efficent epistemic-aleatory uncertainty quantification

NAFEMS World Congress 2015 inc. the 2nd International SPDM Conference | San Diego, CA, 21-24 June 2015

EFFICENT EPISTEMIC-ALEATORY UNCERTAINTY QUANTIFICATION: APPLICATION TO THE NAFEMS CHALLENGE PROBLEM Matteo Broggi, (Virtual Engineering Centre, and Institute of Risk and Uncertainty University of Liverpool, United Kingdom ); Roberto Rocchetta, Edoardo Patelli (Institute of Risk and Uncertainty, University of Liverpool , United Kingdom ) Abstract Nowadays, it is generally accepted that the explicit inclusion of uncertainties in simulations due to variation in parameters and operational conditions is necessary in order to design reliable and robust component and systems. For example, the design of new products require to perform a series of analyses in order to achieve the desired goals and needs. It is important that easy to follow and accurate best practices and solutions are available to facilitate the adoption of uncertainty quantification approaches and techniques by a larger public of engineering practitioners. At the last NAFEMS world congress 2013 held in Salzburg, the NAFEMS Stochastics Working Group has launched a challenge problem [Fortier, 2013]. Open to contributors from Academia, Government and Industry, it aims at promoting different approaches for Uncertainty Quantification and compare dedicated software tools to automate the analysis. The challenge problem is based on an electric equivalent RLC circuit, with different levels of uncertainty estimated around the model input parameters (intervals from multiple sources, finite number of sampled values, incomplete intervals). The scope of the challenge is to evaluate the reliability of the circuit under the different input uncertainties as well as the quantification of the value of information in each case. A solution adopting only traditional probabilistic approaches has been provided [Fortier et al. 2014] by one of the authors of this manuscript.. In this work, the NAFEMS challenge problem is solved by using generalized probabilistic approaches (such as imprecise probabilities, interval analysis and probabilistic boxes [Patelli et al, 2014]) for uncertainty quantification and management. The results obtained by these novel approaches, which account for both epistemic and aleatory

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source of uncertainty, are compared with the solution obtained adopting only traditional tools. A critical analysis of the results allows to show the limitations and shortcomings of the classic approach. On the other hand, it demonstrates the increased flexibility and capabilities in the treatment of uncertainty when generalized probabilistic approaches are employed. In particular it does not confuse epistemic and aleatory uncertainties allowing to obtain more robust design without the necessity to include unsupported simplifications and approximations. The proposed solution has been integrated in OpenCossan [Patelli et al, 2014],an open general-purpose software for uncertainty quantification.

1.

Introduction

Nowadays, it is generally understood that is necessary to include uncertainties in simulations, e.g. due to variation in parameters and operational conditions, in order to achieve goals and needs such as robust design of new products, execute model validation, ensure reliable operation for the whole product-life, etc. It is important that easy to follow and accurate best practices and guidances are available to make uncertainty quantification a standard techniques adopted by a larger public of engineering practitioners. At the last NAFEMS world congress, held in 2013 Salzburg, the NAFEMS Stochastics Working Group has launched a challenge problem open to contributors from Academia, Government and Industry. The aim was to showcase and promote different approaches to Uncertainty Quantification and the use of dedicated software tools to automate the analysis. The system subject to the analysis is a simple RLC series electric circuit with different levels of uncertainty estimated around the model input parameters (i.e., intervals from multiple sources, finite number of sampled values, incomplete intervals). The scope of the challenge is to evaluate the reliability of this circuit under the different input uncertainties as well as to quantify the value of information in each case. The interested readers are reminded to reference NAFEMS challenge problem (2013) for further details about the problem statement and the case study definition. So far, the problem has been addressed adopting only classic probabilistic approaches NAFEMS Bench Mark (2014). In this paper, the NAFEMS challenge problem is tackled by adopting a non-traditional probabilistic approach that fits in the framework of imprecise probabilities. Explanatory examples of such powerful framework are provided by e.g. Beer (2013), Ferson (2003) and Patelli (2014). The results obtained by these novel approaches, which account for both epistemic and aleatory

