Duality

The following duality theorem has been presented in a variety of contexts and in a number of dierent forms. It is the basis of the level-set (or -cut) formulae for implementing the fuzzy number convolutions (see chapter 4). Frank and Schweizer 278]have considered the duality in some detail. Their results are summarised in theorem 7.7.3 of 718]which is restated below.

Denition 3.4.1

For any F in #⁺ let F^{^}be the left continuous quasi-inverse ofF dened in section 3.2.3. Then ^r+ is the set ^fF^{^j}F ²#^+g.

Denition 3.4.2

For any two place function L and any copula C^, C L^{^} is the func-tion C L^{^} :^{r+ r+7!r+} given by

C L^{^} (F^{^}G^{^})(x) = inf_{C(u v)=x}L(F^{^}(u)G^{^}(v))]:

Denition 3.4.3

For any two place function L and any copula C, ^{^}C L is the func-tion ^{^}C L:^{r+ r+ 7!r+} given by

^{^}_{C L}(F^{^}G^{^})(x) = sup

C^d(u v)=xL(F^{^}(u)G^{^}(v))]:

Theorem 3.4.4 (Theorem 7.7.3 of 718])

Let L be a function L:^{<+ <+ 7!}

<+, with ¹ as its null element, and which is continuous everywhere. Let C ^{be a} copula. Then for any FG ²#⁺,

C L(FG) = C L^{^} (F^{^}G^{^})]^{^} _{C L}^{^} (F^{^}G^{^}) = C L(FG)]^{^} C L(FG) = ^{^}C L(F^{^}G^{^})]^{^ ^}C L(F^{^}G^{^}) = C L(FG)]^{^}:

The advantage of using the duality theorem becomes apparent when a specic L is considered. For example, if L = Sum, then the quasi-inverse representation of the lower and upper dependency bounds can be calculated in terms of the quasi-inverses of the distribution functions of the random variables in question by using the following formulae: Let Z = X +Y and let ldb^{^}C^XY(FZ) and udb^{^}_CXY(FZ) denote the quasi-inverses of ldbC^XY(FXFY+) and udbC^XY(FXFY+) respectively. Then

ldb^{^}C^XY(FX^{^}FY^{^}+)(x) = inf_C

XY(u v)=x(FX^{^}(u) + FY^{^}(v)) (3.4.1) udb^{^}C^XY(FX^{^}FY^{^}+)(x) = sup

C^dXY(u v)=x(FX^{^}(u) + FY^{^}(v)): (3.4.2) The standard dependency bounds (with CXY = W) follow with the appropri-ate substitution. Equations 3.4.1 and 3.4.2 are simply the maximum and mini-mum of the pointwise sum of quasi-inverses. The functions ldb^{^}_CXY(FX^{^}FY^{^2}) and udb^{^}_CXY(FX^{^}FY^{^2}) are quasi-inverses of the lower and upper dependency bounds and not lower and upper bounds on the quasi-inverses. That is

ldb^{^}C^XY(FX^{^}FY^{^2})udb^{^}C^XY(FX^{^}FY^{^2}) (3.4.3) is the consequence of

ldbC^XY(FXFY²)udbC^XY(FXFY²):

Note that for L = Sum or Dierence, theorem 3.4.4 holds for any FG ² # (and not just #⁺) because in these cases the point x = 0 has no special signicance.

When CXY =W the formulae are particularly simple, and this gives us a very simple way of calculating lower and upper dependency bounds. Recalling that W(uv) = max(u + v^;10) and W^d(uv) = min(u + v1) we have

ldb^{^}(FX^{^}FY^{^}+)(x) =

8<

:

u²infx 1](FX^{^}(u) + FY^{^}(x^;u + 1)) if x⁶= 0,

u+vinf^;1<0(FX^{^}(u) + FY^{^}(v)) if x = 0. (3.4.4) Since FX^{^} and FY^{^} are non-decreasing, the case forx = 0 becomes

ldb^{^}(FX^{^}FY^{^}+)(0) = FX^{^}(0) +FY^{^}(0):

Similarly,

udb^{^}(F_X^{^}F_Y^{^}+)(x) =

8<

:

usup²0x](FX^{^}(u) + FY^{^}(x^;u)) if x⁶= 1,

FX^{^}(1) +FY^{^}(1) if x = 1. (3.4.5) (The restrictions on the range of the supremum and inmum operations arise from the fact that v²01]because domF^{^}= 01].) Likewise, we have

ldb^{^}(FX^{^}FY^{^} )(x) =

8<

:

u²infx 1](FX^{^}(u) FY^{^}(x^;u + 1)) if x⁶= 0,

FX^{^}(0) FY^{^}(0) if x = 0 (3.4.6) and

udb^{^}(FX^{^}FY^{^} )(x) =

8<

:

usup²0x](FX^{^}(u) FY^{^}(x^;u)) if x⁶= 1,

FX^{^}(1) FY^{^}(1) if x = 1. (3.4.7) Analogous formulae for ldb^{^} and udb^{^} for ^{2 2 f;g} can be determined in a manner similar to that used to derive (3.3.5) and (3.3.7). Consider the quotient rst.

