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. Chapter 1 and Section 2.1.

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.).

When we perform structural optimization with a model of the structure, we typically use software that embodies two stages of moving from reality to a computerized model. It is important to be aware of what is involved in each stage, because it may allow us to estimate the uncertainties in the predictions of the structural response.The first step is model qualification, were we select a mathematical model to represent reality. For example, we may select a 3D model, or a shell model, or a beam model. In each case, it is important that we have some idea of what kind of errors we commit by using such model.The second stage is the numerical evaluation of the conceptual model. Here we mostly need to be aware of discretization error, which can be estimated, for example, by mesh refinement.The developers of the software conduct verification of their code by comparing with analytical solutions, and they rely on experiments for validation of the conceptual model. Again the analyst should be aware of the range of validation experiments, because it is possible to use structural analysis software for extreme conditions that did not get validated properly.3Classification 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 next January 1?

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/

Uncertainties are classified as aleatory or epistemic. Aleatory uncertainties are due to inherent variability. So the price of gas today in Gainesville is a random number that depends on which gas station you go to. I can sample all or part of the gas stations and get an idea of the distribution of uncertainties.

Epistemic uncertainties reflect lack of knowledge, like what will be the average price of gasoline in Gainesville on 1/1/2015. It can also be represented by a proability distribution, but there is some opposition to doing that, because there is only one true value. Knowledge can reduce this uncertainty, for example, an oil analyst may be able to make a more accurate prediction than a layman.

The distinction is not absolute, in that some knowledge reduces variability. So, for example, if I limit myself to Marathon stations, possibly the variability is smaller.4A slightly differentuncertainty classification.

British Airways 737-400Distinction between Acknowledged and Unacknowledged errors

5In terms of structural analysis and design, I prefer the simplified division into error and variability. This is because the epistemic uncertainty that is of interest to us when we optimize a structure is mostly due to the limitations of our analysis model.

When it comes to aircraft structural design, errors affect all the copies of the structure that will be made, while variability will change from one copy to the next.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

Airplanes are safe partly because we compensate for the uncertainties by using conservative loads and material models. However, we mostly make them safe by taking safety measures that reduce the uncertainty.

Before an airplane is put into the market, we do that by improving the models that we use to analyze it, often based on tests. Then we reduce the risk further by certification tests.

At the manufacture level we have quality control by the manufacturer and oversight by the regulator. Similarly, during operations we have maintenance, inspection, training of pilots and maintenance personnel, and licensing by regulators.

Finally, if all fails and there is an accident, we investigate to find the cause, and thereby greatly reduce the chance that the same flaw in design or operations will cause a similar accident in the future.6 Airlines invest in maintenance and inspections. A or E?

FAA certifies aircraft & pilots. A or E?

NTSB, FAA and NASA fund accident investigations.

Boeing performs higher fidelity simulations and high accuracy manufacturing. A or E?

The federal government (e.g. NASA) develops more accurate models and measurement techniques. A or E?

Many players invest to reduce uncertainty in aircraft structures.

The very high safety of aircraft structures reflect the fact that uncertainty is reduced by many players. NASA works on improving analysis models and reducing modeling errors. Aircraft manufacturer invest in high fidelity simulation and quality control in manufacturing. Airlines invest in maintenance and inspection.

FAA invests in certification of aircraft and pilots and together with NTSB in investigating accidents and selecting remedial action.7

problems uncertaintyList 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.

Source: Smithsonian InstitutionNumber: 2004-57325Representation 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)

9Probability 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.

One common way of describing a random variable is by giving the probability for each value it can take. For example, with a single die toss, the random variable may be the number on the top face of the die; the probability of taking 1,2,3,4,5. or 6 is 1/6. If we toss two dice, and the random variable is the sum of the two, then the probabilities of different outcomes are different. For example the probability of 12 is 1/36 (only one combination out of 36) while the probability of getting 11 is 1/18 (two ways 5,6 and 6,5). The function that gives the probability for each possible value is called probability mass function.For continuous variables we have instead a probability density function (PDF), whose integral over any range gives the probability of falling in that range. For example, the figure shows the PDF of a normal (Gaussian) distribution. The integral of the PDF over the center region (darker blue) is 0.5 for the top figure and 0.6827 for the bottom one (Figure from Wikipedia). Since the integral over a single point is zero, the probability of taking any given value is zero. For example, If the random variable is the top sustained wind speed of a hurricane hitting Gainesville Florida, the probability of that speed being exactly 85 MPH is zero, while the probability of its being between 84.5 and 85.5 is the integral of the PDF from 84.5 to 85.5.10HistogramsProbability 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.

If the PDF is not known in advance we can get an idea of its shape from a sample by plotting a histogram. The histogram is obtained by dividing the overall range of the sample into a finite number of intervals (usually of equal size) and showing the number of samples in each interval.For example, the Matlab sequence z=randn(1,50)+10; hist(z,8); generates 50 samples from the normal distribution (see lecture on random variable distributions) and then divides the range to eight intervals. The resulting plot on the left, only vaguely resembles a normal distribution. With z=randn(1,500000)+10 (500,00 samples) we get a better resemblance, but we may have benefitted from more intervals (boxes).It is also worth noting that random variables can infinite range, and many common distributions, like the normal distribution do. For those distributions the probability of getting very large or very small values is usually small, so the larger the sample the larger the range of the sample. This is shown in the two figures in that the range of values for 500,000 samples is almost twice as large as the range for 50 samples.

11Number of boxes

12Histograms and PDFHow do you estimate the PDF from a histogram?Only need to scale.

13

Cumulative distribution functionIntegral of PDF

Experimental CDF from 500 samples shown in blue, compares well to exact CDF for normal distribution.

14Problems CDFOur 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 plotA more powerful way to compare data to a possible CDF is via a probability plot (500 points here)

16

MomentsMean

Variance

Standard deviation

Coefficient of variationSkewness

A compact way to give information about samples and distributions is to provide some of their moments. The first moment is the mean, and it is commonly denoted by the letter . The operation of integrating a function of a random variable times its PDF is also called calculating the expectation of the function. So the mean is also the expected value of the random variable.The square deviation of the random variable from its mean, or its second moment is called the variance. We normally use the square root of the variance, known as the standard deviation. Alternatively, we use the coefficient of variation, which is the ratio of the standard deviation to the mean.Higher order moments are also used, especially normalized central moments. They are centralized by taking them around the mean, and normalized by the standard deviation. The third normalized central moment is called skewness and it measures the asymmetry in the distribution. 17