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source of uncertainty, will be compared with the previous solution pointing out the limitations and shortcomings of the classic probabilistic approaches. This will show the increased flexibility and capabilities in uncertainty treatment that can be found when imprecise probabilities are employed. Increased flexibility can be achieved by improving the uncertainty models with provided information of different quality and by efficiently including evidence from wide range of sources. Finally, easy to follow best practice and methods for uncertainty management will be laid out. OpenCOSSAN an open general purpose software for uncertainty quantification, has been employed in all the steps of the analysis, Patelli (2014). This paper is structured as follows: section 2 presents some theoretical background and a brief introduction to Dempster-Shafer theory of evidence. Section 3 shows the NAFEMS challenge problem, the reliability requirements and computational challenges. In section 4 the proposed solution for the uncertainty characterization, propagation and reliability analysis of the NAFEMS challenge problem is presented. In section 5 discussions and conclusions are drawn. 2.

Theoretical Background

It is nowadays well understood in the scientific community that uncertainty can generally affect systems in two different forms, the so called aleatory and epistemic uncertainties, Kiureghian, Ditleven, (2009). The aleatory, also indicated in literature as Type I or irreducible uncertainty, is related to stochastic behaviours and randomness in events and variables. Hence, due to its intrinsic random nature is normally regarded as not reducible, that means the degree of uncertainty cannot be decreased even if the knowledge of the system or of the physical phenomena is improved. The epistemic, also called Type II or reducible uncertainty, is commonly related to lack of knowledge about a particular behaviour, imprecision in measurement and poorly designed models. It is considered as reducible since further information gathering can reduce the level of uncertainty, but this is not always practical or feasible. In the last decades efforts were focused in the treatment of imprecise knowledge, non-consistent information and both epistemic and aleatory uncertainty by efficient approaches. The methodologies are discussed in literature by different mathematical concepts: Dempster-Shafer Evidence theory Dubois (1986) Shafer(1976), interval probabilities Augustin (2004), level two probably approach Oberkampf (2004), Zio (2013), Fuzzy-based approaches Blaria (2001) and Bayesian updating approaches Faber (2005), Kiureghian (2008), Kiureghian (2009),

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Veneziano (2009), Ching (2007), are some of the most intensively applied concepts. Considering all the available methodologies, the Dempster-Shafer approach based on the theory of Evidence has been selected to solve the challenge problem. The main reasons are that this approach require little assumption and can be easily implemented and coupled to classical probabilistic frameworks, moreover is a straightforward approach to deal with imprecise and multiple intervals which are of interest in this problem. Moreover, it can be easily applied to solve all the tasks proposed in the challenge. Kolmogorov-Smirnov test and Kernel Density Estimator have been used to represent the sample uncertainty for extremely small sample size, one of the challenge tasks. a.

Dempster-Shafer Structures and Probability Boxes

One of the largely used frameworks of subjective probability is the Dempster–Shafer theory which is a well-suited framework to represent both aleatory and epistemic uncertainty and it can be seen as a generalization of Bayesian probability. In the Dempster-Shafer theory, numerical measures of uncertainty (a degree of belief also referred to as a mass) may be assigned to overlapping sets and subsets of hypothesis, events or propositions as well as individual hypothesis, Beer (2013). Probability values are assigned to sets of possibilities rather than single events. In the Shafer theory the sets are represented as intervals, bounded by two values, belief and plausibility. In order to characterize both epistemic and aleatory uncertainty, probability boxes (often referred as P-boxes) are often used. Probability boxes can be seen as a further generalization of the Dempster-Shafer structures where the sets are represented by distributions. The Dempster-Shafer structures are similar to discrete distribution but rather than precise points, the locations where the probability mass resides are set of real values. A Dempster-Shafer structure can be expressed as set of focal elements as presented by Ferson (2003): {

