OWA aggregation with an uncertainty over the arguments

Accepted Manuscript

OWA Aggregation with an Uncertainty Over the Arguments Ronald R. Yager PII: DOI: Reference:

S1566-2535(18)30606-7 https://doi.org/10.1016/j.inffus.2018.12.009 INFFUS 1067

To appear in:

Information Fusion

Received date: Revised date: Accepted date:

30 August 2018 26 December 2018 31 December 2018

Please cite this article as: Ronald R. Yager , OWA Aggregation with an Uncertainty Over the Arguments, Information Fusion (2019), doi: https://doi.org/10.1016/j.inffus.2018.12.009

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

HIGHLIGHTS

AC

CE

PT

ED

M

AN US

CR IP T

• Introduce idea of weight generating function & describe use in OWA aggregation • OWA aggregation when we have a probability distribution over the arguments • Aggregation using OWA when there is a measure based uncertainty over arguments • Obtain weight generating function from given set of OWA weights • Use of Choquet Integral to model combined OWA & uncertain aggregation

1

ACCEPTED MANUSCRIPT

Ronald R. Yager

CR IP T

OWA Aggregation with an Uncertainty Over the Arguments

Machine Intelligence Institute Iona College

AN US

New Rochelle, NY 10801

M

[email protected]

AC

CE

PT

ED

Technical Report #MII-3901R

2

ACCEPTED MANUSCRIPT

ABSTRACT We discuss the OWA aggregation operation and the role the OWA weights play in determining the type of aggregation being performed.

We introduce the idea of a weight

generating function and describe its use in obtaining the OWA weights. We emphasize the importance of the weight generating function in prescribing the type of aggregation to be

CR IP T

performed. We consider the problem of performing a prescribed OWA aggregation in the case when we have a probability distribution over the argument values. We show how we use the weight generating function to enable this type of aggregation. Next we consider the situation when we have a more general measure based uncertainty over the argument values. Here again

AN US

we show how we can use the weight generating function to aid in performing the prescribed OWA aggregation in the face of this more general type of uncertainty. Finally we look at the task of obtaining a weight generating function from a given set of OWA weights.

Keywords: Measure-Based Uncertainty; Choquet Integral; Weight Generating Function;

1. Introduction

ED

M

Aggregation

The aggregation of a collection of values is a pervasive problem in the modern

PT

technologically focused world it is central to the such fields as data science [1, 2], data mining

CE

[3-5] and decision making [6, 7]. The ordered weighted averaging (OWA) operator is extensively used to perform this aggregation operation [8]. One reason for the popularity of the OWA

AC

operator is its ability to implement many different types of aggregation by simply changing the values of the parameters associated with the OWA operator, these parameters are called the OWA weights. In a given application these OWA weights can be provided directly or indirectly via a weight generating function f, which, as the name implies can be used to obtain the weights used in the aggregation. As discussed by Yager [9, 10] the weight generating function can, in some situations, be expressed in a linguistic manner which can make it easier for users to specify the type of aggregation they want to be performed. However, the important point we want to emphasize here is that the type of aggregation to be performed is implicit in the mathematical 3

ACCEPTED MANUSCRIPT

form of the weight generating function. A frequent situation where we want to aggregate a collection of values is when we want to find one representative value of the collection of values. One prominent case where we want to find a representative value of a collection values occurs in decision-making under uncertainty. Here the collection of values is the possible outcomes of a decision alternative and we need a representative value to compare different decision alternatives.

CR IP T

Our interest here is in the extension of the OWA aggregation to the situation in which there is some uncertainty over the values being aggregated, the collection of possible outcomes of a course of action. Formally the problem that we focus on is the OWA aggregation, guided by a weight generating function f, of collection of values over which there is an uncertainty. While we

AN US

initially consider a probabilistic uncertainty as we did in our earlier work [11-13] and our more recent work [14] we then we look at a more general fuzzy measure based formulation of the uncertainty,  [15].

2. The Ordered Weighted Average (OWA) Aggregation

M

In [16] we introduced the ordered weighted average (OWA) operator, which provides a general class of mean like aggregation operators. These operators are defined as follows.

ED

Definition: An OWA operator of dimension n is a mapping F: R n  R which has an associated

å w j = 1 with j=1

n

F(a1, …, an) =

å w j a(j) j=1

CE

PT

collection of n weights wj for j = 1 to n such that wj  [0, 1] and

n

where  is an index function so that (j) is index of the j largest ai,

AC

The key feature of this aggregation operation is the ordering of the arguments by value,

via the index function . This ordering process introduces nonlinearity into the operation. It can be

shown

this

is

a

mean,

it

is

commutative,

monotonic

and

bounded

Mini[ai]  F(a1, …, an)  Maxi[ai]. It is also idempotent, if all ai = a then F(a1, …, an) = a. At times we shall find it convenient to denote the collection of weights as W = (w 1, …, wn) and denote F(a1, …, an) and FW(a1, …, an). We refer to W as the weighting vector. The generality of the OWA operator lies in the fact that by appropriately selecting the weights we can implement different aggregation operations. Consider the situation where the 4

ACCEPTED MANUSCRIPT

weight are such that w1 = 1 and wj = 0 for j  1 which we denote as W*. In this case we get that F(a1, …, an) = Maxj[aj]. If the weights are such that wn = 1 and wj = 0 for j  n, denoted as W*, we get, F(a1, …, an) = Minj[aj]. If the weights are such that wj = 1/n for all j, denoted as WAVE, in this case we get F(a1, …, an) =

1 n åa j . n j=1

An interesting OWA operation is the Olympic aggregation operator. For the Olympic

CR IP T

n-1

1 1 aggregator w1 = wn = 0 and wj = for j  1 or n. In this case F(a1, …, an) = å a( j) . n-2 j=2 n-2 Here we eliminate the biggest and smallest arguments and take the average of the rest.

