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Dependent Possibilistic Arithmetic Using Copulas

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Dependent Possibilistic Arithmetic Using Copulas Proceedings of Machine Learning Research 147:169–179, 2021 ISIPTA 2021 Dependent Possibilistic Arithmetic Using Copulas Ander Gray [email protected] Institute for Risk and Uncertainty, University of Liverpool, UK Dominik Hose [email protected] Institute of Engineering and Computational Mechanics, University of Stuttgart, Germany Marco De Angelis [email protected] Institute for Risk and Uncertainty, University of Liverpool, UK Michael Hanss [email protected] Institute of Engineering and Computational Mechanics, University of Stuttgart, Germany Scott Ferson [email protected] Institute for Risk and Uncertainty, University of Liverpool, UK Abstract the form of combination operations for constructing multi- We describe two functions on possibility distribu- variate possibility distributions from univariate marginals, tions which allow one to compute binary operations allowing one to make such constructions with stochastic with dependence either specified by a copula or par- independence and unknown interaction (Fréchet). It has tially defined by an imprecise copula. We use the fact been shown in [12] that when these combination methods that possibility distributions are consonant belief func- are used with the extension principle, probability measures tions to aggregate two possibility distributions into with the specified dependence are correctly propagated. a bivariate belief function using a version of Sklar’s Probability box arithmetic, or probability bounds analy- theorem for minitive belief functions, i.e. necessity sis, is based on three convolutions which were originally in- measures. The results generalise previously published troduced by Schweizer and Sklar [21] as triangle functions, independent and Fréchet methods, allowing for any stochastic dependence to be specified in the form of a or solutions to the triangle inequality in probabilistic metric (imprecise) copula. This new method produces tighter spaces. Williamson and Downs [24] describes how these extensions than previous methods when a precise cop- convolutions may be used in an arithmetic of random vari- ula is used. These latest additions to possibilistic arith- ables with a partially defined dependence structure, given metic give it the same capabilities as p-box arithmetic, as a lower bound of a copula. They also describe a general and provides a basis for a p-box/possibility hybrid numerical method for computing robust outer solutions to arithmetic. This combined arithmetic provides tighter these convolutions, in the form of an upper and lower dis- bounds on the exact upper and lower probabilities than crete approximation of quantile functions. These structures either method alone for the propagation of general were originally named dependency bounds, but are now belief functions. labeled probability boxes, or p-boxes, and have been gener- Keywords: Possibility Theory, P-box, Copulas, Prob- alised to include uncertainties other than dependency errors abilistic Arithmetic, Probability Bounds Analysis, Im- [11]. The three convolutions from [24] are the following. precise Probabilities For a non-decreasing binary operator f : R+ R+ R+, × ! two random variables X and Y with distribution functions 1. Introduction (df) FX and FY and copula CXY : Z Due to its simplicity in formulation and calculus, possibility sCXY ; f (FX ;FY )(z) = dCXY (FX (x);FY (y)); (1) f z theory [9] is a popular model for bounding sets of probabil- f g ity measures. Early in the formulation of the theory, many tCXY ; f (FX ;FY )(z) = sup [CXY (FX (x);FY (y))]; (2) authors [8] argued that Zadeh’s classical min aggregation f (x;y)=z [26], and its implied levelwise interval arithmetic, was suf- d ficient for computing functions of sets of probabilities, and rC ; f (FX ;FY )(z) = inf [CXY (FX (x);FY (y))]; (3) XY f (x;y)=z that it corresponds to a non-interactivity between variables. More recently however [2], it has been shown that such where the set f z = (x;y) x;y R+; f (x;y) < z , and f g f j 2 g a direct application of fuzzy set theory is not consistent where Cd is the dual of a copula: Cd(u;v) = u+v C(u;v). − with probability theory in most circumstances, with modi- The s convolution is a Lebesgue-Stieltjes integral which fications being proposed [12]. These modifications are of gives the df resulting from a binary operation f between © 2021 A. Gray, D. Hose, M.D. Angelis, M. Hanss & S. Ferson. DEPENDENT POSSIBILISTIC ARITHMETIC USING COPULAS two dfs with a known copula. The t and r convolutions Bounds on a probability measure PX on the measurable compute the lower and the upper cdf of a p-box respectively space (R;B) where B is the Borel s-algebra can be ob- when only the copula’s lower bound is known. That is, tained from pX by tCXY ; f and rCXY ; f propagate all copulas more positive than C . For example t and r compute a p-box which NecX (U) PX (U) PX (U) U B; (5) XY W; f W; f ≤ ≤ 8 2 bounds all stochastic dependencies, where W is the lower where P is the possibility measure given by Fréchet bound. tuv; f and ruv; f propagate all dependencies X more positive than independence, that is, all copulas which PX (U) = suppX (x); (6) are positive quadrant dependent [16], where the product x U copula C(u;v) = uv gives stochastic independence. 