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,

,

1 ,

,

,

,…,

,

,

}

(1)


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Where x , x is the i-th focal element having upper bound x and lower bound x , m is the probability mass associated with the i-th focal element. P-boxes are set of cumulative distribution functions (CDFs) for which , . Note that the probability lower and upper bounds are assigned distribution associated to the random variable of interest can be either defined or not. As summarized by Alvarez (2014) the first are generally named distributional P-boxes (or parametric P-boxes) the latter are called distribution-free P-Boxes (or non-parametric P-Boxes). Fig.1 shows an illustrative example of distributional P-Box, the parent distribution is the normal distribution; the red dashed line is the lower bound , black solid line is the upper bound .

Figure 1:

Illustrative example of distributional P-Box

The wider the distance between upper and lower bound the higher epistemic uncertainty is associated to the random variable. The bounds for the CDFs mean bounds on probabilities, upper and lower probability bounds can be hence obtained as in Klir (2006):

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=

≤ =

=1−

≤

≤

(2)

(3)

In literature the lower bound on probability is sometime referred as plausibility and the upper bound as belief. Dempster-Shafer structures can be always translated into a P-box but without forgetting that isn’t an information preserving procedure. These approaches are straightforward to deal with some of the cases proposed in the NAFEMS challenge, such as multiple and overlapping intervals and inconsistent sources of information. The drawback is that the propagation of intervals and P-boxes through the system can results computational expansive. On the other hand the quantification approaches are generally not-intrusive and hence applicably to any model. For further acknowledgment the reader is reminded to Laura (2009). This is an interesting feature of the approach which is therefore suitable to be used in parallel to classical uncertainty quantification method to give different prospective. b.

Kolmogorov-Smirnov Test

One possible non-parametric approach that can be use to characterize the uncertainty of a process starting from samples with a reference probability distribution is the so called one-sample Kolmogorov-Smirnov test (Massey (1951)). The Kolmogorov-Smirnov (KS) test is a distribution-free statistical test based on the maximum difference between an empirical CDF and a hypothetical CDF. It returns upper and lower bound of CDFs assuming a predefined confidence level. The bounds can be computed by use of equation (4) as shown by Ferson (2003): #$ 1,

% 0, '

, ' (, $

(4)

where ' denotes the best estimate of the distribution function and ' (, $ is the one-sample KS critical statistic for confidence level 100(1()% where ( is the selected significance level and $ the sample size. The KS critical statistic can be therefore used to obtain different confidence limits on the CDFs by choosing different critical values of the statistic test. Different level of confidence lead to different confidence

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bounds on the CDFs which, when propagated, produce boarder or narrow bounds of the resulting P-boxes. c.

The Kernel Density estimator

It is in general difficult to get the true distribution from a small number of samples using parametric methods. This is because there is no enough information to estimate the PDF when only few data points are available. Kernel density estimator is another non-parametric approach that can be used to estimate the probability density function of a random variable Pradlwater (2008). The approach does not need any assumptions regarding the underlying distribution. A commonly used univariate parametric kernel is the Gaussian or normal Kernel: )*

=

1

/0 12

$+√2. 35

−

− 2+

3

4

(5)

where )* represents the estimated probability density function of $ samples 3 drawn from an unknown density function f. The variance (or bandwidth) + is the only parameter that needs to be estimated. The best bandwidth can be estimated using for instance the Silverman's rule of thumb Silverman (1986) or in case of very small sample sizes the approach proposed by Pradlwater (2008). 3.

