1

Institute of Engineering MechanicsInstitute of Engineering Mechanics

Leopold-Franzens UniversityLeopold-Franzens University

Innsbruck, Austria, EUInnsbruck, Austria, EU

H.J. Pradlwarter and G.I. Schuëllermechanik@uibk.ac.at

Confidence in the Range of VariabilityConfidence in the Range of Variability

2

Problem definitionProblem definition

Suppose, only few measured values of an uncertain

quantity are available:

Is it under such circumstance possible to establish a credible

probability distribution for the reliability assessment?

- Without any strong assumptions, certainly not !

- There is an infinite number of options !

- Some physical background information is needed to

proceed any further.

3

Problem definition (cont.)Problem definition (cont.)

Among the infinite set of options, the choice should reflect

the needs of the analysis

The uncertainty due to the insufficient amount of data points

should be considered For estimating the performance (safety assessment)

confidence in the estimates will be crucial.

Some options:

4

Problem definition (cont.)Problem definition (cont.)

Only few measured values of an uncertain quantity are

available:

Is it under such circumstance possible to establish a credible

probability distribution for the reliability assessment?

Yes if:

- Some physical background information can be safely assumed

- We are not looking for the best estimate (e.g. a Bayesian approach)

but for an conservative PDF estimate for a required confidence level.

5

OverviewOverview

Bootstrap procedure Statistical results, e.g. confidence intervals

Probability of observation Probability of lying outside the observed domain

Probability density estimation Extended bootstrap procédure Marginal distributions for calibration data Joint distributions

Results Conclusions

6

Bootstrap procedureBootstrap procedure

Bootstrap procedure:

Modern, computer-intensive, general purpose approach to statistical inference.

Approach to compute properties of an estimator (e.g. variance, confidence intervals, correlations).

Advantage: Straightforward also for complex estimators and complex distributions.

Disadvantage: Tendency to be too optimistic for small sample sizes.

7

Bootstrap procedureBootstrap procedure

Bootstrap procedure (cont.): Given the data set

Resampling: Generate artificially a large number N of sets

from data by random sampling

Determine for each of the N sets the estimator (e.g. mean, variance, etc.) , establish the histogram and derive confidence intervals:

1Probability [ ( , ) ,1 ]P I i j k k n

n

( )(1, ) (2, ) ( 1, ) ( , )( , ,..., , )jI j I j I n j I n jx x x xx

1 2 1( , ,..., , )n nx x x xx

( )

1

Nj

jx

Estimator (e.g.variance)

8

Bootstrap procedureBootstrap procedure

Bootstrap procedure (cont.): Resampling corresponds to sampling from the discrete

probability distribution

The inference is only justified in case the sample represent the underlying unknown distribution well.

The method is not reliable if only very few data are availble, i.e. in case n is small.

The case n < 30 will be investigated in the following:

x

9

Probability mass outside the observation rangeProbability mass outside the observation range

N > 1 data points specify the observed range

Define interval [a,b] max minmin 2 2

x xa x

N

max min

max 2 2

x xb x

N

a bminx maxx

(1)x( )Nx ( )jx

Assume independent data points ( ) with ( )

bj N

X

a

P a x b q q f x dx Suggestion: Interpret qN as level of significance

confidence level = 1-

10

Probability mass outside the observation rangeProbability mass outside the observation range

Probability

( , )p N

1/ /1 ( , ) 1- NN p Nq

( )[ [ , ]]N kP x a b ( )[ [ , ]] 1N kp P x a b q

11

Probability densityProbability density

Density outside the observed domainUntil now we just have an estimate for the probability, not the density!

