Dependent Possibilistic Arithmetic Using Copulas

Proceedings of Machine Learning Research 147:169–179, 2021 ISIPTA 2021

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 σCXY , f (FX ,FY )(z) = dCXY (FX (x),FY (y)), (1) f z theory [9] is a popular model for bounding sets of probabil- { } ity measures. Early in the formulation of the theory, many τCXY , 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 ρC , 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 { } { | ∈ } 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 σ 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

two dfs with a known copula. The τ and ρ 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 σ-algebra can be ob- when only the copula’s lower bound is known. That is, tained from πX by

τCXY , f and ρCXY , f propagate all copulas more positive than C . For example τ and ρ compute a p-box which NecX (U) PX (U) ΠX (U) U B, (5) XY W, f W, f ≤ ≤ ∀ ∈ bounds all stochastic dependencies, where W is the lower where Π is the possibility measure given by Fréchet bound. τuv, f and ρuv, f propagate all dependencies X more positive than independence, that is, all copulas which ΠX (U) = supπX (x), (6) are positive quadrant dependent [16], where the product x U copula C(u,v) = uv gives stochastic independence. ∈ 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(πX (x)) = 1 ΠX (U ). (7) − x/U − to one of the variables and then evaluating the convolutions ∈ with a non-decreasing operator. For example subtraction A probability measure PX and a possibility distribution πX 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. σCX, 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 (πX ), which are consistent with πX : therefore τuv, and ρuv, give p-boxes which bound nega- − − tive quadrant dependence [10]. P-box arithmetic has been C(πX ) = PX : PX (U) ΠX (U) U B . (8) { ≤ ∀ ∈ } 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 πX . In particular, a probability and a possibility In this contribution we present analogous functions to measure are consistent iff the probabilities of the superlevel α (1)-(3) for possibility distributions, giving a dependent pos- sets, the so-called α-cuts, C = x R : πX (x) > α πX { ∈ } sibilistic arithmetic which allows for any stochastic depen- for α [0,1] are bounded from below by 1 α [4]: ∈ − dence to be precisely defined as a copula, or imprecisely α PX C 1 α α [0,1]. (9) as a copula’s lower bound. This generalisation has perfect πX ≥ − ∀ ∈ 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 τ and ρ 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 σ 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 variables X ,...,X with possibility distributions ,..., cies, the results of this paper bring possibilistic arithmetic 1 N πX1 πXN 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 πY (y) = sup J(πX1 (x1),...,πXN (xN)) (10) the context of imprecise probabilities. N (x ,...,xN ) R : 1 ∈ y= f (x1,...,xN )

2. Possibility Theory for all y R. Therein, the operator J accounts for the joint ∈ dependency structure of the marginal input variables. Hose A possibility distribution is any measurable function πX : and Hanss [12] describe three possible choices: R [0,1] which satisﬁes the normality condition → Zadeh: min(α ,...,α ) = min(α ,...,α ), supπX (x) = 1. (4) J 1 N 1 N x R ∈ Strong Independence: In this paper we consider only continuous π . The mea- SI N X J (α1,...,αN) = 1 (1 min(α1,...,αN)) , surability is required for the super and sub level sets of πX − − 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 (α1,...,αN) = min(1,N α1,...,N αN), · ·

