Liu R. and Singh K. (1992) Ordering Directional Data Concepts of Data Depth on Circles and Spheres

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Ordering Directional Data: Concepts of Data Depth on Circles and Spheres

Author(s): Regina Y. Liu and Kesar SinghReviewed work(s):Source: The Annals of Statistics, Vol. 20, No. 3 (Sep., 1992), pp. 1468-1484Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2242022 .

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The Annals of Statistics1992, Vol. 20, No. 3, 1468-1484

ORDERING DIRECTIONAL DATA: CONCEPTS OF DATADEPTH ON CIRCLES AND SPHERES'

BYREGINA . Liu ANDKESAR INGH

RutgersUniversity

Threenotions f depth ordirectional ata,angular implicial epth(ASD), angularTukey'sdepth ATD) and arc distance epth ADD), aredeveloped nd studied. he empiricalersions f thesedepths iverisetocenter-outwardankings f ngular ata whichmaybe regarded s exten-sions of the usual center-outwardanking n the line. Threemediansderived rom hese depths re examined nd compared. pplicationsnnonparametriclassificationnd in implementinghebootstrapo con-struct onfidenceegions or irectionalarametersre brieflyiscussed.

1. Introduction. Thepurpose fthisarticles todevelop hree onceptsofdata depth fordirectional ata, namely, ngular simplicial epth ASD),angularTukey'sdepth ATD) and arcdistance epth ADD). ASD extends henotionof simplicial epth SD) in Liu (1988, 1990) fromRP to circles ndspheres.ATD is an analog ofTukey'sdepth TD) [Tukey 1975)] on RPforpopulationsnd dataon circles nd spheres.A notion quivalent oATD hasbeen ntroduced ySmall 1987).

Lidistance ntheEuclidean pacegives ise

tothenotion fADD for pheres ndcircles.The concept f depth on spheres eads to a propernotionof center or

median) and a rankingof directional ata in the orderof centrality.nparticular,uchrankingeadstodetection f extreme ata values, naturaldefinitionf nterquartileange on thecircle) nd analogsof inear ombina-tionsoforder tatistics fdirectional ata in general.The rankings erivedfromASD and ATD can be ustified s natural xtensionsf theusual linearranking ythefollowingrgument.When theentiredistributions concen-trated n a semicircle,hedistributionouldbe regarded s beingonthe inesegment- r/2, 7/2]. In sucha case one wouldnaturallyxpect heangulardepths being zero throughout he othersemicircle) o coincidewith theirparentnotions fdepthon the line.Both ASD and ATD possess this consis-tencyroperty. s a result, hecenter-outwardankings ased on thedecreas-ingvaluesofthesedepths ompletelygreewith hatbasedon theusualorderstatistics n the ine.As an illustrativexample, uppose he angulardata indegrees) re 62, 73, 85, 96, 97; thentheranking n the orderofcentrality

ReceivedMarch1989;revised une1991.'Research upported yNSF Grants MS-88-02558ndDMS-90-04658.AMS 1980subject lassifications.rimary2H05, 62H12;secondary 0D05.Key words ndphrases.Angular implicialepth, ngularTukey'sdepth, rc distance epth,

median, irectionalata,center-outwardrdering,onMisesclass ofdistributions.

1468



ORDERINGDIRECTIONALDATA 1469

provided y the inearranking, SD or ATD is 85, (73, 96), (62, 97),wherepair , ) indicates tie.

The center-outwardanking ivenby ASD orATD has several nteresting

applications.pecifically,emention he followingwo:1. On classificationproblems: Suppose two training amples (X1, . . ., Xm) and

(Y1, .. ,Yn)fromwodifferentpherical opulationsregiven.Considerheproblem fclassifying newdata point Z to one of the two populations.First, ompute, espectively,he center-outwardanksof Z with espect oXi's and Yi's.Let theseranksbe denoted yrxandry.The proposed ule sto classify to the X populationfrx/m ry/n, nd to the Y populationotherwise. his classificationule s studiednGross nd Liu (1989).

2. On implementinghe bootstrap:Let 0 be theparameterf nterestn Sd, ad-dimensionalnitsphere, nd let 6n be its associated stimate.A center-outwardranking s essentialfor mplementinghe percentilemethod oform bootstrap onfidenceegion or [seeEfron1979)for hepercentilemethod]. he procedure s as follows: irst,obtain a certainnumber fbootstrap eplicas f6n;second, ssignthe center-outwardank accordingto ASD orATD) to each replica;finally, elete OOa% ofthe outmostreplicas.The smallest onvex atchon Sd containinghe remainingepli-cas is thena (1 - a) bootstrap onfidence egion f 0. Properties fthisbootstrap onfidenceegionwillbe reported lsewhere.

