Transcript Fuzzy arithmetic is not a conservative way to do risk analysis

Fuzzy arithmetic is not a
conservative way to do risk analysis
Scott Ferson and Lev Ginzburg
Applied Biomathematics
www.ramas.com, [email protected]
Abstract
The three chief disadvantages of Monte Carlo methods are computational burden,
sensitivity to uncertainty about input distribution shapes and the need to assume
correlations among all inputs. Fuzzy arithmetic, which is computationally simple,
robust to moderate changes in the shapes on input distributions and does not
require the analyst to assume particular correlations among inputs, might
therefore be considered a prime alternative calculus for propagating uncertainty
in risk assessments. Theorists suggest that fuzzy measures are upper bounds on
probability measures. By selecting fuzzy inputs that enclose the analogous
probability distributions, one might therefore expect to be able to obtain a
conservative (bounding) analysis more cheaply, conveniently and reliably than is
possible with Monte Carlo methods. With a simple counterexample, however, we
show that fuzzy arithmetic is not conservative to uncertainty about input shapes
or correlations. While it may have uses in specialized analyses, fuzzy arithmetic
may not be appropriate for routine use in risk assessments concerned primarily
with variability and incertitude (measurement error).
Monte Carlo
• Used to propagate uncertainty and variability
through risk assessments
• But you have to specify
– precise input distributions
– particular correlation and dependency assumptions
• If you’re not sure about these, the assessment
could be wrong
Fuzzy arithmetic might be useful
• Distributions don’t have to be precise
• Requires no assumption about correlations
• Fuzzy measures are upper bounds on probability
• Fuzzy arithmetic might be a conservative way to do
risk assessments that is more reliable and less
demanding than Monte Carlo
Fuzzy numbers and their arithmetic
– Fuzzy sets of the real line
– Unimodal
– Reach possibility level one
Possibility
• Fuzzy numbers
1
0
2 3 4 5 6 7 8
• Fuzzy arithmetic
– Interval arithmetic at each possibility level
Level-wise interval arithmetic
1
Possibility
1
1
+
0
2 3 4 5 6 7 8
a
=
0
0 1 2 3 4 5 6
b
0
4 6 8 10 12
a+b
Features of fuzzy arithmetic
• Fully developed arithmetic and logic
– Plus, minus, times, divide, min, max,
– Log, exp, sqrt, abs, powers, and, or, not
– Backcalculation, updating, mixtures, etc.
•
•
•
•
Very fast calculation and convenient software
Very easy to explain
Distributional answers (not just worst case)
Results robust to choice about shape
How does shape of X affect aX+b?
1
Possibility
X
0
0
1
2
0
4
5
6
1
a
0
X
Possibility
Possibility
1
3
2 a
4
6
b
0
-20
-10
b
0
7
a = (d * e) / (h + g)
b = f *e
where
d = [0.3, 1.7, 3]
e = [ 0.4, 1, 1.5]
f = [ 0.8, 6, 10]
g = [ 0.2, 2, 5]
h = [ 0.6, 3, 6]
Robustness of the answer
Different choices for the fuzzy number X all yield
very similar distributions for aX + b
Possibility
1
aX + b
0.5
0
-20
-10
0
10
aX + b
20
30
40
Fuzzy seems to bound probability
• A fuzzy number F is said to “enclose” a probability
distribution P iff
– the left side of F is larger than P(x) for each x,
– the right side of F is larger than 1P(x) for each x
• For every event X < x and x < X, possibility is larger
than than the probability, so it is an upper bound
P
0
2
Poss(X < x)
Prob(X < x)
0
2
4
6
x
8
0
10
4
6
x
1
8
Poss(X > x)
Probability
1
Possibility
Probability
1
Possibility
F
Prob(X > x)
0
2
4
6
x
0
10
1
Possibility
F encloses P
1
Prob. density
1
8
0
10
The lazy risk analyst conjecture
If F and G enclose P and Q resp., F+G encloses P+Q,
where
F, G are fuzzy numbers,
P, Q are probability distributions,
F+G is obtained by fuzzy arithmetic, and
P+Q is obtained by probabilistic convolution
such as Monte Carlo simulation.
It’d be nice
• If the lazy risk analyst conjecture were true, we
could do risk assessments by
– getting fuzzy numbers that enclose each probability
distribution
– using fuzzy arithmetic to obtain results that bound the
probabilistic answer
• Easy to get inputs, easy to get answers
• Results conservative (but not hyperconservative)
A, B
Counterexample
1
0
2 3 4 5 6 7 8 9 10
A and B are identically distributed; their
distribution is in red above (they are not
independent)
Distributions (in red) for the
sum A+B under different
correlations and dependencies
are not enclosed by the (blue)
sum of fuzzy numbers
CCDF, Possibility
CCDF, Possibility
1
A+B
0.5
0
0
10
The red parallelogram
is the tightest region
that encloses all of the
possible distributions
for A+B that could
arise under different
dependencies between
A and B.
20
Conclusion
• Like many ideas that would be really cool, the
lazy risk analyst conjecture is false.
• Fuzzy arithmetic does not seem to allow us to
conveniently and conservatively estimate risks
from bounded probabilities