The depth function of a population distribution

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1 Metrika (1999) 49: 213±244 > Springer-Verlag 1999 The depth function of a population distribution Peter J. Rousseeuw1, Ida Ruts2 1 Department of Mathematics, UIA, Universiteitsplein 1, B-2610 Antwerpen, Belgium ( rousse@uia.ua.ac.be) 2 Faculty of Applied Economic Sciences, UFSIA, Prinsstraat 13, B-2000 Antwerpen, Belgium ( ida.ruts@uia.ua.ac.be) Received: April 1999 Abstract. Tukey (1975) introduced the notion of halfspace depth in a data analytic context, as a multivariate analog of rank relative to a nite data set. Here we focus on the depth function of an arbitrary probability distribution on R p, and even of a non-probability measure. The halfspace depth of any point y in R p is the smallest measure of a closed halfspace that contains y. We review the properties of halfspace depth, enriched with some new results. For various measures, uniform as well as non-uniform, we derive an expression for the depth function. We also compute the Tukey median, which is the y in which the depth function attains its maximal value. Various interesting phenomena occur. For the uniform distribution on a triangle, a square or any regular polygon, the depth function has ridges that correspond to an `inversion' of depth contours. And for a product of Cauchy distributions, the depth contours are squares. We also consider an application of the depth function to voting theory. Key words: Depth contour, halfspace depth, location depth, Tukey median, voting 1. Introduction The notion of halfspace depth was introduced by Tukey (1975) in a dataanalytic setting, i.e. for a nite data set. For any data cloud X n ˆ fx 1 ; x 2 ;... ; x n g H R p and any point y A R p, the depth of y relative to X n is k=n where k is the smallest number of observations in a closed halfspace containing y. Without loss of generality, we may assume that the boundary of this halfspace passes through y. Formally, the halfspace depth of y relative to X n is de ned as depth y; X n ˆ 1 n min kukˆ1 a H y; u X X n 1:1

2 214 P. J. Rousseeuw, I. Ruts where H y; u ˆ fx A R p ; u 0 x V u 0 yg is a closed halfspace. Points outside the convex hull con X n have depth y; X n ˆ 0 whereas the depth increases when y is more `centrally located' in X n. In this sense, depth can be seen as a multivariate analog of rank. The depth function is de ned everywhere in R p. It maps any y A R p to a number in 0; 1Š. For any a in 0; 1Š we de ne the depth region D a ˆ fy A R p ; depth y; X n V ag 1:2 which is a closed convex set (possibly empty), see Green (1981, 1985). The boundary of D a will be referred to as the depth contour of depth a. Some distribution theory for the sets D a can be found in Eddy (1983, 1985). The deepest y relative to X n (or, when there are several such y, the center of gravity of the deepest nonempty region D a ) is called the Tukey median. It has good robustness properties as a multivariate location estimator ( Donoho and Gasko 1992). For increasing sample size n, the asymptotic behavior of the depth function was studied by He and Wang (1997) and Masse (1999), and that of the Tukey median by Bai and He (1998). An excellent survey of statistical applications of depth can be found in (Liu, Parelius and Singh 1999). Until recently, no algorithms were available to compute the depth of y relative to a point cloud X n H R p. For p ˆ 2, we constructed an exact algorithm for depth y; X n which runs in O n log n time (Rousseeuw and Ruts 1996). Based on this, we then developed exact algorithms for depth contour lines ( ISODEPTH, Ruts and Rousseeuw 1996) and for the bivariate Tukey median (Rousseeuw and Ruts 1998). In higher dimensions, the computations become harder. When p ˆ 3, the algorithm for depth y; X n constructed by (Rousseeuw and Struyf 1998) takes O n 2 log n time, and for p V 4 (and/or large n) they provided an approximate algorithm. Struyf and Rousseeuw (1998) also devised an iterative algorithm to approximate the multivariate Tukey median for any p. In two dimensions, Rousseeuw, Ruts, and Tukey (1999) proposed the bagplot, which is a bivariate boxplot based on contours of the halfspace depth. Implementations of all these algorithms can be downloaded from Our aim in the present paper is to explore the notion of depth for population distributions, i.e. for any probability distribution (with the empirical distribution of a data set X n being a special case.) In fact we'll go a little further and consider all positive measures, which may be in nite, in Section 2. We then specialize these properties to probability distributions in Section 3. In Section 4 we consider probability distributions with a density, i.e. probability distributions that are absolutely continuous relative to the Lebesgue measure l. Section 5 lists examples of such distributions (uniform as well as nonuniform). Section 6 lists examples of distributions without a density. The domains of these degenerate measures have empty interior. In the examples where a density exists, the depth function is more peaked than the density function. This is discussed in Section 7. Section 8 considers an application of depth to voting theory. 2. The depth function of a positive measure Let us consider any positive measure m on R p (with its Borel s-algebra), where m R p may be nite or in nite. For any y in R p, we then de ne its depth as

3 The depth function of a population distribution 215 depth m y ˆ inf kukˆ1 m H y; u if there exists some u for which m H y; u < y ˆ y otherwise: 2:1 We cannot replace the in mum by a minimum because m H y; u is not necessarily continuous as a function of u on the unit sphere. In the special case of an empirical probability distribution we nd (1.1) again (and there the in mum is over a nite set of numbers, so we can write it as a minimum). Obviously, for any measure m we always have depth m y U m R p. Remark 1. For an example where (2.1) is not a minimum, take the probability measure m ˆ 1 2 N 0; I 1 2 D b on R 2 where b ˆ 1; 1 0 and put y ˆ 1; 0 0. [Here D b denotes the point measure in b, i.e. D b B :ˆ I b A B.] Then m R 2 ˆ 1 and depth m y ˆ 1 2 F 1 but this in mum is not attained: the direction u ˆ e 1 ˆ 1; 0 0 yields m H y; u ˆ 1 2 F whereas halfspaces H y; u not containing the atom in b have m H y; u > 1 2 F 1. It is important to note that depth is a½ne invariant. An a½ne transformation is of the form g x ˆ Ax b where A A R pp with det A 0 0 and b A R p. Then we always have depth g y ; m g ˆ depth y; m where m g is the law of g, i.e. m g B :ˆ m g 1 B for any Borel set B H R p. The a½ne invariance property will simplify many computations. Proposition 1. For any positive measure m the function depth m is quasi-concave, i.e. for y 1 and y 2 in R p and 0 U g U 1: depth m gy 1 1 g y 2 V minfdepth m y 1 ; depth m y 2 g (even if some of these depths are y). Proof. Put 0 < g < 1 and y 1 0 y 2 (otherwise the result is trivial) and write z :ˆ gy 1 1 g y 2. For any kuk ˆ 1 it holds that y 1 A H z;u or y 2 A H z;u or both. If y 1 A H z;u we have H y1 ;u H H z;u hence m H z;u V m H y1 ;u V depth m y 1 V minfdepth m y 1 ; depth m y 2 g. The reasoning for y 2 A H z;u is analogous. r From now on we will drop the subscript m for convenience. For any 0 U a U y we de ne the depth region D a as in (1.2). Always D 0 ˆ R p whereas D a may be empty for some a > 0. The depth regions are clearly nested, i.e. a < a 0 implies D a 0 J D a. From Proposition 1 it follows that: Corollary. For any 0 U a U y the set D a is convex. Proof. Let y 1 and y 2 be in D a and 0 U g U 1. Then depth gy 1 1 g y 2 V minfdepth y 1 ; depth y 2 g V a hence gy 1 1 g y 2 belongs to D a. r Proposition 2. For any a > 0 it holds that D a ˆ 7fH; H is an open halfspace with m H c < ag H 7fH; H is a closed halfspace with m H c < ag:

4 216 P. J. Rousseeuw, I. Ruts Proof. We only have to prove the equality, since the inclusion is then a trivial consequence. We nd y A D a, inf y; u V a kukˆ1, Ekuk ˆ 1 : m H y; u V a, E open halfspace H with m H c < a it holds that y A H where only the last, needs to be proved. For ) we take an open halfspace H with m H c < a. Suppose that y B H. Then there exists a vector u (orthogonal to the boundary of H) such that H y; u H H c hence m H y; u U m H c < a, a contradiction. For ( we suppose there is a vector u with m H y; u < a. Then put H :ˆ H y; u c hence m H c < a, but then by assumption y A H ˆ H y; u c which is a contradiction. r Note that this proposition holds for arbitrary positive measures m on R p, and that it provides another insight to why the sets D a are convex and nested. Also note that D a is the intersection of uncountably many open sets, so we cannot conclude whether D a is open, closed, or neither. For an example where D a is open, consider the uniform measure m on the in nite strip A ˆ R Š0; 1. Then depth x; y ˆ y if x; y A A and depth x; y ˆ 0 otherwise. So for a ˆ 7 we nd the open region D a ˆ A which is indeed the intersection of the open halfspaces in Proposition 2. The intersection of the closed halfspaces in Proposition 2 is the strictly larger set R 0; 1Š, which shows that this inclusion is in general not an equality. (This example also illustrates that D a may be unbounded.) Remark 2. For arbitrary measures m, the function depth y is not continuous in y. In fact, it does not have to be upper semicontinuous (u.s.c.) nor lower semicontinuous (l.s.c.). A function h : R p! R is called u.s.c. i for any a A R it holds that h 1 Š y; a is open, whereas for l.s.c. we need h 1 Ša; y to be open. For an example where depth y is not u.s.c., take again the uniform measure m on A ˆ R Š0; 1. If depth were u.s.c. then depth 1 Š y; 7 would be open, but depth 1 Š y; 7 ˆ R 2 na is closed instead. For an example where depth y is not l.s.c. take the distribution m ˆ D 0; 0. If depth were l.s.c. then depth 1 Š0:6; y would be open, but depth 1 Š0:6; y ˆ f 0; 0 g is closed. For any measure m we can de ne sup y depth y which is either y or a nite real number. For instance, for the Lebesgue measure on R p we nd sup y depth y ˆ y which is attained e.g. at y ˆ 0. But in general, the supremum need not be attained. For instance, let m be the uniform measure on A ˆ f x; y ; y < x < y and 0 U y U g x g H R 2 where g x ˆ 1 x 2 1=2 is a strictly positive function with y y g x dl x ˆ y. For any x; y with x A R and y > 0 it holds that depth x; y is nite [take u ˆ 0; 1 0 Š. Now lim y#0 depth x; y ˆ y but depth x; 0 ˆ 0 0 y (which again illustrates that depth is not u.s.c.) and in fact there is no y A R 2 with depth y ˆ y. Even when p ˆ 1 the supremum depth is not necessarily attained. Take the in nite measure m on R de ned by the density function f x ˆ 1 x for x V 1

5 The depth function of a population distribution 217 and f x ˆ 0 elsewhere. For y < 1 we thus have depth y ˆ 0, and for y V 1 we obtain depth y ˆ ln y. Therefore the supremum depth is in nite, but the supremum is not attained because the depth in any y is nite. 3. The depth function of a probability distribution From now on we will assume that m is a nite measure, and normalize it to the probability measure P B :ˆ m B =m R p. The expression (2.1) of depth y now simpli es to depth y ˆ inf kukˆ1 P H y; u 3:1 and we still have a½ne invariance and quasi-concavity. The in mum in (3.1) need not be a minimum (see Remark 1), but now we know more: Proposition 3. For any P on R p and any y A R p : depth y ˆ inf kukˆ1 P H y; u over all u with P qh y; u nfyg ˆ 0; where qh y; u stands for the boundary of H y; u. Note that in the RHS (right hand side) of the proposition, the inf is still over in nitely many u A S. For instance, take a great circle C on S, where S ˆ S 0; 1 is the unit sphere in R p, and put A ˆ fu A S; P qh y; u nfyg 0 0g. Then we can show that A is countable. Indeed, for any k ˆ 1; 2;... put A k ˆ fu A S; 1 2 k U P qh y; u nfyg < 1 2 k 1 g. Then each A k has at most 2 k elements because the total probability P u A A k P qh y; u nfyg ˆ P 6 u A Ak qh y; u nfyg Š U P R p nfygš U 1. Therefore A ˆ 6 k A k is countable, so P qh y; u nfyg ˆ 0 for all u A C except for a countable set. Since S is a union of great circles through 0;... ; 0; 1 and 0;... ; 0; 1 we have P qh y; u nfyg ˆ 0 for almost all u (relative to the uniform distribution on S). Proof of the proposition. Without loss of generality we put y ˆ 0. Clearly depth 0 U RHS which is an inf over a subset. To prove V we take any ~u A S so possibly P qh 0; ~u nf0g > 0. Now consider a closed halfspace D whose boundary contains both 0 and ~u, and a great circle C through ~u which is orthogonal to qd. On C we take a point w with w 0 ~u > 0, and de ne the arc Šw; ~u. Since (as before) only a countable number of v A Šw; ~u have P qh 0; v nf0g > 0 we can take a monotone sequence v n in Šw; ~u with v n! ~u and all P qh 0; vn nf0g ˆ 0. Now P H 0; vn ˆ P H 0; vn X D P H 0; vn X D c where 7 n # H 0; vn X D ˆ H 0; ~u X D and 6 n " H 0; vn X D c ˆ H 0; ~u X D c. Therefore P H 0; ~u ˆ lim n P H 0; vn V RHS, hence depth 0 ˆ inffp H 0; ~u ; ~u A Sg V RHS as well. r Remark 3. This result also holds in the special case when P is the empirical distribution of a data set X n H R p. This special case was used by Rousseeuw and Struyf (1998) in the construction and proof of an algorithm to compute depth y; X n.

