Klir G.J. Uncertainity and Information. Foundations of Generalized Information Theory

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This equation may help, in principle, to determine the value of a in Eqs. (9.9)

and (9.10) for which the probability–possibility transformation,p ´ r, is uncer-

tainty-invariant in the given sense. However, it is known (see Note 9.5) that

no value of a exists for the some possibility proﬁles that satisfy Eq. (9.11). It

is also known that we do not encounter this limitation under a different, but

closely connected type of scales—the log-interval scales. It is thus reasonable

to abandon interval scales and formulate probability–possibility transforma-

tions in terms of the more satisfactory log-interval scales.

4. Log-Interval Scales. Log-interval scale transformations have the form

= bp

for all i Œ ⺞

, where a and b are positive constants. Determining

the value of b from the possibilistic normalization (d), we obtain

(9.12)

for all i Œ ⺞

. The value of a is then determined by applying Eq. (9.11), which

expresses the requirement of uncertainty invariance. This equation assumes

the form

(9.13)

where S(p) is the Shannon entropy of the given probability distribution p.

After solving the equation (numerically) and applying the resulting value of

a to Eq. (9.12), we obtain the desired possibility proﬁle r, one that is connected

to p via a log-interval scale and contains the same amount of uncertainty as p

in the sense of Eq. (9.11).

For the inverse transformation, from r to p, we determine the value of b via

the probabilistic normalization (c), and obtain

(9.14)

for all i Œ ⺞

, where

Equation (9.11) now assumes the form

(9.15)

()

log ,

1 a

()

log ,

392 9. METHODOLOGICAL ISSUES

where GH(r) is the generalized Hartley measure of the given possibility proﬁle

r. Solving this equation for a and applying the solution to Eq. (9.14) results in

the sought probability distribution p.

Figure 9.8 is a convenient overview of the procedures involved in proba-

bility–possibility transformation that are uncertainty-invariant (in the sense of

Eq. (9.11)) and are based on log-interval scales.

EXAMPLE 9.13. Let p =·0.7, 0.2, 0.075, 0.025Ò. The uncertainty invariant log-

interval scale transformation of p into r is determined as follows. First, we cal-

culate S(p) = 1.2379. Then, we solve Eq. (9.13), which assumes the form

We obtain (by a numerical method) a = 0.2537. Using Eq. (9.12), we calculate

components of the desired possibility proﬁle r =·1, 0.728, 0.567, 0.429Ò.

EXAMPLE 9.14. Let r =·1, 0.7, 0.5Ò. To determine p connected to r via a log-

interval scale and the principle of uncertainty invariance expressed by Eq.

(9.15), we need to calculate GH(r) = 0.99281 ﬁrst. The equation then assumes

the form

where

Solving this equation results in a = 0.262215.Applying this value to Eq. (9.14),

we readily obtain

Observe that S(p) = 0.992482, which is virtually the same as GH(r), except for

a tiny difference at the sixth decimal place due to rounding.

Let us now take the resulting probability distribution p as input and convert

it to the associated possibility proﬁle by the same variation of the principle of

uncertainty invariance and log-interval scales. We now apply Eq. (9.13), which

has the form

0 992482 1 5

. log . .=

p =· Ò0 753173 0 193264 0 0535633.,.,. .

s =+ +107 05

...

0 99281

1 1 07 07 05 05

. log

log

,=- - -

sss s s s

aaaa

1 2379 0 2857 0 585 0 1071 0 415 0 0357. . .. ...=+¥+¥

aaa

9.5. PRINCIPLE OF UNCERTAINTY INVARIANCE 393

Its solution is a = 0.262215, which, as expected, is the same value as the one

obtained for the inverse transformation r Æ p. Applying this value to Eq.

(9.12), we readily obtain

which is the same as the initial possibility proﬁle in this example. This again

conforms to our expectation.