The constancyfASD and ATD throughout circle r a sphere resentsninterestingituation. f course, heir arent epths re never onstant nRWe havefully tudied he constancy,nd themain results re distributionalcharacterizationsnterms f constant epth cf.Sections and 4).

Allthreenotions f ngular epth iverisetomedians n Sd. Somedetailedcomparisonsfthosemedians re presentedater.The definitionf a medianon a circle ivennMardia (1972),page 28] is inspirit he sameas themedianderived romADD, althoughn some unusualcases thedefinitionan lead to

only local maximum fADD ifthe definitions followediterally.For simplicity e restrict urselves o continuous istributionsn theunit

circle nd absolutely ontinuous istributionsn theunitsphere;neachcasewetakethe origin denoted y0 and 0, respectively)s the center. hrough-outthisarticle, 0 is used toindicate he diametricallypposite oint f0.

Section2 contains asic definitionsndnotation.In Section3 we present he following ropertiesf ASD w.r.t. spherical

distribution:

1. Computational simplicity of ASD: Checking that a point belongs to aspherical riangle s equivalent o solving 3 x 3 linearsystem fequa-tions.

2. A differential ormula for ASD and its applications: The derivativeofASD(-) on a circlehas a simple xplicitlyormula see (3.1)] whichyieldsmany nterestingropertiesfASD. Thesepropertiesnclude monotonic-



1470 R. Y. LIU AND K. SINGH

itypropertyfthe ASD and a characterizationf an antipodallyymmetricdistributionn the circle s having constant SD value, 1/4.

3. An equation onnecting SD and SD and itsapplications: quation 3.4)

[(3.6)] connectsASD on a circle a sphere)with he SD on a line a plane).This equationcan be used to characterize ntipodallyymmetricistribu-tions on the sphereby the constant alue ofASD = 1/8 throughouthesphere.

We show n Section thatATD(*) has theappropriateropertieso ustifyitself s a notion f data depth.The maximum aluefor his depthdoesnotexceed1/2 and it is attained, or nstance, t themodeof any member fthevon Mises class ofdistributions.

Section5 discusses he robustness spectofMardia's median ndmedians

givenby ASD, ATD and ADD in terms f nfluenceunctionsnd a break-down concept. ome llustrativexamples re also given.

Some concluding emarks re made nSection .

2. Definitions and notation.

Angular implicialdepth. The angular implicial epthwhichweproposeinthis rticle s a natural nalogfor irectional ataofthe simplicial epth ordata on euclidean spaces, introduced n Liu (1988, 1990), which we nowdescribe riefly.

In Rd, a simplexO(x1,... ,xdI1) with (d + 1) vertices x1, .. , Xd+1 isdefined o be theclosed onvexhull with xtremitiest thesepoints. et F( )be a distributionnd x a point n Rd. The simplicial epthof x w.r.t.F,SD(x), is then definedto be the probabilityhat x be in a simplexO(X1,..., Xd+ ), whereX1, .., X1 are (d + 1) i.i.d. observationsromF.In lR',O(X1,X2) iS simply heclosed ine egmentoiningX1 and X2, ayX1X2,and SD(x) PF(x E X1X2) [= 2F(X)(1 - F(x)), assuming that F is continu-ous]. In R2, O(X1, X2,X3) is theclosed rianglewithverticesX1,X2 and X3,say A(X1,X2 X3),and SD(x) PF(x E A(X1,X2,X3)). The simplicialmedianis then hepointwhichmaximizesD(-) (or theaverage f uchpointsf hereare many).Note thatin R' the simplicialmedian divides he line into twohalf-linesfequal probabilitiesnd it agreeswith he usual median. n Liu(1990) it is argued hatSD(-) can be viewed s a measure f data depth, ndthat the simplicialmedianpossesses manydesirablefeatures fa notionofmedian.

The edges of a simplexn Rd are the line segments onnecting airsofpoints vertices).Whenwe move o the sphere, t is natural o replace uch alinesegment y the shortest urve oining pairof pointson thesphere.Let Pi and P2 be twopointson a sphere. t is known hat sucha shortestcurve s the short rc oiningp1 andP2 on the circlewhich asses through 1and P2 andhas the same center s the sphere. Sucha circles referredo as agreat ircle.) videntlyhis hortest urve an be generalizedo spheres f nydimension ndis ambiguous nly nthe nongenericasewherep1 and P2 are



ORDERING DIRECTIONAL DATA 1471

diametricallypposite ach other.The idea of theshortest urve llowsus togeneralize he notionof simplex o the spherical ase. We discussonlythecases ofthe circle nd thetwo-dimensionalphere, lthoughtwillbe clear

that the definitionxtendsnductivelyo anydimension. or anytwopointsP1 andP2 ona circle hecorrespondingimplexs the short rc oiningp1 andP2 [denoted y arc(p1,P2)],and for hree oints 1, q2 and q3 on a spheret sthesphericalriangledenotedyAj(q1, q2, q3)] boundedythethree hortarcsarc(ql, q2),arc(ql, q3) and arc(q2, 3).Wenow define heangular impli-cialdepth o be

