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Transcript of 1 Institute of Engineering Mechanics Leopold-Franzens University Innsbruck, Austria, EU H.J....
1
Institute of Engineering MechanicsInstitute of Engineering Mechanics
Leopold-Franzens UniversityLeopold-Franzens University
Innsbruck, Austria, EUInnsbruck, Austria, EU
H.J. Pradlwarter and G.I. Schuë[email protected]
Confidence in the Range of VariabilityConfidence in the Range of Variability
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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.
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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:
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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.
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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
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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.
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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)
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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
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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-
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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
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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
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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
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Proposed PDFProposed PDF
Kernel densities: N gaussian densitities centered at the data points
-: ( , ) ,) )( , (
a
X Xbf x dx f px Ndx
( ) 2
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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
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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
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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.
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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
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Application to calibration data
5, 0.02cN 5, 0.10cN
( , | , )cPDF f N
Joint distribution as function of Nc and significance level
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Application to calibration data
20, 0.02cN 20, 0.10cN
( , | , )cPDF f N
Joint distribution as function of Nc and significance level
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30, 0.02cN 30, 0.10cN
Application to calibration data
Joint distribution as function of Nc and significance level
( , | , )cPDF f N
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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
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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
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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
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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
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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.
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AcknowledgmentAcknowledgment
This research is partially supported by the European Commission under contract
# RTN505164 (MADUSE)