From Numerical Intervals to Set Intervals (Interval

From Numerical Intervals to Set Intervals (Interval-Related Results Presented at the First International Workshop on Applications and Theory of Random Sets) Hung T. Nguyen and Vladik Kreinovich

On August 22{24, 1996, The Institute of Mathematics and its Applications (IMA) of the University of Minneapolis, Minnesota, organized the rst International Workshop on Applications and Theory of Random Sets. The organizers of this workshop, John Goutsias, Ron Mahler, and Hung T. Nguyen, invited 38 researchers from France, Germany, USA, and the United Kingdom. SpringerVerlag will publish the proceedings [1]. The main emphasis on this workshop was on random sets; due to this emphasis, the presented papers were mainly formulated in statistical terms. Several results and approaches had important non-statistical parts, but they were somewhat dicult to separate from the purely statistical ones. We believe, therefore, that it will be advantageous to the interval community to have these non-statistical parts clearly formulated and explained.

1

Sets

1.1 Main Idea

The workshop started with a survey of mathematical morphology by J. Goutsias from John Hopkins University (this topic was covered by several other talks as well). In interval-related terms, the main idea of morphology is as follows: Suppose that we are interested in determining a set. For example, we want to reconstruct an image of a ying object (or of a distant planet) from its radar observations, or an image of an object from its laser scan. The scanning creates a pixel-by-pixel image. In the ideal situation when we know the exact position of each pixel and when we are 100% sure whether each point belongs to the desired image set or not, these pixels form the desired set (image) S0 . In real life, however, measurements are never absolutely accurate; as a result, we still get a set of pixels S , but the location of each point from the observed set S may 1

be slightly dierent from its location in the (unknown) actual set S0 .

1.2 How to describe the relationship between the observed set and the actual set: Hausdor distance

If we know the upper bound " > 0 on the distance between the actual and mapped locations, then we can formulate the relationship between the known approximate set S and the actual (unknown) set S0 as follows:  each \mapped" point s (i.e., a point from the \mapped" set S ) is "?close to some \actual" point (i.e., to some point from S0 );  each \actual" point is mapped, and is, therefore, "?close to its \mapped" image. Formally, this means that

8s 2 S 9s0 2 S0 (d(s; s0 )  ") and 8s0 2 S0 9s 2 S (d(s; s0 )  "): For example, a two-point set S0 = f(0; 0); (1; 1)g can be represented by a set S = f(0; "); (1 ? "; 1)g. For every two sets S and S0 , the smallest " for which these two properties hold is called the Hausdor distance dH (S; S0 ) between the sets S and S0 . (If we only consider compact sets, i.e., in Rn , bounded and closed sets, then this smallest value always exists.) So, as a result of the imaging, we get a set S that is "?close to the unknown set S0 .

1.3 Description of the set of all points that can belong to the actual set (image): dilation

Since we do not exactly know which points belong to the set S0 , a natural question is: based on the (known) observed set S and the known accuracy ", how can we describe the set D" (S ) of all the points that can, in principle, belong to the unknown actual set S0 . It is easy to show that D" (S ) is the set of all points s that are "?close to some point from S . Indeed:  If d(s ; s)  ", then s is an element of the set S0 = S [fs g that is "?close to S .  Vice versa, if s 2 S0 and S0 is "?close to S , then, by de nition of set distance, we have d(s ; s)  " for some s 2 S . This set D" (S ) is called the "?dilation of the original set S . This dilation contains the same information about the desired (unknown) set S0 as the original set S , but the good thing about the dilation is that it is in some sense \smoother": e.g., if the set S missed a small (diameter < ") circle 0

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inside, this circle will be added; if we have a set of chaotic close pixels lling a box, the dilation will be the box, etc. In other words, the dilation \ lters" the original set S by deleting the features that were caused purely by errors, and highlighting others.