5. Ordinal Scales. Scales of one additional type are applicable to uncer-

tainty-invariant transformations between probabilities and possibilities. These

are known as ordinal scales. As the name suggests, ordinal scales are only

required to preserve the ordering of components in the n-tuples p and r.Their

r =· Ò10705,.,. ,

394 9. METHODOLOGICAL ISSUES

Eg. (9.12):

Eg. (9.13)

S(p)Eg. (9.11): GH(r)

Eg. (9.14):

Eg. (9.15)

From probabilities to possibilities:

r =

From possibilities to probabilities:

()

Figure 9.8. Uncertainty-invariant probability–possibility transformations based on log-interval

scales.

form is r

= f(p

) for all i Œ ⺞

, where f is a strictly increasing function. Con-

trary to uncertainty-invariant transformations p ´ r based on log-interval

scales, those based on ordinal scales are in general not unique. This is not nec-

essarily a disadvantage, since the additional degrees of freedom offered by

ordinal scales allow us to satisfy various additional requirements.

EXAMPLE 9.15. As a simple example to illustrate probability–possibility

transformations under ordinal scales, let the probability distribution

be given and let the aim be to determine the corresponding possibility proﬁle

r by using the principle of uncertainty invariance under ordinal scales. Then,

where a = f(0.15) and b = f(0.1). Now applying the principle of uncertainty

invariance, we obtain the equation

This means that we still have one remaining degree of freedom. Choosing, for

example, a as the free variable, we have

(9.16)

Since it is required that b £ a, we obtain from the last equation the inequality

which can be rewritten as

Since it is also required that a £ 1, the range of a is deﬁned by the inequalities

(9.17)

The set of all possibility proﬁles r for which

GH Srp

()

££

≥

()

log

Sa ap

()

-£log ,

bS a=

()

-p log .

Sab b

()

log log

log .

r =· = Òr aabbb

1,,,,, ,

p =· Ò04 015 015 010101.,.,.,.,.,.

9.5. PRINCIPLE OF UNCERTAINTY INVARIANCE 395

is thus characterized by the set of all pairs that satisfy Eq. (9.16) within the

restricted range of a deﬁned by inequality (9.17). For the given probability dis-

tribution p, the set of possibility proﬁles r for which GH(r) = S(p) = 2.346 is

characterized by equation

where a Œ [0.908, 1]. This set is shown in Figure 9.9. Also shown in the ﬁgure

is the possibility proﬁle based on log-interval scales and the one for which the

probability–possibility consistency index c, deﬁned for any p and r by the

formula

(9.18)

reaches its maximum. Observe that each value of a in the given range deﬁnes

an ordinal scale characterized by a function f

. For example, when a = 0.95,

then f

(0.4) = 1, f

(0.15) = 0.95, and f

(0.1) = 0.84; moreover GH(f

(p)) = 2.346.

Now consider a generalization of Example 9.15, in which the given proba-

bility distribution contains d distinct values (d £ n). Then, clearly, there are

d - 2 free variables, since we have two equations for d distinct values in r:

= 1 (possibilistic normalization) and GH(r) = S(p). In Example 9.15, d = 3,

cpr

ba=-2 346 1 585..,

396 9. METHODOLOGICAL ISSUES

Maximum c

Log-interval scale

1.0

0.9

0.8

0.7

0.9

0.92

0.908

0.94

0.96 0.98

1.0

Figure 9.9. The sets of possibility proﬁles derived in Example 9.15.

and hence, there is only one free variable. The next example illustrates a gen-

eralization to d = 4.

EXAMPLE 9.16. Given the probability distribution

the corresponding possibility proﬁles,

based on ordinal scales can be expressed in the form

where b

, b

represent distinct values of possibilities. Now, the principle

of uncertainty invariance is expressed by the equation

One of the three variable (say b

) can be expressed in terms of the other two

variables, chosen as free variables. After calculating S(p) = 2.493 and inserting

it into the equation, we obtain

The ordinal-scale transformation is characterized by the equalities

After expressing b

in the last two inequalities in terms of b

and b

, we obtain

the following set of four inequalities regarding the two free variables b

and

, which characterize the set of all possibility proﬁles whose nonspeciﬁcity is

equal to S(p):

(a)

(b)

(c)

(d)

This set is illustrated visually by the shaded area in Figure 9.10. The probabil-

ity–possibility index c deﬁned by Eq. (9.18) reaches its maximum at b

= b

2 288 1 669 2 852 0

2 288 0 669 2 852 0

-≥

+-≥

--+≥

...

