Comparing intervals and moments for the quantification of coarse information

1. Comparing intervalsand moments for the quantification ofcoarse information M. Beer University of Liverpool V. Kreinovich University of Texas at El Paso

2. expert assessment / experience linguistic assessments measuring points measuring devices · · · · high medium low 0 10 30 50 D [N/mm²] plausible range x thickness d measurement / observation under dubious conditions 1 Problem description Coarse Information 2 4 6 5.15 ... 5.35

3. 1 Problem description Classification and modeling According to sources aleatory uncertainty epistemic uncertainty · · » irreducible uncertainty » property of the system » fluctuations / variability » reducible uncertainty » property of the analyst » lack of knowledge or perception collection of all problematic cases, inconsistency of information stochastic characteristics no specific model traditional probabilistic models According to information content uncertainty imprecision · · » probabilistic information » non-probabilistic characteristics set-theoretical models traditional and subjective probabilistic models In view of the purpose of the analysis averaged results, value ranges, worst case, etc. ? ·

4. Problem context specification of 2 parameters · 3 Engineering comparison Structural reliability problem performance function further example and detailed discussion · · Beer, M., Y. Zhang, S. T. Quek, K. K. Phoon Reliability analysis with scarce information: Comparing alternative approaches in a geotechnical engineering context Structural Safety 41 (2013), 1–10. » coarse information about the six variables Xi Quantification of uncertain variables Type and amount of available information ? Purpose of analysis ? » moments μ and σ2 probabilistic analysis, response moments, cdf, Pf » interval bounds xil and xiu interval analysis, range, worst case Comparative study assume normal distribution for the variables · relate interval bounds to moments: ·

5. Interpretation of results 3 Engineering comparison Probabilistic analysis Interval analysis · · Given that input information is coarse failure may occur in a moderate number of cases failure may occur comparable magnitude of exceedance of g = 0 rather small, strong exceedance quite unlikely significant exceedance of g = 0 may occur different focus: consider low-probability-but-high-consequence events General relationship bounding property for general mapping XY · » known distribution of X » unknown distribution of X conclusions from interval analysis mostly too conservative probabilistic results may be too optimistic, worst case (which is emphasized in interval analysis) maybe likely

6. Relationship between Results 3 Engineering comparison Probabilistic analysis Interval analysis normal distributions for all variables Xi for all Xi · · histogram for G(.) » estimation of intervals [glP,guP] with from histogram large difference due to low probability density for small g(.), but critical for failure ▪ both-sided ▪ left-sided w.r.t. lower bound gl g(.) 0 10 10 moderate difference due to high probability density at upper bounds interval result is conservative differences controlled by distribution of G(.)

7. Relationship between Results 3 Engineering comparison Probabilistic approximation Interval analysis using estimated moments of G(.) · · » Chebyshev’s inequality with 10 0 10 g(.) interval result shifted towards failure domain, even more conservative than Chebyshev interval analysis Chebyshev interval result reflects tendency of the distribution of G(.) to left-skewness histogram for G(.) for uniform Xi for right-skewed distribution of G(.), Chebychev‘s inequality may lead to the more conservative result 10 10 g(.)

8. Interval or moments ? 3 Engineering comparison General remarks interval analysis heads for the extreme events, whilst a probabilistic analysis yields probabilities for events · for a defined confidence level , interval analysis is more conservative and independent of distributions of the Xi · conservatism of interval analysis is comparable to Chebyshev‘s inequality · difference between interval results and probabilistic results is controlled by the distribution of the response · for a defined confidence level, interval bounds maybe easier to specify or to control than moments · interval analysis can be helpful » to identify low-probability-but-high-consequence events for risk analysis » in case of sensitivity of Pfw.r.t. distribution assumption and very vague information for this assumption » if the first 2 moments cannot be identified with sufficient confidence What to chose in “intermediate” cases ?

9. Information content 4 Information-based comparison Idea compare interval representation and moment representation of uncertainty by means of information content: Which representation tells us more ? · assume that a variable X is represented alternatively (i) by the first two moments μX and σX2 (ii) by an interval [xl, xu] for a given confidence · apply maximum entropy principle to both representations; calculate the least information of the representation without making any additional assumptions · chose the more informative representation; exploit available information to maximum extent (not in contradiction with maximum entropy principle) · Relating intervals and moments analog to the concept of confidence intervals ·

10. Entropy-based comparison 4 Information-based comparison Shannon‘s entropy continuous entropy · » modification for comparison (ease of derivation) Interval representation maximum entropy principle · uniform distribution relating to moments ·

11. Entropy-based comparison 4 Information-based comparison Moment representation maximum entropy principle · normal distribution Comparison of representations check whether under the assumptions made · for k > 2, the moment representation is more informative (ie, for >95% confidence) for k ≤ 2, the interval representation is more informative (ie, for <95% confidence)

12. 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 is more informative if confidence of >95% is needed · for skewed distributions, moment representation is already more informative for smaller confidence · Remark 1: fuzzy sets nuanced consideration of a nested set of intervals · Remark 2: imprecise probabilities enable “intermediate” modeling between interval and cdf · useful if probabilistic models are partly applicable · consider a set of probabilistic models (eg interval parameters) · worst case consideration in terms of probability (bounds) ·