1.4 When are two images dierent?

Another natural question is as follows: Suppose that we have two observation results; are the observed images the same or not? E.g., if we have observed the same object at two dierent moments of time, did the image change? Due to observation errors, the observed images S1 and S2 are, usually, different even if they correspond to the same actual image S0 . Due to the same observation errors, the images can be dierent, but the dierence can be so tiny that our sensors do not notice it (i.e., S1 = S2 ). So, we can never be sure that the two actual images are exactly the same; however, we can sometimes be sure that the two observations correspond to dierent images. How can we characterize this case in terms of S1 and S2 ? If both observed images correspond to the same actual image S0 , this means that dH (S1 ; S0 )  " and dH (S2 ; S0 )  ", so, from the triangle inequality, we can conclude that dH (S1 ; S2 )  2". It turns out that this inequality is a necessary and sucient condition for the existence of the desired image: n Proposition 1. For two compact sets S1 and S2 in R , the following two conditions are equivalent to each other:  There exists a set S0 for which dH (S1 ; S0 )  " and dH (S2 ; S0 )  ".  dH (S1 ; S2 )  2". (The proofs are given in the last section.)

2

Set Intervals

2.1 Main idea

The description of an image by a single set is based on the assumption that the only problem with the image is that it is inaccurate, and that every pixel is absolutely reliable. In other words, this description is based on the assumption that every time we send a signal to investigate a point, from the returned signal, we know for sure whether this investigated point was part of the desired image or not. In real-life imaging (e.g., in radar imaging), in addition to the cases when we are sure that we hit an image, and to the cases when we are sure that we missed, there are intermediate cases when we may have hit the desired object, but we are not sure about that. In this case, in order to adequately represent 3

the observation results, we need not a single observed set S as before, but two sets (S; S ): the set S of all the pixels that have been observed for sure, and the set S of all the pixels that may correspond to observations. Clearly, S  S ; it is natural to call this pair of sets S = (S; S ) a set interval. If we know the set interval, this means that the only information about the observed set S is that it is in between S and S (S  S  S ). We will say that the set S belongs to the set interval S.

2.2 Operations with set intervals

Standard set operations like union, intersection, complement, can be easily generalized to set intervals. For example, if we know that an (unknown) set S1 belongs to the interval S1 = (S 1 ; S 1 ), and that an (unknown) set S2 belongs to the interval S2 = (S 2 ; S 2 ), then the union of S1 and S2 is in between S = S 1 [ S 2 and S = S 1 [ S 2 . Thus, we can take (S; S ) as the union of the set intervals S1 and S2 . To be more precise, the following result is true: Proposition 2. Let S1 = (S 1 ; S 1 ) and S2 = (S 2 ; S 2 ) be two set intervals. Then, for every set S , the following two conditions are equivalent to each other:  S belongs to the interval S = (S; S ), where S = S 1 [ S 2 and S = S 1 [ S 2 .  S can be represented as the union S1 [ S2 of two sets S1 2 S1 and S2 2 S2 . Comment. Similar (easy-to-prove) results justify the other two de nitions:  the de nition of the intersection of two set intervals as S = (S; S ), where S = S 1 \ S 2 and S = S 1 \ S 2 , and  the de nition of the complement S = S1 ? S2 as S = (S; S ), where S = S 1 ? S 2 and S = S 1 ? S 2 .

2.3 When are two set intervals representing the same image?

Similarly to sets, for set intervals, we can ask the question: when do two observations S1 = (S 1 ; S 1 ) and S2 = (S 2 ; S 2 ) describe the same image? In the idealized situation when each pixel from the observed image is placed precisely (i.e., when S0 = S ), the question becomes as follows: when does there exist a set S that belongs to both intervals, i.e., for which S 1  S  S 1 and S 2  S  S 2 ? These two inclusions are equivalent to S 1 [ S 2  S  S 1 \ S 2 , and thus, the existence of such a set S is equivalent to S 1 [ S 2  S 1 \ S 2 . In a more realistic situation, the actual image S0 is "?close to the observed image S . In this case, the answer to the desired question is given by the following result: 4