170 DEPENDENT POSSIBILISTIC ARITHMETICUSING COPULAS

with Jmin corresponding to the original extension princi- 3. Joint Possibilities Using Copulas ple, and maximally correlated inputs, JSI corresponding In this paper, we study dependent binary operations and to inputs which are stochastically independent, and JUI corresponding to inputs which make no assumption about therefore restrict ourselves to bivariate copulas (2-copulas), stochastic dependence (Fréchet). however the generalisation to n-copulas is straightforward. A 2-copula is a function C : [0,1]2 [0,1] with the follow- The possibility measure induced by a possibility distri- → bution may be viewed as a special case of a plausibility ing properties [16]: function [22], likewise the necessity is a special case of 1. Grounded: C(0,v) = C(u,0) = 0, a belief function, and where the α-cuts correspond to the nested focal sets. In particular the belief of an α-cut is 2. Uniform margins: C(u,1) = u; C(1,v) = v, simply 1 α [4]: − 3. 2-increasing: Nec ( α ) = Bel ( α ) = 1 α. (11) C(u2,v2) C(u2,v1) C(u1,v2) +C(u1,v1) 0 X CπX X CπX − − ≥ − for all 0 u1 u2 1 and 0 v1 v2 1. Conversely, not every belief function Bel is conso- ≤ ≤ ≤ ≤ ≤ ≤ X 3 nant and can, therefore, not usually be precisely described Three important 2-copulas are by a possibility measure [7]. That is, if P(BelX ) = PX : { W(u,v) = max(u + v 1,0), BelX (U) PX (U) U B describes the credal set of a − ≤ ∀ ∈ } Ψ(u,v) = uv, belief function BelX (similar to the credal set of a possibility distribution), then it is not generally possible to find M(u,v) = min(u,v), a possibility distribution πX such that P(BelX ) = C(πX ). with W and M being bounds on all 2-copulas: W C M. However, in such situations, it is possible to find a con- ≤ ≤ sonant outer approximation achieving P(Bel ) C(π ) Copulas are mainly used in dependence modelling [14, 16], X ⊆ X via the imprecise probability-to-possibility transform intro- and can be used to construct multivariate dfs given their duced by Hose and Hanss [13]. univariate marginals. This is enabled by a theorem from Given an arbitrarily chosen candidate1 possibility distri- Sklar [23]: bution qX of X, Hose and Hanss show that the possibility Theorem 1 (Sklar’s theorem) Let X and Y be random distribution πX given by variables with joint df H and univariate marginals FX and (x) = ( q ( ) q (x) ) FY . Then there exists a copula C such that for all (x,y) πX sup PX ξ R : X ξ X 2 ∈ PX P(BelX ) { ∈ ≤ } R : ∈ (12) H(x,y) = C(FX (x),FY (y)). (14) = 1 BelX ( ξ R : qX (ξ) > qX (x) ) − { ∈ } qX (x) If FX and FY are continuous, then C is unique; otherwise C = 1 BelX (Cq ), − X is uniquely determined on support F support F . X × Y for x R is consistent with all PX PX , i.e. it provides ∈ ∈ Moreover, X and Y are stochastically independent iff CXY = an outer approximation of PX . Most importantly, it is also Ψ, perfectly positively dependent (perfect) iff CXY = M and plausibility-conform to qX , i.e. from qX (x1) qX (x2) it ≤ perfectly negatively dependent (opposite) iff CXY = W. follows that πX (x1) πX (x2) for all x1,x2 R (but not ≤ ∈ Sklar’s theorem may be applied in reverse, that is a cop- vice versa), and it is maximally specific2. That is, even ula may be constructed from any multivariate df with con- though π does not generally exactly describe P , one may X X tinuous marginals [16]: not find a ‘better’ plausibility-conform possibility distribution that is also an outer approximation. Of course, much Theorem 2 Let H be a bivariate df with continuous 1 1 of the quality of the resulting πX depends on the candidate marginals F and G, with quasi-inverses F− and G− . Then 2 possibility distribution qX , the obvious degree of freedom there exits a copula C such that for all (u,v) [0,1] : in this approach, but Hose and Hanss provide several rea- ∈ 1 1 sonable options for how to choose it. For instance, for the C(u,v) = H(F− (u),G− (v)). (15) outer approximation of belief functions, they suggest the ‘melting transform’ given by the pointwise plausibilities The Gaussian copula is a well known copula family which is defined using (15): q (x) = Pl ( x ) = 1 Bel ( x ), (13) X X X R Φ 1 1 { } − \{ } Cr (u,v) = Φr(Φ− (u),Φ− (v)), (16) for x R. Below, a different (more constructive) way of ∈ finding qX is pursued. where Φr is a zero mean bivariate Gaussian cdf with uni- 1 tary variance and correlation coefficient r, and Φ− is 1. Hose and Hanss call this a (subjective) plausibility function. 2. The concept of specificity is not discussed here. Refer, e.g., to [6] for 3. The standard notation for the product copula is Π(u,v) = uv, however further details. we reserve this for possibility measures

171 DEPENDENT POSSIBILISTIC ARITHMETICUSING COPULAS the inverse cdf of a standard normal Gaussian. Note that 4. Arithmetic with Known Dependence Φ Φ Φ C 1 = W, C0 = Ψ and C1 = M. − Because a possibility distribution specifies a consonant The inequality W C M suggests a partial ordering ≤ ≤ belief function with nested focal elements, i.e. a necessity on the set of all copulas [16]. This ordering is useful in the α measure: given π , π with α-cuts αx and y correlated preceding sections: X Y CπX CπY by copula CXY , the basic mass assignment on the cartesian product of two cuts is computed by the Möbius inverse: Definition 1 (Concordance ordering) If C1 and C2 are copulas, we say that C1 is smaller than C2 (or C2 is larger αx αy m(C Cπ ) = than C1), and write C1 C2 (or C2 C1) if C1(u,v) πX × Y ≺ 2 ≤ αy αy C2(u,v) for all (u,v) [0,1] . Cπ UX + Cπ UY ∈ ∑ ( 1)| X \ | | Y \ |NecXY (UX ,UY ), αx αy − UX C ,UY C Furthermore, two random variables X and Y are positive ⊆ πX ⊆ πY quadrant dependent if Ψ C and are negative quadrant (18) ≺ XY dependent if Ψ C . XY where Nec is the bivariate necessity, which may be de- Sklar’s theorem has been extended to an imprecise set- XY fined in terms of C using Theorem3. Proof of (18) may ting by several authors. Montes et al. [15] describe an im- XY be found in [17]. This mass assignment may be thought of precise version of Sklar’s theorem for the construction of as the joint probability density of the α-cuts under copula multivariate p-boxes from two marginal p-boxes and an C . A non-decreasing function with an interval extension imprecise copula (bounded set of copulas). Schmelzer [18] XY f : B+ B+ B+ may then be performed on the α-cuts: investigates when copulas can be used to describe or model × → the dependence of belief functions. He demonstrates that in αx αy KZ = f (C ,Cπ ), (19) the general case a joint belief measure cannot be related to πX Y its margins by a single copula; except in the special cases where KZ are the focal elements of the output belief func- that the belief function is a p-box (cumulative belief function BelZ of πX and πY under operation f . The mass assign- tion) or a possibility distribution (minitive belief function). ment m of these focal elements is retained by the images He further describes a version Sklar’s theorem for minitive [25]: belief functions [19], stated in the language of possibility αx αy αx αy theory: m(KZ) = m( f (C Cπ )) = m(C Cπ ). (20) πX × Y πX × Y Theorem 3 (Sklar’s theorem for minitive beliefs) Let The necessity of the resulting random set may be found by N : B2 [0,1] be a bivariate necessity measure with computing the belief: XY → marginal necessity measures NecX and NecY . There exists Nec(KZ) = m(U). (21) a copula C such that for all UX BX ,UY BY : ∑ U KZ ∈ ∈ ⊆