The NAFEMS challenge problem

In the NAFEM challenge problem (NAFEMS challenge problem (2013)), the analyst is asked to evaluate the reliability of an electronic resistive, inductive, capacitive (RLC) series circuit on meet three requirements. Four different cases (A, B, C and D) have been proposed, each one having different information regarding the system parameters R, L and C. In the CASE-A single intervals, one upper bound and one lower bound for each parameter of the R L and C are known. In the CASE-B multiple intervals, three upper and lower bounds are given for each parameter. In CASE-C ten sampled points for each parameter are provided. Finally in the CASE-D, imprecise bounds are the only available information about the parameters; it is similar to CASE-A, but one bound is not precisely defined. Uncertainty is characterized using different approaches according to the level of knowledge available for R, L and C. In the challenge problems, the main goals consist in qualify the value of information and evaluate the reliability of the system with respect to three requirements: the first two requirements are on the voltage at the

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capacitance 67 (since the second requirement on rise time can be translated into a voltage requirement as well) and a third one on the underdamped system responses (which have to be discharged). The equations governing the RLC circuit are provided in APPENDIX A: Solution of the RLC circuit. The first and the second reliability requirements are on the voltage values: (6) V9 10ms ; 0.96 V9 t

tr

0.9V where tr C 8 E

(7)

where tr is the rise time in millisecond for the capacitor. Additionally (third requirement), underdamped system configurations must be discharged (Z ; 1 where Z is the damping coefficient).It can be observed that the third requirement imply a monotonic behaviour for V9 , hence it is trivial write the second requirement as in equation (8) V9 t

8ms ; 0.96

(8)

The CDF of a random variable is commonly used in reliability analysis to extract useful knowledge about the probability as follow: F x

P X!x

(9)

It can be therefore evaluated the failure probabilities by use of the CDFs of 67 8 E , 67 10 E and Z. Equations (2)-(9) can be used to evaluate the bounds on probability of fail to meet the requirements as described by the equations (10 -(15 :

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F Vc 8

0.9

P Vc 10 ! 0.9

PtrLM NOPQ

10

F Vc 8

0.9

P Vc 8 ! 0.9

PtrLM NOPQ

11

F Vc 10

0.9

P Vc 10 ! 0.9

PR9

STUVWXYZ

12

F Vc 10

0.9

P Vc 10 ! 0.9

PR9

STUVWXYZ

13


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F Z=1

1"P Z ; 1 F Z=1

where

^

P Z!1

P Z!1

PzLM NOPQ

PzLM NOPQ

_7 Sàbcdef , _7 Sàbcdef g,

hijk3lmno , hijk3lmno

14 15

and

pjk3lmno , pjk3lmno are the probability bounds of fail to meet the first, the second and the third requirement respectively. 4.

Proposed solution

In a previous solution of this challenge problem some of the authors have tackled the problem by using classical probabilistic methods (NAFEMS Bench Mark (2014)). In NAFEMS Bench Mark (2014) only traditional probabilistic approaches were adopted to characterize the uncertainty of the parameters and then evaluate the system reliability. The well-established and versatile Monte Carlo simulation techniques (Ching (2003), Zio (2015)) were adopted for risk assessment and reliability assessment allowing the previous solution of the problem. a.

Limitation of the previous approach

The uniform distribution was considered, since it is a typical assumption for cases in which equally probable values for a random variable are accounted. This assumption is often regarded as a rational hypothesis that found theoretical support in the Laplace’s principle of indifference (or more in general principle of maximum entropy). Artificial model assumptions are, especially in reliability assessment problem, sometime too narrow and leading to expert overconfidence problem. Assuming uniform probability distribution functions can bear an underestimation of the uncertainty. Consider the cumulative distribution function (CDF) of the uniform distribution; the CDF will have a wellknown linearly increasing shape between minimum and maximum values. If these minimum and maximum bounds values are the only available data, the assumption can led to misleading results especially in reliability assessments. Within an imprecise-information scenario (e.g. parent distribution not specified or unknown, or known but with vague parameters, conflicting and limited knowledge, linguistic incomprehension, intervals etc.), it seems more conservative and robust for the reliability assessment to take into account not just one single

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CDF but all the plausible CDFs accordingly to the available data. Indeed, this approach will produce imprecise results in reliability evaluation, but (as highlighted by Beer (2013)) has the undeniable advantage of not introducing artificial model assumption. b.

