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 assignmentS-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
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, 2014?
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/
The main classification of uncertainties is into aleatory and
epistemic uncertainties. Aleatory uncertainties are due to inherent
variability. The example here is of gas prices on a given day in
the city of Gainesville. Somebody doing a survey of gas stations
may summarize it by noting that the average is $2.25 a gallon for
regular, with a standard deviation of 8 cents. When dealing with
engineering systems, aleatory uncertainty is due to variability in
manufacturing and material properties of nominally identical parts,
or variability in operating conditions. It is sometimes called
irreducible uncertainty because additional knowledge does not
reduce it, but manufacturing variability, for example, can be
reduced by investment in more stringent tolerances. Even for the
gas price example, we may reduce the variability by learning which
stations are the cheapest.Epistemic uncertainty reflects lack of
knowledge. An example is predicting the average price of gas a year
ahead. In engineering applications we use analytical and numerical
models to estimate response and performance, and the epistemic
uncertainty mostly reflects the limitations of the models.3A
slightly differentuncertainty classification.
British Airways 737-400Distinction between Acknowledged and
Unacknowledged errorsType of uncertaintyDefinitionCausesReduction
measuresErrorDeparture of average from modelSimulation errors,
construction errorsTesting and model refinementVariabilityDeparture
of individual sample from averageVariability in material
properties, construction tolerancesTighter tolerances, quality
control
4For this course, we assume that our key epistemic uncertainty
is due to errors, and so the main distinction is between error and
variability. Furthermore, since we calculate uncertainty for
engineering systems, we will usually deal with a population of
nominally identical parts or vehicles. In such setting it is useful
to make the distinction between error and variability based on
which apply to the entire population and which apply to the
individuals.Specifically we consider errors the departure of the
average from the model. For example, the population may include 500
737-400 airliners. Our model predicts that they have a range of
3,000 miles, but they average only 2,900. The 100 mile departure is
due to errors in our model that predicts range, or errors in the
input to the model (for example, we have the wrong aircraft
weight).Variability is the departure of the population from the
average. In the case of the airliners it may reflect variability in
the efficiency of their engines due to manufacturing or operating
conditions.Errors are often divided into acknowledged and
unacknowledged errors. Acknowledged errors are often reduced by
refining a model. Unacknowledged ones may be uncovered an corrected
by tests. Variability may be reduced by tighter manufacturing
tolerance or quality control.Modeling and Simulation.
The epistemic uncertainties that we are mostly interested in are
due to our attempts to simulate reality by a computerized model.
This figure, originally from the Society of Computer simulation)
but borrowed directly from Oberkampf et al., illustrates the
processes responsible for simulation errors.The first step is
modeling reality with a conceptual model, such as Newton laws or
Einstein laws of motion to describe the trajectory of a rocket. The
trajectory of the rocket will then require analysis leading to a
set of differential equations that describe the motion. The process
of model qualification requires checks to ensure that the model is
appropriate. For example, if we model the rocket as a point mass,
we will need some checks that its finite size is not likely to have
a large effects on the solution. The next step is to write or
select software to solve the conceptual model, e.g., the
differential equations. The process of checking the fidelity of the
software for providing the solution is called verification. It is
typically done by comparing with analytical solutions or solutions
obtained by other software.Verification only ensures that the
software can provide an accurate solution to the conceptual model.
Comparison with reality, via physical tests, checks the entire
process and is called model validation.
Oberkmapf et al. Error and uncertainty in modeling and
simulation, Reliability Engineering and System Safety, 75, 333-357,
2002
5Error modeling
Having errors in simulations is acceptable when the magnitude of
the errors can be estimated, so that decisions based on the
simulations take them into account. For example, if our errors in
predicting the trajectory of a rocket are excessive, we may
compensate by installing a control system that will measure errors
and correct the trajectory accordingly.The processes of
qualification, verification and validation often provide error
estimates. For example, the validation test may have revealed 3%
difference, and that may have been judged acceptable. Personal or
group experience may provide additional information on error
magnitude. For example, an analyst with experience in trajectory
calculation, may have an idea how well a given model may
approximate reality. However, both sources of error estimation most
often are expressed as bounds on the error rather than more
detailed assessment.We often settle on numerical models that are
coarse or simplified because we need to analyze them many times,
for example for optimization. 6Safety measures Design: Conservative
loads and material properties, building block and certification
tests.
Manufacture: Quality control.
Operation: Licensing of operators, maintenance and
inspections
Post-mortem: Accident investigations
To counter the uncertainties we have a host of safety measures.
At the design stage we add safety factors. For example in aircraft
design, structures need to be designed to withstand loads that are
50% higher than the highest loads expected in service. Variability
in material properties, are compensated for by taking conservative
properties. For example, aircraft component that are not protected
by redundancy are designed for by using A-basis properties. These
have to be lower than 99% of the distrribution of material
strength, with 95% confidence.
Manufacturing errors are compensated in part by quality control
processes, and once a product goes into service manufacturing
glitches and ageing effects are caught by inspections and
maintenance. Operators and maintenance workers are often licensed
by governmental agencies in order to reduce errors.
Finally, failures do happen, and when they are fatal they are
often followed by accident investigations. These help limit the
effect of systematic design or operation errors to a small number
of cases, by leading to recalls or other preventive measures.7
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 and testing. The federal government (e.g. NASA)
invests in developing more accurate models and measurement
techniques.
Many players reduce uncertainty in aircraft.
In general, safety measures often act by reducing uncertainty
about the performance of a product. For aircraft, this uncertainty
reduction is undertaken by a large number of players.
Federal agencies like NASA and the department of defense conduct
or support research into developing more accurate models and
measurement techniques. This are often implemented and refined
further by engineering software developers, like NASTRAN.Aircraft
manufacturers invest in high fidelity simulations, high accuracy
manufacturing, and in rigorous testing of aircraft parts. This
reduces both epistemic and aleatory uncertainties. The FAA oversees
this process via certification.Airlines invest in maintenance and
inspections to reduce uncertainties about the health of their
airplanes, and in pilot training to reduce uncertainty about
operations. The FAA helps airlines resist temptation to cut corners
by overseeing and regulating the process.Finally, accident
investigations reveal the cause of failure and often expose errors
in simulations, design, maintenance and operation. These errors are
then eliminated or reduced by recommended
correctives.8Representation 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 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 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 histogram.Histogram 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.
14Probability plotA more powerful way to compare data to a
possible CDF is via a probability plot
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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. 16
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
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