NecXY (UX ,UY ) = C(NecX (UX ),NecY (UY )). (17) In the case where BelZ is consonant, the focal elements KZ are α-cuts of the corresponding possibility distribution, The copula is uniquely determined on support Nec X × with the α-level found using (11): support NecY . α = 1 m(U). (22) Z − ∑ U KZ The copula C in Theorem3 characterises the joint distri- ⊆ bution of the alpha cuts (focal elements) of two marginal Therefore, for a non-decreasing function f with an inter- possibility distributions (consonant belief functions). For val extension, a dependent possibilistic arithmetic with a example, C = Ψ would give random set independence be- precise dependence may be deﬁned in terms of α-cuts: tween the belief functions corresponding to πX and πY . In that respect, Theorem3 gives a form of random set depen- αz αx αy Cπ = f (C ,Cπ ), (23) dence for consonant random sets. Section7 discusses some Z πX Y implications of this. for z = f (x,y) and with αz given by: Theorem3 may be used in an α-cut-based implementa- tion of possibilistic arithmetic under a given copula, where αZ = 1 ∑ m(UZ), (24) − αx αy interval arithmetic is performed on α-cuts correlated by UZ f (C ,C ) ⊆ πX πY copula C. In Section4 we show how arithmetic operations may be performed when a copula C is known, giving a with the mass assignment given by (18) and Theorem3. possibilistic analogue to (1). Section5 describes how arith- Notice in the above that the α-levels αx, αy may be distinct, αx αy metic may be performed when only bounds to C are known, so therefore f (CπX ,CπY ) may lead to a non-consonant be- giving an analogue to (2)-(3). lief function, for which an outer approximation may be

172 OUTPUT

DEPENDENT POSSIBILISTIC ARITHMETICUSING COPULAS IP-p-Transform

CXY = W F = C 0.8 F = C 0.3 = Y = CF OUTPUT0.3 F = C0.8 = M

IP-p-Transform

CXY = W F = C 0.8 F CXY = W = C 0.3 F = Y Figure= 2: CSum0.8 of the two triangular fuzzy numbers π,π = (1,2,3) with various precise= CF copulas xF y 0.3 = Cspeciﬁed.0.3 = CF 0.8 = Y = M Figure2 hasF some interesting special cases. C = M = C0.3 XY (perfect dependence) matches a levelwise interval arith- F Figure 1: An example of the imprecise probability-to- metic:= C0.8 Cα = f (Cα ,Cα ), (25) possibility transformation. Above shows the fo- = M πZ πX πY cal elements (of equal mass) of a non-consonant and gives the same result as (10) with Jmin, which is belief function. Below shows the possibility dis- Zadeh’s classical extension principle. CXY = W (opposite tribution retrieved from the transformation. dependence) matches an opposite levelwise arithmetic:

α α 1 α CπZ = f (CπX ,Cπ−Y ). (26) CXY = W Although in this paper we provide no proof of (25) and found via the imprecise probability-to-possibility trans- F (26), and= areC conjectured0.8 based on the observed possibility form. Figure1 shows an example of this. The above shows distributions produced by (23), the rationale is as follows. the focal elements of the belief function Bel found from F Z Theorem=3C characterises0.3 the bivariate distribution of the the product of two triangular fuzzy numbers πX = (1,2,3) α-cuts of πX and πY as copula CXY , with probability mass (range = [1,3] and core = 2) and πY = (1,2,10) using (23) = Y given by (18). The copula CXY = W only has non-zero prob- with CXY = Ψ (random set independence). Below in Fig- ability= massCF on the opposing diagonal of the unit square, ure1 shows the possibility distribution πZ retrieved from i.e. when u =0.31 v. Therefore only opposing α-cuts will the imprecise probability-to-possibility transformation of − be assigned= CF a positive (and equal) mass by (18) under W, BelZ. The credal set induced by πZ is an outer approxi- and so are the0.8 only α-cuts needed to be considered in an mation to the credal set from BelZ, i.e. P(BelZ) C(πZ). ⊆ evaluation= M of (23) when CXY = W. A similar argument can Due to (9) it is sufficient to check that the necessity of be made for CXY = M, which only has positive mass on πZ is a lower bound on BelZ on the α-cuts of πZ, i.e. αz αz the diagonal u = v. A proof may be constructed by show- NecX (Cπ ) BelZ(Cπ ), which is a construct of the trans- Z ≤ Z ing that for CXY = W (= M), the Möbius inverse (18) only formation. gives a non-zero mass on α-cuts α = 1 α (= α ). X − Y Y Figure2 shows (23) evaluated for the sum of two identi- Note that although (25) will always produce a consonant cal triangular πx,πy = (1,2,3) with various CXY . A Gaus- structure, (26) may require an imprecise probability-to- sian copula has been used for CXY , varying its parameter possibility transformation. In the case two identical πX = from 1 to 1. Note that any copula may be used for C , π , (26) with f = sum produces an interval, shown in red − XY Y the Gaussian copula family is only used as a convenient in Figure2. It would be interesting to find for which inputs, example, since it contains the copulas W, Ψ, and M. The operators and copulas (23) produces intervals. possibility distributions πX and πY also do not necessarily Like for the convolutions of p-box arithmetic, we extend have to be identical, triangular, nor uni-modal. (23) to non-increasing binary operations by first applying 1

173

1 DEPENDENT POSSIBILISTIC ARITHMETICUSING COPULAS OUTPUT

IP-p-Transform

CXY = W F = C 0.8 F = C 0.3 = Y F = C0.3 F = C0.8 = M

CXY = W F = C 0.8 F = C 0.3 = Y , = ( , , ) Figure 3: Subtraction, multiplication, and division of πx πy 1 2 3 with various precise copulas speciﬁed.F = C0.3 F = C0.8 an appropriate unary operation to one of the inputs and followed by an evaluation of (23). Again the resulting= M be- evaluating the function with a non-decreasing binary opera- lief function may be made consonant with an imprecise tion. For example a dependent subtraction between πX and probability-to-possibility transformation. This correlated πY may be evaluated as random α-cut slicing method produces an inner approximation to the results of Figures2 and3 when ﬁnite samples αx αy C + ( Cπ ), (27) πX − Y are used. where the negation is performed levelwise on α-cuts of πY . Similarly division may be evaluated with a reciprocate and 5. Arithmetic with Partially Known product: Dependence αx αy CπX (1/CπY ). (28) ∗ Sklar’s theorem for minitive beliefs may be used in a de- However when evaluating these non-increasing operations, pendent possibilistic arithmetic where only the bounds C and C must be used, which may be found by a X, Y X,1/Y to copulas are known, whereby interval arithmetic may simple− transformation of C [16]: XY be performed on α-cuts of two possibility distributions, with resulting α-level of the output being found using CXβ(Y)(u,v) = u CXY (u,1 v), (29) − − Theorem3. Because the necessity measure of an α-cut α α α where β is a strictly decreasing function on the support of is: NX (C ) = 1 α, the belief of the image f (C ,C ) πX − πX πY Y. This gives the following mapping between copulas: can be bounded by

α α α α M W, BelZ( f (Cπ ,Cπ )) NecXY (Cπ ,Cπ ) 7→ X Y ≥ X Y (31) W M, = C(1 α,1 α). 7→ − − Ψ Ψ, 7→ Choosing the candidate possibility distribution qZ such Φ Φ α Cr C r. that its α-cuts are given by these images, i.e. Cq = 7→ − Z f (Cα ,Cα ), immediately implies that they are also the 1 The parameter of the Gaussian copula is negated due the πX πY − C(1 α,1 α)-cuts of a robust outer approximation πZ symmetry of the Gaussian copula. Figure3 shows the de- − − of BelZ under the imprecise probability-to-possibility trans- pendent subtraction, multiplication and division between form in Equation (12). So, for some non-decreasing func- 1 πX ,πY = (1,2,3) for various Gaussian copulas. Note that tion with an interval extension f : B+ B+ B+, two for f = ,/ transformation (29) has been applied, and × → α {− } possibility distributions πX and πY with alpha cuts CπX and so CXY = W gives levelwise arithmetic and CXY = M gives α CπY correlated by copula CXY , an outer approximation of opposite levelwise arithmetic. z = f (x,y) may be found as: Equation (18) with Theorem3 also implies a correlated 1 CXY (1 α,1 α) α α random α-cut slicing strategy for propagating πX ,πY under − − − CπZ = f (CπX ,CπY ). (32) CXY , where random values of αX and αY may be simulated from CXY : The image of the marginal input α-cuts of X and Y consti- (α ,α ) C , (30) tutes the output 1 C (1 α,1 α)-cut of π . However, X Y ∼ XY − XY − − Z