CASE-A and CASE-B

In this section uncertainty characterizations for the CASE-A and CASEB are described and method used to propagate uncertainty through the system introduced. Uncertainty Characterization Available information CASE-A summarize the information available in CASE-A. Lower bounds, upper bounds and assigned mass for the three sources in CASE-B shows the information available in CASE-B and probability mass assigned to the sources. The probability masses associated to the different sources have been considered equal and normalized. Assuming equally likely sources it has been accounted the lack of information about how reliable is one source compared to the others. Table 1:

CASE-A

R[Ohm]

L[mH]

C[µF]

Intervals

[40,1000]

[1,10]

[1,10]

Table 2:

CASE-B

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Available information CASE-A

Lower bounds, upper bounds and assigned mass for the three sources in CASE-B

R[Ohm]

L[mH]

C[µF]

q

q

r

r

s

s

Source 1

40

1000

1

10

1

10

Source 2

600

1200

10

100

1

10

Source 3

10

1500

4

8

0.5

4

= 1⁄3

= 1⁄3

u

= 1⁄3


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Uncertainty has been characterized by using Dempster-Shafer structures. It can be defined one Dempster-Shafer structure for each parameter as shown in equations (16)-(18). { R ,R ,m

,

R ,R

,m

,

R u , R u , mu }

(16)

{ L ,L ,m

,

L ,L ,m

,

Lu , Lu , mu }

(17)

{ C ,C ,m

,

C ,C ,m

,

Cu , Cu , m u }

(18)

m + m + mu = 1

(19)

R , R represents the i-th interval bound for the resistance, L , L is the

ith bound for the inductance, C , C is the ith bound for the capacitance and m is the probability mass associated to the i-th source. CASE-B degenerate to CASE-A if the probability mass m and mu are set equal to zero. For this reason CASE-A has been considered and solved together with CASE-B, because CASE-A is a special case of CASE-B.. Uncertainty Propagation The following procedure has been adopted in order to propagate the uncertainty characterized by Dempster-Shafer structures: 1. Probability masses are assigned to the three available sources of information (m , m , mu ).

2. n=3! interval parameter cells are constructed by permutation of the cells. The first cell is built by selecting the first interval of the parameter R and first interval of the parameter L and combining them with the first interval of C. The second interval cell is created selecting the first interval of R and first of L and combining them with the second interval of C, so on in a combinatorial manner.

3. For each interval cell, Latin Hypercube Sampling (LHS) strategy is used to samples parameter values. Then the system responses are evaluated in terms of V9 and Z and the minimum and maximum of the outputs are calculated. An alternative approach based on

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optimization technique can be used to identify the output bounds for interval cells. 4. The results are stored and n min-max intervals of V9 at the 8-th ms, 10-th ms and n min-max interval of Z are saved and combined to computed probability mass. Results are used to create DempsterShafer structures (see Eqs (20)-(22)). 5. Finally intervals the Dempster-Shafer structures are used ,as explained in Ferson et al. (2003), to create probability boxes of V9 8ms , V9 10ms and Z. Equations (10 -(15 are used to compute the bounds on the probability of failure. { Vc 8 , Vc 8 , m9QNN { Vc 10 , Vc 10 , m9QNN

