How do we classify uncertainties? What are their sources? Lack of knowledge vs. variability.

1. Uncertainty and Uncertainty reduction Measures How do we classify uncertainties? What are their sources? • Lack of knowledge vs. variability. What type of measures do we take to reduce uncertainty? • Design, manufacturing, operations & post-mortems • Living with uncertainties vs. changing them How do we represent random variables? • Probability distributions and moments

2. 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/

3. A slightly differentuncertainty classification British Airways 737-400 . Distinction between Acknowledged and Unacknowledged errors

4. Modeling and Simulation .

5. Error modeling • Model qualification, verification, and validation often provide estimates of the errors associated with the use of simulation. • Experience in modeling similar problems may provide additional guidance. • The most common model of the errors is simple bounds. For example . • We often settle for larger errors than possible because of computational costs or analysis complexity.

6. Uncertainty reduction measures Design: Refined simulation models, building block tests. Aleatory or epistemic? Manufacture: Quality control. A or E? Operation: Licensing of operators, maintenance and inspections. A or E? Post-mortem: Accident investigations. A or E? Living with uncertainties by using safety factors

7. 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)

8. 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.

9. 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 how many samples in each range (box) • Histograms below generated from normal distribution with 50 and 500,000 samples.

10. Number of boxes • Each time we sample, histogram will be different. • Standard deviation (from sample to sample) of the height n of a box is approximately . Keep that below change in n from one box to next. • Histograms below were generated with 5,000 samples from normal distribution. • With 8 boxes s.d. relatively small (~25) but picture is coarse. With 20 boxes it’s about right. With 50, s.d. is too high (~10) relative to change from one box to next.

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

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

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

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

15. Questions • 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?