174 OUTPUT

IP-p-Transform

CXY = W F = C 0.8 F = C 0.3 = Y F = C0.3 F = C0.8 DEPENDENT POSSIBILISTIC ARITHMETICUSING COPULAS = M

W C for any C. It also encloses all those shown in Figure ≺ CXY = W 2. It does not tightly enclose all those from Figure2 since F the example only shows several Gaussian copulas, whilst = C 0.8 W F πZ bounds those propagated using any copula (including = C 0.3 non-Gaussian). CXY = Ψ corresponds to positive quadrant = Y dependent arithmetic, and propagates all copulas greater F Ψ = C0.3 than Ψ. As expected, the computed πZ is greater than or F equal to those in Figure2 propagated with the precise copu- = C0.8 Φ Φ las CXY = Ψ,C0.3,C0.8,M , but not those propagated with = M Φ{ Φ } CXY = C 0.3,C 0.8,W . CXY = M specifies a precise cop- { − − } ula CXY = M, and levelwise arithmetic is retrieved. Only the lower bound on CXY plays a role in (36). This is because C C gives πCXY πCXY . That is, the result XY ≺ XY Z ≥ Z computed with the lower bound will always enclose the result from the upper bound. A similar situation occurs with the τ and ρ convolutions in p-box arithmetic. Figure 4: Sum of π ,π = (1,2,3) with various copula x y Note that (36) with CXY = M gives the same results lower bounds specified. min SI as (10) using J , CXY = Ψ gives (10) using J , and UI CXY = W gives (10) using J . This suggests the gener- alised extension principle with aggregation operations of [12] corresponds to an imprecise propagation of dependen- if C1 C2 then ≺ cies, as opposed to a precise one. C1(1 α,1 α) C2(1 α,1 α), Like in Section4, and Williamson and Downs for the τ − − ≤ − − and ρ convolutions in p-box arithmetic, we extend (36) to and using Sklar’s theorem for minitive beliefs: non-increasing binary operations by first transforming one of the inputs and performing a non-decreasing binary oper- NecC1 NecC2 , (33) XY ≤ XY ation. The copula must also be appropriately transformed. Note that if X and Y are positive quadrant dependent, then and which also from (32) gives X and Y (and X and 1/Y) will be negative quadrant de- − C C pendent [16]. The outcome of this is, like for the τ and π 1 π 2 . (34) Z ≥ Z ρ convolutions, that the copula’s upper bound is used in- C2 stead of the lower bound, i.e. τΨ, and ρΨ, compute a Moreover, from (31) if NecXY is a robust approximation of − − C2 C1 p-box which bounds negative quadrant dependence and BelZ , then so is NecXY ; i.e. τM, and ρM, give Fréchet for subtraction. Therefore for − − C1 C2 C2 a non-increasing binary operations, (36) is computed as: NecXY NecXY BelZ , (35) ≤ ≤ 1 C∗ (1 α,1 α) α α − XY − − = f ( , ), (37) for any C C . The copula in (32) may therefore be re- CπZ CπX CπY 1 ≺ 2 placed by a lower bound: where CXY∗ is the appropriate transformation of the copula’s 1 C (1 α,1 α) α α upper bound. For f = ,/ , this transformation is (29): − XY − − {− } CπZ = f (CπX ,CπY ), (36) C∗ (u,v) = u C (u,1 v), XY − XY − since (32) will compute a robust outer approximation belief 1 C∗ (1 α,1 α) = 1 α CXY (1 α,α), functions propagated with all copulas C C. The above XY − − − − − XY ≺ equation gives, for a non-decreasing function, a levelwise and (37) becomes:

α-cut based arithmetic for possibility distributions when α+CXY (1 α,α) α α Cπ − = f (C ,C ). (38) only a lower bound on a copula is known. The α-level of Z πX πY the images is a simple scaling of α using C . As opposed to (36), for πZ produced by (38) C1 C2 gives XY ≺ Figure4 shows (36) evaluated for the sum of π ,π = πC1 πC2 . x y Z ≤ Z (1,2,3) and various CXY . A Gaussian copula has been Figure5 shows subtraction and division evaluated with used for C , varying its parameter from 1 to 1. When (38) and multiplication with (36) for π ,π = (1,2,3) XY − X Y C = W (the lower bound on all copulas), (36) gives with various Gaussian copulas as bounds. For f = ,/ : XY {− } Fréchet, corresponding to the sum under unknown depen- CXY = M gives Fréchet, CXY = Ψ gives negative quad- dence between πX and πY . The computed possibility distri- rant dependence, and CXY = W gives levelwise arith- W bution πZ is greater than (i.e. it encloses) the possibility metic. When comparing the results from the precise cop- C Ψ distributions calculated under any other copula πZ , since ula calculation in Figure3, the computed πZ using CXY

