Transcript Slide 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 & postmortems – Living with uncertainties vs. changing them How do we represent random variables? – Probability distributions and moments 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/UnionTrib/ – 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! British Airways 737-400 A slightly different uncertainty classification . Type of uncertainty Definition Causes Reduction measures Error Departure of average from model Simulation errors, construction errors Testing and model refinement Variability Departure of individual sample from average Variability in material properties, construction tolerances Tighter tolerances, quality control Distinction between Acknowledged and Unacknowledged errors Modeling and Simulation . 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 ±10%. • We often settle for larger errors than possible because of computational costs or analysis complexity. 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 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) 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 how many samples in each range (box) • Histograms below generated from normal distribution with 50 and 500,000 samples. 5 2 x 10 1.8 14 1.6 12 1.4 10 1.2 1 8 0.8 6 0.6 4 0.4 0.2 2 0 0 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 5 6 7 8 9 10 11 12 13 14 15 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 𝑛. 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. 350 800 1800 300 700 1600 1400 600 1200 500 1000 250 200 400 150 800 300 100 600 200 400 50 100 200 0 -4 -3 -2 -1 0 1 2 3 4 0 -4 -3 -2 -1 0 1 2 3 4 0 -4 -3 -2 -1 0 1 2 3 4 Histograms and PDF How do you estimate the PDF from a histogram? Only need to scale. P robability distribution function f : P(a x a da) f ( x)da Cumulative distribution function x F ( x) P( X x) Integral of PDF f (t )dt 1 Experimental CDF from 500 samples shown in blue, compares well to exact CDF for normal distribution. 0.9 0.8 normal CDF 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -4 -3 -2 -1 0 x 1 2 3 4 Probability plot • A more powerful way to compare data to a possible CDF is via a probability plot (500 points here) Probability plot for Normal distribution 0.9999 0.9995 0.999 0.995 0.99 Probability 0.95 0.9 0.75 0.5 0.25 0.1 0.05 0.01 0.005 0.001 0.0005 0.0001 -4 -3 -2 -1 0 Data 1 2 3 4 Moments • Mean • Variance ( X ) xf ( x)dx E[ X ] 2 Var ( X ) ( x ) 2 f ( x)dx E X • Standard deviation • Coefficient of variation • Skewness X 3 E Var ( X ) COV 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?