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The convenient theoretical properties of the support function and the Minkowski addition-based arithmetic have been shown to be useful when dealing with compact and convex sets on Rp. However, both concepts present several drawbacks in certain contexts. The use of the radial function instead of the support function is suggested as an alternative to characterize a wider class of sets—the so-called star-shaped sets—which contains the class of compact and convex sets as a particular case. The concept of random star-shaped set is considered, and some statistics for this kind of variable are shown. Finally, some measures for comparing star-shaped sets are introduced.
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On some concepts related
to star-shaped sets
Ana Bel´en Ramos-Guajardo1, Gil Gonz´alez-Rodr´ıguez1, Ana Colubi1,
Maria Brigida Ferraro2, and ´
Angela Blanco-Fern´andez1
Abstract The convenient theoretical properties of the support function and
the Minkowski addition-based arithmetic have been shown to be useful when
dealing with compact and convex sets on Rp. However, both concepts present
several drawbacks in certain contexts. The use of the radial function instead
of the support function is suggested as an alternative to characterize a wider
class of sets - the so-called star-shaped sets - which contains the class of
compact and convex sets as a particular case. The concept of random star-
shaped set is considered, and some statistics for this kind of variable are
shown. Finally, some measures for comparing star-shaped sets are introduced.
1 Introduction
Random sets, also called set-valued random variables and denoted by RSs
for short, have been used in different fields. For instance, they have been
shown to be useful in spatial data analysis [18], in Econometrics [3] and in
Structural Engineering [25], to name but a few. RSs can also be viewed as
imprecise random variables, as Pedro Gil and his colleagues have pointed out
in [17]. Several results for RSs have been accomplished, such as limit theorems
[1, 19], confidence sets for the (Aumann) expected value [4], hypothesis testing
for the expected value or the (Fechet) variance [11, 13, 14, 15, 20, 21] and
inference on regression models [2, 9, 12].
In the one-dimensional case, the compact intervals Aof Rcan be char-
acterized by either the infima and suprema of A, (inf A, sup A) so that
inf A < sup A, or by the mid-point and radius of A, (mid A, spr A)R×R+.
The usual interval arithmetic is based on the Minkowski addition [16] and
INDUROT/Dept. of Statistics, OR and MD, University of Oviedo, Spain
{blancoangela,colubi,gil,ramosana}@uniovi.es ·Dipartimento di Scienze Statis-
tiche, Sapienza Universit`a di Roma, Italy mariabrigida.ferraro@uniroma1.it
1
2 Ramos-Guajardo, Gonz´alez-Rodr´ıguez, Colubi, Ferraro and Blanco-Fern´andez
the product by a scalar, and it preserves the length of the resulting intervals.
Many statistical results concerning interval data are based on the Minkowski
arithmetic (see, for instance, [2, 7, 9, 12, 15, 21]).
The statistical studies developed until now for the p-dimensional situation
(with p > 1) frequently take advantage of some convenient theoretical prop-
erties of the support function, but they have several drawbacks in certain
situations. To overcome these drawbacks, an alternative to the support func-
tion for characterizing star-shaped sets by means of the so-called radial/polar
function has been investigated in [5, 6].
Basic concepts related to this new representation and some statistical re-
sults are addressed. More concretely, the concept of random star-shaped set
- i.e. a random variable taking star-shaped sets as outcomes -, and those of
expected value and variance are considered. The corresponding sample mo-
ments are defined, and the consistency with respect to their population coun-
terparts is highlighted. In addition, the concept of mean directional length is
introduced and some comparative measures of centered star-shaped sets are
suggested.
The rest of the paper is organized as follows. Section 2 is devoted to the
introduction of some preliminaries regarding compact and convex sets and
star-shaped sets. The concept of random star-shaped sets and their moments
are recalled in Section 3. A basic example illustrating these sample moments
is provided. The notions related to the mean directional length are discussed
in Section 4. Finally, some conclusions and open problems are provided in
Section 5.
2 Preliminaries
Let the space Rpbe endowed with the Euclidean norm k·k and the correspond-
ing inner product ,·i. Let Sp1={uRp:kuk= 1}be the hypersphere
with radius 1. The space of all non-empty compact and convex subsets of
Rpis denoted by Kc(Rp). If A∈ Kc(Rp), then the support function of Uis
defined such that sA(u) = sup
aA
hu, aifor uSp1[10, 19].
The location and the imprecision of a set A∈ Kc(Rp) can be determined
in terms of the support function by the so-called mid-spread representation
in such a way that sA= mid A+ spr A, where mid A(u)=(sA(u)sA(u))/2
and spr A(u)=(sA(u) + sA(u))/2 for all uSp1.