....

-≥

≥

b .

bbb

423

2 852 2 288 0 669=- -.. ..

Sbb bb bp

()

23 4 34 2 4 2

4611log log log .

r =· = Òb bbbbbbbbbb

1 2223344444

1,,,,,,,,,,

r =· Œ Òri

⺞

p =· Ò05010101005005002 002 002 002 002.,.,.,.,.,.,.,.,.,.,. ,

9.5. PRINCIPLE OF UNCERTAINTY INVARIANCE 397

0.964. The log-interval scaling transformation is obtained for b

= 0.778 and

= 0.698 (a = 0.156122).

The examined examples clearly demonstrate that uncertainty–invariant

transformations from probabilities to possibilities under ordinal scales can be

computed without any insurmountable difﬁculties. However, the complexity

of these computations grows extremely rapidly with the number of distinct

values in the given probability distributions. The opposite transformations

from possibilities to probabilities are more difﬁcult. The main obstacle is

Eq. (9.11). It is, in these transformations, a nonlinear equation, that makes any

symbolic treatment virtually impossible. Hence, we need to determine numer-

ically all combinations of values of d - 1 unknowns (one of the unknowns is

eliminated by the probabilistic normalization) that satisfy the equation and all

the inequalities characterizing ordinal scales. More research is needed to deal

with this particular problem.

398 9. METHODOLOGICAL ISSUES

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5

0.6 0.7

0.8

0.9

0.721, 0.721

0.778, 0.698

1, 0.338

1, 0.843

0.964, 0.964 : maximum c

(b)

(c)

(a)

(d)

Log-interval scale

Figure 9.10. Characterization of the set of possibility proﬁles derived in Example 9.16.

9.5.3. Approximations of Belief Functions by Necessity Functions

In this section, the uncertainty–invariance principle is illustrated by the

problem of approximating given belief functions of DST, by necessity func-

tions of the theory of graded possibilities. The primary reason for pursuing

these approximations is to reduce computational complexity.It is assumed that

the uncertainty to be preserved is the total aggregated uncertainty S

. That is,

the principle is expressed by the equation

(9.19)

where Bel and Nec denote, respectively, a given belief function and a neces-

sity function that is supposed to approximate Bel. It is assumed that Bel and

Nec are deﬁned on P(X ).

In addition to Eq. (9.19), it is also required that Bel and Nec satisfy the

inequality

(9.20)

for all A ŒP(X). The motivation for using this requirement stems from the

fact that belief functions are more general than necessity functions.Therefore,

they are capable of expressing given evidence more precisely. This in turn,

means that the set containment

should hold for all A ŒP(X) when Nec approximates Bel. Functions Bel and

Nec that are deﬁned on the same universal set and satisfy inequality (9.20) are

said to be consistent.

The approximation problem addressed in this section is thus formulated as

follows: Given a belief function, Bel, determine a necessity function, Nec, that

satisﬁes the requirements (9.19) and (9.20). The class of necessity functions

that satisfy these requirements is characterized by the following theorem.

Theorem 9.1. Let Bel denote a given belief function on X (with |X|=n), and

let {A

}

i=1

(with s Œ ⺞

) denote the partition of X consisting of the sets A

chosen

by Step 1 of Algorithm 6.1 in the ith pass through the loop of Steps 1 to 5

during the computation of S

(Bel). A necessity function Nec satisﬁes the

requirements (9.19) and (9.20) if and only if

(9.21)

for each i Œ ⺞

Proof. [Harmanec and Klir, 1997]. 䊏

Nec A Bel A

()

Bel A Pl A Nec A Pos A

() ()

[]

() ()

[]

Nec A Bel A

()

S Bel S Nec

()

9.5. PRINCIPLE OF UNCERTAINTY INVARIANCE 399

We see from the theorem that, unless s = n and |A

|=1 for all i Œ ⺞

, the

solution to the approximation problem is not unique. The question is what cri-

teria should be used to choose one particular approximation.Two criteria seem

to be most natural. According to one of them, we choose the necessity func-

tion with the maximum nonspeciﬁcity; according to the other one, we choose

the necessity function that is in some sense closest to the given belief func-

tion. Choices based on the ﬁrst criterion are addressed by the following

theorem.