6 218 P. J. Rousseeuw, I. Ruts Proposition 4. For any probability distribution P, the function depth y is upper semi-continuous. If moreover P has a density relative to the Lebesgue measure, then depth y is also lower semi-continuous, hence continuous. Proof. See Donoho and Gasko (1992, Lemma 6.1). For any probability distribution P and any a V 0 we thus have, by the upper semi-continuity, that D a ˆ depth 1 a; y ˆ depth 1 Š y; a c is closed in contrast with Remark 2 above. Proposition 5. For any probability distribution P and a > 0 the region D a is bounded, and hence compact. Proof. Consider the closed balls B n ˆ fx A R p ; kxk U ng. Since 6 " B n ˆ R p it holds that P B n " 1, so there exists m such that P B m > 1 a. For any y B B m we see (by a separating hyperplane) that depth y < a, hence D a H B m. r The set D a can of course still be written as an intersection of open halfspaces as in Proposition 2. But now the second part of that proposition can be sharpened: Proposition 6. For any probability P and a > 0 we have D a ˆ 7fH; H is a closed halfspace with P H c < ag: Proof. We already have the inclusion H from Proposition 2, so it remains to prove I. Let y belong to the RHS and suppose that y B D a. Then there exists a vector u with kuk ˆ 1 such that P H y; u ˆ a 0 < a. We can write H y; u as the intersection of a decreasing sequence of open halfspaces Hy; n u ˆ fx A R p ; u 0 x > u 0 y 1 n g. From H y; u ˆ 7 # Hy; n u it follows that a 0 ˆ P H y; u ˆ lim # P Hy; n u because P is a nite measure. Therefore, there exists m such that P Hy; m u < a. By putting H :ˆ H y; m u c we nd P H c < a whereas y B H, a contradiction. r This con rms that D a is a closed set (since it is an intersection of closed sets). The proof of Proposition 6 uses the condition that P is nite, whereas for in nite m we saw an example where D a ˆ R Š0; 1 was not closed. Corollary. For any probability distribution P and a > 0 we have D a ˆ 7fH; H is a closed halfspace with P H > 1 ag: Masse and Theodorescu (1994) have made an extensive study of sets obtained by halfplane trimming of probability distributions on R 2. These sets are close relatives of the depth regions D a considered here, but there is a small

7 The depth function of a population distribution 219 di erence. The sets of Masse and Theodorescu, which we will denote by Da 0 here, are de ned as in the corollary but with P H V 1 a. Therefore Da 0 H D a but not necessarily Da 0 ˆ D a. For instance, for the empirical distribution on two points a 0 b in R 2 (i.e. P :ˆ 1 2 D a 1 2 D b) they obtain D0:5 0 ˆ q whereas D 0:5 is the closed line segment a; bš. (Note that by applying our de nition of D a to any empirical distribution P n ˆ 1 P n iˆ1 n D x i we recover the depth regions of Section 1 above.) Proposition 7. For any probability measure P on R p, there exists at least one y with sup y depth y ˆ depth y. Proof. Denote a :ˆ sup y depth y U P R p ˆ 1: For any 0 < a < a we know that D a is nonempty, closed and bounded. For b < a it holds that D b I D a hence D :ˆ 7fD a ; 0 < a < a g 0 q. For any y A D and 0 < a < a we have y A D a hence depth y V a, therefore depth y V a. Combining a U depth y U a yields D ˆ D a : r Therefore, in the case of a probability measure the supremum depth a is a maximum, with a U 1. There may be one deepest point y or several. The Tukey median is de ned as the y of Proposition 7 when there is only one, i.e. when D a is a singleton. When D a is not a singleton, the Tukey median is de ned as the center of gravity of D a (which belongs to D a because the latter is convex and compact). The following proposition gives su½cient conditions for a point y to have maximal depth. Proposition 8 (Ray Basis Theorem). If for a point y J ˆ fu 1 ;...g of unit vectors such that 8 >< E j: P H y ; u j ˆ depth y >: 6 H y ; u j ˆ R p j A J there exists a set 3:2 then depth y ˆ max y depth y. Proof. Suppose that depth y < max y depth y, so there exists some ~ y with depth ~ y > depth y. Since 6 j H y ; u j ˆ R p there exists at least one u j with ~y A H y ; u j hence H ~y;uj HH y ; u j. But then depth ~ y ˆ inf u P H ~y;u U P H ~y;uj U P H y ; u j ˆ depth y, a contradiction. r Note that the set J doesn't need to contain more than p 1 elements. If the conditions of Proposition 8 are satis ed by a set J 0 with aj 0 > p 1, we can always take a subset J with aj U p 1 for which still 6 j H y ; u j ˆ R p. (Note that 6 j H y ; u j ˆ R p is equivalent to 6 j H 0; uj ˆ R p.) We call y ; y u j i ˆ fy lu j ; l A 0; y g a ray emanating from y. We call a set J ˆ fu 1 ;...g with aj U p 1 and 6 j H 0; uj ˆ R p a ray basis because for each x A R p there is at least one u j such that x 0 u j V 0.

8 220 P. J. Rousseeuw, I. Ruts It turns out that there is a nontrivial lower bound on the maxdepth a, which holds for any probability distribution P without further assumptions (e.g. P need not have a density). Proposition 9. For any probability measure P on R p, max y depth y V 1 p 1 : Proof. Put a ˆ max y depth y U 1 and take any small e > 0. As in the proof of Proposition 5, there exists a closed ball B with P B > 1 e. By Proposition 6, q ˆ D a e ˆ 7fH; H is a closed halfspace with P H c < a eg: All closed halfspaces H with P H c < a belong to the set fh; H is a closed halfspace with P H c < a eg, which is therefore nonempty. Now consider C ˆ fb X H; H is a closed halfspace with P H c < a eg: Then C is an in nite family of compact convex sets in R p with 7 C ˆ q. By Helly's theorem (see Valentine 1976, page 70) there exist sets C 1 ;... ; C p 1 in C such that 7 p 1 jˆ1 C j ˆ q, hence 6 p 1 jˆ1 C j c ˆ R p. For each j we nd P Cj c ˆ P B X H c ˆ P B c W H c <a 2e. Therefore 1 ˆ P R p U P p 1 jˆ1 P C j c < p 1 a 2e for any e > 0, hence a V 1= p 1. r This result was rst obtained in R 2 by Neumann (1945), and extended for any p by Rado (1946). The above proof is a modernized variant of Rado's. The result was independently discovered by Birch (1959). For a nite sample X n ˆ fx 1 ; x 2 ;... ; x n g we can apply Proposition 9 to the empirical distribution P n which shows that there exists a centerpoint of X n, i.e. some ~ y with depth ~ y; X n V dn= p 1 e=n. This nite-sample result was rediscovered in combinatorial geometry (see, e.g., Edelsbrunner 1987, pages 63±66). Note that none of the above papers contained the notion of depth: they proved the existence of a `centerpoint' ~ y without considering the depth at other y, and without attempting to obtain the maximal depth a [which may be strictly higher than 1= p 1 ] or the deepest point y. In statistics, the inequality a V 1= p 1 was rediscovered and proved in a di erent way by Donoho and Gasko (1992). 4. Probability distributions with a density The lower bound 1= p 1 on the maxdepth holds for any distribution P. For general P we cannot nd a sharper upper bound than max y depth y U 1, since this bound is attained for P ˆ D a. But we can do better when P has a density, i.e. when P is absolutely continuous relative to the Lebesgue measure l. Proposition 10. For any P with a density we have max y depth y U 1 2.