(2.1) ASD(p) PH(P E arc( W1,W2))

if p is a point and H a distributionn a circle, nd W1 and W2 are i.i.d.

observations romH;(2.2) ASD(p) = PH p E AS WI,W2,W3))

ifp is a point nd H a distributionn a sphere ndW1,W2andW3are i.i.d.observations romH. Note that fH is continuous n a circle ndabsolutelycontinuousn a sphere, hentheambiguous impliciesccurwithprobabilityzero.A maximum ointofASD( ) is defined o be an angular simplicialmedian (ASM). Evidentlythis median is rotation nvariant.

Wedefineheempirical ersion fASD( ) as

(2.3) ASDn,(p)=() '(p E arc(Wil, ))

for a point p on the circle,where W1, . ,Wn is a random sample from acircular istributionnd E * runsoverall possiblepairsof(Wi1l,Wi), nd as

(2.4) ASDn(P) = (n) 1n(P) E (p e S(W1~Wi2'WJ))* *

for a point p on the sphere,where W1, . ,Wn is a random sample from aspherical istributionnd E** runsover ll possible ripletsWi ,Wi2,Wi3).

Angular Tukey's depth. Following Small (1987), we definethe angularTukey's depth for a givensphericaldistributionH as follows:

(2.5) ATDH(O) = inf {PH(S)},{S: Oec-S)

where he nfimums takenoverthesetof all closedhemispheres contain-ing0 intheir oundaries r intheir nteriors.

We call a maximum ointofATD( ) an angularTukey'smedian ATM).Note thatATM is also rotationnvariant. ee Example4.4.4 in Small 1987)formore nvariance ropertiesfATM.

In definingheempirical ersion fATD( ), ATDn( ), we replacePH( ) in(2.5) by tscorrespondingmpirical robability.



1472 R.Y. LIU AND K.SINGH

Arc distance depth. We define DD of a point on the sphere d as

ADD(O) = 7r fl(0, p) dH( p),

where1(6, p) s the Riemannian istance etween and (p;that s,the engthof theshort rc oining0 and *con thegreatcircledeterminedy 0 and (p.Again, heempirical ersion fADD is defined yreplacingH( ) byHn( ). Amaximum ointofADD(-) is referredo as an arc distancemedian ADM).This idea ofminimizing 1 distancewas used byGower 1974) to definegeneralizedmedian n Rd. Its extensiono circleswas given nMardia 1972)and tospheresn Fisher 1985).

3. Properties of angular simplicial depth.

Computational simplicity of ASD. We firstpoint out that determiningwhether r not a pointon a circlea sphere) ieson the short rc oining wodatapointsthe spherical rianglehreedatapoints) anbe reduced osolvinga simple ystem f linearequations.This shows thatcomputing SD(-) isquite straightforward.et H( ) be thepopulation istributionefinedn theunitcircle enteredt theorigin . Given point0 onthe circle ndanytwodatapointsW1 and W2fromH(0), 0 lies on the short rc arc(W,,W2) fand

only if the line segments00 and W1W2ntersect. n otherwords,E6arc(Wi,W2) f and only fthere exist a and /3 uch that0 < a,, < 1 anda6c = /Wc + (1 - 13)W2c. ere the notation *c stands forthe Euclidean coor-dinatesofthepoint*. For the spherical ase this observation ecomes hefollowing: is on the spherical riangleA,(W1,W2,W3)f and only f 00intersects he EuclideantriangleA(W1,W2,W3).his is equivalent o aOc -

,f3Wc yW2c+ (1 - /B 7)W3 forsome a, /3 nd y such that 0 < a, ,B,y < 1and 0 /3+ y < 1. The same observation olds forany general Sd. ThiscomputationalimplicityfASD shouldgreatlynhance tsapplicability.

A differential ormula for ASD and its applications. Let H(Q) be thedistributionn the unitcircle nd leth(-) be itsdensityf texists.Here0 canbe simplyxpresseds an anglebetween and2-7.

PROPOSITION 3.1. Suppose thath(*) exists and is continuous at 0. Then

d(3.1) - ASD(6) = 2(A6-CO)h(0)-

whereAO and Co stand fortheprobabilities of the semicircles oining 0 and- 0 in thecounterclockwise nd clockwisedirections,respectively.

