Multi-argument distances

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1 Fuzzy Sets and Systems 167 (2011) Multi-argument distances J. Martín,G.Mayor University of the Balearic Islands, Cra. Valldemossa km 7.5, Palma, Spain Available online 5 November 2010 Abstract In this paper, we propose a formal definition of a multi-argument function distance. The conventional definition of a distance is extended to apply to collections of more than two elements. Some general properties of multidistances are investigated and significant examples are exhibited. Finally, we introduce the concept of ball centered at a list in order to extend the one of the usual ball in a metric space Elsevier B.V. All rights reserved. Keywords: Distance; Multidistance; Fermat point; Sum-based multidistance; OWA operator 1. Introduction The conventional definition of distance over a space specifies properties that must be obeyed by any measure of how separated two points in that space are. Given a set X, a distance d on X is a function that distinguish between every two different points x and y of X by assigning to the ordered pair (x, y), in a symmetric manner, a single positive real number, d(x, y), in such a way that, given any point z of X, d(x, y) does not exceed the sum of d(x, z)andd(z, y), and we set d(x, y) = 0 if and only if x = y. These axioms arise when one examines the fundamental properties of the point-to-point-along-a straight-line-segment prototype with a view to developing a theory around those properties, a theory that will then be applicable in many situations, some of them very different from that of the prototype. However one often wants to measure how separated the members of a collection of more than two elements are. The conventional way to do this is to combine the pairwise distance values for all pairs of elements in the collection, into an aggregate measure. Thus, given an Euclidean triangle (A, B, C) we can combine the distances AB, AC, BC using, for instance, a three-dimensional OWA operator, F(x 1, x 2, x 3 ) = w 1 y 1 + w 2 y 2 + w 3 y 3. Then, we measure how separated (A, B, C) are by means of the formula D(A, B, C) = F(AB, AC, BC). It is clear that we have to choose the weighting vector (w 1,w 2,w 3 ) such that the multi-argument distance function D satisfies a group of axioms that extend to some degree those for ordinary distance functions. For pairwise distances and related distance matrix see for example [1]. The authors acknowledge the support of the Govern Balear Grant PCTIB2005GC1-07 and the Spanish DGI Grants MTM and MTM They also wish to thank the referees for their valuable comments and suggestions. Corresponding author. address: javier.martin@uib.es (J. Martín) /$ - see front matter 2010 Elsevier B.V. All rights reserved. doi: /j.fss

2 J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) Of course we can consider many other procedures to measure how separated the vertices (A, B, C) are: In Euclidean geometry the Fermat point of a triangle (A, B, C), also called Torricelli point, is the point F for which the sum of the distances from F to the vertices is as small as possible; i.e. the Fermat point is the point F such that FA+ FB + FC is minimized. Then we can define D( A, B, C) = FA+ FB + FC. In this paper the formal definition of a distance function is extended to apply to collections of more than two elements. The measure presented here applies to n-dimensional ordered lists of elements, and it can be directly incorporated into many domains where ad hoc combinations of pairwise metrics are currently used. The main advantage of our proposal is that it allows the general treatment of the problem of measuring the distance for more than two points by means of an axiomatic procedure. This new measure also permits in a natural way to measure how separated from each other two collections are. In the next section the definitions of multidistance and strong multidistance are introduced, and some elementary properties of multi-argument distances are presented. Sections 3 and 4 are devoted to present significant examples of multidistances: Fermat and averaged multidistances. In Section 5 we introduce the fundamental concept of ball centered at a list in order to deal with topological properties in a multidistance space. In 2004, Wolpert [9] presented the definition of a multi-argument metric (multimetric, for short) in a rather different manner. The measure introduced by Wolpert applies even to collections with fractional numbers of elements, however his axiomatic, an extension of the usual conditions defining a metric, is stronger than the one we present here. To be more precise, Wolpert s multimetrics can be viewed as similar to our strong multidistances. In [7,2,4] terms liken-distances and multi-metrics are introduced in certain contexts, but with a very different meaning with respect to those defined by Wolpert and ourselves. Thus, in [4] a multi-metric space is simply a union X = i X i where each X i is a metric space with metric d i, i = 1,..., m. In[7] given a vector of metric functions (d 1,..., d m ) and one of the weights (w 1,...,w m ), w i [0, 1], a (linear) multi-metric is defined as d = w 1 d 1 + +w m d m. 2. Multidistances Definition 1. A function D : n 1 Xn [0, ] isamultidistance on a non-empty set X when the following properties hold, for all n and x 1,..., x n, y X: (m1) D(x 1,..., x n ) = 0 if and only if x i = x j for all i, j = 1,...n. (m2) D(x 1,..., x n ) = D(x π(1),..., x π(n) ) for any permutation π of 1,..., n, (m3) D(x 1,..., x n ) D(x 1, y) + +D(x n, y), We say that D is a strong multidistance if it fulfills (m1), (m2) and: (m3 ) D(x 1,...x k ) D(x 1, y) + + D(x k, y)forallx 1,..., x k, y n 1 Xn. Expressions like D(x, y) in(m3 ), that is, the function D applied to two lists x = (x 1,..., x n ) X n and y = (y 1,..., y m ) X m, mean the multidistance of the joined list: D(x, y) = D(x 1,..., x n, y 1,..., y m ). (1) Remark 1. (1) If D is a multidistance on X, then the restriction of D to X 2, D X 2, is an ordinary distance on X. (2) Any ordinary distance d on X can be extended in order to obtain a multidistance. For example, we can define a function D M in this way: D M (x 1,..., x n ) = maxd(x i, x j ); i < j}. (2) Then, D M is a strong multidistance on X such that D X 2 = d. It will be called maximum multidistance. Example 1. Based on the drastic distance, defined by 1ifx y, d(x, y) = 0ifx = y, (3)