9 The depth function of a population distribution 221 Proof. At any y A R p we have depth y ˆ inf kukˆ1 P H y; u ˆ inf kukˆ1 minfp H y; u ; P H y; u g U 1 2 because H y; u X H y; u is a hyperplane hence its probability is zero, and 1 ˆ P H y; u P H y; u P H y; u X H y; u hence P H y; u ˆ 1 P H y; u, so that always minfp H y; u ; P H y; u g U 1=2. r This upper bound of 1=2 can be attained, e.g. when P is the standard multivariate normal distribution N p 0; I. More generally, it is attained at any distribution P which is angularly symmetric about some point y 0 A R p. The latter notion is de ned as follows. Consider the measurable mapping on R p given by h x ˆ y 0 x y 0 =kx y 0 k ˆ y 0 if x 0 y 0 if x ˆ y 0 and let P h be the law of h. Then P is called angularly symmetric about y 0 i for any Borel set B in R p it holds that P h y 0 B ˆ P h y 0 B : 4:1 Let us consider the set A ˆ B X S where S is the sphere around the origin with unit radius. Then the Borel set C ˆ fsa; a A A and 0 < s < yg is called a cone emanating from the origin. An equivalent de nition of angular symmetry is that P y 0 C ˆ P y 0 C 4:2 for any such cone. Formula (4.2) shows that the notion of angular symmetry is a½ne invariant, because any nonsingular linear transformation maps Borel cones emanating from the origin to Borel cones emanating from the origin. A special case of angular symmetry is centrosymmetry, which requires that (4.2) holds not only for such cones but for any Borel set C. Also centrosymmetry is a½nely invariant. Proposition 11. If P has a density and is angularly symmetric about some y 0 then max y depth y ˆ depth y 0 ˆ 1 2 : Proof. Because of Proposition 10 we only need to show that depth y 0 ˆ 1 2. By invariance, we can assume that y 0 ˆ 0. Take any unit vector u and consider

10 222 P. J. Rousseeuw, I. Ruts H 0; u. Because P has a density, the hyperplane qh 0; u has zero probability. Therefore P H 0; u ˆ P int H 0; u where the interior of H 0; u is a cone as in (4.2). By angular symmetry P int H 0; u ˆ P int H 0; u ˆ P int H 0; u, so nally P H 0; u ˆ P H 0; u ˆ 1 2. Since this holds for any unit vector u we obtain depth 0 ˆ 1 2. r Under the following conditions, the converse of Proposition 8 holds. Proposition 12 (Inverse Ray Basis Theorem). Let P have a continuous density f with f x > 0 for all x A R p (or for all x in an open convex set). Let y be a point with maximal depth. Then there exists a set J ˆ fu 1 ;...g of unit vectors, with aj U p 1, such that 8 >< E j: P H y ; u j ˆ depth y >: 6 H y ; u j ˆ R p j A J Conversely, any y satisfying (4.3) has maximal depth. 4:3 Proof. The existence of J follows from part of a proof in (Donoho and Gasko 1992, p. 1818±1820). For the other direction see Proposition 8 above. r Remark 4. Under the conditions of Proposition 12 we recover the lower bound a V 1= p 1 of Proposition 9. This follows immediately from 1 ˆ P R p U X j A J P H y; uj ˆ aj a U p 1 a : Note that Proposition 9 is much more general because it does not require that P has a density. In two dimensions, the Inverse Ray Basis Theorem states that we can always nd a set J of unit vectors with aj U 3 for which (4.3) holds. In Section 5.3, we will construct a ray basis fu 1 ; u 2 ; u 3 g for the uniform distribution on an equilateral triangle. Note that aj < 3 ˆ p 1 can only happen for J ˆ fu 1 ; u 1 g. But then we have a V 1 2, so a ˆ 1 2. This implies that for every u A R2 with kuk ˆ 1 we have P H y; u ˆ P H y; u ˆ a. (This happens e.g. for the uniform distribution on a square.) Note that in this situation we can still take some J ˆ fu 1 ; u 2 ; u 3 g with aj ˆ 3. This implies that for all u with kuk ˆ 1 there exists j such that u 0 u j > 0, hence angle u; u j < p 2. The halfspace depth also appears in the paper of Caplin and Nalebu (1988) on voting, which is concerned with the maximal value of a for which D a 0 q for a distribution with a density f relative to the Lebesgue measure l p on R p. We already know by Proposition 9 above that this maximal depth a 1 is at least. Caplin and Nalebu prove that this bound is higher if we p 1 assume that the density f is a concave function over its support B (which

