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

2. 2 / 12 Michael Beer, Vladik Kreinovich 1 Problem description 4 2 6 5.15... 5.35 measuring devices  d thickness measuring points  010 3050  D [N/mm²] low medium high linguistic assessments  x measurement / observation under dubious conditions plausible range expert assessment / experience  COARSE INFORMATION

3. 3 / 12 Michael Beer, Vladik Kreinovich CLASSIFICATION AND MODELING » reducible uncertainty » property of the analyst » lack of knowledge or perception According to sources aleatory 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 content uncertainty  » probabilistic information traditional and subjective probabilistic models imprecision  set-theoretical models no specific model traditional probabilistic models In view of the purpose of the analysis averaged results, value ranges, worst case, etc. ?  1 Problem description

4. 4 / 12 Michael Beer, Vladik Kreinovich PROBLEM CONTEXT 3 Engineering comparison Structural reliability problem  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. Comparative study assume normal distribution for the variables   performance function » coarse information about the six variables X i Quantification of uncertain variables specification of 2 parameters   further example and detailed discussion » interval bounds x il and x iu interval analysis, range, worst case Type and amount of available information ? Purpose of analysis ? » moments μ and σ 2 probabilistic analysis, response moments, cdf, P f relate interval bounds to moments:

5. 5 / 12 Michael Beer, Vladik Kreinovich INTERPRETATION OF RESULTS Probabilistic analysis  failure may occur in a moderate number of cases Interval analysis  failure may occur magnitude of exceedance of g = 0 rather small, strong exceedance quite unlikely significant exceedance of g = 0 may occur comparable different focus: consider low-probability-but-high-consequence events Given that input information is coarse » known distribution of X General relationship  bounding property for general mapping XY conclusions from interval analysis mostly too conservative » unknown distribution of X probabilistic results may be too optimistic, worst case (which is emphasized in interval analysis) maybe likely 3 Engineering comparison

6. 6 / 12 Michael Beer, Vladik Kreinovich RELATIONSHIP BETWEEN RESULTS Probabilistic analysis  Interval analysis  normal distributions for all variables X i for all X i histogram for G(.) » estimation of intervals [g lP,g uP ] with from histogram differences controlled by distribution of G(.) interval result is conservative g(.) 10 010 ▪ both-sided ▪ left-sided w.r.t. lower bound g l large difference due to low probability density for small g(.), but critical for failure moderate difference due to high probability density at upper bounds 3 Engineering comparison

7. 7 / 12 Michael Beer, Vladik Kreinovich RELATIONSHIP BETWEEN RESULTS Probabilistic approximation  Interval analysis  using estimated moments of G(.) » Chebyshev’s inequality with interval result shifted towards failure domain, even more conservative than Chebyshev interval result reflects tendency of the distribution of G(.) to left-skewness 0g(.) 10 10 interval analysis Chebyshev for right-skewed distribution of G(.), Chebychev‘s inequality may lead to the more conservative result g(.) 10 10 histogram for G(.) for uniform X i 3 Engineering comparison

8. 8 / 12 Michael Beer, Vladik Kreinovich 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 3 Engineering comparison  for a defined confidence level, interval analysis is more conservative and independent of distributions of the X i  difference between interval results and probabilistic results is 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 P f 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 specify or to control than moments What to chose in “intermediate” cases ?

9. 9 / 12 Michael Beer, Vladik Kreinovich INFORMATION CONTENT 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 μ X and σ X 2 (ii) by an interval [x l, x u ] for a given confidence 4 Information-based comparison  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)  analog to the concept of confidence intervals Relating intervals and moments

10. 10 / 12 Michael Beer, Vladik Kreinovich ENTROPY-BASED COMPARISON Shannon‘s entropy  continuous entropy 4 Information-based comparison Interval representation  maximum entropy principle » modification for comparison (ease of derivation) uniform distribution  relating to moments

11. 11 / 12 Michael Beer, Vladik Kreinovich ENTROPY-BASED COMPARISON 4 Information-based comparison Moment representation  maximum entropy principle normal distribution Comparison of representations  check whether for k > 2, the moment representation is more informative (ie, for >95% confidence) under the assumptions made for k ≤ 2, the interval representation is more informative (ie, for <95% confidence)

12. 12 / 12 Michael Beer, Vladik Kreinovich 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 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 sets nuanced consideration of a nested set of intervals  enable “intermediate” modeling between interval and cdf 