,…,

Vc{ 8 , Vc{ 8 ], m9QNN{ }

, … , [ Vc{ 10 , Vc{ 10 ], m9QNN{ }

{ [Z , Z ] m9QNN , [ Z , Z ] m9QNN ,…, [ Z{ , Z{ ] m9QNN{ }

20

21

22

Where m9QNN = m m m , m9QNN = m m m , … , m9QNN{ = m{ m{ m{ are the probability mass computed for n parameters cells. Results and discussions The Dempster-Shafer structures obtained by propagating uncertainty have been plotted creating Plausibility and Belief bounds on the CDFs. Applying the procedure to the CASE-A, the resulting P-boxes gave no valuable information on the probability of failure associated to the three system requirements. The probability of failure is in fact just bounded in the interval [0, 1] for all three the requirements. In CASE-B equal probability mass have been assigned to the three distinct sources of information, such that m = m = mu = u. The resulting P-boxes of Vc and Z are shown in Figs.2-4. It can be seen that results still have high uncertainty associated, but an increased precision if compared to the Case-A. Fig.2 show that PR9 STUVWXYZ lay within the interval [0, 0.9]. In Fig.4 it can be observed that PzLM NOPQ upper and lower bounds are [0, 0.78]. Fig. 14 shows that PR9•TUVWXYZ lay within the interval [0, 1]. The CASE-B includes all the information available for the CASE-A (information in source 1 coincide with information of CASE-A) plus two

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additional sources of information. In this particular case, the additional sources of information contribute on reducing the uncertainty on the system performance. By comparing the results, it can be observed that CASE-B produces more precise intervals for the probability of fail to meet requirement one and three. On the other hand, failure probability for requirement two don’t show any uncertainty reduction compared to CASE-A. It can be therefore argued that, with respect to the first and third requirements, the quality of the information given in CASE-B is higher compared to the quality of the information given in CASE-A. On the other hand, results have the same quality with respect to the second reliability requirement.

Figure 2:

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P-Box67 10 E , CASE-B.


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Figure 3:

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P-Box 67 8 E , CASE-B.


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Figure 4:

c.

P-Box €, CASE-B.

CASE-C

For the analysis of case C, two different approaches have been investigated; first, confidence bounds have been obtained by applying the Kolmogorov-Smirnov (KS) approach, Dempster-Shafer structures and P-box are used to propagate the uncertainty as presented in the previous cases. In the second approach, the Kernel Density Estimator (KDE) is used to fit probability distribution to small sample data. Uncertainty characterization propagation and discussion of the results are presented. KOLMOGROV-SMIRNOV APPROACH Uncertainty Characterization and propagation TabAvailable information CASE-C shows the available information for CASE-C. The available data consists of 10 sampled values for each parameter of R, L and C, respectively.

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Table 3:

TabAvailable information CASE-C

CASE-C

R[Ohm]

L[mH]

C[µF]

Sampled

{861,87,430,798,

{4.1,8.8,4.0,7.6,0.7,3.

{9.0,5.2,3.8,4.9,2.9,8

Data

219,152,64,361,2

9,7.1,5.9,8.2,5.1}

.3,7.7,5.8,10,0.7}

24,614}

No additional information about the parameter such as, sampling procedure or parent probability distribution functions has been provided. In order to characterize the uncertainty, Kolmogorov-Smirnov critical statistic is used as shown in equation (4). The results are bounds on the empirical distribution function of the sampled parameters. Fig.5 display upper and lower bounds for the empirical CDF of the inductance L. Fig.6 shows upper and lower bounds for the empirical CDF of R and in Fig.7 upper and lower bounds for the empirical CDF of C are presented. Three different confidence levels are been accounted and the different confidence bounds have been propagated. Some assumptions have to be made in order to apply Kolmogorov-Smirnov test and draw the upper and lower bound for the empirical CDF. The samples are regarded as independent and identical distributed and ranges of variation of the parameter values have been assumed in order to truncate the CDF bounds. Positivity is assumed as the lower bound while the upper bound is in first instance assumed equal to the sample mean plus three times the sample standard deviation. Very long tails if underestimated can lead to overconfidence especially in reliability studies. Hence, following the considerations made, it has been selected a relatively high upper bound to truncate the CDF, as shown in Figs.5-7.In order to better investigate the goodness of the assumptions, sensitivity of the results to the assumed upper bounds variation have been analysed. Further analysis hasn’t shown any significant changes in the resulting probability bounds.

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Figure 5:

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P-Box for inductance L, CASE-C.


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Figure 6:

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P-Box for resistance R, CASE-C.


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Figure 7:

P-Box for Capacitance C, CASE-C.

