Slide 1

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-mortemsLiving with uncertainties vs. changing them

How do we represent random variables?Probability distributions and moments Uncertainty and Safety Measures

Reading assignmentOberkmapf et al. Error and uncertainty in modeling and simulation, Reliability Engineering and System Safety, 75, 333-357, 2002

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=320007&page=1

Source: www.library.veryhelpful.co.uk/ Page11.htm

Modeling uncertainty (Oberkampf et al.).

Classification of uncertaintiesAleatory uncertainty: Inherent variabilityExample: What does regular unleaded cost in Gainesville today?

Epistemic uncertainty Lack of knowledgeExample: What will be the average cost of regular unleaded January 1, 2013?

Distinction is not absoluteKnowledge often reduces variabilityExample: 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/

A slightly differentuncertainty classification.

British Airways 737-400Distinction between Acknowledged and Unacknowledged errorsType of uncertaintyDefinitionCausesReduction measuresErrorDeparture of average from modelSimulation errors, construction errors, Testing and model refinementVariabilityDeparture of individual sample from averageVariability in material properties, construction tolerancesTighter tolerances, quality control

5Safety measures Design: Conservative loads and material properties, accurate models, certification of design

Manufacture: Quality control, oversight by regulatory agency

Operation: Licensing of operators, maintenance and inspections

Post-mortem: Accident investigations

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. The federal government (e.g. NASA) invests in developing more accurate models and measurement techniques.

Many players reduce uncertainty in aircraft.

Representation of uncertaintyRandom variables: Variables that can takemultiple values with probability assigned toeach value

Representation of random variablesProbability distribution function (PDF)Cumulative distribution function (CDF)Moments: Mean, variance, standard deviation, coefficient of variance (COV)

Probability density functionProbability density function is for continuous variables that the probability of a single value is zero. For example, with the function rand, the probability of getting exactly 0.5 is zero.If the variable is discrete, it is like concentrated masses. For example, when you toss a single die, the probability of getting 6 is 1/6; so is the probability of getting 3.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.

HistogramsProbability density functions have to be inferred from finite samples. First step is histogram.z=randn(1,50)+10; hist(z,8); z=randn(1,500000)+10; hist(z,8)

Number of boxesSample of 500 from randn

Histograms and PDFHow do you estimate the PDF from a histogram?

SOURCE: http://schools.sd68.bc.ca/ed611/akerley/question.jpg

Cumulative distribution functionIntegral of PDF

X = [-3:0.1:3];p=normcdf(x,0,1)plot(x,p)

MomentsMean

Variance

Standard deviation

Coefficient of variationSkewness

problemsList at least six safety measures or uncertainty reduction mechanisms used to reduce highway fatalities of automobile drivers.

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 InstitutionNumber: 2004-57325