As shown in Figure 1, the support function identifies the boundary of
the corresponding set, but the obtained result is not easy to relate with the
original shape of the set. Actually, it is very difficult to identify which is the
original set associated with a function verifying the properties of the support
function (if any). This could be a drawback in some applied problems in
On some concepts related to star-shaped sets 3
1.5 10.5 0 0.5 1 1.5
1
0.5
0
0.5
1
1.5
1.5 10.5 0 0.5 1 1.5
1
0.5
0
0.5
1
1.5
Fig. 1 Support function (distance from (0,0) to the contour of the gray line, marked in
dashed-dotted black) of a line and a square in R2(in black)
which it is necessary to clearly identify the shape of the sets (as, for instance,
in image processing).
To overcome this disadvantage, other characterizations of sets can be taken
into account. For instance, a useful tool in this framework is the so-called
radial function [24]. It is defined on the class of star-shaped sets of Rp, denoted
by Ks(Rp), which is an extension of Kc(Rp) - i.e. Kc(Rp)⊂ Ks(Rp). A star-
shaped set A∈ Ks(Rp) with respect to kA, where kAis a center of A, is a
nonempty compact subset of Rpsuch that for all aA,λkA+ (1 λ)aA
for all λ[0,1]. The radial function of a star-shaped set A is defined as
ρA:Sp1R+so that ρA(u) = sup {λ0 : kA+λu A}. In this context,
kAcan be viewed as a location point of the star-shaped set Awhereas ρAis
related to the imprecision of the set. The formal definition of kAand ρAto
be used in statistical problems is not trivial. This problem has been recently
addressed in [6]. From now on, this representation of sets will be called center-
radial characterization.
In contrast to the support function, the radial function identifies the shape
of the sets in an intuitive way, as it is shown in Figure 2, because it is simply
based on the well-known polar coordinates over the unit sphere. In the case
of the line (left side image in Figure 2), the radial function is equal to 0 for
all uSexcept for u1= (1,0) and u2= (1,0), with ρA(u1) = ρA(u2) = 1.
Further advantages of the radial function with respect the support function
are pointed out in [6].
The space Ks(Rp) can be embedded into a cone on the Hilbert space
Hr=Rp× L2(Sp1) through the center-radial characterization. For the the-
oretical developments, from now on, star-shaped sets in K
s(Rp) will be con-
sidered, where
K
s(Rp) = A∈ Ks(Rp)|ρA∈ L2(Sp1).(1)
4 Ramos-Guajardo, Gonz´alez-Rodr´ıguez, Colubi, Ferraro and Blanco-Fern´andez
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5
0
0.5
1
1.5
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5
0
0.5
1
1.5
Fig. 2 Radial function of a line (black dots) and a square (which corresponds exactly to
the square) in R2
Regarding the arithmetic, we could consider the Minkowski addition be-
tween two star-shaped sets Aand B,A+B={a+b|aA, b B}. How-
ever, it has been shown that the Minkowski addition is not always meaningful
(see [5, 18]), and it does not agrees with the natural arithmetic induced by
the center-radial characterization from the Hilbert space. That is, A+rλB
should be the element in K
s(Rp) satisfying that kA+rλB =kA+λkBand
ρA+rλB =ρA+λρB, where + denotes the usual sum of two points in Rpand
the usual sum of two functions in L2(Sp1). An example of the differences
between the Minkowski sum and the center-radial sum of two star-shaped
sets in K
s(Rp) is provided in Figure 3. We observe that the center-radial
sum preserves, directionally, the lengths, whereas the Minkowski sum dilates
them.
321 0 1 2 3
3
2
1
0
1
2
3
321 0 1 2 3
3
2
1
0
1
2
3
Fig. 3 Minkowski (left) and radial (right) sums (in black) of the gray quadrilaterals
Regarding the metric structure in K
s(Rp), the center-radial characteriza-
tion induces a natural family of distances from the corresponding one in the
associated Hilbert space. Thus, for any two star-shaped sets A, B ∈ K
s(Rp),
On some concepts related to star-shaped sets 5
the τ-metric is defined as
dτ(A, B) = qτkkAkBk2+ (1 τ)kρAρBk2
p,(2)
where τ(0,1) determines the importance given to the location in contrast
to the imprecision, k·kdenotes the usual norm in Rpand k·kpis the usual
L2-type norm in L2(Sp1) [6].
3 Random star-shaped sets
Given a probability space (Ω, A, P ), a mapping X:Rp× K
s(Rp) is a
random star-shaped set if it is a Borel measurable mapping with respect to
Aand the Borel σ-field generated by the topology induced by the metric dτ
on Rp× K
s(Rp). Equivalently, Xcan be decomposed in terms of its center-
radial characterization, that is, X= (kX, ρX), and Xcan be defined to be a
random star-shaped set iff kXand ρXare random elements in the real and
functional framework respectively [6].
