Bi-directional dominance for measure modeled uncertainty
Introduction
A monotonic measure μ provides a very general structure for the representation of variables having uncertain values [9], [20], [23]. Here we use the measure of a set to provide the anticipation that the value of the variable lies in the set. Probability and possibility distributions can very naturally be modeled using these structures. One difficult task with uncertain information is the ordering of uncertain values with regard to which of two values is bigger. No natural ordering exists, as is the case with scalar values, we must select some reasonable methodology for ordering uncertain information. One commonly accepted approach for ordering probability distributions is stochastic dominance [2], [5], [15], [19]. Here we say probability distribution A dominates probability distribution B if the probability that A is bigger than any value y is at least as large as the probability that B is bigger than the value y. Here we look to generalize this methodology to the case of measure-based representation of uncertainty. One implicit feature of a probability distribution must be amended to the definition of dominance when porting this idea to the more general case of a measure. To conclude that A dominates B in addition to requiring that for all y we have Prob(A ≥ y) ≥ Prob(B ≥ y) we also require for all y that Prob(A < y) ≤ Prob(B < y). The special nature of the probability distributions, its additivity, implies that if the first condition is satisfied then the second is automatically satisfied, uncertainty distributions having this property are called self-dual, as such in expressing the dominance requirement for probability there is no need to explicitly indicate this second requirement. In generalizing our concept of dominance to the more general structure of a measure representation of uncertainty, which is typically not self-dual, we must explicitly indicate both aspects of the requirement for dominance, here we use the term bi-directional dominance to express this formulation containing the two aspects.
There exists one fundamental problem with the use of dominance, both in the classical case of stochastic dominance and the new measure based method of bi-directional dominance proposed here, it is not an easy condition to satisfy. In particular there are often cases where there exists no dominance relationship between two uncertainty representations. In order to circumvent this problem we introduce the idea of surrogates for bi-directional dominance. Here we associate with each measure representing an uncertain value a scalar value and then compare these scalar values. The special feature of scalar values that make them surrogates for bi-directional dominance is that if there exists dominance relationship between two measures the one that is dominant will have a larger scalar value.
Section snippets
Measures and the modeling of uncertainty
Assume Y = {y1,…, yn} is a finite space, a fuzzy measure [10], [16], [17], [18] on the space Y is a set mapping μ: 2Y → [0, 1] such that
- 1
μ(ø) = 0
- 2
μ(Y) = 1
- 3
μ(A) ≥ μ(B) if B ⊆ A
We see μ maps subsets of Y into the unit interval. In the following we shall simply refer to μ as a measure.
We shall say μ1 ≥ μ2 if μ1(A) ≥ μ2(A) for all A. If μ1 ≥ μ2 and there is at least one of subset B such that μ1(B) > μ2(B) we shall denote this as μ1 > μ2.
We define the dual of a measure μ to be the set mapping
Comparing uncertain values
Our interest here is going to be on situations where the space Y is ordered. In particular we shall assume that there exists an ordering on Y = {y1,…, yn} such that yi > yi+1. Thus here we have y1 > y2 > …. > yn. Here the meaning of > is “bigger than” or more abstractly “preferred to.”
Let X = {x1,…, xr} be a collection of alternatives. Let V be an attribute associated with the objects in X such that V(xk) is a variable that takes its value in the space Y. A task that arises in many environments
Bi-directional dominance
If V(xi) = μi then we understand that is essentially providing the anticipation that V(xi) is at least yj and not less than yj. We begin to see that Qi(j) can provide a reasonable way for ordering the V(xi) [21]. Toward this application we introduce the idea of Bi-directional dominance.
Assume V(x1) = μ1 and V(x2) = μ2. We shall say that μ1 Bi-directionally dominates μ2 if
We shall denote this as μ1 > Bidμ2.
Here then if μ1 >
Functionally modified bi-directional surrogates
Assume that f is a primal monotonic function, that is a function f: [0, 1] → [0, 1] having f(0) = 0, f(1) = 1 and strict monotonicity, f(a) > f(b) if a > b. In the following we show that if f is a primal monotonic function, PM function, is a strict Bid surrogate in the case where D is real numbers.
Assume that μ is a certain measure focused as yk. Here μ(A) = 0 if y ∉ A and μ(A) = 1 if yk ∈ A. Here then μ(Hj) = 0 for j < k and μ(Hj) = 1 if j ≥ k and for
Weak surrogates
The definition of a bi-directionally based surrogate can be slightly modified to provide for a weak bi-directionally based surrogate.
Definition Assume V is a variable with domain D and μ is a measure on the range Y ⊆ D. A mapping is a called a weak Bid based surrogate for μ if it satisfies the following conditions:
If μ is a certain type measure focused at yk the If μ1 and μ2 are two measures on Y such that μ1 > Bid μ2 then
Here we have replaced condition 2 by 2ʹ.
We now show
Universal integrals and surrogates
In Ref. [6], [7], [8] Klement and Mesiar provided two classes of discrete universal integrals, they can provide structures for defining bi-directional surrogates for measures μ that are defined on Y = {y1,…, yn} where Y is a subset of the unit interval with yj > yj+1.
The first class of universal integrals is based on semi-copulas.
Definition A function S: [0, 1] × [0, 1] → [0, 1] is called a semi-copula if
One is a neutral element: S(x, 1) = S(1, x) = x,. S is monotonic: S(x1, y1) ≤ S(x2, y2) if x1 ≤ x2
Conclusion
Our objective here was to provide a formulation for a dominance between variables having uncertain values represented by measures that can be used in ordering these uncertain values. We first discussed the use of monotonic set measures, fuzzy measures, for the representation of uncertain information. We considered the issue of comparing, ordering, variables whose values are uncertain and represented via a measure and suggested the use of bi-directional dominance, which can be seen as a
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