If wk = 1 and wj = 0 for j  k, denoted W(k), we get F(a1, …, an) = a(k), the kth largest ai.

AN US

Another notable aggregation, is the so-called Hurwicz aggregation. Here with 0    1 we let w1 =  and wn = 1 -  and we get F(a1, …, an) =  Maxj[aj] + (1 - ) Min[aj]. In [16, 17] Yager associated with a weighting vector W a characterizing feature called its n

attitudinal character. AC(W) =

n- j

å n -1 w j .

In [16] it was shown AC(W)  [0, 1]. The closer

M

j=1

AC(W) to one the more Max like the aggregation and the closer AC(W) is to zero the more Min

ED

like the aggregation. It can be shown that AC(W*) = 1 and AC(W*) = 0 and AC(WAVE) = 0.5. n-k For the Hurwicz aggregation we have AC(W) = . For W(k) we have AC(W) = . Typically n-1

PT

the larger the attitudinal character the bigger the aggregated value. In [18] we suggested a way of obtaining the OWA weights with the use of a

CE

weight-generating function f.

Here f: [0, 1]  [0, 1] is a function having the properties

AC

1) f(0) = 0, 2) f(1) = 1 and 3) monotonicity, f(a)  f(b) if a > b. Using this weight-generating j j-1 function we get the wj = f( ) - f( ). We see that these generated weights have the required n n j j-1 properties of OWA weights. Since f is monotonic the f( )  f( ) and hence wj > 0. Since n n n n j j-1 j-1 j f(1) = 1 then f( ) - f( )  1 and hence wj  [0, 1]. Furthermore å w j = å (f( ) - f( )) = n n n n j=1 j=1 j j-1 f(1) - f(0) = 1. With wj = f( ) - f( ) it is interesting to see that n n j j j j k k-1 å w k = å (f( n ) - f( n ) ) = f( n ) - f(0) = f( n ). k=1 k=1

5

ACCEPTED MANUSCRIPT

If we let Ff(a1, …, an) denote the OWA aggregation of the ai based on the weight-generating function f we have n

j

å (f( n ) - f(

Ff(a1, …, an) =

j=1

j-1 ))a(j) n

j j-1 Consider attitudinal character in the case where w j = f( ) - f( ), using this we see that n n n j n- j n- j j-1 ) wj = å ( ) (f( ) - f ( )) n -1 n -1 n n j=1 j=1

å(

CR IP T

n

AC(W) =

Here we see that AC(W) depends on n. In [17, 18] we suggested a formulation for the attitudinal character in the case of a weight-generating function f independent of n. In particular it can be shown that n

n-j

j

j=1

1

1 j-1 ) ) = ò f (x)dx 0 n

AN US

å ( n-1)(f( n ) – f( n¥ lin

ò0 f (x)dx provides a cardinality free formulation of the attitude character

Thus here the AC(f) = of f.

M

In [10] Yager described a technique for obtaining a weight-generating function based on a quantifier expressing the aggregation imperative of the a i. Consider a linguistic qualifier Q such is

ED

most, at least about . Using Zadeh’s idea of computing with words [19] we can represent the linguistic qualifier Q as a proportional fuzzy subset Q of the unit interval. If Q is a monotonic

PT

quantifier, then the membership function of Q, fQ, has the form of a weight-generating function.

CE

3. The OWA Aggregation with Probability Distribution Over the Arguments In [13] we extended the weight-generating-function based OWA aggregation of the a i to

AC

the case where we have an additional probability distribution over the a i. Thus here pi  [0, 1] is n

the probability associated with ai and

å pi = 1.

Here we desire Ff((a1, p1), (a2, p2), …, (an, pn)).

i=1

Again letting (j) be the index of jth largest ai in [13] we indicated that n

Ff((ai, pi) for i = 1 to n) =

å (f (Tj) - f(Tj - 1))a(j) j=1

j

where Tj =

å p(k) .

We see that Tj is the probability of the set of arguments with the j largest

k=1

6

ACCEPTED MANUSCRIPT

values for ai, thus if Hj = {a(1), …, a(j)} then Tj = Prob(Hj). We can now indicate some properties of the f(Tj). 1) f(Tj)  [0, 1], since Tj  [0, 1] 2) f(Tn) = 1, since Tn = 1 3) f(T0) = 0, since T0 = 0

CR IP T

4) f(Tj)  f(Tj - 1), since Tj  Tj - 1 5) f(Tj) - f(Tj - 1)  [0, 1], follows from the above n

6)

å (f (Tj) - f(Tj - 1)) = 1 j=1

Here we see that the f(Tj) - f(Tj - 1) have the properties of OWA weights. Let us denote

AN US

= f(Tj) - f(Tj - 1) and we shall refer to these as joint OWA-Probability weights, O-P weights. Thus here we have Ff((ai, pi)) =

j j 1 1 j for all i. In this case Tj = å p(k) = å = . From this we get n n n k=1 k=1 n

Ff((ai, pi)) =

M

Assume pi =

ap(j). Let us look at some special cases.

å (f (Tj) - f(Tj - 1))a(j) =

n 1 j- 1 ( ) – f( )) a = (f å n (j) å w j a(j) n j=1 j=1

ED

j=1

n

It is just the OWA aggregation of the ai as determined by the weight generating function f.