Proposition 3. Let S1 = (S 1 ; S 1 ) and S2 = (S 2 ; S 2 ) be two set intervals. Then, the following four conditions are equivalent to each other: (i) There exists a set S0 that is "?close to some set S1 from the set interval S1 , and that is also "?close to some set S2 from the set interval S2 . (ii) Some set S1 from the interval S1 is 2"?close to some set S2 from the set interval S2 . (iii) S 1  D2" (S 2 ) and S 2  D2" (S 1 ). (iv) dH (S~1 ; S~2 )  2", where  S~1 = S 1 [ (D" (S 2 ) \ S 1 ) and  S~2 = S 2 [ (D" (S 1 ) \ S 2 ). Comment. Due to condition (iii), it is natural, for arbitrary two set intervals S1 and S2 , to de ne the Hausdor distance between them as the smallest  > 0 for which S 1  D (S 2 ) and S 2  D (S 1 ), i.e., for which:  each point s1 2 S 1 is ?close to some point from S 2 , and  each point s2 2 S 2 is ?close to some point from S 1 . It is easy to check that this is indeed a metric, and that for degenerate set intervals Si = (Si ; Si ) thus de ned distance coincides with the standard Hausdor distance between sets.

3

Intervals of probabilities

In the previous two sections we described interval aspects of the topics presented at the workshop, but, of course, the main emphasis was on random sets. The corresponding probability topics, however, also had an interesting interval aspect caused by the fact that in many real-life situations, we do not know the exact values of the probabilities, only intervals of possible values. The related problems were described in a talk given by Teddy Seidenfeld from the Carnegie-Mellon University. Among other interesting things, he showed that many problems encountered by random sets with interval-values probabilities can be actually formulated in basic probability terms, without any mention of sets. One of these problems is the so-called dilation. In traditional statistics, if we make additional observations or measurements, we can use the well-known Bayes rule to update our probabilities. There are many results showing that, basically, no matter what a priori probability we assume, after suciently many updates, we are getting closer and closer to the actual probabilities. It turns out that if, instead of the exact probabilities, we start with intervals of possible 5

values of probability, then, instead of getting better and better estimates, we can have situations in which, paradoxically, the updated interval is worse (much wider) than the original estimate. This dilation phenomenon can be illustrated on a very simple example: Let A be an event whose probability P (A) is completely unknown to us, so that the interval of possible values of probability is P(A) = [P (A); P (A)] = [0; 1], and let (H; T ) be two possible results of tossing a fair coin. It is natural to assume that the event A and the coin toss are independent, so that P (A&H ) = P (A)
P (H ) = 0:5
P (A). In this case, the probability P (E ) of the event E = (A&H ) _ (:A&T ) is precisely known: it is equal to 0.5. This is an a priori probability of E , known before any observations are made. Let us now toss the coin. Intuitively, after tossing the coin, we gain some extra knowledge, so our initial uncertainty can only decrease. However, according to Bayes formula, if, e.g., the coin falls on heads, the updated probability P (E ) will be equal to P (E jH ) = P (E &H )=P (H ) = P (A&H )=P (H ) = (0:5
P (A))=0:5 = P (A). Thus, the new interval of possible values of probability is equal to [0; 1]. So, after the experiment, the a priori degenerate (maximally narrow) probability interval P(E ) = [0:5; 0:5] changes into an a posteriori maximally wide interval [0; 1]. Seidenfeld has shown that this \dilation" is a reasonably general phenomenon, and that, crudely speaking, the only way to avoid it is to keep the intervals suciently narrow. The complete paper with these results will appear in the Proceedings [1]. Preliminary results were published in [2].

4

The city and the workshop

The state of Minnesota is near the Northern border of the US. In the winter, it grows very cold here. It was originally populated by immigrants from Scandinavia, to whom Minnesota's climate reminded their native countries. Even now, many people are blond, and many last names remind of Sweden. Minneapolis' downtown is pretty and unusual: all major buildings are connected by skywalks so that in winter, one can walk anywhere without going out into the cold. The campus is located right on the Mississippi river. This great river is just starting here, so it is reasonably narrow and students cross it to get from one class to another. In Minneapolis, \West Bank" is an ocial name of the west-bank part of the city, so, on the local TV, \news from the West Bank" was not about Middle East. Scienti cally, the workshop was very well organized. During the workshop, every participant was given a key to a computer-equipped oce at the Institute, and email, fax, xerox, library access, and other services were at our disposal. 6