The uncertainty has been propagated through the system as explained for CASE-A and CASE-B. Results are shown in Figs.8-10, bounds for three confidence levels, α=0.01, α= 0.1 and α=0.2 are displayed. Each confidence level lead to different probability bounds on the CDFs and resulting P-Boxes. The resulting lower and upper probability bounds are shown in the Probability Bounds for the three requirements, three confidence levels. It can be noticed that, even if the distance between Plausibility and Belief still high, typical of case denoted by high epistemic uncertainty, CASE-C shows an increased precision compared to the CASE-B and A. In Figs.8-10 P-Boxes for the voltage at the 10th ms, 8th ms and for damping coefficient are presented, red blue and black colour lines refer to α=0.01 α=0.1 and α=0.2 respectively. Fig.10 has been zoomed around the value Z=1 for improve the graphical output.

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Table 4:

Probability Bounds for the three requirements, three confidence levels

CASE-C _7 Sàbcdef _7S•àbcdef

pjk3lmno

α=0.01

α=0.1

α=0.2

[0, 0.87]

[0, 0.7]

[0, 0.63]

[0, 0.92]

[0, 0.77]

[0, 0.7]

[0, 0.83]

[0, 0.7]

[0, 0.64]

Applying Dempster-Shafer approach to CASE-C, the resulting failure probability bounds appear less uncertain compared to all the four cases. This increased precision find intuitive explanations in the assumptions made in order to apply the Kolmogorov-Smirnov approach.

Figure 8:

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P-Boxes for 67 10 E , three confidence levels, CASE-C.


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Figure 9:

Belief and plausibility P-Boxes for67 8 E , three confidence levels, CASE-C.

Figure 10: Belief and plausibility P-Boxes for€, three confidence levels, CASE-C.

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Kernel Density Estimation Approach CASE-C have been analysed using also a classical probabilistic approach. Probability distribution functions for the given samples have been fitted using the Gaussian Kernel Density estimation as explained in section 2. Then, Monte Carlo samples are generated form the fitted distribution and the system performance evaluated. The results obtained in the previous classical approach, which adopted uniform probability distribution to represent the uncertainty are PR9 STUVWXYZ = 0.15, PtrLM NOPQ = 0.23 and PzLM NOPQ = 0.71. Kernel Density fitting and Monte Carlo sampling have led to the similar results for two failure probabilities (PR9 STUVWXYZ = 0.232, PtrLM NOPQ = 0.292) which find relatively good agreement with the previous, major difference is observed in the PzLM NOPQ which result lower accounting kernel distribution( PzLM NOPQ = 0.121). The bounds for the three failure probabilities obtained by Dempster-Shafer approach include the results found by assuming uniform distribution and by using the Kernel Density fitting. Figs.11-13 show the resulting CDF, Fig. 13 has been zoomed around the value Z=1 for graphical reasons.

Figure 11: CDF for Vc (10ms) using Kernel estimation.

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Figure 12: CDF for Vc (8ms) using Kernel estimation.

Figure 13: CDF for Z using Kernel estimation.

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d.

CASE-D

In CASE-D, similarly to CASE-A, parameter bounds are the available information. Here, just one bound is precisely defined. The imprecisely defined bounds are the upper bounds of R and L and the lower bounds of C. Information available have been given as shown in Available information CASE-D. The lower and upper bounds are defined as major of or less than a nominal value. Table 5:

Available information CASE-D

CASE-D

R[Ohm]

L[mH]

C[µF]

Interval

[40, RU1]

[1, LU1]

[CL1,10]

Other information

RU1>650

LU1>6

CL1 1

Vc t = V + A1 + A2t eŒM•

Vc t = V + A1eŽ

•

+ A2eŽ

•

Where S1,2 = †−

R 1 R ‡ ± ‘2† ‡ 4 − 2L LC 2L

Damping factor ζ, values of ω and α are determined as follow: α=

R , 2L

ω=

1

√LC

,

ζ=

α ω

Coefficients A1 and A2 are determined from the initial conditions Vc 0 = 0 ,

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dVc 0 =0 dt


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