PT

Consider now the case where f is linear, f(x) = x, here

CE

Ff((ai, pi) for i = 1, n) =

n

å (f (Tj) - f(Tj - 1))ap(j) = j=1

n

å (Tj - Tj - 1)ap(j) j=1

AC

Since Tj - Tj - 1 = p(j) then n

Ff((ai, pi) for i = 1, n) =

å p( j) a(j) = j=j

n

å pi ai i=1

It is simply the expected value of the ai under the probability distribution pi. In passing we note that when f is linear then the regular OWA weights are 1/n. We see this proposed calculation for Ff((ai, pi) for i = 1, n) includes both the formulations for the expected values of ai under the probability distribution pi and the ordinary OWA aggregation based on the weight-generating function f. Let us look at Ff((ai, pi) for i = 1, n) for some other notable cases of f and probability 7

ACCEPTED MANUSCRIPT

distribution P. Assume we have the case where one element, a r, has probability 1. Here pr = 1 and pi = 0 for us i  r. In this case Tj = 1 if ar  Hj and Tj = 0 for if ar  Hj. Assume ar is the j* largest argument. Here then Tj = 0 for j < j* and Tj = 1 for j  j*. In this case f(Tj) = 0 for j < j* and F(Tj) = 1 for j  j*. From this we see n

Ff((ai, pi)) =

å (f(Tj ) - f(Tj - 1))a(j) j -1

Ff((ai, pi)) =

å (f(Tj ) - f(Tj - 1))a(j) + (f( T * ) - f( T * )) a j=1

*

Ff((ai, pi)) =

j -1

j

j -1

n

j=1

j=j*+1

å 0 a(j) + a( j* ) + å

0 a(j) = a

( j* )

CR IP T

j=1

*

n

( j* )

= ar

+

å *

(f(Tj ) - f(Tj - 1))a(j)

j= j +1

AN US

Thus here independent of f if pr = 1 then the aggregation of ff((ai, pi)) = ar.

Consider now the complementary situation here pr = 0. Again assume ar is the j* largest j*

argument. Here we see that T * = j

j*-1

å p( j) = å p( j) j=1

= T * . Thus here f( T * ) = f( T * ) and j -1

j=1

weight associate with a

( j* )

M

å (f(Tj )

- f(Tj - 1))a(j) we easily see that the

j=1

is zero and since a

ED

j -1

n

here f( T * ) - f( T * ) = 0. Since Ff((ai, pi)) = j -1 j ( j* )

j

= ar and if ar has pr = 0 then it does not

contribute to the aggregation.

PT

We have already shown if f is linear we get the simple expected value. Let us now look at some other notable weight generating functions f.

Consider a step like weight generating

CE

function, f(x) = 0 for x < b and f(x) = 1 for x  b.

AC

f(Tj) = 1 for Tj  b. We recall that Ff((ai, pi)) =

Here f(Tj) = 0 for Tj < b and

n

å (f(Tj ) - f(Tj - 1))a(j).

Assume j* is such that

j=1

T *  b and Tj*-1 < b. Here we see f(Tj) = f(Tj - 1) for all j  j* and f( T * ) - f( Tj*-1) = 1. Thus j j we have Ff((ai, pi)) = a * . Here the aggregated value is the value of the ordered element that is ( j )

the position where Tj transitions from being less than b to greater or equal b. Actually the step like weight generating function at b corresponds to an ordinary OWA aggregation with wk = 1 and wj = 0 for j  k where (k - 1) < bn  k Two special cases of step-like weight generating function are the max and min. For the min we have b = 1 and for the max we have b = , where  is some arbitrary small value near zero. 8

ACCEPTED MANUSCRIPT

In the case of min, b = 1, we can show that Ff(ai, pi) is the smallest argument with a non-zero probability and in the case of max, b = , the Ff(ai, pi) is the largest argument with a non-zero probability. Actually the max is obtained from weight generating function f* where f*(0) = 0 and f*(x) = 1 for x > 0. The min is obtained from f* where f*(x) = 0 for x < 1 and f*(1) = 1.

function with step at b we have AC(f) =

1

1

CR IP T

We see that AC(f*) = 1 and AC(f*) = 0. We note that for a step-like weight generating

ò0 f(x) dx = òb dx = 1 - b.

the larger the AC. For the linear function f(x) = x we get 1

ò0 x dx =

x2 = | 2 0

1 2

AN US

AC(f) =

1

Thus the smaller value of b,

Thus we see that the attitudinal character of f locates it on a scale, from 0 to 1 where 0 is the minimum and one is the max. Here the linear function is 0.5.

M

An interesting class of weight generating functions are f(x) = x  for  > 0. In this case 1 1 AC(f) = ò x dx = . Here we see that the smaller  the more max like aggregation. In 0 +1 particular as   0 then AC  1 while as   ¥, AC  0. In this case f(Tj) = (Tj) and using this we see

ED

f(Tj) - f(Tj - 1) = (Tj) - (Tj - 1) =

n

å ((Tj )  - (Tj - 1))a(j) j=1

PT

Ff((ai, pi)) =

where  is an index function so that (j) is the index of the argument with the jth largest ai and j

å p(k) , the sum of the probabilities of j largest arguments.

k=1

CE

Tj =

AC

Let us provide an illustrated example of the calculation n

Ff((ai, pi)) =

å (f(Tj ) - f(Tj - 1))a(j) j=1

for some examples of f. Example: Assume n = 4, with a1 = 20, a2 = 10, a3 = 40, a4 = 30 and p1 = 0.4, p2 = 0.2, p3 = 0.3 and p4 = 0.1. Here since a3 > a4 > a1 > a2 we have (1) = 3, (2) = 4, (3) = 1 and (4) = 2. In this example a(1) = 40, a(2) = 30, a(3) = 20 and a(4) = 10. We also see that p(1) = 0.3,

9

ACCEPTED MANUSCRIPT

j

p(2) = 0.1, p(3) = 0.4, and p(4) = 0.2. With Tj =

å p(k) we get T1 = 0.3, T2 = 0.4, T3 = 0.8

k=1

and T3 = 1. = f(Tj) - f(Tj - 1) and

F((ai, pi)) =

ap(j) =

(40) +

(30) +

Let us look at this for different weight generating functions

(20) +

(10)