11 The depth function of a population distribution 223 is a convex subset of R p with 0 < l B < y), i.e. that for 0 U g U 1 and x 1 ; x 2 A B we have f gx 1 1 g x 2 V g f x 1 1 g f x 2 : In this situation they prove that a V p p : p 1 4:4 The proof is based on the process of Schwarz symmetrization ( Bonnesen and Fenchel 1934). GruÈnbaum (1960) obtained (4.4) when f is constant on B. For p ˆ 2, the maximal depth is larger than ˆ 4 9 ˆ 0:444 instead of 1 3 ˆ 0:333. For p ˆ 3, the new bound is ˆ ˆ 0:422 instead of 1 4 ˆ 0:25. p p Since decreases to 1 for p! y, we nd for any dimension p that p 1 e a V 1 e ˆ 0:368: p p The lower bound in (4.4) is attained at a solid simplex. For p 1 p ˆ 2, consider the uniform distribution on a triangle (see Section 5.3). The deepest point is then the center of gravity, and its depth is indeed ˆ 49 V 1 e. Caplin and Nalebu (1991) relax the requirement of a concave density as follows. Take n A y; yš. For n > 0 a strictly positive function f with convex support B H R p is called n-concave if for 0 U g U 1 and x 1 ; x 2 A B it holds that f gx 1 1 g x 2 V g f x 1 n 1 g f x 2 n Š 1=n : Let us consider some special values of n. For n ˆ y we nd f gx 1 1 g x 2 V max f x 1 ; f x 2, which is only the case when f is constant over its support. The case n ˆ 1 corresponds to the standard de nition of concavity. The case n ˆ 0 corresponds to logconcavity, where it holds for all x 1 ; x 2 A B that ln f gx 1 1 g x 2 Š V g ln f x 1 1 g ln f x 2 : For n ˆ y we nd f gx 1 1 g x 2 V min f x 1 ; f x 2, which is called quasi-concavity as in Proposition 1. Higher values of n correspond to more stringent versions of concavity. For instance, each positive concave function is also logconcave, but not vice versa. When n V 1 and the density function f is n-concave over its support p 1 B (which is a convex subset of R p with 0 < l B < y), Caplin and Nalebu (1991) prove that 0 p 1 1 a B V n p 1 1 A n p 1=n : 4:5

12 224 P. J. Rousseeuw, I. Ruts The proof is based on the theorem of PreÂkopa (1973) and Borell (1975), which says the following. Let f be a probability density function on R p with convex support B. Take any measurable sets A 1 and A 2 in R p with A 1 X B 0 q and A 2 X B 0 q. For 0 U g U 1, de ne A g ˆ ga 1 1 g A 2. If f x is a n-concave function with n V 1 p, then f x dl x A g n= 1 pn n= 1 pn 1 pn =n V g f x dl x 1 g f x dl x : A 1 By applying (4.5) with n ˆ 0 we obtain a new bound for the maximal depth in the case of logconcave densities, namely A 2 a V 1 e ˆ 0:368 This bound doesn't depend on the dimension p, and for p ˆ 2 it is only slightly better than the previous bound 1 3, but it is a big improvement for p V 3. Caplin and Nalebu (1988) provide two other interesting results. Firstly, let f be a concave density function over a convex support, and let g be an arbitrary probability density on R p. Then maxdepth g V p p k f gk p 1 1 4:6 where k f gk 1 ˆ B j f x g x j dl x. Secondly, a bound is given for the probability mass of any depth region: P D a U a p 1 p a U a a e U 1 p 2 a e: 4:7 In Sections 5 and 6 we derive, for various measures m on R 2, an expression for the depth in an arbitrary point, the maximal depth, and the Tukey median. Moreover, we study and interpret the depth contour D a graphically. 5. Examples with a density In this section the positive measure m on R 2 is assumed to be absolutely continuous relative to the Lebesgue measure l 2, i.e. it has a density f. In the examples of this section, f will be strictly positive on the interior of a closed convex domain Q, and f ˆ 0 outside Q. The domain Q may be bounded or unbounded. Domains with empty interior cannot support an absolutely continuous m, and will be discussed in Section 6 on degenerate measures. In the examples 5.1 to 5.6 the measure m is uniform on the domain Q, which we will denote as m ˆ U Q. This means that we can take f z ˆ

13 The depth function of a population distribution 225 Fig The depth of x 0 ; y 0 is the minimal area of the shaded triangle ci z A Q where c is a constant. The computations will be simpli ed by noting that an a½ne transformation t turns m into the uniform measure on t Q, so we can restrict ourselves to ``standard'' sets Q Uniform measure on a convex 1-gon Here Q is the convex hull of two rays emanating from the same point. Therefore Q has only one vertex and one interior angle, which by convexity is less than p. By a½ne invariance, we may assume w.l.o.g. that the corner point is 0; 0 and the rays are the positive x-axis and the positive y-axis, so that m ˆ U Q with Q ˆ f x; y ; x V 0 and y V 0g: Since l Q ˆ y, this m is a positive measure with m Q ˆ y. Let us now consider a point x 0 ; y 0 A Q and compute its depth. For this we draw a line L b which passes through x 0 ; y 0 and makes an angle b > p=2 with the positive x-axis (Figure 5.1). Let H b be the halfplane with boundary line L b that cuts o a bounded triangle from Q. The depth a is then the minimal value of m H b ˆ area H b X Q ˆ x 0 tan b y 0 2 = 2 tan b. Using calculus we nd that the angle minimizing this area is b ˆ Arctan y 0 =x 0. The corresponding minimal area is thus 2x 0 y 0. For each point x; y A Q it therefore holds that depth x; y ˆ 2xy so the depth regions for any a > 0 are given by n D a ˆ x; y A Q; y V a o : 2x For a ˆ 0 we nd D a ˆ Q, whereas the depth contours for a > 0 are hyperboles. Figure 5.2 shows some depth contours for various values of a. The circles in the gure have radius equal to one fth of the radius of curvature of the contour they are in contact with. For a small depth the circles vary quickly in

14 226 P. J. Rousseeuw, I. Ruts Fig Depth contours of the uniform measure on a 1-gon size, whereas for a larger depth the size varies less rapidly. In words, the depth contours are smoothing the corner of Q. If we apply an a½ne transformation, we obtain the contours in an arbitrary convex 1-gon. For instance, consider the transformation x! A x y y with A ˆ 1 cos g 0 sin g for g < p, which maps Q to the convex 1-gon A Q ˆ f x; y ; y V 0 and x V y=tan gg with internal angle g. The depth of a point x; y inside A Q is the area of the image of the original triangle D ˆ H X Q: depth x; y ˆ area A D ˆ jdet A jarea D ˆ 2y x y : tan g The depth regions become D a ˆ x; y A A Q ; 2y x y V a. tan g 5.2. Uniform measure on a convex 2-gon with two parallel edges Consider a line segment with two endpoints from which two rays emanate, both in the same direction. The convex hull of these three edges forms the 2-gon Q, which has only two vertices. (In other words, Q is a parallellogram of which one edge has moved to in nity.) By a½ne invariance, we can assume that Q has the simple expression m ˆ U Q with Q ˆ f x; y ; x V 0 and 0 U y U 1g:

15 The depth function of a population distribution 227 Fig Depth contours of the uniform distribution on a 2-gon Also here m Q ˆ y. When computing depths, we can use the fact that a 2-gon is the intersection of two 1-gons, and extend the result of the previous section. We then nd depth x; y ˆ and for any a > 0: 2xy if y U 0:5 2x 1 y if y V 0:5 D a ˆ f x; y A Q; 2x min y; 1 y V ag: Figure 5.3 shows some depth regions. Note that D 0 ˆ Q whereas the deeper regions (that is, with larger a) move to the inside and to the right. The outer contours have approximately the shape of the original domain, but with smoothed corners. The contours consist of two curves. The lower curve has equation y ˆ a a and the upper curve has y ˆ 1. The angle between these 2x 2x two curves gets smaller as a increases. A point on the intersection of the two curves is of the form a; 1=2. The slope of the tangent line along the lower curve in this point is thus y0 0 ˆ a 2x0 2 ˆ 1. The slope of the tangent line 2a along the upper curve in a; 1=2 is y0 0 ˆ 1. Both slopes tend to zero as a 2a increases, so the inner contours become very peaked Uniform distribution on a triangle (3-gon) Since all triangles are a½ne images of a single triangle, we can restrict ourselves top an equilateralp triangle Q ˆ D formed by the vertices 1; 0, 1; 0, and 0; 3. Its area is 3. Therefore, we can de ne m by the density f x; y ˆ p 1 I x; y A D 3 so that m is a probability distribution (since m R 2 ˆ m D ˆ 1).