175 OUTPUT OUTPUT

IP-p-Transform IP-p-Transform

CXY = W CXY = W F F = C 0.8 = C 0.8 F F = C 0.3 = C 0.3 = Y = Y F F = C . = C . 0 3 OUTPUT 0 3 = CF = CF 0.8 0.8 OUTPUT = M = M IP-p-Transform OUTPUT OUTPUT IP-p-Transform CXY = W C = W CXY = W XY IP-p-Transform OUTPUT = CF F = CF 0.8 = C 0.8 0.8 = F IP-F p-Transform = F C 0.3 = C 0.3 C 0.3 CXY = W = Y = Y CXY = W = Y F F F F F = C 0.8 = C0.3 = C0.3 = CIP-p=-TransformC0.3 CXY = W 0.8 DEPENDENT= CF POSSIBILISTIC ARITHMETICUSINGFCOPULAS = CF F 0.8 = C0.8 = F 0.8 = C 0.3 F C 0.3 = M = M = C 0.8 = M = Y = Y F = C 0.3 F CXY F= W = C0.3 = C0.3 = Y F F F CXY = W CXY = W = CXY = W OUTPUT C0.8 = C0=.8 C 0.8 F F F F = C 0.8 = C 0.8 = C0.3 = C 0.8 = M = M F F F F = C = C 0.3 = C 0.3 F = C 0.3 0.3 = C0.8 = Y = Y IP-p-Transform = Y = M = Y = CF = CF = CF 0.3 0.3 C = W 0.3 F F F XY F = = C . = C0.8 = C . OUTPUTCXY = W C0.3 0 8 CXY = W F 0 8 = C 0.8 = M = M = M F F FF = C =0.8C ==CC 0.8 0.8 CXY = W 0.3 F = CF IP-p-Transform F ==CY0.3 =0.3M = C = W C 0.8 XY ==YCF = Y F F 0.3 = C 0.8 = C 0.3 F F = C F = C F CXY = W = C00.3.8 0.3 = C 0.3 = Y F ==CMF F = Y = C 0.8 0.8 OUTPUT= C0.8 F C = W F = CF0.3 = M XY = C0.3 = C = M F0.3 F = C F = C0.8 = Y 0.8 = C 0.8 IP-p-Transform = M = MFCXY = W F Figure 5: Subtraction, multiplication, and division of πx,πy = (1,2,3) with various copulas copula= C bounds. specified. 0C3 = W F = C 0.3 F XY = C 0.8 = C CXY = W 0.8 = CFF = Y CXY ==WC 00.8.3 F = M = C F 0.8 for f = ,/ encloses those propagated with C = C = W ==CY 0.3 F XY XY 0.8 F= C . Φ {−Φ } = C 0 3 , , , = F = YFF 0.3 Ψ C 0.3 C 0.8 W , but not those propagated with CXY = C ==CC0.03.3 F { − − } 0.8 F Φ Φ = C F = Y= C0.8 C0.3,C0.8,M , as expected for negative quadrant depen- C = W F ==YC00.3.8 XY = C 0.3 F { } F = dence. F ==CM0.8 C0=.3 M ==CY0.8 0.3 1 = MF F 1 F = C = C0.8 cumulative = C F0.3 0.8 = C0.3 Sum with CXY= Y = M = M 6. Relationship to p-box Arithmetic = CF Sum with CXY F0.8 = C0.3 cumulative C = W sCXY ,+ = MCXY = W XY In this section we compare our dependent possibilistic arith- F , = C FSum with C tCXY ,+ rCXY ,+ 0.8 = C XY F metic using precise CXY and imprecise C with the σ, τ, CXY = W 0.8 = C 0.8 XY cumulative = M Sum with CXY = CFF and ρ convolutions of p-box arithmetic. The method out- = C s00..38 F Sum with CXY CXY ,+ = C = YF 0.3 lined in Section4 may be considered a possibilistic ana- Sum with CXY = C t0C.3XY ,+,rCXY ,+ F = Y logue to a σ convolution, and the methods of Section5 are sC ,+ = CY0.3 XY CXY 1= W tC ,+,rC ,+ = CFF F a possibilistic analogue to the combination of a τ and ρ XY XY F = C00..83 = C . = C 0.8 0 3 convolution (one for each p-box bound). = MF F = C0.8 F = C 0.3 = C0.8 Consider performing similar calculations with p-box cumulative = Y = M arithmetic. A possibility distribution π may be converted = M X Sum with CXY F = C0.3 to a p-box by accumulating ΠX and NX : Sum with CXY F sC ,+ = C0.8 cumulative XY CXY = W tC ,+,rC ,+ FX (x) = PX (X x) = ΠX (( ∞,x]), XY XY = M FSum with CXY ≤ − = C 0.8 F (x) = P (X x) = NX (( ∞,x]), (39) FSum with CXY X X cumulative = C 0.3 ≤ − s Sum with CXY = Y CXY ,+ 1 with the credal set defined by π , C(π ), being contained Sum with C X X XY FtCXY ,+,rCXY ,+ s = C0.3 in credal set defined by the p-box: C(π ) C([F ,F ]) CXY ,+ X X X F ⊆ tC ,+,rC ,+ = [1]. Figure6 is an illustration of the similarity between XY XY C0.8 = M the dependent arithmetic operations presented in this pa- 1 per to those from [24]. Beginning with the fuzzy numbers cumulative πX ,πY = (1,2,3) in the centre left, they may be converted Sum with CXY 1 into the p-box in the centre right. Several evaluations of Sum with CXY s ,+ τ and ρ with different copula lower bounds will yield the CXY tCXY ,+,rCXY ,+ enclosing set of p-boxes on the top right, with Fréchet in red and perfect in purple. A similar set of σ convolutions yield the non-enclosing p-boxes on the bottom left, with the Figure 6: Illustration of the similarity between the depen- 1 purple being perfect and matching the purple p-box from dent possibilistic operations derived in this paper the τ-ρ convolution. Red (opposite) is an interval. The τ-ρ to those from p-box arithmetic. The colours and copulas are the same as Figures2 and4. p-boxes calculated with copula CXY enclose all p-boxes produced with copulas C which are greater CXY C, for both ≺ 1 σ and τ-ρ calculations. An identical behaviour can be seen in the dependent possibilistic arithmetic. Arithmetic with 1 176