3 94 J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) several multidistances can be defined extending it. For instance, 0ifxi = x j i, j, D 1 (x 1,..., x n ) = 1otherwise, D 2 (x 1,..., x n ) = x 1,..., x n } 1. Both of them are strong. Proposition 1. Let D and D be multidistances on X. (1) D + D is a multidistance on X. (2) If k > 0, then kd is a multidistance on X. (3) D/(1 + D) and min1, D} are also multidistances on X, with values in [0, 1]. Remark 2. If D is a multidistance on X, then the function D : n 1 Xn n 1 Xn [0, + ) definedby D(x, y) = D(x 1,..., x n, y 1,..., y m ) (4) for all x = (x 1,..., x n ) X n and y = (y 1,..., y m ) X m is not a distance on the set n 1 Xn. However, pseudodistances (distance functions allowing d(x, y) = 0 for distinct values x, y) on this set can be constructed by means of D and an ordinary distance δ on R in this way: D δ (x, y) = δ(d(x), D(y)). (5) 3. Fermat multidistances Let (X, d) be a metric space and (x 1,..., x n ) X n. Proposition 2. The function f : X [0, + ) defined by f (x) = n d(x i, x) is continuous and it reaches a minimum. Definition 2. The Fermat set F x of a list x = (x 1,..., x n ) X n is the set of points where the function f given in Proposition 2 reaches the minimum value: F x = x X: n d(x i, x) } n d(x i, x ), x X. (6) Example 2. In the metric space (R m, d), where d is the Manhattan distance: d((a 1,..., a m ), (b 1,..., b m )) = the set F associated to the points x i = (a i1,..., a im ), i = 1,..., n is m a i b i, (7) m F (x1,...,x n ) = [a ( (n+1)/2 )i, a ( (n+2)/2 )i ], (8) where the parentheses in the subindexes mean that a (1)i,..., a (m)i is a non-decreasing rearrangement of the components i of the points, x is the floor function and is the usual Cartesian product of sets. If m is odd, the extreme points of the intervals coincide and the set F (x1 reduces to a point, whose coordinates,...,x n ) are the median of the respective coordinates of the points x 1,..., x n :seefig.1.

4 J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) F F Fig. 1. Fermat sets F associated to two families of points in R 2. Proposition 3. Let D F : n 1 Xn [0, ) be the function defined by n } D F (x 1,..., x n ) = min d(x i, x). (9) x X (1) D F is a multidistance onx. (2) D F is maximum in the sense that any other multidistance D such that D X 2 = D F X 2 takes values not greater than D F, i.e. D(x) D F (x), x X n. Proof. Condition (m1): if x 1 = = x n,then n d(x i, x 1 ) = 0andsoD F (x 1,..., x n ) = 0. Conversely, let us suppose that D F (x 1,..., x n ) = 0 and that there exist two different elements, x r, x s, such that d(x r, x s ) > 0; hence, for all x X, we have a contradiction: d(xi, x) d(x r, x) + d(x s, x) d(x r, x s ) > 0. Properties (m2) and (m3), and also (2), follow immediately from (9). This function D F will be called the Fermat multidistance associated to an ordinary distance d. Example 3. Let us consider the drastic distance (3). We denote by x 1,..., x m the elements of the list (x 1,..., x n ) X n without repetitions, arranged in such a way that the respective number of repetitions n 1,..., n m, verifies n 1 n m. The elements of the Fermat set associated to (x 1,..., x n ) X n are the most repeated elements of the list: F (x1,...,x n ) =x 1,..., x k : n 1 = =n k > n k+1 }. (10) If all of the elements of the list are different, then F (x1,...,x n ) =x 1,..., x n }. Then the resulting Fermat multidistance is D F (x 1,..., x n ) = n n 1. (11) The distance D F is not strong in general, as the following example shows. Example 4. In R 2, with the Euclidean distance: d((a 1, a 2 ), (b 1, b 2 )) = (a 1 b 1 ) 2 + (a 2 b 2 ) 2, (12) the set F associated to two points A, B is the segment AB. The Fermat set of three points is the Fermat point (see, for example, [3]), and the Fermat Weber point,orgeometric median, for more than three points [8]. Let us see that the associated D F is not strong. Consider the points A = (1, 0), B = (0, 1), C( 1, 0), D = (0, 1) and E = (0, 0) in R 2. The Fermat set of the square (A, B, C, D)isF (A,B,C,D) =E}, andsod F (A, B, C, D) = 4.