Now we are in a position to define some summarizing measures for random
star-shaped sets. On one hand, if E(kkXk)<and E(kρXkp)<, then
the expected value of Xis defined as the element E(X)Rp×K
s(Rp) so that
kE(X)=E(kX) and ρE(X)=E(ρX) - this last expectation being considered
in terms of the Bochner integral in L2(Sp1).
From an empirical point of view, given Xa random star-shaped set
and {Xi}n
i=1 an i.i.d. sequence of random star-shaped sets drawn from X,
the sample expectation of Xcan be defined in terms of the arithmetic in
Rp× K
s(Rp) as follows:
X=1
n
n
X
i=1
Xi.(3)
It is easy to show that (kX, ρX)=(kX, ρX).
If E(kkXk2)<and E(kρXk2
p)<, then E(X) is the unique element
in Rp× K
s(Rp) satisfying that
E(d2
τ(X, E(X))) = min
(k, A)Rp× K
s(Rp)E(d2
τ(X, (k, A))).(4)
Thus, by following the Fr´echet approach, the (scalar) variance of a random
star-shaped set X, denoted by σ2
X, is defined as
σ2
X=E(d2
τ(X, E(X))).(5)
The sample variance is also defined in terms of the distance dτ, or equiva-
lently, in terms of the corresponding variances in Rpand L2(Sp1), as follows:
6 Ramos-Guajardo, Gonz´alez-Rodr´ıguez, Colubi, Ferraro and Blanco-Fern´andez
bσ2
X=1
n
n
X
i=1
d2
τ(Xi, X) = (1 τ)bσ2
kX+τbσ2
ρX.(6)
The consistency of the estimators (3) and (5) for the mean and the variance
of random star-shaped sets, respectively, is provided in the following result. It
is an immediate consequence of the Strong Law for Large Numbers in Banach
spaces.
Theorem 1. [6] Let Xbe a random star-shaped set and {Xi}n
i=1 be an i.i.d.
sequence of random star-shaped sets drawn from X. Then,
(a) If E(kkXk)<and E(kρXkp)<, then
kX
a.s.P
E(kX)and ρX
a.s.P
E(ρX). Therefore, Xa.s.P
E(X).
(b) If E(kkXk2)<and E(kρXk2
p)<, then
bσ2
kX
a.s.P
σ2
kXand bσ2
ρX
a.s.P
σ2
ρX. Therefore, bσ2
X
a.s.P
σ2
X.
Example 1. Let Xbe a random rectangle-shaped set (a particular case of a
random star-shaped set) so that the upper right vertex is generated by fol-
lowing real normal distributions of means 2 and 3, respectively, and variance
equal to 1; the longest side is distributed as an U(1,3) and the shortest one
as an U(3,5). A sample {Xi}10
i=1 of rectangle-shaped sets i.i.d. as Xis gen-
erated. The rectangles are centered on their center of gravity. The centered
sample and the corresponding sample mean are represented in Figure 4.
21 0 1 2
2
1
0
1
2
Fig. 4 Sample mean (in black) of a sample of 10 (gray-culoured) rectangles
It should be noticed that the sample mean is not a rectangle, as the corners
are rounded due to the directional averaging.
The sample variance is computed for τ= 1 (the sets are centered so that
the importance is given to the imprecision) providing bσ2
X=bσ2
ρX=.1192.
On some concepts related to star-shaped sets 7
4 Comparison of centered star-shaped sets
Let A, B ∈ K
s(Rp) be two centered star-shaped sets (i.e., two star-shaped sets
with common center which, without lack of generality, can be assumed to be
0) and let f
K
s(Rp) be the space of, either non-empty or empty, centered star-
shaped sets of Rp. In the same way that the so-called length of intervals and
(fuzzy) sets has been used previously to develop statistics to compare convex
and compact sets (see [21, 22]), the analogous concept can be considered for
star-shaped sets. Thus, the mean directional length of Ais defined in terms
of the radial function by
S(A)=2ZSp1ρA(u)Sp1(u),(7)
where λSp1denotes the normalized Lebesgue measure on the sphere. It
should be noted that S(A) is not the area of A, but an average of the magni-
tude of ρAover the unit sphere. It generalizes, in this way, the length of the
intervals directionally, as it is always the case for the radial function.
Regarding the intersection, it is clear that ABf
K
s(Rp). The mean
directional length of ABcan be expressed as follows:
S(AB)=2ZSp1min (ρA(u), ρB(u)) Sp1(u).(8)
Based on the ideas in [23], the degree of inclusion of Ain B, denoted by
Inc(A, B), is a value in [0,1] which can be defined by considering the quotient
between the mean directional length of the intersection of Aand Band the
the mean directional length of the reference set A, i.e.