1 DEPENDENT POSSIBILISTIC ARITHMETICUSING COPULAS

CXY produces a set of enclosing fuzzy numbers. Precise the belief and plausibility. P-boxes return tight probability propagation produces results which are not self enclosing, bounds in their tail regions, whilst return vacuous proba- but match the imprecise propagation when CXY = M and bility intervals [0,1] in their central regions. Conversely give an interval when CXY = W. The CXY calculation also possibility distributions return vacuous intervals in their encloses all possibility distributions calculated with copu- tails and tight probability bounds in their centre. Since las C which are greater C C, for both the precise and these two imprecise structures bound a belief function in XY ≺ imprecise cases. opposing ways, their arithmetics may be combined to give

CXY CXY a tighter propagation of a general belief function than either Furthermore, when the πZ and πZ possibility distributions are converted to p-boxes, the σ and τ-ρ p-boxes method individually. The results of this paper provide a are retrieved. Note that here we are not stating that the means to perform the dependent binary operations on pos- p-boxes produced both ways are equivalent, nor that depen- sibility distributions previously only available to p-boxes. dent arithmetic and ‘building a p-box’ commute. Figure6 only serves as a comparison. However from the conducted numerical example the p-box bounds from both approaches 7. Discussion are very similar. Performing a more rigorous mathematical Theorem3 gives a form of random set dependence between or numerical study about the agreement of the two methods possibility distributions. It should be noted however that it should be pursued. Particularly a comparison of the credal is unclear how the dependence between the α-cuts relates to sets produced by both methods is interesting. A situation the dependence of the distributions in the credal set C(π ). similar to Figure6 can be observed for the other binary XY That is to say, Theorem3 does not imply that operations.

The numerical approaches for computing dependent π C(FX ,FY ) C(πXY ) (40) and p-box operations are also quite similar. For example in ∈ both approaches it is computationally simpler to calculate for all F C(π ) and F C(π ), and where C(π ) is X ∈ X Y ∈ X XY with an imprecise copula. A σ convolution requires a cumu- the credal set of bivariate distributions induced by Theorem lative integration of a joint probability measure over an in- 3. Further investigation is required to find if (40) holds, and creasing domain defined by f . As outlined in [24], this may its implication for dependent possibilistic arithmetic. Couso be done in terms of a number of discrete quasi-inverses of et al. [3] do however show that for the case of independence, the quantiles of a p-box. The probability measure induced a set of joint dfs for variables that are strongly independent by a copula on this discretisation is found, followed by an (i.e. H(u,v) = F(u)G(v)) is a subset of the distributions operation f of these inverses and an outer approximation under random set independence [10]. of the integral. Similarly, our precise CXY operation may be Schmelzer further makes the distinction between bivari- performed by finding the probability measure induced by ate minitive belief functions and joint minitive belief func- a copula on some discrete number of α-cuts, followed by tions [20], where the later are belief functions defined on an interval operation on these cuts and a outer approxima- product spaces of focal elements, and also account for the tion of the resulting belief function, which involves large shape of the joint focal elements. He argues that joint belief sums over mass assignments. The τ and ρ convolutions functions provide an upper bound on a set of joint probabil- on the other had only require the infimum and supremum ity measures, and studies the relationship between bivariate of the joint cdf to be found in some domain, which may and joint belief functions, and their relation to copulas. also be outer approximated in term of the quasi-inverses. Note that we only show examples of f = +, , ,/ , { − ∗ } Our imprecise propagation of CXY only requires a scaled however the described methods may be ready expanded to levelwise arithmetic, and thus requires less interval opera- any non-increasing and non-decreasing binary operation. tions and no (explicit) imprecise probability-to-possibility transformation. In the context of propagating dependencies, these latest Software additions to possibilistic arithmetic give it the same capabilities as p-box arithmetic, allowing for any dependence The methods developed in this paper are made available in or partial dependence to be specified as a copula. This an open-source Julia package for performing dependent provides a basis for a p-box/possibility hybrid arithmetic possibilistic arithmetic: https://github.com/ which gives tighter bounds on the exact upper and lower AnderGray/PossibilisticArithmetic.jl. probabilities than either method independently for the prop- The figures of this paper may be reproduced by running agation of general belief functions. Baudrit and Dubois the scripts found in examples/ISIPTA2021. The p-box [1] discuss how well possibility distributions and p-boxes arithmetic used in this paper was performed using the bound general belief functions, and on which events do open-source Julia package: https://github.com/ these two imprecise representations return a tight bound on AnderGray/ProbabilityBoundsAnalysis.jl