5 96 J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) On the other hand, the Fermat points of (A, B, E)and(C, D, E) are ((3 3)/6, (3 3)/6) and ( (3 3)/6, (3 3)/6), respectively, and That is, D F (A, B, E) = D F (C, D, E) < 2. D F (A, B, C, D) > D F (A, B, E) + D F (C, D, E), and D F is not a strong multidistance. Example 5. The multidistance D F associated to the Manhattan distance of Example 2 is defined in this way: D F (x 1,..., x n ) = m n/2 j=1 4. Sum-based multidistances j(a (n j+1)i a ( j)i ). (13) Let (X, d) be a metric space. Let us consider the function D : n 1 Xn [0, ]definedby D(x1 ) = 0, D(x 1,..., x n ) = i< j d(x i, x j ), n 2, (14) that is, the sum of the pairwise distance values for all pairs of elements of the list. It is not a multidistance: it fulfills (m1) and (m2) but not (m3). For example, let (A, B, C) be an equilateral triangle of side length 1 in the Euclidean plane. Then, condition (m3) is not fulfilled for y = A: D(A, B, C) = 3πD(A, A) + D(B, A) + D(C, A) = 2. In order to obtain multidistances based on this sum, we have to multiply it by a factor λ which depends on the length of the list: λ = λ(n), n 1. Proposition 4. The function D λ : n 1 Xn [0, ] defined by Dλ (x 1 ) = 0, D λ (x 1,..., x n ) = λ(n) i< j d(x i, x j ), n 2 (15) is a multidistance if and only if: (i) λ(2) = 1, (ii) 0< λ(n) 1/(n 1) for any n > 2. Proof. The conditions are necessary: (i) because D λ X 2 = d and the first inequality of (ii), 0 < λ(n), because D λ is non-negative and fulfills (m1). Let us see that the second inequality of (ii), λ(n) 1/(n 1), also is: if not, if there exists n 0 such that λ(n 0 ) > n 0 1 }} 1/(n 0 1), we can consider the drastic distance, two different points x, y X and ( x,..., x, y) X n 0 : n 0 1 }} D λ ( x,..., x, y) = λ(n 0 ) (n 0 1) > 1. }} But the sum of distances from the components of the list ( x,..., x, y)tox is 1, and (m3) does not hold. n 0 1

