1. Introduction
The theory of aggregation functions and related functions is an explosively growing branch of mathematics. This fact is directly related to the increase in computer-based applications in fields such as multicriteria decision support, fuzzy-rule-based systems, fuzzy logic, and image processing. For more details, see books [
1,
2,
3,
4,
5]. Recall that the most considered framework for aggregation functions deals with the domain
, i.e., with the input
n-tuples from
, and with the co-domain
for outputs, boundary conditions
and
, and the monotonicity of
A. We denote
as the class of all such
n-ary aggregation functions.
Recently, inspired by potential applications, several modifications of aggregation functions have been considered. In particular, Wilkin and Beliakov [
6] have proposed the concept of weak aggregation functions, where the monotonicity of aggregation functions was replaced by
the weak monotonicity, i.e.,
for any
and
such that
. As a typical weak aggregation function, we recall the well-known (minimal and maximal) modal value from statistics. We denote by
the class of all
n-ary weak aggregation functions.
Weak monotonicity is a particular case of
the directional monotonicity [
7]. For a vector
, a function
is
-increasing (increasing in direction
) if
for any
and
such that
. Clearly, weakly increasing functions are simply those that are
-increasing. Lucca et al. [
8] proposed the concept of
pre-aggregation functions, i.e., functions
satisfying the boundary conditions for aggregation functions, where the monotonicity of
F is considered to be
-increasingness with respect to some direction
. We denote by
the class of all
n-ary pre-aggregation functions.
In addition, some other classes of n-ary real functions are considered in this paper, namely
The class
of all
n-ary
fusion functions [
7], i.e., functions
;
The class of all n-ary bounded functions on , i.e., bounded functions ;
The class of all n-ary semi-aggregation functions, i.e., fusion functions satisfying the boundary conditions for aggregation functions.
The next inclusions are obvious:
(If
, all inclusions are strict; if
, then
.)
One of the most important possible properties of (aggregation) functions is their
shift invariance characterized by
for any real constant
c (it is enough to consider
) and
such that
. Obviously, the shift invariance (also called the difference scale invariance in measurement theory) ensures weak monotonicity.
The aim of this paper is a generalized look at shift stability as inspired by the way the directional monotonicity generalizes the weak monotonicity. Considering a fixed non-negative vector , we propose and discuss (aggregation) functions with increments solely dependent on the real constant c, independent of the input n-tuple . Clearly, such functions are then necessarily -directionally monotone. Subsequently, we expect to see the applications of our results in fuzzy classification, image processing, and all other fields where directionally monotone functions are successfully applied.
The paper is organized as follows: In the next section, we introduce the notion of directional shift stability for the above-discussed functional classes and give some basic examples. In
Section 3, we recall some known characterizations and add some new characterizations of binary shift-invariant functions from the above-introduced classes.
Section 4 deals with general binary directional shift-stable functions. Finally, some concluding remarks are presented.
2. Directional Shift-Stable Functions
The following property of functions was inspired by shift invariantness and directional monotonicity.
Definition 1. Let and . Function F is deemed -directional shift stable wheneverfor all and such that . Example 1. Define by Obviously, (moreover, it can be shown that ). Let Then, for any and such that , it holdsindependently of , and, hence, F is -directional shift-stable. The next result brings an alternative view on the directional shift stability, which, in several cases, simplifies the study of directional shift-stable functions.
Theorem 1. Let be an -directional shift-stable function, where . Then, and only then, there is a real constant k such that for each and satisfying it holds Proof. The sufficiency is obvious as, then,
does not depend on
. Consider that
F is
-directional shift stable. The difference
, if well defined, does not depend on
. Denote this difference as
. Note also that the domain of possible constants
c is a closed subinterval
containing 0 (e.g., if
, then
). Hence,
is a bounded real function (this follows from the boundedness of
F). Moreover,
for any
such that also
, i.e.,
is an additive function. Hence, the classical Cauchy equation
, together with the boundedness of
, means that there is some constant
such that
; see, e.g., [
9]. Hence, the result
follows. □
For simplicity, we introduce the notation for the -directional shift-stable function F related to the constant k by . Directional shift stability ensures directional monotonicity, as formulated in the next result, which is a trivial consequence of Theorem 1.
Corollary 1. Let and let be an -directional shift-stable function characterized by constant k, i.e., . Then:
- (i)
If , F is -increasing;
- (ii)
If , F is -constant;
- (iii)
If , F is -decreasing.
Based on Theorem 1, we also see that if
and
, then it is also
-directional shift stable for any non-zero constant
, and then
F is linked to the constant
, i.e.,
,
Recall that for a function
, its dual
is given by
Evidently, . Moreover, the next result could be of interest.
Corollary 2. Let and . Then, also .