CR IP T

Here we have

1. f(x) = x2 here f(T1) = 0.09, f(T2) = 0.16, f(T3) = 0.64 and f(T4) = 1. For this weight generating function = 0.09 - 0 = 0.09

AN US

= 0.16 - 0.09 = 0.07 = 0.64 - 0.16 = 0.48 = 1 - 0.64 = 0.36 For this example Ff(ai, pi)) = 18.9

M

2. f(x) = x1/2 here f(T1) = 0.55, f(T2) = 0.63, f(T3) = .089 and f(T4) = 1. For this weight

= 0.55 = 0.08

= 0.1

PT

= 0.27

ED

generating function

CE

For this example Ff((ai, pi)) = 30.7. 3. f(x) = x. Here f(Ti) = Ti and hence

AC

= 0.3

= 0.1 = 0.4 = 0.2

For this example Ff((ai, pi)) = 25. 4. Consider a step-like weight generating function f(x) = 0 if x < a f(x) = 1 if x  a 10

ACCEPTED MANUSCRIPT

where a = 0.5. Here we get =0 =0 =1 =0

CR IP T

and hence Ff((ai, pi)) = 20 n

Consider Ff((ai, pi), i = 1 to n) =

å (f (Tj) - f(Tj - 1))a(j) where (j) is the index of jth j=1

j

largest ai and Tj =

å p(k) the

sum of probabilities of the j largest arguments.

k=1

Denoting

AN US

= f(Tj) - f(Tj - 1), the, OWA-probability, O-P, weights we see Ff((ai, pi), i = 1 for n) =

ap(j)

This formulation inspires us to consider a joint OWA-probability attitudinal character AC(f, P). n

=

M

AC(f, P) =

n-j

å ( n-1)(f(Tj) - f(Tj - 1)) j=1

n

AC(f) =

ED

j j-1 We note n the case where we have no probability we get w j = f( ) - f( ) and hence n n n-j

å ( n-1)wj , the normal value of AC.

PT

j=1

In [14] we developed a related approach for combining OWA aggregation with

CE

probabilistic uncertainty called the Crescent method. In that approach instead of using the weight

AC

generating function we described the OWA aggregation directly in terms of the OWA weights.

4. Uncertainty Modeling with Fuzzy Measure Assume X = {x1, …, xn} are a set of elements. A fuzzy measure [20-22] on the set X is a

set function on X, : 2X  [0, 1] having the properties: 1) () = 0, 2) (X) = 1 and 3) if A  B then (A)  (B). We see that  associates with each subset of X a value in the unit interval. Assume 1 and 2 are two fuzzy measures on X such that for any subset A of X, 1(A)  2(A) we say 1 is larger than 2 and denote it 1  2. 11

ACCEPTED MANUSCRIPT

Assume  is a fuzzy measure of X the set function ˆ defined such that ˆ (A) = 1 - ( A ) is also a measure on X and is called the dual of . We easily see that

= .

A measure  is called self-dual ˆ (A) = (A). We observe that if  is self-dual then (A) = 1 - ( A ) and hence (A) + ( A ) = 1. This is a very special property. In [23] we discussed how to obtain new fuzzy measures from other fuzzy measures. Let

CR IP T

h: [0, 1]  [0, 1] be a monotonic function on the unit interval, if b 1 > b2 then h(b1)  h(b2), having the properties h(0) = 0 and h(1) = 1 Assume  is a fuzzy measure on X, then the set function h defined such that h(A) = h((A)) is also a fuzzy measure on X, 1) n() = h(()) = h(0) = 0 2) h(X) = h((X)) = h(1) = 1

AN US

3) If A  B, then (B)  (A), consider h here h(B) = h((B))  h((A)) = n(A). We observe that if h is linear, h(x) = x, then h = . We also observe two notable cases of h, h and h. These are defined so that h(x)  x for all x and h(x)  x for all x. We easily see that

 h   hÙ .

M

Assume V is an uncertain variable that takes its value in the space X, a fuzzy measure  can provide a very general formulation for modeling the uncertainty associated with V [15].

ED

Under this interpretation of A for any subset A  X, (A) indicates the anticipation that the value of V lies in A.

PT

If  is such that (A) = 1 if xk  A and (A) = 0 if xk  A then this is the fuzzy measure representation of the information that V = x k. If  is such that for any subset B such that xk  B

CE

we have (B  {xk}) = (B) then this is a representation of the information that V  xk.

AC

 is a probability measure if  is such that ({xi}) = pi and (A) = that since (X) = 1 then

å

p i = 1.

å  ({xi}).

We note

x iA

We can easily show that the probability measure is

all x i

self-dual and ( A ) = 1 - (A). If (xi) = i with (A) = Max [i] then  is a possibility measure. We note here that at i1x iA

least one i = 1. For a possibility measure (A  B) = Max[(A), (B)]. Another class of measures useful for modeling uncertainty are the cardinality-based measures. If i for i = 0 to n are parameters where 0 = 0, n = 1 and i + 1  i then  defined 12

ACCEPTED MANUSCRIPT

such that (A) = |A| is a cardinality-based measure. Here the anticipation that V  A just depends on the number of elements in A, independent of which elements they are. A special case i of cardinality-based measure is one in which i = , we shall call this the neutral measure and n |A| denote this neutral measure $. We note for the neutral measure $(A) = . n

CR IP T

5. OWA Aggregation with Fuzzy Measure Based Uncertainty Over the Arguments

Assume X = {x1, …, xn} where each xi is a numerical values. Let  be a fuzzy measure

defined [24] as

AN US

on X. The Choquet integral of the collection X with respect to the measure , Choq(X), is n

Choq(X) =

å ((H j ) - (Hj - 1))x(j) j=1

where  is an index function so that (j) is the index of the jth largest of the xi and Hj = {x(k) for k = 1 to j}, it is the subset of the j largest xi. Since each ((Hj) - (Hj - 1))  [0, 1] and

M

n

å ((H j ) - (Hj - 1)) = 1 we see that Choq(X) is a weighted average of the xi.

