Transcript Slide 1

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 & postmortems
– 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. Available on-line from UF library
http://www.springerlink.com/content/w62672/#section=3200
07&page=1
Source: www.library.veryhelpful.co.uk/ Page11.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/UnionTrib/
– Example: What will be the average cost of regular unleaded
January 1, 2013?
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
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 reduce uncertainty in
aircraft.
The federal government (e.g. NASA) invests in
developing more accurate models and measurement
techniques.
Boeing invests in higher fidelity simulations and
high accuracy manufacturing.
Airlines invest in maintenance and inspections.
FAA invests in certification of aircraft & pilots.
NTSB, FAA and NASA fund accident investigations.
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
• Probability density function is for continuous variables
that the probability of a single value is zero. For
example, with the function rand, the probability of getting
exactly 0.5 is zero.
• If the variable is discrete, it is like concentrated masses.
For example, when you toss a single die, the probability
of getting 6 is 1/6; so is the probability of getting 3.
• 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.
Histograms
• Probability density functions have to be inferred
from finite samples. First step is histogram.
z=randn(1,50)+10; hist(z,8);
z=randn(1,500000)+10; hist(z,8)
5
14
2
x 10
1.8
12
1.6
10
1.4
1.2
8
1
6
0.8
4
0.6
0.4
2
0.2
0
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
0
5
6
7
8
9
10
11
12
13
14
15
Number of boxes
• Sample of 500 from randn
140
20
18
120
16
100
14
12
80
10
8
60
6
40
4
2
20
0
0
6
7
8
9
10
11
12
13
6
7
8
9
10
11
12
13
Histograms and PDF
How do you estimate
the PDF from a histogram?
SOURCE:
http://schools.sd68.bc.ca/ed
611/akerley/question.jpg
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
0.9
X = [-3:0.1:3];
p=normcdf(x,0,1)
plot(x,p)
0.8
0.7
CDF
0.6
0.5
0.4
0.3
0.2
0.1
0
-3
-2
-1
0
x
1
2
3
Moments
• Mean
 ( X )   xf ( x)dx  E[ X ]
• Variance Var ( X )   ( x   ) 2 f ( x)dx
• Standard deviation
  Var ( X )
2


 X   
• Coefficient of variation COV   E 
 

   


• Skewness
 X   3 
E 
 

 

problems
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.
3. Let x be a standard normal variable N(0,1).
Calculate the mean and standard deviation of
sin(x)