Fuzzy arithmetic in risk analysis

Scott Ferson

Applied Biomathematics

scott@ramas.com

Fuzzy numbers

• Fuzzy set that’s unimodal and reaches 1

• Nested stack of intervals

1

0.5

0

no YESno

Fuzzy addition

• Subtraction, multiplication, division, minimum, maximum, exponentiation, logarithms, etc. are also defined.

• If distributions are multimodal, possibility theory (rather than just simple fuzzy arithmetic) is required.

0 2 4 6 8

1

0.5

0

A A+BB

Kinds of numbers

• Scalars are well known or mathematically defined integers and real numbers

• Intervals are numbers whose values are not know with certainty but about which bounds can be established

• Fuzzy numbers are uncertain numbers for which, in addition to knowing a range of possible values, one can say that some values are more plausible, or ‘more possible’ than others

What is possibility?

• No single definition• Depends on your applications• Many definitions could be used

– Subjective assessments– Social consensus – Measurement error– Upper betting rates (Giles)– Extra-observational ranges (Gaines)

How to get fuzzy inputs

• Subjective assignments– Make them up from highest, lowest and best-guess

estimates

• Objective consensus– Stack up consistent interval estimates or bridge

inconsistent ones

• Measurement error– Infer from measurement protocols

• Other special ways

Subjective assignments

• Triangular fuzzy numbers, e.g., [1,2,3]

• Trapezoidal fuzzy numbers, e.g., [1,2,3,4]

0 1 2 3 40

0.5

1

0 1 2 3 4 50

0.5

1

Pos

sibi

lity

Objective consensus

[1000, 3000]

[2000, 2400]

[500, 2500]

[800, 4000]

[1900, 2300]P

ossi

bili

ty

0 2000 40000

0.5

1

Measurement error

46.8 0.3

[46.5, 46.8, 47.1]

[12.32]

[12.315, 12.32, 12.325]

46.5 46.7 46.9 47.10

1

Pos

sibi

lity

12.31 12.32 12.330

1

Pos

sibi

lity

When the data are inconsistent

• Find and emphasize regions of consonance– Let possibility flow to intersections– Doesn’t work for totally disjoint data sets– May have counterintuitive features

• Use (agglomerative hierarchical) clustering– Single linkage, complete linkage, UPGMA, etc.– Can define ‘similarity’ between intervals in

various ways– Even works for totally disjoint data sets

Examples

(Donald 2003)

Betting definition

• By asserting a A, you agree to pay $1 if A is false. • If the probability of A is P, then a Bayesian rational

agent should agree to assert A for a fee of $(1-P), and should equally well assert not-A for a fee of $P. Although refusing to bet is not irrational, Bayesians don’t allow this.

• Possibility of A can be measured as the smallest number [0,1], such that, for $, a rational agent will agree to pay $1 if A is found to be false.

• Possibility is thereby an upper bound on probability.

Extra-observational ranges• Theoretical ranges are often very wide• The range between the minimum and maximum

observed values (where the data is) should be modeled by probability theory

• Fuzzy/possibility is about the range within the theoretical range but beyond observations

minimumobserved

maximumobserved

theoreticalminimum

theoreticalmaximum

0

1

Poss

ibil

ity

Robustness

Triangular fuzzy numbers are robust characterizations

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

XP

ossi

bili

ty X

-20-10 0 10 20 30 400

1

Xde/(h+g)fe

Pos

sibi

lity de

h + gX fe

Distributional results

• Tails describe possible extremes

• More comprehensive than intervals

• Full distribution of various magnitudes

Comparison

Probability theoryAxioms

0 P() 1P() = 1P(AB) = P(A) + P(B)

whenever AB=

ConvolutionC(z) = A(x) B(y)

Possibility theoryAxioms

() = 0() = 1(A) (B)

whenever ABConvolution

C(z) = V A(x) B(y)z=x+y z=x+y

v

Max-min convolutions

B = 1 = 0.2

A = 1 = 0.3A + B

B = 2 = 0.8

B = 3 = 1.0

B = 4 = 0.2

A = 2 = 0.7

A = 3 = 1.0

A = 4 = 0.6

A = 5 = 0.4

A+B = 2 = 0.2

A+B = 3 = 0.2

A+B = 4 = 0.2

A+B = 5 = 0.2

A+B = 6 = 0.2

A+B = 3 = 0.3

A+B = 4 = 0.7

A+B = 5 = 0.8

A+B = 6 = 0.6

A+B = 7 = 0.4

A+B = 4 = 0.3

A+B = 5 = 0.7

A+B = 6 = 1.0

A+B = 7 = 0.6

A+B = 8 = 0.4

A+B = 5 = 0.2

A+B = 6 = 0.2

A+B = 7 = 0.2

A+B = 8 = 0.2

A+B = 9 = 0.2

Result of convolution

If the inputs are fuzzy numbers (unimodal, reach 1), then possibilistic convolution is the same as level-wise interval arithmetic (Kaufmann and Gupta)