ED

j=1

The Choquet

integral is a mean or averaging function that has the following properties It is bounded by the min and max of the

PT

1) Mini(xi) ≤ Choq(X) ≤ Maxi(xi). arguments.

CE

2) If Y = {y1, …, yn} where xi ≥ yi for all i then Choq(X) ≥ Choq(Y). The Choquet integral is monotonic.

AC

3) Choquet integral is idempotent, if x i = a for all i then Choq(X) = a.

With some algebraic manipulation we can show that n

Choq(X) =

å (H j )

(xp(j) - xp(j + 1))

j=1

where xp(n + 1) = 0 by convention. From this it follows that if 1 and 2 are two measures on X such that 1 ≥ 2 then Choq  (X)  Choq  (X). 1

2

Recalling the neutral measure, $(A) =

|A| , consider the Choq  (X). Here $ n

13

ACCEPTED MANUSCRIPT

n

Choq  (X) = $

n

j j-1 1 n 1 n (x = = å ( n - n ) (j) n å x( j) n å xi j=1 j=1 i=1

å ($ (H j ) - $(Hj - 1))x(j) = j=1

It is the simple average of the elements in X. Observation: Choq  ({x1, …, xn}) is the simple average of {x 1, …, xn}. $

Assume f is a weight-generating function associated with an OWA aggregation. We recall

CR IP T

f(0) = 0, f(1) = 1 and f is monotonic, if a  b then f(a)  f(b). Consider now the set function  = f$ defined such that (A) = f($(A)). In this case  is also a fuzzy measure on X. Consider now the Choquet integral of X with respect to  = f$, here n

Choq(X) =

å (f($ (H j )) - f($(Hj - 1)) x(j) j=1

AN US

|A| j and Hj has j elements then $(Hj) = and hence n n

Since $ is cardinality based with $(A) =

n

Choq(X) =

j

å (f( n ) – f( j=1

j-1 )) x(j) n

Observation: If  = f$ then Choq(X) is the OWA aggregation of the arguments {x 1, …, xn}

M

with respect to the weights obtained using the weight-generating function f. P(A) =

å

i,x iA

Here pi is the probability of xi and p i , it is the probability of the subset A. Consider a measure P such that

å pi .

x iA

Here P is a probability measure.

PT

p({xi}) = pi and P(A) =

ED

Assume P is a probability distribution on X.

Consider the Choquet integral of {x 1, …, xn} with respect to this measure P. In this case n

CE

Choq  ({x1, …, xn}) = P

å (p (H j ) - P(Hj - 1)) x(j). j=1

AC

Here  is an index function so that x(j) is the index of jth largest xi and Hj = {x(k), k = 1 to j}. Since

Hj

=

{x(k),

P(Hj-1) = Prob(Hj-1) =

k

=

1

to

j}

then

P(Hj)

j

=

Prob(Hj)

=

å p(k)

and

k=1 j-1

å p(k) so in this case P(Hj) - P(Hj - 1) = p(j).

k=1

n

Choq  ({x1, …, xn}) = P

å p( j) xp(j) = j=1

n

å p ix i , i=1

this is the expected value of {x1, …, xn} with the probability distribution P. 14

From this we see that

ACCEPTED MANUSCRIPT

Observation: The Choquet integral of X with respect to the measure P is the expected value of the elements in X. Consider now the measure  = fp where f is the weight generating function associated with an OWA aggregation. Let us look at the Choquet integral of X with respect to the measure  = fp n

j=1

j=1

å ((H j ) - (Hj - 1))x(j) = å (f( P (H j ) - f(P(Hj - 1))x(j)

CR IP T

Choq({x1, …, xn}) =

n

Since Hj = {x(k) for k = 1 to j}, the set of elements with the j largest values for x i Choq({x1, …, xn}) =

n

j

j-1

j=1

k=1

k=1

å (f(å p(k) ) - f( å p(k) )) x(j)

AN US

We see here that Choq(x1, …, xn) is the OWA aggregation of the xi with this probability distribution P on X.

Let 1 be some arbitrary uncertainty measure of the space X.

Let f be the weight

generating function associated with an OWA aggregation on X. Consider now the measure

M

 = f1, with (A) = f1(A). Consider now the Choquet integer Choq ({x1, …, xn}) this is the OWA aggregation of the collection X = {x1, …, xn} where there is some measure  on elements

ED

in X. In this case

n

å (f((H j ) ) - f((Hj - 1))) x(j) j=1

PT

Choq({x1, …, xn}) =

Here we note that (Hj) is the measure of subset of X consisting of the elements with the j largest

CE

values for xi.

Thus we see we have developed as formulation for the OWA aggregation of a collection

AC

of arguments X where there is some uncertainty associated with the elements in X. Assume f1 and f2 are two weight generating functions so that f1(a)  f2(a) for all a. In this

case for any uncertainty measure on X, f1((Hj))  f2((Hj)) and we have 1 = f1  f2 = 2. In this case we see Choq  (X) = 1

n

n

j=1

j=1

å 1(H j ) (xp(j) - xp(j + 1)) ≥ å  2 (H j ) (xp(j) - xp(j + 1)) = Choq 2 (X)

Consider the special case of possibility measure  over X.