16 228 P. J. Rousseeuw, I. Ruts Fig Depth contours of the uniform distribution on a triangle To compute the depth of a point x; y A D it is convenient to assume p that x; y belongs to the region G ˆ f x; y A D; x V 0; y V 0; x y 3 U 1g. Through x; y we draw a line L b which makes an angle 0 < b < p=2 with the positive x-axis, and consider the halfplane H b to its right. Then m H b ˆ p 1 area H b X D ˆ 1 y 1 x tan b 2 p tan b tan 2 b : p p p Calculus yields the optimal angle b ˆ arctan y 3 = 3 x 3 2y. Therefore, depth x; y ˆ p 23 y 3 1 x y 2 : 5:1 The same result can be obtained by using the 1-gon of Section 5.1. For this we apply the a½ne transformation x! y 0 cos 2p B sin 2p 3 1 cos p C A sin p x y 1 0 which turns the edges of the original 1-gon and moves its vertex to the position 1; 0, yielding the right lower part of D. For the depth p itself we need to divide the 1-gon depth (which was the usual area) by 3. This again yields (5.1). In the remainder of the triangle D we nd the depth by symmetry considerations (invariance under re ection and 2p=3 rotation). Figure 5.4 shows

17 The depth function of a population distribution 229 Fig Depth contours of the uniform distribution on a square, with depths 0.05, 0.15, 0.25, 0.35 and 0.45 some p depth contours. The maximal depth a is 4=9 and the Tukey median is 0; 1= 3, which coincides with the center of gravity of D. For an arbitrary triangle we can obtain the depth function and its contours by a½nely transforming D. A ray basis for Figure 5.4 (see Proposition 12) consists of unit vectors u 1, u 2 and u 3 with starting point at the Tukey median 0; p 1 3. For instance, let u 1 be the unit vector which lies along the y-axis, let u 2 be the unit vector making a 7p 6 angle with the positive x-axis, and let u 3 be the unit vector making a p 6 angle with the positive x-axis Uniform distribution on a square (regular 4-gon) We take Q to be the square Q ˆ 0; 1Š 0; 1Š with unit area. Therefore, the density f x; y :ˆ I x; y A Q describes the probability distribution m. To compute the depth we can consider the square Q as the intersection of four separate 1-gons, and extend the result of Section 5.1. This yields depth x; y ˆ 2 min x; 1 x min y; 1 y for x; y A Q, and n D a ˆ x; y A Q; min x; 1 x min y; 1 y V a o : 2 The maximal depth a is 1=2 and the Tukey median is 1=2; 1=2. The depth contours (Figure 5.5) again smoothen the corners of Q. The gure illustrates that depth is very di erent from density: The density f has no contours inside the square because it is constant there, but the depth does.

18 230 P. J. Rousseeuw, I. Ruts Fig Depth function in all points of the square Figure 5.6 shows a three-dimensional graph of the depth function in all points of Q. For points in the subsquare 0; 0:5Š 0; 0:5Š the expression of the depth function becomes z ˆ 2xy, which means that this portion of the depth function is part of a hyperbolic paraboloid. (In Figure 5.6 we see that this portion of the graph is indeed composed of many straight line segments.) The same holds in the three other quadrants of the square. The three-dimensional graph looks like a strange roof, the edges (ridges) of which are not at the corners of the house but point to the middle of each wall Uniform distribution on a regular m-gon Here m ˆ U Q where Q is the m-gon with vertices cos j 2p m ; sin j 2p for j ˆ 0; 1;... ; n 1. Therefore, m f x; y :ˆ I x; y A Q m 2 sin 2p m For points near the boundary of the m-gon, the minimal area that can be cut o by a halfplane will be a triangle, which means that we can again use the m 2 p expression of the depth function in an arbitrary 1-gon. We put g ˆ m, m 2 p rotate over b ˆ and move the 1-gon to position 1; 0. This is 2m achieved by the transformation x! y cos b sin b x sin b cos b y : 1 : 0 To obtain an expression for the depth in a point, we divide by the total area of

19 The depth function of a population distribution 231 the m-gon, which is m 2 sin 2p. For points near the boundary of the m-gon, we m then nd 0 1 depth x; y ˆ 1 B 1 x 2 y 2 m sin p 2 cos p 2 A: m m In other parts of the m-gon the contours for small depths can be found by symmetry considerations. For points closer to the center of the m-gon, the area cut o from G will be a q-gon (with q U m 1). The equation of the corresponding depth contours then becomes more complicated. The maximal depth a is attained in 0; 0. For an m-gon with m even, the maximal depth is always 0.5, because any line through 0; 0 divides the plane in two halfplanes with probability mass 0.5. For odd m, the maximal depth is found as follows. We need to nd a halfplane with boundary line through 0; 0 which has a minimal probability mass. Obviously, if we consider such a halfplane with boundary line through a vertex of the m-gon, the probability mass of this halfplane is 0.5. We veri ed numerically that the optimal choice for the halfplane is a halfplane with boundary line parallel to an edge of the m-gon. Let us choose the vertical edge (to the left hand side) of the m-gon. The resulting boundary line is then the y-axis (see Figure 5.7). We nd that for m ˆ 3 modulo 4 the halfplane with the minimal amount of probability mass lies to the right of the y-axis, and for m ˆ 1 modulo 4 it lies to the left. The area of the intersection of the `bad' halfplane (this is the one with probability mass 1 a ) and the m-gon can then be seen as the sum of the areas of m 1 =2 2 triangles. The rst kind of triangles have one vertex at 0; 0 and the two other vertices are adjacent vertices of the m-gon (see the dashed lines in Figure 5.7). The other kind of triangles have the same area as a triangle of which one edge is part of the positive x-axis, another is part of the edge of the m-gon that connects 0; 0 to cos 2p m ; sin 2p, and the angle m between the two edges at 0; 0 is p (see the full line in Figure 5.7). We have 2m m 1 =2 triangles of the rst kind and two of the second kind, which results in an area area ˆ m 1 sin 2p m tan p 2m sin 2p m sin 2p p tan m 2m cos 2p m 1 The use of some goniometric formulas gives the maximal depth p 2 a ˆ 1 tan 2 2m : 2m For a regular pentagon this results in a maximal depth of , and for a regular heptagon it becomes If we let m go to in nity in the ex-