Proof. The result is a matter of a direct computation. Indeed, if
, then also
. Then,
□
Note also that
- (i)
if and for , then also any linear combination , belongs to , and it is -directional shift stable with constant , .
In particular, if and , then is a self-dual function such that ;
- (ii)
if and , , then F is also -directional shift stable with constant , ;
- (iii)
if
and
and
is a permutation, then
is
-directional shift stable with constant
k.
Directional shift stability can also be seen as a weakening of linearity. The next important result is presented below.
Theorem 2. Let . Then, F is -directional shift stable for any directional vector if and only if F is an afinne function, i.e., for some real constants .
Proof. Suppose
. Then, for any
it holds
where
. Due to Theorem 1,
F is
-directionally shift stable.
Concerning the necessity, it is enough to consider that F is -directionally shift stable, where , .
Let
for any
and
such that
. Then
proving the desired result with
. □
Observe that a linear function given by satisfies:
whenever and ;
if , and , then F is a weighted arithmetic mean and (thus and ). Hence, this F is a semi-aggregation function if and only if it is an aggregation function.
3. Binary Shift-Invariant Functions
Shift-invariant (difference scale invariant) functions were studied and characterized in several works; see, e.g., [
4,
10,
11,
12,
13] (Chapter 7).
Definition 2. Let . Function F is deemed shift invariant wheneverfor any and . When considering the shift invariantness, we deal with directional vector , i.e., .
For the sake of transparency, hereon, we fix
, i.e., we deal with binary functions only. We introduce an equivalence relation ∼ on
by
if and only if
i.e.,
and
are linearly dependent vectors. Obviously, the set of all equivalence classes of ∼ forms a partition of
. Consider a set
of single representative points of all such equivalence classes. Then, an arbitrary shift-invariant function
is determined by its value in points from
. Indeed, knowing
for any
, for each
, there is
such that
, and then
We introduce three typical sets :
;
;
.
These sets are depicted in the
Figure 1.
The usual approach to characterize binary shift-invariant functions is related to
. Then, introducing two unary functions
and
,
,
is given by
The only constraint on
to generate
by (
3) is
and the boundedness of
f and
g. We summarize the properties of
yielding
F from the remaining classes of the above-introduced functions. Note that for shift-invariant functions, fusion functions, semi-aggregation functions, pre-aggregation functions, and weak aggregation functions coincide.
Proposition 1. Let be shift invariant and given by (3). Then, - (i)
(, , ) if and only if and for all ;
- (ii)
if and only if both are non-decreasing and 1-Lipschitz (i.e., ), and .
Proof. Part (ii) can be found in [
13].
Concerning (i), for any , and, for each for some . Suppose . Then, necessarily , i.e., and . Hence, . Similarly, can be shown.
Vice-versa, suppose
F is given by (
3) and
and
for any
. Then,
If , .
Similarly, if , .
Summarizing, for any and, thus, . □
When considering the set
to study binary shift-invariant functions, we have to consider functions
,
and
. Using similar arguments as in the previous case (when we have considered the set
), for any
it holds
This F belongs to if and only if and for all . Moreover, F belongs to () if and only if . Finally, this F belongs to (i.e., it is a shift-invariant binary aggregation function) if and only if and both h and q are non-decreasing and 1-Lipschitz. Note also that, if , necessarily, and for all .
Finally, when considering the set
, for each
it holds
Let
be given by
. Due to (
5), we can derive that
Note that there is, in general, no constraint on
t (clearly, up to the boundedness of
t) to determine
F by Formula (
6). Our
f belongs to
if and only if
for all
, (note that, then, necessarily,
). Concerning the belongingness of our
F to
or
, it is equivalent to
. Finally,
F is a shift-invariant binary aggregation function if and only if
and
t is 1-Lipschitz.
Example 2. Let us consider the set and let the function be defined as Then, the corresponding function can be expressed by The functions and are illustrated in Figure 2. Observe that we have no monotonicity constraint concerning t generating a shift-invariant binary aggregation function. The greatest possible t is given by and then . The smallest t is given by and then . Finally, if t is constant for all , then ; i.e., F is the arithmetic mean. Observe also that if t is linear, , then F is the weighted arithmetic mean, , where .
4. General Binary Directional Shift-Stable Functions
Based on representative sets
, or
(or their subsets, in dependence of
), for any
and
k, one can find
such that
F is
-directional shift stable with constant
k, i.e.,
. This is no longer the case if
F belongs to the above-considered proper subclasses of
. Thus, for example, if
, and both
and
belong to
(clearly, this is the case if
), then
and, thus,
. Moreover, if
and
(this is surely the case if
),
is the only possible value for
k once
F is
-directional shift stable with
.
Due to Formula (
2), if, for a given
, there is a constant
such that
and
, for any
,
. Then, to determine all possible non-zero constants
k such that there exist
F from some of discussed classes of functions, which
, it is enough to consider vectors
such that
. For such vectors, and for a subclass
of functions from
, we denote by
the set of all constants
k such that there is
satisfying
. As previously mentioned,
.