Theorem 9.2. Among the necessity functions that satisfy Eq. (9.21) for given

belief function Bel, the one that maximizes nonspeciﬁcity is given by the

formula

(9.22)

for all B ŒP(X), where i is the largest integer such that , and it is

assumed, by convention, that .

Proof. [Harmanec and Klir, 1997]. 䊏

EXAMPLE 9.17. To illustrate the meaning of Eq. (9.22), consider the belief

function Bel deﬁned in Table 9.11. When applying Algorithm 6.1 to this belief

function, we obtain

Hence, the unique necessity function, Nec

, that approximates Bel (i.e., that

satisﬁes requirements (9.19) and (9.20)) and maximizes nonspeciﬁcity is the

one speciﬁed in Table 9.11, together with the associated functions Pos

and m

This approximation may conveniently be represented by the possibility proﬁle

In order to apply the second criterion operationally, we need a meaningful

way of measuring the closeness of a necessity function to a given belief func-

tion. For this purpose, it is reasonable to use functional D

Bel

deﬁned by the

formula

(9.23)

for each necessity function Nec that is consistent with a given belief function

Bel. We want to minimize D

Bel

(Nec) for all necessity functions that are con-

sistent with Bel and satisfy Eq. (9.19) or, according to Theorem 9.1, satisfy Eq.

(9.21).

D Nec Bel A Nec A

Bel

()

[]

()

1070702=· Ò,.,.,. .

Bel A Bel a

Bel A Bel b c

Bel A Bel d

()

{}

()

{}

()

{}

()

,.,

=∆

Nec B Bel A

()

400 9. METHODOLOGICAL ISSUES

Table 9.11. Necessity Approximations of a Belief Function (Examples 9.17 and 9.18)

X Given Approximation 1 Approximation 2

a b c d Bel(A) Pl(A) m(A) Nec

(A) Pos

(A) m

(A) Nec

(A) Pos

(A) m

(A)

A:0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

1000 0.3 0.6 0.3 0.3 1.0 0.3 0.3 1.0 0.3

0100 0.0 0.4 0.0 0.0 0.7 0.0 0.0 0.7 0.0

0010 0.1 0.4 0.1 0.0 0.7 0.0 0.0 0.6 0.0

0001 0.0 0.2 0.0 0.0 0.2 0.0 0.0 0.2 0.0

1100 0.4 0.9 0.1 0.3 1.0 0.0 0.4 1.0 0.1

1010 0.4 1.0 0.0 0.3 1.0 0.0 0.3 1.0 0.0

1001 0.5 0.6 0.2 0.3 1.0 0.0 0.3 1.0 0.0

0110 0.4 0.5 0.3 0.0 0.7 0.0 0.0 0.7 0.0

0101 0.0 0.6 0.0 0.0 0.7 0.0 0.0 0.7 0.0

0011 0.1 0.6 0.0 0.0 0.7 0.0 0.0 0.6 0.0

1110 0.8 1.0 0.0 0.8 1.0 0.5 0.8 1.0 0.4

1101 0.6 0.9 0.0 0.3 1.0 0.0 0.4 1.0 0.0

1011 0.6 1.0 0.0 0.3 1.0 0.0 0.3 1.0 0.0

0111 0.4 0.7 0.0 0.0 0.7 0.0 0.0 0.7 0.0

1111 1.0 1.0 0.0 1.0 1.0 0.2 1.0 1.0 0.2

401