Here ({xi}) = i and (A) = Max (i). Since Hj = {x(k) for k = 1 to j} then (Hj) = Max [(k)] and x iA

k=1to j

15

ACCEPTED MANUSCRIPT

f(Hj) = f( Max [(k)] thus for the case where we have an a possibility measure  over X and we k=1to j

are interested in OWA aggregation of X guided by the weight generating function f then the aggregated value is n

Choqf(X) =

Max [(k)]) - f( Max [(k)])) x(j) å (f ( k=1to j k=1to j-1 j=1

over a set of element X having a possibilistic uncertainty.

CR IP T

Thus the above formula provides the OWA aggregation guided by a weight generating function f Consider the case where  is a cardinality-based measure with parameters j where j ≤  j + 1,  0 = 0 and  n = 1. First for any subset A of X, (A) = Card(A). In this situation

with the OWA aggregation then

AN US

since Hj has j elements then (Hj) = j. Here if f is the weight generating function associated n

Choqf(X) =

å (f (j) - f(j - 1))x(j) j=1

Here H0 = 

ED

H1 = {x1} = {40}

M

Example: Assume x1 = 40, x2 = 30, x3 = 20, x4 = 10. Here p(j) = j.

H2 = {x1, x2} = {40, 30}

PT

H3 = {x1, x2, x3} = {40, 30, 20} H4 = {x1, x2, x3, x4} = {40, 30, 20, 10}

AC

CE

Assume  is a possibility measure with 1 = 0.7, 2 = 0.4, 3 = 1.0 and 4 = 0.2 We see that since (Hj) = Max [k] we have x k H j

(H0) = 0, (H1) = 0.7, (H2) = 0.7, (H3) = 1, (H4) = 1

We see that

4

Choq(40, 30, 20, 10) =

å (f ((Hj) - f((Hj - 1))xj j=1

Since (H1) = (H2) and (H3) = (H4) we have Choqf(X) = f((H1)x1 + (f((H3)) - f((H2))x3= f(0.7)(40) + (f(1) - f(0.7))20 Choq(X) = f(0.7)(40) + (1 - f(0.7))20 16

ACCEPTED MANUSCRIPT

Consider the following cases of weight generating function f i) f(y) = y2. Here f(0.7) = 0.49 and (1 - 0.49) = 0.51 Choq f (y) = 29.8 ii) f(y) = y1/2. Here f(0.7) = 0.84 and (1 - 0.84) = 0.16 Choq(y) = 30.7

Choq(y) = 34

6. Obtaining a Weight Generating from OWA weights

CR IP T

iii) f(y) = y. Here f(0.7) = 0.7 and (1 - 0.7) = 0.3

AN US

Many of the procedures developed in the preceding took advantage of the availability of a weight generating function. A natural question that arises is that given a set of n OWA weights, wj for j = 1 to n, can we associate with these OWA weights a weight generating function f. Clearly the function f should have the following properties.

1) f is a mapping from the unit interval into the unit interval, f: [0, 1]  [0, 1]

M

2) f(0) = 0 3) f(1) = 1

ED

4) f is monotonic, if a > b then f(a)  f(b)

PT

5) The wj should be obtained from f, j j-1 wj = f( ) - f( ) for j = 1 to n n n

CE

It is clear that there are many functions f that have these properties. Let us look at some further features that we can associate to any f and some notable special cases in f. We can associate with any f a number of fixed points in addition to f(0) = 0 and f(1) = 1.

AC

j

j If by convention we let w0 = 0 we see that for j = 0 to n we have f( ) = å w k . From this we n k=0 j

easily see that for j = 1 to n we have f( f is fixed at the n + 1 points f(

j-1

j ( j-1) )-f = å w k - å w k = wj. Thus we see that any n n k=1 k=0

j ) for j = 0 to n. We see any freedom we have in choosing f lies in n

our determination of the value of f between these fixed points. Thus f may be described in a j-1 j precise manner by expressing its value in the open intervals ( , ) for j = 1 to n. Because of n n 17

ACCEPTED MANUSCRIPT

the required monotonicity of f we see two properties of f in these intervals are j-1 j j-1 j f( )  f(y)  f( ) for y  ( , ) n n n n j-1 j f(y2)  f(y1) if y2  y1 and y1 and y2  ( , ) n n We note that any function f that satisfies these two conditions will have monotonicity in the whole [0, 1] interval, for a, b  [0, 1] and a > b then f(a)  f(b).

CR IP T

Using these bounds we can obtain the smallest, f+, and largest, f+, weight-generating functions associated with a set of OWA weights. Here f+ is defined as j-1

f+(y) =

å wk

for y  [

k=0

j-1 j , ) for each j = 1 to n n n

AN US

f+(1) = 1

Thus f+ is essentially a step function takes a step wj at the end of jth interval. At the other extreme is the largest function f+ defined so that f+(0) = 0 j

å wk for y  (

k=0

j-1 j , ] for each j = 1 to n n n

M

f+(y) =

ED

This is also a step function it takes a step of w j at the beginning of the jth interval. We see that any weight generating function f associated the OWA weights will be such

PT

that f+(y)  f(y)  f+(y).

AC

CE

Another notable example of weight generating function associated with a set of OWA j-1 j weights is the piecewise linear function fL. In this case for y  [ , ] we have n n j-1

fL(y) =

å wk + wj (ny - j + 1).

k=0

We note here that j-1

j-1

j

j nj fL( ) = å w k + wj( - j + 1) = å w k + wj = å w k n n k=0 k=0 k=0

We see in this case that f+(y)  fL(y)  f+(y). We can consider a family of weight generating functions f  based on a generalized fL. Here for 0 <  < ¥ we have

18

ACCEPTED MANUSCRIPT

j-1

f(y) =

å wk + wj(ny - j + 1)

for y  [

k=0

j-1 j , ] n n

We see here that if 1 < 2 then f (y)  f (y). 1 2 Thus we see that we have an ordered family of weight generating functions associated with a set of OWA weights. Furthermore as   0 then f  f+, as   ¥ then f  f+ and for

CR IP T

 = 1 we get fL. We see that fL with  = 1 is in the middle. The following discussion leads us to believe fL is a good choice.

