Uncertainty and Safety Measures How do we classify

Uncertainty and Safety Measures 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

Reading assignment Oberkmapf 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. Chapter 1 and Section 2. 1. Available on-line from UF library http: //www. springerlink. com/content/w 62672/#section=3200 07&page=1 Source: www. library. veryhelpful. co. uk/ Page 11. htm

Modeling uncertainty (Oberkampf et al. ).

Classification of uncertainties Aleatory uncertainty: Inherent variability – Example: What does regular unleaded cost in Gainesville today? Epistemic uncertainty Lack of knowledge Source: http: //www. ucan. org/News/Union. Trib/ – Example: What will be the average cost of regular unleaded next January 1? 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!

British Airways 737 -400 A slightly different uncertainty classification . Distinction between Acknowledged and Unacknowledged errors

Safety 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

Many players invest to reduce uncertainty in aircraft structures. The federal government (e. g. NASA) develops more accurate models and measurement techniques. A or E? Boeing performs higher fidelity simulations and high accuracy manufacturing. A or E? Airlines invest in maintenance and inspections. A or E? FAA certifies aircraft & pilots. A or E? NTSB, FAA and NASA fund accident investigations.

roblems uncertainty 1. List at least six safety measures or uncertainty reduction mechanisms used to reduce highway fatalities of automobile drivers. Source: Smithsonian Institution Number: 2004 -57325 2. Give examples of aleatory and epistemic uncertainty faced by car designers who want to ensure the safety of drivers.

Representation of uncertainty Random variables: Variables that can take multiple values with probability assigned to each value Representation of random variables – Probability density function (PDF) – Cumulative distribution function (CDF) – Moments: Mean, variance, standard deviation, coefficient of variance (COV)

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 of 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.

Histograms • Probability density functions have to be inferred from finite samples. First step is a histogram. • Histograms divide samples to finite number of ranges and show many samples in each range (box) • Histograms below generated from normal distribution with 50 and 500, 000 samples.

Number of boxes •

Histograms and PDF How do you estimate the PDF from a histogram? Only need to scale.

Cumulative distribution function Integral of PDF Experimental CDF from 500 samples shown in blue, compares well to exact CDF for normal distribution.

Problems CDF • Our random variable is the number seen when we roll one die. What is the CDF of 2? • Our random variable is the sum on a pair of dice. What is the CDF of 2? Of 13?

Probability plot • A more powerful way to compare data to a possible CDF is via a probability plot (500 points here)

Moments • Mean • Variance • Standard deviation • Coefficient of variation • Skewness