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**Fuzzy arithmetic in risk analysis**

Scott Ferson Applied Biomathematics

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**Fuzzy numbers Fuzzy set that’s unimodal and reaches 1**

Nested stack of intervals

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Fuzzy addition 1 A B A+B 0.5 2 4 6 8 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.

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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

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**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)

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**How to get fuzzy inputs Subjective assignments Objective consensus**

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

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**Subjective assignments**

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

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**Objective consensus [1000, 3000] [2000, 2400] [500, 2500] [800, 4000]**

[1900, 2300] 2000 4000 0.5 1 Possibility

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**Measurement error 46.8 0.3 [46.5, 46.8, 47.1] [12.32]**

46.7 46.9 47.1 1 Possibility [12.32] [12.315, 12.32, ] 12.31 12.32 12.33 1 Possibility

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**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

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Examples (Donald 2003)

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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.

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**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 theoretical minimum maximum 1 Possibility minimum observed maximum observed

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**Robustness X de X fe h + g 1 2 3 4 5 6 7 1**

1 2 3 4 5 6 7 X Possibility Triangular fuzzy numbers are robust characterizations -20 -10 10 20 30 40 1 Xde/(h+g)fe Possibility de h + g X fe 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]

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**Distributional results**

Tails describe possible extremes More comprehensive than intervals Full distribution of various magnitudes

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**Comparison Probability theory Possibility theory Axioms 0 P() 1**

P(AB) = P(A) + P(B) whenever AB= Convolution C(z) = A(x) B(y) Possibility theory Axioms () = 0 () = 1 (A) (B) whenever AB Convolution C(z) = V A(x) B(y) v z=x+y z=x+y

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**Max-min convolutions A + B A = 1 = 0.3 A = 2 = 0.7 A = 3 = 1.0**

= 0.6 A = 5 = 0.4 A + B B = 1 = 0.2 A+B = 3 = 0.2 A+B = 4 A+B = 5 A+B = 6 = 0.3 = 0.7 = 0.8 = 0.6 A+B = 7 = 0.4 = 1.0 A+B = 8 A+B = 9 A+B = 2 = 0.2 B = 2 = 0.8 B = 3 = 1.0 B = 4 = 0.2

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Result of convolution 1 2 3 4 5 A B 8 A+B 6 If the inputs are fuzzy numbers (unimodal, reach 1), then possibilistic convolution is the same as level-wise interval arithmetic (Kaufmann and Gupta)

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**Probability Possibility X Y X+X Y+Y X+…+X Y+…+Y 1 1 0.5 0.5 2 4 6 8 10**

2 4 6 8 10 2 4 6 8 10 1 1 X+X Y+Y Probability Possibility 0.5 0.5 2 4 6 8 10 2 4 6 8 10 0.4 1 X+…+X 0.2 0.5 Y+…+Y 2 4 6 8 10 2 4 6 8 10

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**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, ]

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**Data needs Worst case Interval analysis Fuzzy arithmetic Monte Carlo**

extreme values ranges ranges or distributions distributions and dependencies

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Backcalculations Deconvolutions in fuzzy arithmetic are completely straightforward level-wise generalizations of interval deconvolutions Easy, fast When impossible, yields no answer

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**Software FuziCalc Fuzzy Arithmetic C++ Library Cosmet (Phaser)**

(Windows 3.1) FuziWare, Fuzzy Arithmetic C++ Library (C code) anonymous ftp to mathct.dipmat.unict.it and get \fuzzy\fznum*.* Cosmet (Phaser) (DOS, soon for Windows) Risk Calc (Windows) ;

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Risk analysis example

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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 = [ , , ] m Hydraulic gradient i = [0.0003, , ] m m-1 Hydraulic conductivity K = [ 300, , ] m yr-1 Effective soil porosity n = [ , , ] Soil bulk density BD = [ 1500, , ] kg m-3 Fraction of organic carbon in soil foc = [0.0001, , ] Octanol-water partion coefficient Koc = [ , , ] m3 kg-1

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**Time until contamination**

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**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

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**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

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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 Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3-28.

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Applications Bardossy, A., I. Bogardi and L. Duckstein 1991 Fuzzy set and probabilistic techniques for health-risk analysis. Applied Mathematics and Computation 45: 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 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 Millstein, J.A Propagation of measurement errors in pesticide application computations. International Journal of Pest Management 40: Simulating extremes in pesticide misapplication from backpack sprayers. 41:

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