Assume X = {x1, …, xn} is a bag of data and we want to aggregate this data using an OWA aggregation with weights wj for j = 1 to n. However we have some uncertainty over the collection X as expressed by a measure . To perform this aggregation we can select a weight

AN US

generating function f associated with the given OWA weights and calculate n

Cho q f  (x1, …, xn) = 

å (f ((Hj)) - f((Hj - 1)))x(j) j=1

Here we recall (j) is the index of the xi with the jth largest value and Hj is the subset of X with

M

the j largest values for xi. We observe that Choq (x1, …, xn) will depend on our choice of f. In particular we have earlier shown that the bigger the weight generating function the larger the aggregation. Thus if f > f then Cho q f  (X)  Cho q f  (X). Since fL is in the middle of 2

ED

1

1

2

the ordered weight generating functions the use of fL is a reasonable choice. In addition its being

PT

linear is always a good feature. Thus a good solution to the aggregation problem is

CE

Cho q f  (X) = L

n

å (fL ((Hj)) - fL((Hj - 1)))x(j) j=1

Here we provide an illustrative example.

AC

Example: Let x1 = 40, x2 = 30, x3 = 20, x4 = 10, here (j) = j. Assume we have as the given OWA weights w1 = 0.3, w2 = 0.1, w3 = 0.4 and w4 = 0.2. Here we shall assume a probability measure on the X with p1 = 0.2, p2 = 0.3, p3 = 0.4 and p4 = 0.1. In this case since H1 = {x1}, H2 = {x1, x2}, H3 = {x1, x2, x3} and H4 = X we have (Hj) =

j

å pk and hence (H1) = 0.2, (H3) = 0.5, (H3) = 0.9 and (H4) = 1.

k=1

j

In this example we shall use fL. Here fL(j/n) =

å wk

k=0

19

ACCEPTED MANUSCRIPT

fL(0) = 0, fL(1/4) = f(0.25) = 0.3 fL(2/4) = f(0.5) = 0.4 fL(3/4) = f(0.75) = 0.8 fL(n/L) = f(1) = 1

CR IP T

Here we see that

0 1 , ) 4 4 1 1 (H2) = 0.5  [ , ) 2 2 3 (H3) = 0.9  [ , 1) 4 (H1) = 0.2  [

AN US

(H4) = 1  [1, 1]

j-1

j-1 j , ] we have fL(y) = å w k + wj (ny - j + 1). In this example n n k=0 0.2(0.3) 4 fL((H1) = = 0.3 = 0.24 0.25 5 fL((H2) = f(0.5) = 0.4 fL((H3) = 0.92

M

We recall that for y  [

Since

ED

fL((H4) 1.00

= f((Hj)) - f((Hj - 1)) we obtain

PT

= 0.24,

CE

and hence

= 0.52,

= 0.08

xj = (0.24)40 + (0.16)(30) + (0.52)20 + (0.08) 10 = 25.6

AC

Choq(X) =

= 0.66,

7. The Attitudinal Character of Weight Generating Functions We recall that the attitudinal character of a set of n OWA weights W, w j for i = 1 to n is n

AC(W) =

n- j

å n -1 wj.

We also recall that the attitudinal character associated with a weight

j=1

generating function f is AC(f) =

1

ò0 f(y) dy.

In the preceding we suggested a procedure for obtaining a family, f  of weight generating 20

ACCEPTED MANUSCRIPT

functions associated with the OWA weights wj for j = 1 to n. Since for any f we have f+(y)  f(y)  f+(y) for all y hence for any f AC( f+ )  AC(f)  AC(f+) j

å wk

k=0

1

j

1

j

n

å ò( j-1)/n f( n )dy = n å f( n )= n (å ( å w k )) =

(y) dy =

1 +

ò0 f

j

j/n

n

j=1 n

(y) dy =

j=1

j=1 k=0

n+1-j wj n j=1

å

While AC(W) and

1 +

ò0 f

(y) dy are similar it is clear that 1 +

ò0 f More specifically since

1 (nw1 + (n -1) w2 + … + wn) n

n+1-j n-j  for all j and hence we have n-1 n

(y) dy  AC(W).

AN US

1 +

ò0 f

n

j j-1 j = f( ) for y  ( , ]. Thus n n n

CR IP T

Let us look at these bounding cases. We recall f +(y) =

n n+1-j n-j j-1 j-1 = then AC(f) - AC(W) = å wj n(n-1) n-1 (n)(n-1) n j=1

j-1

j-1 j-1 j ) = å w k for y [ , ) we have n n n k=0

M

Further recalling that f+(y) = f(

n j-1 n n ( j-1) 1 n j-1 1 (n-j) )dy = = = wj. f f( ) ( ( w ) ) j-1 k 0 n n n n n j=1 n j=1 j=1 j=1 k=0 + n-j n-j Since  for all j then AC(f+) = f (y) dy  AC(W). 0 + n

f+ (y) dy =

åò

n-1

å

åå

å

ò

n n-j n-j n-j n-j = we have AC(W) - AC(f+) = å wj (n)(n-1) n-1 (n)(n-1) n j=1

PT

n

ò

+

ED

AC(f+) =

More specifically since

AC

CE

Thus we see for any f we have AC(f+)  AC(f)  AC(f+) and hence n

n (n-j) n +1-j å n wj  AC(f)  å ( n ) wj j=1 j=1

While in general the calculation of AC(f) is rather complex because of the piecewise

linearity of FL we can obtain a closed form for AC(fL). In this case we shall let f(mj) denote the j-1 j middle portion for the interval [ , ]. In this case because of piecewise linearity n n AC(fL) =