Almost everthing is possible without any physical background information

Reasonable (physical) assumptions: The density is high in the neighbourhood of any observations The density decreases with its distance from observations The density has a single domain with PDF(x)>0

12

Proposed PDFProposed PDF

Extended bootstrap distribution: Replace the underlying discrete bootstrap probability distribution

by continuous Gaussian kernel density functions

a bminx maxx

(1)x( )Nx ( )jx

a bminx maxx

(1)x( )Nx ( )jx

13

Proposed PDFProposed PDF

Kernel densities: N gaussian densitities centered at the data points

-: ( , ) ,) )( , (

a

X Xbf x dx f px Ndx

( ) 2

21

1 ( )( , ) exp( )

22

jN

Xj

x xf x

N

Justification:

+ each data point has equal weight and provides identical information

+ each data point has the same variability

+ the probability of occurrence decreases with the distance

is used to specify the standard deviation ( , )p N

14

Application to dataApplication to data

[ ]cL mm

( / 2)[ ]cE L MPa

Calibration experiments: Young's modulus elongation

Three data sets

5cN

20cN

30cN

Notation:

inverse Young's modulus

average inverse Young's modulus

over the length

( )j( )j

( )( ) ( ) ( )

( ) 0

1 1( / 2) , ( )

( / 2)

cj

Lj j j c cc j

c c c c

A LL x dx

E L L L F

15

Application to calibration data

5

20

30

c

c

c

N

N

N

5

20

30

c

c

c

N

N

N

0.02

0.10

Inverse Young's modulus

( )( )

1, 1,2, ,

( / 2)j

cjc

j NE L

Exceptionally large dispersion

for Nc=5 when compared with

Nc=30.

The distribution is function of

the amount of data points Nc

and the required confidence

level 1-a.

16

Application to calibration data

5

20

30

c

c

c

N

N

N

0.10

5

20

30

c

c

c

N

N

N

0.02 Average inverse

Young's modulus

( ) ( )

0

1( )

cLj j

c

x dxL

Exceptionally small dispersion

for Nc=5 when compared with

Nc=30.

The distribution is function of

the amount of data points Nc

and the required confidence

level 1-a.

( )jc c

c c

A L

F L

17

Application to calibration data

5, 0.02cN 5, 0.10cN

( , | , )cPDF f N

Joint distribution as function of Nc and significance level

18

Application to calibration data

20, 0.02cN 20, 0.10cN

( , | , )cPDF f N

Joint distribution as function of Nc and significance level

19

30, 0.02cN 30, 0.10cN

Application to calibration data

Joint distribution as function of Nc and significance level

( , | , )cPDF f N

20

Random field calibrationRandom field calibration

Random field model simple piecewise linear

i.i.d inverse Young's moduli distribution of (maximum entropy principle)

derives from mechanics

1

Mc

c i iic

FL

A

1

M

c ii

L

1ˆ ˆ( ) / 2i i i

i

i

min( ) 1 exp( )FD

min

minE[ ]D

21

Random field calibrationRandom field calibration

Fitting of

The average correlation length

D is selected such that it fits

the joint distribution

best.

1[ , ][ ] [ , ] [ , ] [i,1] [ ] [ , ]

1 1

1, with ,

i i

i

M Mi Mi i k i k i i k

ck kc

LL

[ ] [ ] [ ] [ ] [ ] [ ]

1

1 1( , ) [ ]

2 1

MCNi i k i k i

kMC MC

K IN N

[ ] [ ] [ ] [ ] 2

1

1( ) ( ( , ) ( , ))

MCNi i i i

iMC

F K MinN

minE[ ]D

( , )F

Simple Monte Carlo search

22

Application: Static Challenge problemApplication: Static Challenge problem

Prediction of Exceedance probability

Young's modulus in all four bars modelled as random field Challenge: Estimation of exceedance probability

[ 3.0 mm] = ( )P fP y p

Py

23

Application: Static Challenge problemApplication: Static Challenge problem

Prediction of Exceedance probability

Consistent results Severe underestimation without

introducing a low level of Py

cPrediction for = 0.02,0.10 and N 5,20,30

[ 3.0 mm] = ( )P fP y p

24

Summary and ConclusionSummary and Conclusion

The spread of the assumed probability distribution is a function of the number of data points and the required confidence level.

The introduction of confidence level provides a suitable safeguard against a severe underestimation of the variability of the parameters derived from a small data set.

Consistent results can be obtained although the small data set might be misleading.

25

AcknowledgmentAcknowledgment

This research is partially supported by the European Commission under contract

# RTN505164 (MADUSE)