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Acknowledgments In Proceedings of International Conference on Infor- mation Processing and Management of Uncertainty We would like to thank the ISIPTA reviewers, whose com- in Knowledge-Based Systems, pages 75–84. Springer, ments and questions greatly improved the quality of this 1986. doi: 10.1007/3-540-18579-8\_6. paper. We also thank Bernhard Schmelzer for clarifica- tion on the interpretation of his theorem. We are also very [7] Didier Dubois and Henri Prade. Consonant ap- grateful to those who have attended the virtual Fuzzy and proximations of belief functions. International Possibility interest group hosted by the University of Liver- Journal of Approximate Reasoning, 4(5):419–449, pool, for the many fruitful discussions which lead to this 1990. ISSN 0888-613X. doi: https://doi.org/10.1016/ paper. Ander Gray would like to thank the support from 0888-613X(90)90015-T. the EPSRC iCase studentship award 15220067. We also [8] Didier Dubois and Henri Prade. When upper proba- gratefully acknowledge funding from UKRI via the EP- bilities are possibility measures. Fuzzy Sets and Sys- SRC and ESRC Centre for Doctoral Training in Risk and tems, 49(1):65–74, 1992. doi: 10.1016/0165-0114(92) Uncertainty Quantification and Management in Complex 90110-P. Systems. This research has also received funding by the Engineering Physical Sciences Research Council (EPSRC) [9] Didier Dubois and Henri Prade. Possibility theory: an with grant no. EP/R006768/1, which is greatly acknowl- approach to computerized processing of uncertainty. edged for its funding and support. This work has been Springer Science & Business Media, 2012. carried out within the framework of the EUROfusion Con- sortium and has received funding from the Euratom re- [10] Scott Ferson, Janos Hajagos, Daniel Berleant, search and training programme 2014-2018 and 2019-2020 Jianzhong Zhang, W Troy Tucker, Lev Ginzburg, and under grant agreement No 633053. The views and opin- William Oberkampf. Dependence in dempster-shafer ions expressed herein do not necessarily reflect those of theory and probability bounds analysis. Sandia Na- the European Commission. Dominik Hose and Michael tional Laboratories, 2004. Hanss gratefully acknowledge the support provided by the [11] Scott Ferson, Vladik Kreinovich, Lev Grinzburg, German Research Foundation (DFG) in the framework of Davis Myers, and Kari Sentz. Constructing proba- the research project no. 319924547 (HA2798/9-2). bility boxes and dempster-shafer structures. Sandia National Lab.(SNL-NM), Albuquerque, NM (United References States), 2015. [1] Cédric Baudrit and Didier Dubois. Practical represen- [12] Dominik Hose and Michael Hanss. Possibilistic cal- tations of incomplete probabilistic knowledge. Com- culus as a conservative counterpart to probabilistic putational Statistics & Data Analysis, 51(1):86–108, calculus. Mechanical Systems and Signal Processing, 2006. 133:106290, 2019. [2] Cédric Baudrit, Didier Dubois, and Dominique Guy- [13] Dominik Hose and Michael Hanss. A universal ap- onnet. Joint propagation and exploitation of proba- proach to imprecise probabilities in possibility theory. bilistic and possibilistic information in risk assess- Submitted to International Journal of Approximate ment. IEEE Transactions on Fuzzy Systems, 14(5): Reasoning, 2021. 593–608, 2006. [14] Harry Joe. Multivariate models and multivariate dependence concepts. CRC Press, 1997. [3] Inés Couso, Serafín Moral, and Peter Walley. A sur- vey of concepts of independence for imprecise prob- [15] Ignacio Montes, Enrique Miranda, Renato Pelessoni, abilities. Risk, Decision and Policy, 5(2):165–181, and Paolo Vicig. Sklar’s theorem in an imprecise 2000. setting. Fuzzy Sets and Systems, 278:48–66, 2015. [4] Inés Couso, Susana Montes, and Pedro Gil. The neces- [16] Roger B Nelsen. An introduction to copulas. Springer sity of the strong α-cuts of a fuzzy set. International Science & Business Media, 2007. Journal of Uncertainty, Fuzziness and Knowledge- Based Systems, 9(02):249–262, 2001. [17] Bernhard Schmelzer. Characterizing joint distributions of random sets by multivariate capacities. In- [5] Miguel Delgado and Serafın Moral. On the concept ternational Journal of Approximate Reasoning, 53(8): of possibility-probability consistency. Fuzzy Sets and 1228–1247, 2012. Systems, 21(3):311–318, 1987. [18] Bernhard Schmelzer. Joint distributions of random [6] Didier Dubois and Henri Prade. The principle of min- sets and their relation to copulas. International Jour- imum specificity as a basis for evidential reasoning. nal of Approximate Reasoning, 65:59–69, 2015.

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