Let Y⁰ = 1=Y . If FY has a quasi-inverse FY^{^}, what is the quasi-inverse of FY⁰? It is known thatFY^{^}(x) = y ⁾x = FY(y) and FY⁰(x) = 1^;FY(1=x). If x = (1^;FY(1=y)), then FY^{^0}(x) = y. But 1^;x = FY(1=y). Therefore FY^{^}(1^;x) = 1=y and so

FY^{^0}(x) = 1

F_Y^{^}(1^;x) = y:

Using this and (3.4.6{3.4.7), it can be shown that ldb^{^}(FX^{^}FY^{^})(x) =

8<

:

u²infx 1](FX^{^}(u)=FY^{^}(u^;x)) if x⁶= 0,

F_X^{^}(0)=F_Y^{^}(1) if x = 0, (3.4.8) and

udb^{^}(F_X^{^}F_Y^{^})(x) =

8<

:

usup²0x](FX^{^}(u)=FY^{^}(1 +u^;x)) if x⁶= 1,

FX^{^}(1)=FY^{^}(0) if x = 1. (3.4.9) Similarly, it can be shown that if Y⁰=^;Y then FY^{^0}(x) =^;FY^{^}(1^;x), and so using (3.4.4) and (3.4.5)

ldb^{^}(FX^{^}FY^{^;})(x) =

8<

:

u²infx 1](F_X^{^}(u)^;F_Y^{^}(u^;x)) if x⁶= 0,

FX^{^}(0)^;FY^{^}(1) if x = 0, (3.4.10) and

udb^{^}(FX^{^}FY^{^;})(x) =

8<

:

usup²0x](F_X^{^}(u)^;F_Y^{^}(1 +u^;x)) if x⁶= 1,

FX^{^}(1)^;FY^{^}(0) if x = 1. (3.4.11) These formulae for ldb^{^} and udb^{^} can be seen to be slightly simpler than the corresponding formulae for ldb and udb (3.3.4{3.3.7). The one disadvantage is the

Figure 3.10: The copula ^W.

Figure 3.11: The numerical representation of probability distributions.

special case for x = 0 (for + and ) or for x = 1 (for^; and ). The reason these special cases are necessary becomes apparent upon inspection of gure 3.10 which shows the copula W. When W(uv) = x ⁶= 0, the values u and v can take are constrained by a linear relationship. However when W(uv) = 0, u and v can take any values within the cross-hatched region. When a numerical representation of the quasi-inverses is used these special cases actually disappear.

3.4.2 Numerical Representation and the Calculation of

ldb^{^}

and

udb^{^} In order to use equations (3.4.4{3.4.11) to calculate lower and upper dependency bounds numerically, it is necessary to have a discrete approximation to probability distribution functions dened on a continuum. The approach taken is illustrated in gure 3.11. The distributionF is approximated by lower and upper discrete approx-imations denoted F~ and ~F respectively. These are formed by uniformly quantising F along the vertical axis (see gure 3.11). This denes the points x~ⁱ and ~xi. For a single valued F, x~ⁱ = ~xi+1 for i = 1:::n^;1. We retain the two sets of numbers

fx~^igand^fx~i^gfor clarity. Interval valued distributions (i.e.lower and upper bounds on F) can result in x~^{i 6}= ~xi+1. The end points are

inf suppF = ~x1

sup suppF = x~ⁿ: (3.4.12)

The above approximation ofF motivates the following numerical approximation to F^{^}. Both in anticipation of the exclusive use of upper and lower bounds (to contain both dependency and representation error) and to make the ideas clearer, the discrete approximationsF~^{^}and ~F^{^}toF^{^}andF^{^}are presented in gure 3.12. In the following discussion it is not assumed that F^{^}=F^{^}, although the same results do apply to this situation which is often the case at the beginning of a calculation using probabilistic arithmetic.

Figure 3.12: The quasi-inverse representation of a distribution function.