20 232 P. J. Rousseeuw, I. Ruts Fig Computing the maximal depth of a regular m-gon with m ˆ 3, m ˆ 5, m ˆ 7, and m ˆ 9 pression above we nd 0.5, which is indeed the maximal depth for the uniform distribution on a disk (see the following section). Figure 5.8 shows some data-based depth contours obtained by applying the exact algorithm ISODEPTH (Ruts and Rousseeuw 1996) to 3000 points generated uniformly on the 5-gon. These empirical contours are very close to the population contours. Note that the at edges of the m-gon for m V 2 yield `cracks' in the depth contours, whereas the (convex) corners of the same m-gon are `rounded o ' by the depth contours. Figure 5.8 also shows that contours near 0; 0 have the shape of a 5-gon with edges parallel to the original vegon. In between, we nd an `inverted' 5- gon which has vertices that point to the edges of the original 5-gon. The depth contours of the 5-gon thus show two inversions, whereas those of the 3-gon in Figure 5.4 had only one (so the contours near the median formed a triangle that was the inverse of Q). For a general m-gon with odd m there are m 1 = 2 inversions. As m goes to in nity, the m-gon becomes a circle and the inversions disappear Uniform distribution on a disk (y-gon) For increasing m the regular m-gon tends to the disk Q ˆ f x; y ; x 2 y 2 U 1g. Then m ˆ U Q has the density

21 The depth function of a population distribution 233 Fig Empirical depth contours (for depths 0.001, 0.025, 0.05, 0.075, 0.15, 0.225, 0.3, 0.375, and 0.45) of 3000 points generated uniformly on a regular 5-gon f x; y ˆ 1 p I x; y A Q : Due to symmetry, the depthp of an arbitrary point x; y A Q equals the depth of the point r; 0 with r ˆ x 2 y 2. The depth of r; 0 is the mass cut o from the disk by the halfplane H through r; 0 and parallel to the y-axis. Therefore, depth x; y ˆ m H ˆ 2 p 1 r p 1 x 2 ˆ p Arcsin p x2 y dl y dl x q x 2 y 2 1 x 2 y 2 : The maximal depth is 1=2 and the Tukey median is 0; 0. The depth region of depth p a is then D a ˆ f x; y ; x 2 y 2 U r 2 g with r such that Arcsin r r 1 r 2 ˆ p 0:5 a, which coincides with the ndings of Masse and Theodorescu (1994). We now consider some non-uniform densities Bivariate gaussian distribution The standard bivariate gaussian density f x; y ˆ 2p 1 exp x 2 y 2 =2 is strictly positive on R 2. By symmetry, the depth of an arbitrary point x; y equals the pmass of the halfplane H to the right of the vertical line through r; 0 ˆ x 2 y 2 ; 0. Therefore p depth x; y ˆ 1 F x 2 y 2

22 234 P. J. Rousseeuw, I. Ruts Fig Depth function of the bivariate gaussian distribution and D a ˆ f x; y ; x 2 y 2 U F 1 1 a 2 g: The maximal depth is 1=2 and the Tukey median is 0; 0. The depth function of the bivariate standard gaussian distribution ( Figure 5.9) does not attain zero since it is strictly positive in all points of R 2. The depth function of a bivariate gaussian distribution with arbitrary center and shape can be obtained through an a½ne transformation Bivariate Cauchy distribution Let X and Y be independent variables with the univariate Cauchy distribution. The density function of X; Y in R 2 is then f x; y ˆ 1 1 p 2 1 x y 2 Figure 5.10 shows the density contours (dashed lines). We will prove that the depth contours are squares around the origin (the full lines in Figure 5.10). Lemma. For X and Y independent and Cauchy distributed we have P ax by ˆ P jaj jbj X for any positive real numbers a and b.

23 The depth function of a population distribution 235 Fig Depth contours of depths 0.05, 0.15, 0.25, 0.35 and 0.45 (full lines) and density contours (dashed lines) of the bivariate Cauchy distribution Proof. Using the characteristic function j X t ˆ e jtj j ax t j by t ˆ e jatj jbtj ˆ e jaj jbj jtj ˆ j jaj jbj X t. yields j ax by t ˆ r Proposition 13. For the bivariate Cauchy distribution the depth function is constant on squares around the origin. Proof. Consider a point u; v A R 2 with 0 U u < v. Take a halfplane through u; v of the form H a; b ˆ f u; v ; ax by V au bvg for positive real numbers a and b. The depth in u; v is the minimal amount of probability mass in a halfplane H a; b. We calculate this mass: P H a; b ˆ P ax by V au bv ˆ P a b X V au bv by the lemma au bv ˆ P X V ˆ P X V ~au 1 ~a v for ~a ˆ a a b a b Here ~au 1 ~a v is maximal for ~a ˆ 0 (since u < v), so depth u; v ˆ min a; b H a; b ˆ P X V v : Analogous reasonings hold for other quadrants and for a and b not necessarily positive. For an arbitrary point x; y A R 2 the depth function becomes

24 236 P. J. Rousseeuw, I. Ruts depth x; y ˆ 1 P X V maxfjxj; jyjg ˆ 1 F k x; y k y ˆ Arctan maxfjxj; jyjg ; p which shows that the depth contours are constant on squares around the origin, and that the maximal depth is 1=2. r The expression depth x; y ˆ 1 F k x; y k y is analogous to the expression for the depth function of the bivariate Gaussian distribution in the previous paragraph, which can be written as depth x; y ˆ 1 F k x; y k 2. The expression 1 F k x; y k 1 can be obtained as the depth function of the distribution which is obtained when the bivariate Cauchy distribution is rotated over p=4. In that case the resulting contours will be diamond-shaped. A ray basis for this depth function is of the form J ˆ fu; ug since for every u A R 2 with kuk ˆ 1 we have P H y; u ˆ P H y; u ˆ a. This is also the case for the uniform distribution on a square (Section 5.4) and on a circle (Section 5.6), and for the gaussian distribution (Section 5.7). 6. Examples without a density We will now consider a uniform distribution on a one-dimensional domain in R 2. Instead of minimizing an area, we will have to minimize a length. Note that the depth remains a½ne invariant, although length is not a½ne equivariant. This can be explained by noting that an a½ne transformation of such a uniform distribution may be non-uniform Uniform measure on the boundary of a 1-gon The convex 1-gon Q ˆ f x; y ; x V 0 and y V 0g is as in Section 5.1, but we put m ˆ U qq with qq ˆ f x; 0 ; x V 0g W f 0; y ; y V 0g: Again consider a point x; y A Q and a line L b through it which makes an angle b with the positive x-axis, as in Figure 5.1. Now m H b becomes the total length of the two line segments cut o by L b from qq. This length is p y x tan b x y cotan b. Minimizing this expression yields b ˆ Arctan y=x, so we nd p p depth x; y ˆ x y 2 and for any a > 0 the depth region is p p p D a ˆ f x; y A Q; x y V ag: The depth contours are shown in Figure p 6.1. Contrary to Figure 5.2, p the contour of D a touches the x-axis [at a ; 0 ] and the y-axis [at 0; a ].