2 Convolutions (1)-(3) are defined for non-decreasing bi- and where the necessity measure NecX is nary operations, but may be extended to non-increasing op- C erations by first performing an appropriate unary operation NecX (U) = 1 sup(pX (x)) = 1 PX (U ): (7) − x=U − to one of the variables and then evaluating the convolutions 2 with a non-decreasing operator. For example subtraction A probability measure PX and a possibility distribution pX may be performed by negating one of the variables and eval- are said to be consistent [5] if inequality (5) holds, that uating (1)-(3) with sum: i.e. sCX; Y ;+(FX ;F Y ). The copula is if the probability measure is a bounded above by the − − CX; Y may be found from a simple transformation of CX;Y . − possibility measure, and below by the necessity measure. Note that if X and Y are positive quadrant dependent, then This defines a bounded set of probability measures, a credal X and Y will be negative quadrant dependent [16], and C − set (pX ), which are consistent with pX : therefore tuv; and ruv; give p-boxes which bound nega- − − tive quadrant dependence [10]. P-box arithmetic has been C(pX ) = PX : PX (U) PX (U) U B : (8) f ≤ 8 2 g extended to p-boxes defined on R and many of the base binary and unary operations that are required in a program- Alternatively, consistency may be defined in terms of the ming language [11]. level sets of pX . In particular, a probability and a possibility In this contribution we present analogous functions to measure are consistent iff the probabilities of the superlevel a (1)-(3) for possibility distributions, giving a dependent pos- sets, the so-called a-cuts, C = x R : pX (x) > a pX f 2 g sibilistic arithmetic which allows for any stochastic depen- for a [0;1] are bounded from below by 1 a [4]: 2 − dence to be precisely defined as a copula, or imprecisely a PX C 1 a a [0;1]: (9) as a copula’s lower bound. This generalisation has perfect pX ≥ − 8 2 dependence (Zadeh), independence, and Fréchet as special cases. The results of this paper suggest that the propagation The propagation of possibility distributions is usually de- methods of [12] correspond to a t and r convolution in fined in terms of the extension principle. However it has p-box arithmetic (an imprecise propagation of dependen- been argued by several authors [1, 2, 12] that the prop- cies), as opposed to s convolution (a precise dependence). agated possibility distributions are only consistent with We further show how precise copulas may be propagated, probability theory in limited circumstances. Therefore a giving tighter results than when only the lower bounds are variant of the extension principle has been defined in order used. In the context of propagating (imprecise) dependen- to preserve consistency [12]: given N random input vari- ables X ;:::;X with possibility distributions ;:::; cies, the results of this paper bring possibilistic arithmetic 1 N pX1 pXN and a function f : RN R, then the possibility distribution in line with the capabilities of p-box arithmetic, and is the ! motivation of this work. of the output variable Y = f (X1;:::;XN) is given by In the following section we discuss possibility theory in pY (y) = sup J(pX1 (x1);:::;pXN (xN)) (10) the context of imprecise probabilities. N (x ;:::;xN ) R : 1 2 y= f (x1;:::;xN ) 2. Possibility Theory for all y R. Therein, the operator J accounts for the joint 2 dependency structure of the marginal input variables. Hose A possibility distribution is any measurable function pX : and Hanss [12] describe three possible choices: R [0;1] which satisfies the normality condition ! Zadeh: min(a ;:::;a ) = min(a ;:::;a ), suppX (x) = 1: (4) J 1 N 1 N x R 2 Strong Independence: In this paper we consider only continuous p . The mea- SI N X J (a1;:::;aN) = 1 (1 min(a1;:::;aN)) , surability is required for the super and sub level sets of pX − − to be measurable, and for the probability of these sets to Unknown Interaction: UI be well defined; which will be discussed in the following. J (a1;:::;aN) = min(1;N a1;:::;N aN), · · 170 DEPENDENT POSSIBILISTIC ARITHMETIC USING COPULAS with Jmin corresponding to the original extension princi- 3.
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