Fuzzy arithmetic is not aconservative way to do risk analysis

Scott Ferson and Lev Ginzburg

Applied Biomathematics

www.ramas.com, scott@ramas.com

AbstractThe 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 numbers– Fuzzy sets of the real line

– Unimodal

– Reach possibility level one

• Fuzzy arithmetic– Interval arithmetic at each possibility level

2 3 4 5 6 7 80

1

Pos

sibi

lity

Level-wise interval arithmetic

a0 1 2 3 4 5 6

0

1

b a+b0

1

4 6 8 10 122 3 4 5 6 7 80

1

Pos

sibi

lity

+ =

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?

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]

00 1 2 3 4 5 6 7

1

X

Pos

sibi

lity X

0 2 4 6a

Pos

sibi

lity

0

1

a

-20 -10 0b

Pos

sibi

lity

0

1

b

Robustness of the answer

Different choices for the fuzzy number X all yield very similar distributions for aX + b

-20 -10 0 10 20 30 400

0.5

1

aX + b

aX + b

Pos

sibi

lity

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 1P(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

F encloses P

02 4 6 8 10

1

Pro

babi

lity

0

Pos

sibi

lity

Prob(X > x)

Poss(X > x)1

x

Prob(X < x)

Poss(X < x)

02 4 6 8 10

1

Pro

babi

lity

0

1

Pos

sibi

lity

x

02 4 6 8 10

1

PPro

b. d

ensi

ty

F

0

1

Pos

sibi

lity

x

The lazy risk analyst conjecture

If F and G enclose P and Q resp., F+G encloses P+Q,

whereF, G are fuzzy numbers,P, Q are probability distributions,F+G is obtained by fuzzy arithmetic, andP+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)

Counterexample

0 10 200

0.5

1

CC

DF

, Pos

sibi

lity

2 3 4 5 6 7 8 9 100

1

CC

DF

, Pos

sibi

lity

A, B

A+B

Distributions (in red) for the sum A+B under different correlations and dependencies are not enclosed by the (blue) sum of fuzzy numbers

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.

A and B are identically distributed; their distribution is in red above (they are not independent)

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