2011). Un-certainty quantification and model validation of fatiguecrack growth prediction. Engineering Fracture Me-chanics, 78(7), 1487–1504.

Sankararaman, S., Ling, Y., Shantz, C., & Mahadevan, S.(2011). Uncertainty quantification in fatigue crack

13


growth prognosis. International Journal of Prognos-tics and Health Management, 2(1), 15 pages.

Sankararaman, S., & Mahadevan, S. (2011). Likelihood-based representation of epistemic uncertainty due tosparse point data and/or interval data. Reliability En-gineering & System Safety, 96(7), 814–824.

Sankararaman, S., & Mahadevan, S. (2013a). Distributiontype uncertainty due to sparse and imprecise data. Me-chanical Systems and Signal Processing, 37(1), 182–198.

Sankararaman, S., & Mahadevan, S. (2013b). Separating thecontributions of variability and parameter uncertaintyin probability distributions. Reliability Engineering &System Safety, 112, 187–199.

Sankararaman, S., Saxena, A., & Goebel, K. (2014). Are cur-rent prognostic performance evaluation practices suf-ficient and meaningful? In Proceedings of the 2014annual conference of the prognostics and health man-agement society.

Saxena, A., Goebel, K., Simon, D., & Eklund, N. (2008).Damage propagation modeling for aircraft engine run-to-failure simulation. In Prognostics and health man-agement, 2008. phm 2008. international conference on(pp. 1–9).

Saxena, A., Sankararaman, S., & Goebel, K. (2014). Perfor-mance evaluation for fleet-based and unit-based prog-nostic methods. In Second European conference of thePrognostics and Health Management society.

Swanson, D. C. (2001). A general prognostic tracking algo-rithm for predictive maintenance. In Aerospace confer-ence, 2001, ieee proceedings. (Vol. 6, pp. 2971–2977).

Tang, L., Kacprzynski, G., Goebel, K., & Vachtsevanos, G.(2009, march). Methodologies for uncertainty manage-ment in prognostics. In Aerospace conference, 2009IEEE (p. 1 -12).

Van Zandt, J. R. (2001). A more robust unscented trans-form. In International symposium on optical scienceand technology (pp. 371–380).

Zio, E., & Peloni, G. (2011). Particle filtering prognostic esti-mation of the remaining useful life of nonlinear compo-nents. Reliability Engineering & System Safety, 96(3),403–409.

BIOGRAPHIES

Shankar Sankararaman received his B.S.degree in Civil Engineering from the In-dian Institute of Technology, Madras in In-dia in 2007; and later, obtained his Ph.D. inCivil Engineering from Vanderbilt Univer-sity, Nashville, Tennessee, U.S.A. in 2012.His research focuses on the various aspects

of uncertainty quantification, integration, and managementin different types of aerospace, mechanical, and civil engi-neering systems. His research interests include probabilis-tic methods, risk and reliability analysis, Bayesian networks,system health monitoring, diagnosis and prognosis, decision-making under uncertainty, treatment of epistemic uncertainty,and multidisciplinary analysis. He is a member of the Non-Deterministic Approaches (NDA) technical committee at theAmerican Institute of Aeronautics and Astronautics (AIAA),the Probabilistic Methods Technical Committee (PMC) at theAmerican Society of Civil Engineers (ASCE), the Institute ofElectrical and Electronics Engineers (IEEE), and the Prog-nostics and Health Management (PHM) Society. Currently,Shankar is a researcher at NASA Ames Research Center,Moffett Field, CA, where he develops algorithms for uncer-tainty assessment and management in the context of systemhealth monitoring, prognostics, and decision-making.

Kai Goebel is the Deputy Area Lead forDiscovery and Systems Health at NASAAmes where he also directs the Prognos-tics Center of Excellence. After receivingthe Ph.D. from the University of Californiaat Berkeley in 1996, Dr. Goebel worked atGeneral Electrics Corporate Research Cen-

ter in Niskayuna, NY from 1997 to 2006 as a senior researchscientist before joining NASA. He has carried out applied re-search in the areas of artificial intelligence, soft computing,and information fusion and his interest lies in advancing thesetechniques for real time monitoring, diagnostics, and prog-nostics. He holds 17 patents and has published more than250 papers in the area of systems health management.

14

Uncertainty in Prognostics and Systems Health Management€¦ · Uncertainty in Prognostics and...

Documents

Transcript of Uncertainty in Prognostics and Systems Health Management€¦ · Uncertainty in Prognostics and...