Geoscience Reference
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10.2
The Ordered Median Function
As mentioned above, the structure of Ordered Median Functions involves a nonlin-
earity in the form of an ordering operation that introduces a degree of complication
but at the same time gives an extra freedom which allows a lot of flexibility in
modeling. In this section, we will review interesting properties of these functions in
a first step to understand their behavior and then, we shall give a characterization of
this objective function.
We start defining the ordered median function. This function is a weighted
average of ordered elements. For any x 2
n denote x ord D .x .1/ ;:::;x .n/ / where
x .1/ x .2/ :::x .n/ . We consider the function:
R
n
x ! x ord :
n !
sort n W
R
R
(10.1)
n !
Definition 10.1 The function f W
is an ordered median function, for
short f 2 OMf .n/,iff .x/ Dh ; sort n .x/ i for some D . 1 ;:::; n / 2
R
R
n ,
R
n .
where h ; i denotes the usual scalar product in
R
It is clear that ordered median functions are nonlinear. Whereas the nonlinearity
is induced by the sorting. One of the consequences of this sorting is that the pseudo-
linear representation given in Definition 10.1 is pointwise defined. Nevertheless,
one can identify its linearity domains. (See Puerto and Fernández 2000 ;Nickel
and Puerto 2005 ; Rodríguez-Chía et al. 2000 .) The identification of these regions
provides us with a subdivision of the framework space where in each of its cells
the function is linear. Obviously, the topology of these regions depends on the
space and on the lambda vector. A detailed discussion can be found in Puerto
and Fernández ( 2000 ). As mentioned in Sect. 10.1 , different choices of lambda
lead also to different functions within the same family: D .1=n;:::;1=n/
is the mean average, D .0;:::;0;1/ is the center, D .Ǜ;:::;Ǜ;Ǜ;1/
is the Ǜ-centdian, Ǜ 2 Œ0;1, D .0;:::;0;1; :::;1/ is the k-centrum or
D .Ǜ;0;:::;0;1 Ǜ/ is Hurwicz's criterion, see Chaps. 1 , 2 and 4 for further
details.
These functions are not new and some operators related to them have been
developed by other authors independently. This is the case of the ordered weighted
operators (OWA) studied by Yager ( 1988 ) to aggregate semantic preferences in
the context of artificial intelligence; as well as SAND functions (isotone and
sublinear functions) introduced by Francis et al. ( 2000 ) to study aggregation errors
in multifacility location models.
First, we recall some simple properties and remarks concerning ordered median
functions. Most of them are natural questions that appear when a family of functions
is considered. Partial answers are summarized in the following proposition.
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