1

n

j/n

n

1

ò0 fL (y) dy = å ò( j-1)/n f(m j )dy = å n f(mj) j=1

In this case 21

j=1

ACCEPTED MANUSCRIPT

j-1

j-1

j

1 j-1 j 1 1 j-1 1 f(mj) = (f( ) + f( )) = ( å w k + å w k ) = å w k + wj = f( ) + wj 2 n n 2 k=0 2 n 2 k=0 k=0 Here then n n 1 1 1 1 1 j-1 (n - j+ 0.5) (f( ) + w ) = AC(f ) + w = AC(f ) + = wj ån n 2 j ån 2 j å + + n 2n j=1 j=1 j=1

CR IP T

n

AC(fL) =

8. Conclusion

We discussed the OWA aggregation operation and the role the OWA weights play in determining the type of aggregation being performed. We introduced the idea of a weight generating function and described its use in obtaining the OWA weights. We emphasized the

AN US

importance of the weight generating function in prescribing the type of aggregation to be performed. We considered the problem of performing a prescribed OWA aggregation in the case when we have a probability distribution over the argument values. We showed how we could use the weight generating function to enable this type of aggregation. Next we considered the

M

situation when we have a more general measure based uncertainty over the argument values. Here again we showed how we can use the weight generating function to aid in performing the

ED

prescribed OWA aggregation in the face of this more general type of uncertainty. Finally we

9. References

AC

[2] [3]

Yager, R. R., Reformat, M. Z., and To, N. D., "Drawing on the iPad to input fuzzy sets with an application to linguistic data science," Technical Report MII-3805 Machine Intelligence Institute, Iona College, New Rochelle, NY, 2018. Cady, F., "The Data Science Handbook," John Wiley: Newark, 2017. Ulutagay, G., Yager, R. R., De Baets, B., and Allahviranloo, T., "Forefront of fuzzy logic in data mining: Theory, algorithms, and applications," Advances in Fuzzy Systems 2016, pp. 1-2, 2016. Tan, P. N., Steinbach, M., and Kumar, V., Introduction to Data Mining. Boston: Addison Wesley, 2006. Ruan, D., Chen, G., Kerre, E., and Wets, G., Intelligent Data Mining: Techniques and Applications. Heidelberg: Springer-Verlag, 2005. Wu, J., Xiong, R., and Chiclana, F., "Uninorm trust propagation and aggregation methods for group decision making in social network with four tuples information," KnowledgeBased Systems 96, pp. 29-39, 2016. Ventre, A. G. S., Maturo, A., Hoskova-Mayerova, S., and Kacprzyk, J., "Multicriteria and Multiagent Decision Making with Applications to Economics and Social Sciences," Springer, Heidelberg, 2013.

CE

[1]

PT

looked at the task of obtaining a weight generating function from a given set of OWA weights.

[4] [5] [6] [7]

22

ACCEPTED MANUSCRIPT

[14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

AC

[24]

CR IP T

[13]

AN US

[12]

M

[11]

ED

[10]

PT

[9]

Yager, R. R., Kacprzyk, J., and Beliakov, G., Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice. Berlin: Springer, 2011. Pasi, G. and Yager, R. R., "A majority guided aggregation operator in group decision making," In Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice, Edited by Yager, R. R., Kacprzyk, J., Beliakov, G., Springer, Berlin, pp. 111-134, 2011. Yager, R. R., "Quantifier guided aggregation using OWA operators," International Journal of Intelligent Systems 11, pp. 49-73, 1996. Yager, R. R., "On the evaluation of uncertain courses of actions," International Journal of Fuzzy Optimization and Decision Making 1, pp. 13-41, 2002. Yager, R. R., "On the valuation of alternatives for decision making under uncertainty," International Journal of Intelligent Systems 17, pp. 687-707, 2002. Yager, R. R., "Generalizing variance to allow the inclusion of decision attitude in decision making under uncertainty," International Journal of Approximate Reasoning 42, pp. 137-158, 2006. Jin, L. S., Mesiar, R., and Yager, R. R., "Melting probability measure with OWA operator to generate fuzzy measure: the Crescent Method," IEEE Tranactions on Fuzzy Systems, (To Appear). Yager, R. R., "Uncertainty modeling using fuzzy measures," Knowledge-Based Systems 92, pp. 1-8, 2016. Yager, R. R., "On ordered weighted averaging aggregation operators in multi-criteria decision making," IEEE Transactions on Systems, Man and Cybernetics 18, pp. 183-190, 1988. Yager, R. R., "On the cardinality index and attitudinal character of fuzzy measures," International Journal of General Systems 31, pp. 303-329, 2002. Yager, R. R., "Belief structures, weight generating functions and decision-making," Fuzzy Optimization and Decision Making 16, pp. 1-21, 2017. Zadeh, L. A., "A computational approach to fuzzy quantifiers in natural languages," Computing and Mathematics with Applications 9, pp. 149-184, 1983. Sugeno, M., "Fuzzy measures and fuzzy integrals: a survey," in Fuzzy Automata and Decision Process, Gupta, M.M., Saridis, G.N. & Gaines, B.R. (eds.), Amsterdam: NorthHolland Pub, pp. 89-102, 1977. Wang, Z. and Klir, G. J., Generalized Measure Theory. New York: Springer, 2009. Wang, Z. Y. and Klir, G. J., Fuzzy Measure Theory. New York: Plenum Press, 1992. Yager, R. R., "A measure based approach to the fusion of possibilistic and probabilistic uncertainty," Fuzzy Optimization and Decision Making 10, pp. 91-113, 2011. Beliakov, G., Pradera, A., and Calvo, T., Aggregation Functions: A Guide for Practitioners. Heidelberg: Springer, 2007.

CE

[8]

23