1 2 3 4 50

1

A

1 2 3 40

1

B

2 80

1

A+B

4 6

0

0.5

1

0 2 4 6 8 10

0

0.5

1

0 2 4 6 8 10

0

0.2

0.4

0 2 4 6 8 10

0

0.5

1

0 2 4 6 8 10

0

0.5

1

0 2 4 6 8 10

0

0.5

1

0 2 4 6 8 10

Pro

babi

lity

Pos

sibi

lity

X

X+X

X+…+X

Y

Y+Y

Y+…+Y

Computational cost

Analysis Operations

Deterministic F

Interval analysis 4F

Fuzzy arithmetic MF

Monte Carlo NF

Second-order Monte Carlo N2F

where M ~ [40,400], and N ~ [1000, 100000]

Data needs

• Worst case

• Interval analysis

• Fuzzy arithmetic

• Monte Carlo

• extreme values

• ranges

• ranges or distributions

• distributions and dependencies

Backcalculations

• Deconvolutions in fuzzy arithmetic are completely straightforward level-wise generalizations of interval deconvolutions

• Easy, fast

• When impossible, yields no answer

Software

• FuziCalc – (Windows 3.1) FuziWare, 800-472-6183

• Fuzzy Arithmetic C++ Library– (C code) anonymous ftp to mathct.dipmat.unict.it and

get \fuzzy\fznum*.*

• Cosmet (Phaser)– (DOS, soon for Windows) acooper@sandia.gov

• Risk Calc – (Windows) 800-735-4350; www.ramas.com

Risk analysis example

exposure =

conctimefillups52 weeks year 1

intakedriving1 day/ (24 hours

60 minutes hour 1) / 1e6 mg kg 1

0 1 2 3 4 0 2 4 6 8 10 12 14 0 2 4

0 0.002 0.004 0.006

conc = [0, 3.2, 4] mg m 3 time = [1, 10, 14] minutes fillup 1 fillups = [0.5, 1, 3] fillups week 1

intake = [22] m3 day 1

48 49 50 51 5220 21 22 23 24

driving = [66 years] [16 years]

Another example

Consider a simple example model of octanol contamination of groundwater due to Lobascio (1993 Uncertainty analysis tools for environmental modeling. ENVIRONews 1:6-10). Its assumptions include one-dimensional constant uniform Darcian flow, homogeneous material properties, linear retardation, no dispersion, and the governing equation T = (n + BD foc Koc ) L / (K i).

Distance from source to receptor L = [ 80, 100, 120] m

Hydraulic gradient i = [0.0003, 0.0005, 0.0008] m m-1

Hydraulic conductivity K = [ 300, 1000, 3000] m yr-1

Effective soil porosity n = [ 0.2, 0.25, 0.35]

Soil bulk density BD = [ 1500, 1650, 1750] kg m-3

Fraction of organic carbon in soil foc = [0.0001, 0.0005, 0.005]

Octanol-water partion coefficient Koc = [ 5, 10, 20] m3 kg-1

Time until contamination

0 100000 200000

1.0

0.5

0

Contaminant plume traveling time (years)

Pos

sibi

lityFull distribution

0 500 1000 1500

1.0

0.5

0

Pos

sibi

lity

Contaminant plume traveling time (years)

Detail of left side

0 20 40 60 80 100

1.0

0.5

0Contaminant plume traveling time (years)

Pos

sibi

lity

Detail of first 100 years

Reasons to use fuzzy arithmetic

• Requires little data• Applicable to all kinds of uncertainty• Fully comprehensive• Fast and easy to compute• Doesn’t require information about correlations• Conservative, but not hyperconservative• In between worst case and probability• Backcalculations easy to solve

Reasons not to use it

• Controversial• Are alpha levels comparable for different variables?• Not optimal when there're a lot of data• Can’t use knowledge of correlations to tighten answers• Not conservative against all possible dependencies• Repeated variables make calculations cumbersome

References

• Dubois, D. and H. Prade 1988 Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, New York.

• Kaufmann, A. and M.M. Gupta 1985 Introduction to Fuzzy Arithmetic: Theory and Applications. Van Nostrand Reinhold, New York.

• Zadeh, L. 1978 Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3-28.

Applications

• Bardossy, A., I. Bogardi and L. Duckstein 1991 Fuzzy set and probabilistic techniques for health-risk analysis. Applied Mathematics and Computation 45:241-268.

• Duckstein, L., A. Bardossy, T. Barry and I. Bogardi 1990 Health risk assessment under uncertainty: a fuzzy risk methodology. Risk-based Decision Making in Water Resources. Y.Y. Haimes and E.Z. Stakhiv (eds.), American Society of Engineers, New York.

• Ferson, S. 1993 Using fuzzy arithmetic in Monte Carlo simulation of fishery populations. Management Strategies for Exploited Fish Populations, T.J. Quinn II (ed.), Alaska Sea Grant College Program, AK-SG-93-02, pp. 595-608.

• Millstein, J.A. 1994. Propagation of measurement errors in pesticide application computations. International Journal of Pest Management 40:149-165. 1995 Simulating extremes in pesticide misapplication from backpack sprayers. 41: 36-45.