25 The depth function of a population distribution 237 Fig Depth contours of the uniform distribution on the boundary of the 1-gon Q ˆ f x; y ; x V 0 and y V 0g, and the unit ``circle'' of k x; y k 1=2 Each contour consists of a curved segment, extended by parts of the x-axis and the y-axis. (Note that the depth function is not continuous at the boundary of Q!) Note that the depth of a point x; y in Q can be written as depth x; y ˆ jxj 1=2 jyj 1=2 2 ˆ k x; y k 1=2 where the notation k... k 1=2 stands for the p-pseudonorm with p ˆ 1=2 (the triangle inequality is only satis ed when p V 1). The unit ``circle'' for k... k 1=2 is given by jxj 1=2 jyj 1=2 ˆ 1 and is shown in Figure 6.1. It is star-shaped but not convex. In Section 5.1 we also considered the convex 1-gon A Q ˆ f x; y ; y V 0 and x V y=tan gg with internal angle 0 < g < p. The depth of a point x; y A A Q relative to the uniform distribution on q A Q can be computed directly as p r y 2 depth x; y ˆ x y=tan g sin g : 6:1 We cannot obtain (6.1) by means of a½ne invariance because the uniform distribution on q A Q di ers from the image of the uniform distribution on qq through A (the latter is not uniform).

26 238 P. J. Rousseeuw, I. Ruts Fig Depth region D 0:2 of the uniform distribution on the boundary of a triangle 6.2. Uniform distribution on the boundary of a triangle Using the same equilateral triangle D as in Section 5.3, we put m ˆ U qd with p p qd ˆ 1; 0 ; 1; 0 Š W 1; 0 ; 0; 3 Š W 0; 3 ; 1; 0 Š where a 1 ; a 2 ; b 1 ; b 2 Š denotes the line segment with endpoints a 1 ; a 2 and b 1 ; b 2. We can compute the depth in a point x; y A D by extending the result for the 1-gon where we put g ˆ p=3, and then rotating the corner of the 1-gon to position 1; 0 as in Section 5.3. (Note that length is rotation invariant.) p For a point x; y in the region G ˆ f x; y A D; x V 0; y V 0; x y 3 U 1g the depth then becomes depth x; y ˆ 1 q p p 2 p 1 x 3 y 2y : 6 3 For an arbitrary point in DnG we nd the depth by symmetry considerations. p The maximal depth is 4=9 and the Tukey median is the center 0; 1= 3. Figure 6.2 shows the approximate contour of depth 0.2. The plot was obtained as follows. First, we divide each side of the triangle into 50 segments of equal size. The total length of the boundary is thus 150 segments. If we want the contour of depth 0.2, we draw all lines which cut o 0:2 150 ˆ 30 segments from the boundary. The region D 0:2 is then (approximately) the white region inside the triangle. A similar construction is possible for a contour of the 1-gon, the 2-gon, the square or a general m-gon.

27 The depth function of a population distribution Uniform distribution on a circle Denoting the unit disk as Q ˆ f x; y ; x 2 y 2 U 1g we now consider m ˆ U qq with qq ˆ f x; y ; x 2 y 2 ˆ 1g: The depth of an arbitrary point x; y A Q is 1= 2p ptimes the length of qq to the right of the vertical line through r; 0 with r ˆ x 2 y 2. This yields depth x; y ˆ 1 p Arccos p x2 y 2 and D a ˆ f x; y A Q; x 2 y 2 U cos 2 pa g: The maximal depth is 1=2, and the Tukey median is 0; Uniform distribution on a halfcircle We now put m ˆ U C where C is the halfcircle C ˆ f x; y ; x 2 y 2 ˆ 1 and y V 0g: Clearly, the depth will be zero for points outside the convex hull of the halfcircle C. Consider a point x; y inside the convex hull of C. We want to nd the halfplane through x; y which cuts o the smallest length of C. Two ways to cut o a piece of C are illustrated in Figure 6.3. In the rst situation we take the halfplane whose boundary also contains the other end point of the halfcircle. The length l 1 is given by 0 1 B l 1 x; y ˆ 2 Arccos@ q x 1 C A: x 1 2 y 2 In the second situation, let l 2 b be the length of C cut o by a halfplane whose boundary makes an angle b with the horizontal axis. We nd that l 2 b ˆ Arccos 1 2 y x tan b 2 = 1 tan 2 b, which is minimized for Fig Two ways to cut o a piece of a halfcircle

28 240 P. J. Rousseeuw, I. Ruts Fig Depth contours of the uniform distribution on a halfcircle b ˆ Arctan x=y, hence l 2 x; y ˆ Arccos 1 2 x 2 y 2 : For an arbitrary point x; y inside the convex hull of C we have to consider both situations and use the one which provides the smallest length: depth x; y ˆ 1 >< p min 2 Arccos x 1 >= B q A; Arccos 1 2 x 2 y 2 >: x 1 2 y 2 >; : Note that l 1 x; y ˆ l 2 x; y if and only if x 1 2 ˆ 2 x 2 y 2 x 1 2 y 2. For positive x, the p boundary of the depth region D a consists of the graphs with equations y ˆ 1=2 x 2 1=2 cos pa and y ˆ x 1 tan pa=2. Some contours p are shown in Figure 6.4. The maximal p depth is p 1=p Arccos 5 2 A 0:424, and the Tukey median is 0; t where t ˆ 5 1 = 2 is the golden ratio Uniform distribution on a line segment In this case we can w.l.o.g. take the line segment 0; 0 ; 1; 0 Š yielding depth x; y ˆ min x; 1 x if x; y A 0; 0 ; 1; 0 Š 0 else and for 0 U a U 1=2 we nd D a ˆ 1 a; 0 ; a; 0 Š. The maximal depth is 1=2 and the Tukey median is 1=2; 0, the midpoint of the segment.