6 J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) Conditions (i) and (ii) are also sufficient: (i) and the first inequality in (ii) ensure D λ 0 and (m1); condition (m2) follows from the definition of D λ, and finally, condition (m3) follows from λ(n) 1/(n 1): for any y X, D λ (x 1,..., x n ) = λ(n) d(x i, x j ) i< j λ(n) i< j (d(x i, y) + d(x j, y)) = λ(n)(n 1) i d(x i, y) i d(x i, y). Some of these multidistances are like averaged sums of the all ordinary distances between all pairs of points. In fact, if λ(n) = 1/( n 2 ) 1/(n 1), D λ is the arithmetic mean of these distances OWA-based multidistances Let (X; d) be a metric space and W = (W n ) n 3 a collection of OWA operators W n with weighting vector (w1 n,...,wn n ). We consider the multi-argument function D : n 1 Xn [0, ) defined as follows: D(x 1 ) = 0, D(x 1, x 2 ) = d(x 1, x 2 ), (16) D(x 1,..., x n ) = W n (a 1,..., a n ) whenever n 3, where a i = D(x 1,..., x i,..., x n ), i = 1,..., n,and(x 1,..., x i,..., x n ) means the (n 1)-ordered list obtained from the n-list (x 1,..., x i,..., x n ) once removed the element placed in the i-position. To be precise, we will write D W to indicate that this function comes from the collection W. We will write W n = (w1 n,...,wn n ) for the sake of brevity. Example 6. (i) The maximum multidistance D M givenby(2) is nothing else that D W with W n = (0,..., 0, 1), for all n 3. (ii) We can also introduce the median multidistance D med,where (0,..., 0, 1, 0,..., 0) if n is odd, W n = (0,..., 0, 2 1, 2 1, 0,..., 0) if n is even. (17) (ii) Of course we can consider many other collections (W n ) n 3 ; thus we can combine two different classes of OWA operators: (0,..., 0, 1, 0,..., 0) if n is odd, W n = ( 2 1, 0,..., 0, 2 1 ) ifn is even. (18) Proposition 5. The multi-argument function D W, where W = (W n ) n 3 is a collection of OWA operators W n = (w1 n,...,wn n ) is a multidistance if and only if wn 1 < 1, for all n 3. Proof. Let us prove first the only if part. Suppose W n = (1, 0,..., 0) for some n 3, then D W (x 1,..., x n ) = W n (a 1,..., a n ) = mina 1,..., a n }=0 does not imply x 1 = = x n. Thus we see that D W does not satisfy condition (m1) in Definition 1 and consequently it is not a multidistance.

7 98 J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) Reciprocally, let us suppose now W n = (w n 1,...,wn n ) with wn 1 < 1foralln 3. We will demonstrate that D W is a multidistance: Condition (m1), that is, D W (x) = 0 x i = x j for all i, j = 1,..., n, holds for n=1 and 2. Let us suppose that (m1) is true for an n 3 and let us prove it for n + 1. If x i = x j for all i, j = 1,..., n + 1, n 3, then D W (x 1,..., x n+1 ) = W n+1 (a 1,..., a n+1 ), where a i = D W (x 1,..., x i,..., x n+1 ), i = 1,..., n+1. Therefore a i = 0foralli = 1,..., n+1andd W (x 1,..., x n+1 ) = W n+1 (0,..., 0) = 0. Conversely, if D W (x 1,..., x n+1 ) = 0then W n+1 (a 1,..., a n+1 ) = w1 n+1 a (1) + +wn+1 n+1 a (n+1) = 0 (a (1) a (n+1) is an increasing reordering of a 1,..., a n+1 ) thus wi n+1 a (i) = 0foralli = 1,..., n + 1, therefore a (r) = 0forsomer, 2 r n + 1 (recall w1 n+1 < 1). Let us suppose that a (r) = a s, then we can write a (r) = a s = D W (x 1,..., x s,..., x n+1 ), therefore D W (x 1,..., x s,..., x n+1 ) = 0, which implies, by the induction hypothesis, x i = x j for all i, j = 1,..., n + 1, i, j s. On the other hand, we know that a (1) a (r 1) a (r) = 0 which finally implies x i = x j for all i, j = 1,..., n + 1. Condition (m2) is D W (x π(1),..., x π(n) ) = D W (x 1,..., x n ) for all permutation π of 1, 2,..., n. Of course this is true for n = 1 and 2. Let us consider that it is also true for an n 3 and let us prove it for n + 1: D W (x π(1),..., x π(n+1) ) = W n+1 (b 1,..., b n+1 ), where b i = D W (x π(1),..., x π(i),..., x π(n+1) ), i = 1,..., n + 1. On the other side, D W (x 1,..., x n+1 ) = W n+1 (a 1,..., a n+1 ), where a i = D W (x 1,..., x i,..., x n+1 ), i = 1,..., n + 1. From the induction hypothesis, we can write b i = a π(i), i = 1,..., n + 1. Finally, from the symmetry of the OWA operators W n,wehave D W (x π(1),..., x π(n+1) ) = W n+1 (b 1,..., b n+1 ) = W n+1 (a π(1),..., a π(n+1) ) = W n+1 (a 1,..., a n+1 ) = D W (x 1,..., x n+1 ). And condition (m3), D W (x) n D W (x i, y) forally X, again by induction. It is true for n = 1 and 2. Let us consider that (m3) is true for an n 3 and let us prove it for n + 1. D W (x 1,..., x n+1 ) = W n+1 (a 1,..., a n+1 ) maxa 1,..., a n+1 }. Let us suppose maxa 1,..., a n+1 }=a 1, for short. Then we can write, for all y X, n n D W (x 1,..., x n+1 ) a 1 = D W (x 2,..., x n+1 ) D W (x i, y) D W (x i, y). 5. Balls centered at a list i=2 Given a nonempty set X, a multidistance D defined on it and a list x = (x 1,..., x n ) X n, the following sets can be defined: X (x;>) =y X : D(x, y) > D(x)}, X (x;=) =y X : D(x, y) = D(x)}, X (x;<) =y X : D(x, y) < D(x)} (similar definitions for X (x; ) and X (x; ) ). (19)