Inc(A, B) = S(AB)
S(A).(9)
If Ais included in B, then it is clear that S(AB) = S(A) and Inc(A, B) =
1; Otherwise, S(AB)< S(A) and I nc(A, B )<1.
It is also possible to define the degree of similarity of A and B, denoted by
Sim(A, B ), by following the ideas in [8], as the quotient between the shape
of the intersection of Aand Band the shape of the union of Aand B, i.e.
Sim(A, B ) = S(AB)
S(AB),(10)
where
S(AB) = 2 ZSp1max (ρA(u), ρB(u)) Sp1(u).
8 Ramos-Guajardo, Gonz´alez-Rodr´ıguez, Colubi, Ferraro and Blanco-Fern´andez
In this case, if Ais equal to B, then S(AB) = S(AB) and Sim(A, B ) =
1; Otherwise, S(AB)< S(AB) and Sim(A, B)<1. Moreover,
Sim(A, B )< Inc(A, B) in all the situations.
Two illustrative examples concerning rectangle-shaped sets are shown in
Figure 5. On the left part of the graphic, two partially overlapping centered
rectangles A(in gray) and B(in black) are depicted. If we compute both
the inclusion degree of Ain Band the similarity degree between Aand B,
we obtain that Inc(A, B ) = .6822 whereas Sim(A, B) = .459. On the right
part of the graphic, the rectangle A(in gray) is completely contained in the
rectangle B(in black). The computation of both indexes in this case leads
us to the following results: Inc(A, B ) = 1 and Sim(A, B) = .5509.
42 0 2 4
3
2
1
0
1
2
3
4
42 0 2 4
3
2
1
0
1
2
3
4
Fig. 5 Comparison of two rectangle-shaped sets in two different situations
The measures presented in this section might be greatly useful in the
context of image processing. Therefore, it would be interesting to develop a
deep statistical analysis about these measures in the near future.
5 Conclusions
An alternative representation for the class of star-shaped sets, called center-
radial characterization, has been described. It has been shown to be useful
for identifying intuitively the original shape of the sets. On the basis of this
representation, some descriptive statistics for random star-shaped sets have
been provided. Additionally, comparison measures based on the concept of
mean directional length have been proposed. These measures are expected
to be the starting point of an interesting research line in the area of image
analysis. Furthermore, all the concepts provided in this work will be extended
to the case of fuzzy subsets of Rpin a near future.
On some concepts related to star-shaped sets 9
Acknowledgements
The research in this paper has been partially supported by MTM2013-44212-
P, GRUPIN14-005 and the COST Action IC1408.
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To Pedro, our research father and grandfather, a good man who encouraged
us to follow this road. Thanks for giving us so much. Always in our hearts.
... Inspired by the concept of random convex sets, for which the recourse to tools like support functions and Minkowski addition is common, the authors of [421] propagated the use of radial functions instead, and the study of the more general concept of random starshaped sets. They introduced the analogous concepts of expected value and variance, various further notions (such as mean directional length) and suggested also some comparative measures for centered starshaped sets. ...
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A hypothesis test for the inclusion of the expected value of a fuzzy-valued random element in a given fuzzy set is provided. The exclusion, or the empty intersection of the expected value of a random fuzzy set and a fuzzy set, is also tested, that allows us to define one-sided tests for this expected value without the need of considering any restrictive ordering. Both tests are developed by using a measure of the intersection between fuzzy sets. Asymptotic and bootstrap techniques are established. Some simulations are included to show the performance of the bootstrap approaches. Finally, the methodology proposed is applied on a real-life situation related to the field of sensorial analysis.
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We provide a tractable characterization of the sharp identification region of the parameters θ in a broad class of incomplete econometric models. Models in this class have set valued predictions that yield a convex set of conditional or unconditional moments for the observable model variables. In short, we call these models with convex moment predictions. Examples include static, simultaneous move finite games of complete and incomplete information in the presence of multiple equilibria; best linear predictors with interval outcome and covariate data; and random utility models of multinomial choice in the presence of interval regressors data. Given a candidate value for θ, we establish that the convex set of moments yielded by the model predictions can be represented as the Aumann expectation of a properly defined random set. The sharp identification region of θ, denoted Θ1, can then be obtained as the set of minimizers of the distance from a properly specified vector of moments of random variables to this Aumann expectation. Algorithms in convex programming can be exploited to efficiently verify whether a candidate θ is in Θ1. We use examples analyzed in the literature to illustrate the gains in identification and computational tractability afforded by our method. This paper is a revised version of CWP27/09.