Additionally,
. Consider
, and for
,
Then
i.e.,
. Thus
.
For the remaining discussed function classes, we have , and, thus, the related functions are shift invariant, and they are discussed in the previous section.
Similarly, for any
, it holds
. Now, we discuss some particular directional vectors
. Let
and suppose
such that
. Then, necessarily,
F also belongs to
due to the fact that
and, hence,
. Consider the set
and related functions
f and
g,
and
. Then,
means that
Necessarily, and , for any . Note that, then, means that .
Define
by
Then, and . Then, for any , and . Thus, .
Moreover, if we suppose F to be a binary aggregation function, necessarily, , i.e., . It is not difficult to check that given by satisfies , which proves .
Now, we characterize all
-directional shift-stable binary aggregation functions, considering Formula (
7).
Proposition 2. Let be given by Formula (7). Then, F is a -directional shift-stable binary aggregation function if and only if there is such that , f is non-decreasing and -Lipschitz, and , g is non-decreasing and k-Lipschitz. Proof. The proof follows from the previously shown fact that
, and showing the monotonicity of
F given by (
7) in the first coordinate and in the second coordinate. For example, the constraint of
k-Lipschitzianity of
g follows from the necessity of non-decreasingness of
g (i.e., of
) and the inequality
valid for any
such that
(
). Then,
means that
i.e.,
g is
k-Lipschitz. Similarly, for appropriate
we have
i.e.,
showing the
-Lipschitzianity of
f. The sufficiency is a matter of an easy processing and is thus omitted. □
Example 3. Consider and define by and . Then, satisfy the constraints of Proposition 2 and, applying (7), we obtain the aggregation function , given by Using similar argument as we have considered in the case , the next results can be shown.
Proposition 3. Let and . Then,
;
;
;
.
Proposition 4. Let and . For , denote and .
- (i)
If , then , if and only ifand, then, F is an aggregation function if and only if , f is non-decreasing, and . - (ii)
If , then , if and only ifand then F is an aggregation function if and only if , are non-decreasing, , , f is -Lipschitz and g is k-Lipschitz.
Note that results for directional vectors are the same; we only need to exchange the corresponding functions f and g. Therefore, for example, based on Example 3, we see that is an aggregation function .
For any () with , we can fix the constant and find related extremal ()-directional shift-stable aggregation functions related to k. Obviously, these extremal aggregation functions are related to extremal possible functions f and g. Based on the constraints given in Proposition 4, the next interesting result is presented below.
Proposition 5. Under the constraints of Proposition 4, fix . Then,
- (i)
If and , then F is the greatest (smallest) aggregation function if it is determined by () given byand, then, it is given by Both families and are pairwise incomparable, i.e., if , then and ( and ) are incomparable aggregation functions.
- (ii)
If and , then is the greatest ( is the smallest) aggregation function if it is determined by and given by (by and given byand then it is given bysee Figure 3. For any fixed , both families and are pairwise incomparable.
5. Conclusions
Inspired by generalization of monotone functions into directionally monotone functions and shift invariantness (difference scale invariantness) of
n-ary real functions defined on
, we introduced a new notion of directional shift-stable functions. This new type of stability is related to a non-zero directional
n-ary vector
and a real constant
k expressing the output increment of a function
F once the increment of its argument
is just
, and proportionally if the argument
is just
or if the argument increment is
,
. Observe that the class of
-directional shift-stable aggregation (pre-aggregation) functions does not change if we replace the vector
by a vector
; here,
u is an arbitrary positive real constant. This fact allows us to fix the considered directional vectors
by the constrain
, as exemplified in
Section 4. As an important advantage of
-shift-invariant aggregation functions, one can consider the fact that instead of their description on the full domain
, it is enough to have them fixed on a significantly smaller subdomain. This fact is stressed in
Section 3; see the subdomains
exemplified there. Our approach can also be seen as a special generalization of the linearity of functions. Indeed, if
is an
-directional shift-stable function, then, for any points
such that vectors
and
are linearly dependent,
F is linear on the segment determined by
and
.
Although this paper is purely theoretical, we expect to see applications of our results in measurement theory but also in image processing, classification systems, and related domains where directional monotonicity is successfully considered; see, e.g., [
8,
14,
15]. Note that for any successful application, a proper choice of considered theoretical tools is necessary. This fact opens several new problems for the future study of our approach. In particular, based on the real data, we aim to focus on fitting appropriate directional vector
to be fixed. Based on our theoretical results, we can then deduce possible values of constants
k characterizing the increments of
-directionally shift-stable functions. Fixing the direction
and knowing the possible domain for constants
k, another task is the appropriate determination of the constant
k.