5

Proofs

5.1 Proof of Proposition 1

Since we have already shown that the rst condition implies the second one, it is sucient to show that if dH (S1 ; S2 )  2", then there exists a set S0 for which dH (S1 ; S0 )  " and dH (S2 ; S0 )  ". Indeed, the fact that dH (S1 ; S2 )  2" for the compact sets S1 and S2 means that:  for every s1 2 S1 , there exists a point f1 (s1 ) 2 S2 for which d(s1 ; f (s1 ))  2", and that  for every s2 2 S2 , there exists a point f2 (s2 ) 2 S1 for which d(s2 ; f2 (s2 ))  2". As S0 , we take the closure of the following union U : U = f(1=2)(s1 + f1 (s1 )) j s1 2 S1 g [ f(1=2)(s2 + f2(s2 )) j s2 2 S2 g: Let us show that dH (S0 ; S1 )  " and dH (S0 ; S2 )  ". Since S0 is the closure of U , it is sucient to show that dH (U; S1)  " and dH (U; S2 )  ". Without loss of generality, it is sucient to prove the rst of these inequalities. This rst inequality means two things:  For every s 2 U , there exists a point s~1 2 S1 for which d(s; s~1 )  ". Indeed: { If s = (1=2)(s1 + f1 (s1 )) for some s1 2 S1 , then s is a mid-point of a segment of length d(s1 ; f1 (s1 ))  2", and hence, the desired inequality holds for s~1 = s1 2 S1 . { Similarly, if s = (1=2)(s2 + f2 (s2 )) for some s2 2 S2 , then the desired inequality holds for s~1 = f2(s2 ) 2 S1 .  For every s1 2 S1 , there exists a point s 2 U for which d(s; s1 )  ". Indeed, we can take s = (1=2)(s1 + f1 (s1 )) 2 U . Q.E.D.

5.2 Proof of Proposition 3

(i)$(ii) follows from Proposition 1. (ii)!(iii) From dH (S1 ; S2 )  2", it follows that every point s1 2 S1 is 2"?close to some point from S2 . Since S 1  S1 , it follows that every point s1 2 S 1 is 2"?close to some point s2 2 S2 . Since S2  S 2 , we conclude that every point from S 1 is 2"?close to some point from S 2 . Thus, S 1  D2" (S 2 ). The second inclusion S 2  D2" (S 1 ) is proven similarly. (iii)!(iv). Let us assume (iii), and let us prove that: 7

 for every element s1 2 S~1 , there exists a 2"?close point s2 2 S~2 , and  vice versa, for every element s1 2 S~2 , there exists a 2"?close point s1 2 S~1 . Without loss of generality, it is sucient to prove the rst statement, because the second one is absolutely similar. Let s1 2 S~1 = S 1 [ (D" (S 2 ) \ S 1 ); let us nd an 2"?close element from S~2 . Since S~1 is the union of two sets, we must consider two possible cases:  s1 2 S 1 ;  s1 2 D2" (S 2 ) \ S 1 . In the rst case, from (iii), we get S 1  D2" (S 2 ), and thus, from s1 2 S 1 , we conclude that s1 2 D2" (S 2 ). This means that there exists an s2 2 S 2 for which d(s1 ; s2 )  2": This point s2 belongs to S 2 and is 2"?close to some point from S 1 . Thus, s2 2 D2" (S 1 ) \ S 2 . Hence, s2 2 S~2 . So, in the rst case, we have found the desired point s2 2 S~2 that is 2"?close to s1 . In the second case, when s1 2 D2" (S 2 ) \ S 1 , we conclude that s1 2 D2" (S 2 ) and therefore, there exists a point s2 2 S 2 that is 2"?close to s1 . Since S~2 is the union of S 2 and one more set, this same point s2 also belongs to S~2 . In both cases, the implication is proven. (iv)! (ii) is easy: indeed, from (iv), it immediately follows that S~1 2 S1 , S~2 2 S2 , and dH (S~1 ; S~2 )  2": Q.E.D.

References [1] J. Goutsias, R. Mahler, and H. T. Nguyen (eds.), Applications and Theory of Random Sets, Springer-Verlag, 1997 (to appear). [2] T. Seidenfeld and L. Wasserman, \Dilations for sets of probabilities", Annals of Statistics, 1993, Vol. 21, pp. 1139{1154. Authors' addresses: Hung T. Nguyen, Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, email [email protected]; Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, email [email protected].

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