8 J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) These three sets form a partition of X, with possibly one or two of them empty. Example 7. For the Maximum multidistance D M, the sets above are n X (x;>) = X B(x i, D M (x)), X (x;=) = n B(x i, D M (x)), X (x;<) =, where the balls on the right sides are ordinary balls in (X, d). For the Fermat multidistance D F,wehave X (x;>) = X \ F x, X (x;=) = F x, X (x;<) =. (21) The set X (x;<) can be non-empty, as the following example shows. Example 8. Consider the Euclidean plane R 2 with the multidistance D(x 1,..., x n ) = 1/( n 2 ) i< j d(x i, x j ). For a list x = (x 1, x 2 ), the set X (x;=) is the ellipse with focus at x 1, x 2 and semimajor axis d(x 1, x 2 ). The interior of the ellipse is X (x;<) and the exterior, X (x;>). Note that if x 1 = x 2 = x, thenx (x;=) =x}, X (x;<) = and X (x;>) = X \x}. Definition 3. Given a multidistance D and a list x = (x 1,..., x n ) n 1 Xn,theclosed ball of center x and radius r R is the set: B(x, r) =y X : D(x, y) D(x) + r}. (22) If n = 1andr 0 the ball is B(x, r) =y X : D(x, y) D(x) + r}. AsD(x) = 0, B(x, r) is an ordinary ball in the metric space X. Remark 3. (1) The closed ball centered at a list x and radius 0 is the set X (x; ). (2) There exist non-empty balls with negative radius if and only if X (x;<). Example 9. For a given metric space (X, d), the balls corresponding to the distance D M (x) = max i, j d(x i, x j )} are n B(x, r) = B(x i, D M (x) + r), (23) where the balls on the right side are ordinary balls in (X, d)andr D M (x). Let us prove it y B(x, r) D(x, y) D(x) r D(x i, y) D(x) r i y B(x i, D M (x) + r) i n y B(x i, D M (x) + r). (20) Example 10. The balls corresponding to the Fermat drastic multidistance (see Example 3)are if r < 0, B(x, r) = F x =x 1,..., x k } if 0 r < 1, X if r 1. Balls with negative radius are empty because D F (x, y) D F (x) always holds for any Fermat multidistance. (24)

9 100 J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) The element x l, l k, which is repeated n l = n 1 times, belongs to any ball with non-negative radius, because D(x, x l ) D(x) = n + 1 (n l + 1) (n n 1 ) = 0. On the other hand, if y / F rmx then D(x, y) D(x) = n + 1 n 1 (n n 1 ) = 1, and so y B(x, r) if and only if r Conclusions We have proposed a formal definition of multi-argument function distance, which captures the idea of distance for collections of more than two points. We think that this new tool can be useful every time you want to measure how separated are the members of an n-dimensional list. In this paper, some general aspects of multidistances have been investigated and several root examples exhibited. Several topics and research lines arise in this new framework. For instance, aggregation of pairwise distance values in order to construct multidistances [6], connections between pseudo-multidistances and multi-indistinguishability operators [5], asymmetric multidistances and aggregation of multidistances. References [1] L.M. Blumenthal, Theory and Applications of Distance Geometry, second ed., Chelsea Publishing Co, New York, [2] B. Bustos, T. Skopal, Dynamic similarity search in multimetric spaces, in: Multimedia Info Retrieval, 2006, pp [3] H.S.M. Coxeter, Introduction to Geometry, John Wiley& Sons, New York, [4] L. Mao, On multi-metric spaces, Scientia Magna 2 (1) (2006) [5] J. Martín, G. Mayor, An axiomathical approach to T-multi-indistinguishabilities, in: IFSA-EUSFLAT 2009, [6] J. Martín, G. Mayor, Aggregating pairwise distance values, in: EUROFUSE 2009, 2009, pp [7] J. Rintanen, Distance estimates for planning in the discrete belief space, in: Planning & Scheduling, 2004, pp [8] E. Weiszfeld, Sur le point pour lequel la somme des distances de n points donnes est minimum, Tohoku Mathematical Journal 43 (1937) [9] D.H. Wolpert, Metrics for more than two points at once, in: Y. Bar-Yam (Ed.), International Conference on Complex Systems ICCS 2004, Perseus Books, in press.