The discrete approximations ~F^{^} and F~^{^} are dened by

~F^{^}(p) = F^{^}(pi) p²pipi+1) (3.4.13) F~^{^}(p) = F^{^}(pi+1) p²pipi+1) (3.4.14) for i = 0:::N ^;1. Whether the range in these denitions is pipi+1) or (pipi+1] doesn't really make any dierence: as long as one choice is used consistently there are no problems. N is the number of points required to represent either F~^{^} or

~F^{^}. In gure 3.12, N=6. Note that pN = 1. The quantisation is uniform and so pi =i=N for i = 0:::N^;1. Explicit array representations of F~^{^}and ~F^{^} suitable for computer implementation are given by

~F^{^]}i] = F^{^}(pi) =F^{^}_Nⁱ (3.4.15) and F~^{^]}i] = F^{^}(pi+1) =F^{^i+1}_N (3.4.16) for i = 0:::N^;1. Notice the dierent conventions used for deningF~^{^]} and ~F^{^]} in terms ofF^{^}andF^{^}. This is necessary because of the nature of the approximation.

Two special values worth noting are

~F^{^]}0]= F^{^}(p0) =F^{^}(0) and F~^{^]}N ^;1]=F^{^}(pN) =F^{^}(1):

These two values correspond to the points a and b in gure 3.12. Regarding F~^{^]}0], recall the convention of redening the value of F^{^} at the end points of the range (see section 3.2.3). This convention resulted in F^{^}(0) = inf suppF, and so ~F^{^]}0]

is set equal to this.

An important advantage of this representation is that any representation error (dierence betweenF and F~ or betweenF and ~F) isrigorously containedwithin the bounds. That is, it is perfectly correct to state that F~ F ~F or F~^{^}F^{^} ~F^{^}

although this does not bound F as tightly as stating F F F. By always performing any necessary rounding approximations in a manner such that the width between F~ and ~F is made larger (\outwardly directed rounding"), this property is preserved. This is referred to below as the preservation of the representation error containment property. The representation error for any given approximation can be made arbitrarily small by increasingN. The method of approximation proposed here is better than simply using a single functionFapproxto approximateF for which

jFapprox(x)^;F(x)^jis \small" for \most" x.

Now that a numericalrepresentation has been chosen, it is necessary to derive the appropriate formulae for ldb^{^]}and udb^{^]}, the lower and upper dependency bounds in terms of the numerical representation. We explicitly derive the four cases of the lower and upper dependency bounds for sums and dierences.

(i) ldb^{^]}(^F~^{X^] F}~^{Y^]} +)

Equations 3.4.4 and 3.3.22 give ldb^{^}(F^{^}XF^{^}X+)(x) = inf_u

2x 1](F^{^}X(u) + F^{^}Y(x^;u + 1)) (x⁶= 0):

Let x = ⁱ⁺¹_N and u = ^j+1_N . Then ldb^{^}(F^{^}XF^{^}Y+)ⁱ⁺¹_N = j+1inf

N ^{2i+1N 1}]

F^{^}Xj+1 N

+F^{^}Y i+1 N ^;j+1

N +1 ⁱ⁺¹_N ⁶=0

= inf_j

2i N^;1]

F^{^}_X^j+1_N +F^{^}_Y ^i;j+N_N (i⁶=^;1):

Using the correspondences (3.4.15) and (3.4.16), this can be written as ldb^{^]}(F~^{X^]}F~^{Y^]}+)i] = inf_{j=i ::: N}

;1(F~^{X^]}j]+ F~^{Y^]}i^;j + N ^;1]) (i⁶=^;1): (3.4.17) Since it is only required to calculate ldb^{^]}i]for i = 0:::N ^;1, the special case for x = 0 in (3.4.4) has been avoided (as we mentioned was possible at the end of section 3.4.1).

(ii) udb^{^]}(^F~^{X^] F}~^{Y^]} +)

Equations (3.4.5) and (3.3.22) give udb^{^}F^{^}XF^{^}Y+(x) = sup_u

20x]

F^{^}X(u) + F^{^}Y(x^;u) (x⁶= 1) Let x = _Nⁱ andu = _N^j. Then

udb^{^}F^{^}XF^{^}Y+ _Nⁱ =_j sup

N^2h0 iNⁱ

F^{^}Xj N

+F^{^}Y i N ^; j

N

i

N ⁶= 1:

Using the correspondences (3.4.15) and (3.4.16) gives

udb^{^]}~F_X^{^]} ~FY^{^]}+i] = sup_{j=0 ::: i}F~^{X^]}j] + F~^{Y^]}i^;j] (i⁶=N) (3.4.18) for i = 0:::N ^;1 and the special case (x = 1) has been avoided.

