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Slide 2 How do we classify uncertainties? What are their sources? Lack of knowledge vs. variability. What type of safety measures do we take? Design, manufacturing, operations & post- mortems Living with uncertainties vs. changing them How do we represent random variables? Probability distributions and moments Uncertainty and Safety Measures Slide 3 Reading assignment S-K Choi, RV Grandhi, and RA Canfield, Reliability-based structural design, Springer 2007. Available on-line from UF library http://www.springerlink.com/content/w62672/#section=3200 07&page=1 http://www.springerlink.com/content/w62672/#section=3200 07&page=1 Source: www.library.veryhelpful.co.uk/ Page11.htm Slide 4 Classification of uncertainties Aleatory uncertainty: Inherent variability Example: What does regular unleaded cost in Gainesville today? Epistemic uncertainty Lack of knowledge Example: What will be the average cost of regular unleaded January 1, 2014? Distinction is not absolute Knowledge often reduces variability Example: Gas station A averages 5 cents more than city average while Gas station B 2 cents less. Scatter reduced when measured from station average! Source: http://www.ucan.org/News/UnionTrib /http://www.ucan.org/News/UnionTrib / Slide 5 A slightly different uncertainty classification. British Airways 737-400 Distinction between Acknowledged and Unacknowledged errors Type of uncertainty DefinitionCausesReduction measures ErrorDeparture of average from model Simulation errors, construction errors Testing and model refinement VariabilityDeparture of individual sample from average Variability in material properties, construction tolerances Tighter tolerances, quality control Slide 6 Modeling and Simulation. Slide 7 Error modeling Slide 8 Safety measures Design: Conservative loads and material properties, building block and certification tests. Manufacture: Quality control. Operation: Licensing of operators, maintenance and inspections Post-mortem: Accident investigations Slide 9 Airlines invest in maintenance and inspections. FAA invests in certification of aircraft & pilots. NTSB, FAA and NASA fund accident investigations. Boeing invests in higher fidelity simulations and high accuracy manufacturing and testing. The federal government (e.g. NASA) invests in developing more accurate models and measurement techniques. Many players reduce uncertainty in aircraft. Slide 10 Representation of uncertainty Random variables: Variables that can take multiple values with probability assigned to each value Representation of random variables Probability distribution function (PDF) Cumulative distribution function (CDF) Moments: Mean, variance, standard deviation, coefficient of variance (COV) Slide 11 Probability density function (PDF) If the variable is discrete, the probabilities of each value is the probability mass function. For example, with a single die, toss, the probability of getting 6 is 1/6.If you toss a pair dice the probability of getting twelve (two sixes) is 1/36, while the probability of getting 3 is 1/18. The PDF is for continuous variables. Its integral over a range is the probability of being in that range. Slide 12 Histograms Probability density functions have to be inferred from finite samples. First step is histogram. Histogram divide samples to finite number of ranges and show how many samples in each range (box) Histograms below generated from normal distribution with 50 and 500,000 samples. Slide 13 Number of boxes Slide 14 Histograms and PDF How do you estimate the PDF from a histogram? Only need to scale. Slide 15 Cumulative distribution function Integral of PDF Experimental CDF from 500 samples shown in blue, compares well to exact CDF for normal distribution. Slide 16 Probability plot A more powerful way to compare data to a possible CDF is via a probability plot Slide 17 Moments Mean Variance Standard deviation Coefficient of variation Skewness Slide 18 problems 1.List at least six safety measures or uncertainty reduction mechanisms used to reduce highway fatalities of automobile drivers. 2.Give examples of aleatory and epistemic uncertainty faced by car designers who want to ensure the safety of drivers. 3. Let x be a standard normal variable N(0,1). Calculate the mean and standard deviation of sin(x) Source: Smithsonian Institution Number: 2004-57325