1 / 12 Michael Beer, Vladik Kreinovich COMPARING INTERVALS AND MOMENTS FOR THE QUANTIFICATION OF COARSE INFORMATION M. Beer University of Liverpool V

1 / 12Michael Beer, Vladik Kreinovich

COMPARING INTERVALSAND MOMENTS FOR THE QUANTIFICATION OFCOARSE INFORMATION

M. BeerUniversity of Liverpool

V. KreinovichUniversity of Texas at El Paso


1 Problem description

42

65.15 ... 5.35

measuring devices

dthickness

measuring points

0 10 30 50 D [N/mm²]

low

medium

high

linguistic assessments

x

measurement / observationunder dubious conditions

plausible range

expert assessment / experience

COARSE INFORMATION


CLASSIFICATION AND MODELING

» reducible uncertainty» property of the analyst» lack of knowledge or perception

According to sourcesaleatory uncertainty» irreducible uncertainty» property of the system» fluctuations / variability

stochastic characteristics

epistemic uncertainty

collection of all problematic cases, inconsistency of information

» non-probabilistic characteristics

According to information contentuncertainty» probabilistic information

traditional and subjectiveprobabilistic models

imprecision

set-theoretical models

no specific modeltraditionalprobabilistic models

In view of the purpose of the analysisaveraged results, value ranges, worst case, etc. ?

1 Problem description


PROBLEM CONTEXT

3 Engineering comparison

Structural reliability problem

Beer, M., Y. Zhang, S. T. Quek, K. K. PhoonReliability analysis with scarce information:Comparing alternative approaches in ageotechnical engineering contextStructural Safety 41 (2013), 1–10.

Comparative studyassume normal distribution for the variables

c o

0 o

C H P pN

1 e Plog

cG . c 6 35 cm.

performance function

» coarse information about the six variables Xi

Quantification of uncertain variables

specification of 2 parameters

further example and detailed discussion

» interval bounds xil and xiu interval analysis, range, worst case

Type and amount of available information ? Purpose of analysis ?

» moments μ and σ2 probabilistic analysis, response moments, cdf, Pf

i i i iil iu X X X Xx x 3 3, ,relate interval bounds to moments:


INTERPRETATION OF RESULTS

Probabilistic analysis

failure may occur in amoderate number of cases

Interval analysis l ug g 9 66 6 24 9 66 0 0 6 24, . , . . , , .

failure may occur

magnitude of exceedanceof g = 0 rather small, strongexceedance quite unlikely

significant exceedance of g = 0may occur

comparable

different focus: consider low-probability-but-high-consequence events

Given that input information is coarse

» known distribution of X

General relationship l u l uP Y y y P X x x, ,bounding property for general mapping XY

conclusions frominterval analysis mostlytoo conservative

4fP P G 0 8 94 10. .

» unknown distribution of Xprobabilistic results may be too optimistic, worst case (which is emphasized in interval analysis) maybe likely



RELATIONSHIP BETWEEN RESULTS

Probabilistic analysis

Interval analysis

i il iuP X x x i 1 6 0 98391, , , .., .

normal distributions for allvariables Xi

i i i iil iu X X X Xx x 3 3, , for all Xi

l uP G . g g 0 98391, .

histogram for G(.)

l uP G . g g 0 99993, .

» estimation of intervals [glP,guP]

with from histogram

lP uPP G . g g 0 98391, .

differences controlled by distribution of G(.)interval result is conservative

g(.)10 0 10

l ug g 9 66 6 24, . , .

▪ both-sided

lP uP centralg g 1 15 5 53, . , .

▪ left-sided w.r.t. lower bound gl

l uP leftg g 9 66 5 39, . , .

large difference due to lowprobability density for small g(.),but critical for failure

moderate differencedue to high probabilitydensity at upper bounds



RELATIONSHIP BETWEEN RESULTS

Probabilistic approximation

Interval analysisusing estimated moments of G(.)

l uP G . g g 0 98391, .

l ug g 9 66 6 24, . , .

2G G3 795 0 822. , .

lP uP ChebyP G . g g 0 98391, .

» Chebyshev’s inequality with

lP uP Chebyg g 3 35 10 94, . , .

interval result shifted towardsfailure domain, even more conservative than Chebyshev

interval result reflects tendencyof the distribution of G(.)to left-skewness

0 g(.)10 10interval analysis

Chebyshev

for right-skewed distribution of G(.), Chebychev‘s inequality may leadto the more conservative result g(.)10 10

histogram for G(.)for uniform Xi



INTERVAL OR MOMENTS ?

General remarks interval analysis heads for the extreme events,

whilst a probabilistic analysis yields probabilities for events

» to identify low-probability-but-high-consequence events for risk analysis


for a defined confidence level , interval analysis is more

conservative and independent of distributions of the Xi

i il iuP X x x,

difference between interval results and probabilistic resultsis controlled by the distribution of the response

conservatism of interval analysis is comparable to Chebyshev‘s inequality

interval analysis can be helpful

» in case of sensitivity of Pf w.r.t. distribution assumption and very vague information for this assumption» if the first 2 moments cannot be identified with sufficient confidence

for a defined confidence level, interval bounds maybe easier to specifyor to control than moments

What to chose in “intermediate” cases ?


INFORMATION CONTENT

Idea compare interval representation and moment representation

of uncertainty by means of information content:Which representation tells us more ?

l uP X x x,

assume that a variable X is represented alternatively(i) by the first two moments μX and σX

2

(ii) by an interval [xl, xu] for a given confidence

4 Information-based comparison

apply maximum entropy principle to both representations;calculate the least information of the representationwithout making any additional assumptions

chose the more informative representation;exploit available information to maximum extent(not in contradiction with maximum entropy principle)

l u X X X Xx x k k, ,

analog to the concept of confidence intervals

Relating intervals and moments


ENTROPY-BASED COMPARISON

Shannon‘s entropy continuous entropy


2S ff x f x dxlog

Interval representation maximum entropy principle

u

l

x

m intx

u l u l

1 1S dx

x x x x, ln

2

f xf x

2

lnlog

ln

» modification for comparison (ease of derivation)

m

1S f S ff x f x dx

2ln

ln

uniform distribution u l

1f x

x x

m int X XS 2 k 2 k, ln ln ln

relating to moments

u lx xln

u l Xx x 2 k


ENTROPY-BASED COMPARISON


Moment representation maximum entropy principle

m momS f x f x dx, ln

normal distribution

2

X

2

XX

x1f x

22exp

X

12

2ln ln

Comparison of representations

1

2 2 k2

ln ln

X X

12 2 k

2ln ln ln ln

check whether m mom m intS S, ,

2 e 2 k

ek 2 066

2.

for k > 2,the moment representationis more informative(ie, for >95% confidence)

under the assumptions made

for k ≤ 2,the interval representationis more informative(ie, for <95% confidence)


CONCLUSIONS

Comparing intervals and moments for the quantification of coarse information

Interval or moments

depends on the problem and purpose of analysis

for symmetric distributions, moment representation ismore informative if confidence of >95% is needed

for skewed distributions, moment representation is alreadymore informative for smaller confidence

Remark 2: imprecise probabilities

consider a set of probabilistic models (eg interval parameters)

worst case consideration in terms of probability (bounds)

useful if probabilistic models are partly applicable

Remark 1: fuzzy setsnuanced consideration of a nested set of intervals

enable “intermediate” modeling between interval and cdf

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1 / 12 Michael Beer, Vladik Kreinovich COMPARING INTERVALS AND MOMENTS FOR THE QUANTIFICATION OF COARSE INFORMATION M. Beer University of Liverpool V