(iii) ldb^{^]}(^F~^{X^] F}~_Y^{^] ;})

From (3.4.10) and (3.3.23),

ldb^{^}F^{^}_XF^{^}_Y^;(x) = inf_u

2x 1]

F^{^}_X(u)^;F^{^}_Y(u^;x) (x⁶= 0):

Set x = ⁱ⁺¹_N and u = ^j+1_N . Then ldb^{^}F^{^}XF^{^}Y^{; i+1}_N =_j+1 inf

N ^2hi+1 N 1ⁱ

F^{^}Xj+1 N

;F^{^}Y j+1 N ^;i+1

N

i+1

N ⁶= 0: Using the correspondences (3.4.15) and (3.4.16)

ldb^{^]}F~^{X^]} ~FY^{^];}i] = inf_{j=i ::: N}

;1

F~^{X^]}j]^; ~F_Y^{^]}j^;i] (i⁶=^;1) (3.4.19) which is all that is required for i = 0:::N ^;1.

(iv) udb^{^]}( ~^F_X^{^] F}~^{Y^] ;})

Using (3.4.11) and (3.3.23),

udb^{^}F^{^}_XF^{^}_Y^;(x) = sup_u

20x]

F^{^}_X(u)^;F^{^}_Y(u^;x + 1) (x ⁶= 1):

Setting x = _Nⁱ and u = _N^j gives udb^{^}F^{^}XF^{^}Y^; _Nⁱ =_j sup

N^2h0 iNⁱ

F^{^}X j N

;F^{^}Y j N ^; i

N + 1 _N^{i 6}= 1 and thus

udb^{^]}~FX^{^]}F~^{Y^];}i] = sup_{j=0 ::: i}~FX^{^]}j]^;F~^{Y^]}j^;i + N^;1] (i⁶=N) (3.4.20) which holds for i = 0:::N^;1.

The analogous formulae for product and quotient are ldb^{^]}(F~^{X^]}F~^{Y^]} )i] = inf_{j=i ::: N}

;1(F~^{X^]}j] F~^{Y^]}i^;j +N ^;1]) (i⁶=^;1): (3.4.21) udb^{^]}~F_X^{^]} ~FY^{^]} i] = sup_{j=0 ::: i}~F_X^{^]}j] ~FY^{^]}i^;j] (i⁶=N) (3.4.22)

ldb^{^]}F~^{X^]} ~F_Y^{^]}i] = inf_{j=i ::: N}

;1

F~^{X^]}j]= ~F_Y^{^]}j^;i] (i⁶=^;1) (3.4.23) udb^{^]}~F_X^{^]}F~^{Y^]}i] = sup_{j=0 ::: i}~F_X^{^]}j]=F~^{Y^]}j^;i + N ^;1] (i ⁶=N) (3.4.24) All these hold fori = 0:::N^;1. The use of lower and upper bounds alone (instead of distributions within the bounds), and whether F~^{X^]} or ~FX^{^]} should be used in a particular instance, is discussed at the end of section 3.3.4.

Two signicant points to note about the above formulae for ldb^{^]} and udb^{^]} are their low computational complexity and their lack of approximation error. To calculate a dependency bound with N points in terms of two N point discrete approximations requires only O(N²) operations. The results are free from error in the two senses of any representation error being rigorously bounded byF~^{X^]}and ~FX^{^]}

(see above), and the supremum and inmum operations are exact (compared with the numerical implementation of the formulae developed in section 3.3 for which approximation is required).

The nal point which needs to be settled is how to generate F~^{X^]} and ~F_X^{^]} from a given exact distribution FX. This is in fact quite simple. One just sets F^{^}X(p) = F^{^}X(p) = FX^{^}(p) for p ² 01) and uses the denitions (3.4.15) and (3.4.16). The only diculty is nding FX^{^}(p) given a formula for FX(x). While for some simple distributions (such as the uniform and triangular distributions) an exact formula for FX^{^}(p) can be derived, in general one has to perform a numerical search. This will give FX^{^}(p) to any desired accuracy. This method is described, along with various techniques for numericallycalculatingFX for some common distributions, in chapter 5 of 451]. Some examples are given later. When the distributionFX has unbounded support, it is necessary to curtail the distribution so that ~FX^{^]}0]and F~^{X^]}N^;1]are nite.

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