1 Institute of Engineering Mechanics Leopold-Franzens University Innsbruck, Austria, EU H.J. Pradlwarter and G.I. Schuëller Confidence in the Range of Variability

2 Problem 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.)  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.)  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 Overview  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 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 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: Estimator (e.g.variance)

8 Bootstrap 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:

9 Probability mass outside the observation range  N > 1 data points specify the observed range  Define interval [a,b]  Assume independent data points  Suggestion : Interpret q N as level of significance confidence level = 1-

10 Probability mass outside the observation range  Probability

11 Probability density  Density outside the observed domain Until 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 PDF  Extended bootstrap distribution:  Replace the underlying discrete bootstrap probability distribution  by continuous Gaussian kernel density functions

13 Proposed PDF  Kernel densities: N gaussian densitities centered at the data points 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

14 Application to data  Calibration experiments: Young's modulus elongation Three data sets Notation: inverse Young's modulus average inverse Young's modulus over the length

15 Application to calibration data  Inverse Young's modulus Exceptionally large dispersion for N c =5 when compared with N c =30. The distribution is function of the amount of data points N c and the required confidence level 1- a.

16 Application to calibration data  Average inverse Young's modulus Exceptionally small dispersion for N c =5 when compared with N c =30. The distribution is function of the amount of data points N c and the required confidence level 1- a.

17 Application to calibration data  Joint distribution as function of N c and significance level

18 Application to calibration data  Joint distribution as function of N c and significance level

19 Application to calibration data  Joint distribution as function of N c and significance level

20 Random field calibration  Random field model  simple piecewise linear  i.i.d inverse Young's moduli  distribution of (maximum entropy principle) derives from mechanics

21 Random field calibration  Fitting of The average correlation length D is selected such that it fits the joint distribution best.  Simple Monte Carlo search

22 Application: Static Challenge problem  Prediction of Exceedance probability  Young's modulus in all four bars modelled as random field  Challenge: Estimation of exceedance probability

23 Application: Static Challenge problem  Prediction of Exceedance probability  Consistent results  Severe underestimation without introducing a low level of

24 Summary 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 Acknowledgment This research is partially supported by the European Commission under contract # RTN (MADUSE)