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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 (Fr´echet) 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 h·,·i. Let Sp−1={u∈Rp: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

a∈A

hu, aifor u∈Sp−1[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 u∈Sp−1.

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 −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. 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 a∈A,λkA+ (1 −λ)a∈A

for all λ∈[0,1]. The radial function of a star-shaped set A is deﬁned as

ρA:Sp−1→R+so that ρA(u) = sup {λ≥0 : 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 u∈Sexcept 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(Sp−1) 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(Sp−1).(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|a∈A, 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(Sp−1). 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.

−3−2−1 0 1 2 3

−3

−2

−1

0

1

2

3

−3−2−1 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) = qτkkA−kBk2+ (1 −τ)kρA−ρBk2

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(Sp−1) [6].

3 Random star-shaped sets

Given a probability space (Ω, A, P ), a mapping X:Ω→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(Sp−1).

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(Sp−1), 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.

−2−1 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)=2ZSp−1ρA(u)dλSp−1(u),(7)

where λSp−1denotes 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 A∩B∈f

K∗

s(Rp). The mean

directional length of A∩Bcan be expressed as follows:

S(A∩B)=2ZSp−1min (ρA(u), ρB(u)) dλSp−1(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(A∩B)

S(A).(9)

If Ais included in B, then it is clear that S(A∩B) = S(A) and Inc(A, B) =

1; Otherwise, S(A∩B)< 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(A∩B)

S(A∪B),(10)

where

S(A∪B) = 2 ZSp−1max (ρA(u), ρB(u)) dλSp−1(u).

8 Ramos-Guajardo, Gonz´alez-Rodr´ıguez, Colubi, Ferraro and Blanco-Fern´andez

In this case, if Ais equal to B, then S(A∩B) = S(A∪B) and Sim(A, B ) =

1; Otherwise, S(A∩B)< S(A∪B) 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.

−4−2 0 2 4

−3

−2

−1

0

1

2

3

4

−4−2 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.