PROOF. For a 0 and a positivencrement0,thedifferenceASD(0 + 30) -ASD(6)] involves nlythosepairsofobservationsW1,W2}fromH(-) which



ORDERINGDIRECTIONAL DATA 1473

havethefollowingroperty:

{0 E arc(Wl,W2)and 0 + 80) e arc(W1,W2)}

or(0 C arc(W1,W2) and(O + 80) E arc(Wl,W2)}.

These two situationswill occur if and onlyif either W1 or W2 lies onarc(0,0 + 80). Usingthis fact ndtheequality

P(E1) - P(E2) = P(E1 - E2) - P(E2 -El)

for nytwo eventsE1 and E2,weobtain

(3.2) ASD(0 + 80) - ASD(0) = 2(AO - Co)f| '6 h(a) da + o(80).

Thepropositionollows.El

We define point0 to be regularw.r.t. distribution (-) ifH(-) has acontinuous ensityn a neighborhoodf 0. We also define pointto be amedian axis on the circle f the diameter assing hrough and -0 dividesthecircle nto twosemicircles ith qual probabilities.he followingroposi-tionasserts hatASM is always median xis.

PROPOSITION 3.2. If 00 is a median axis withh(0o) > h(-00) and thepoints 00 and -00 are regular,then 00 is a local maximumof ASD.Conversely, f 00 is a local maximum of ASD and 00 and - 0 are regularwithh(00) > 0, then00 is a median xis and h(00) ? h(- 00).

REMARK .1. (i) On the circleADD(-) also allows a simpledifferentialequation,namely,

d- ADD(0) = (AO - Co)do

provided hat h(Q)exists t 0 and -0. The proofs given nMardia (1972),page31].

(ii) The equation in (i) immediatelympliesthat statements imilartoProposition .2 hold forADD.

COROLLARY.1 [MonotonicityfASD( )]. Supposeh( ) is symmetricboutits maximumoint00and decreasesmonotonicallynboth idesof 00until tsdiametricallypposite oint - 00.Then ASDQ is also monotonic onincreas-ing n both irectionsrom00 to - 00. In particular, 00 is a maximumpoint ofASD(-).

The next property f ASD(-) on a circle will allow us to characterizeantipodallyymmetricistributions.hese are defined s follows.




DEFINITION. Let H be the distribution f a random variable W ond-dimensionalphereSd. H is said to be antipodally ymmetricabout theorigin) if the distribution of (-W) is also H, where (-W) stands for the

diametrically ppositepointof W. If H has a continuousdensity, henantipodal ymmetrys equivalent o h(0) = h(- 0) for ll 0 on Sd.

PROPOSITION .3. Assume hath(-) is continuous. hen ASD(6) = c, forpositive onstant and all 0 E [0,2wr),fand only fh(6) = h(- 0) for ll 0.Moreover, he onstant must henbe 1/4.

PROOF. (= ) Suppose ASD(6) = c throughout.Then by Proposition 3.1,eitherA6 = C6 or h(0) = h( - 0) = 0 holdsfor very0. Thus, h(0) = h( -0) for

all0 in

viewofthecontinuityfh( ).(4=) If h(0) = h(- 6) for all 0, then Aa = Co and d ASD(6)/d6 = Ofor all0, which mpliesASD(W) c for omeconstant .

To showthat c mustbe 1/4, we need only howthatASD(O)= 1/4 sincethe sameargumentpplies oa general oint0 after rotation fthe axesby0. Due to antipodal ymmetryi.e., h(0) = h(- 0) for ll 0],we have

ASD(0) = 2f(2 - H(a))h(a) da.

It sufficeso check hat

(3.3) |fH(a)h(a)da =8

This is doneby ettingH(a) = y andconverting2.3) intoJo 2ydy. O

Note that n alternativeroof fProposition.3 can begiven infact nderthe weaker condition hat H(-) is continuous nly] using the connectingequation nProposition.4.

REMARK .2. On a circle his characterizationytheconstancyfASD( )also holdsforADD(-). The constant there s 7/2.

REMARK .3. It maybe of nterest o note that ftheunderlyingistribu-tionhas itsprobability ass concentratedn a semicirclenly, hen hedepthASD( ) is zero for ll points n thecomplementaryemicircle.

Anequation onnectingSD with D and itsapplications. Equation 3.1)forthe rate ofchangeof the ASD was the main tool in our studyforthecircular ase. For the sphere, heredoes not seemto be any such simpleequation. nsteadwe focus na reduction f hesphere o theplanetangentothesphere t a givenpoint. uch a process s often alled n exponentialmap.Thiswill allow us to apply ome known ropertiesf SD on R2. To makethediscussion learer,webeginwith he constructionn thecase ofcircles.




Let H(-) be the distributionnd 0 somefixed oint ntheunit ircle. hereis a natural ength-preservingapping gfrom he circlewithouthepoint- 0) to the segment - iT,w) of the tangent ine Lo at 0. For a point (pon the

unit ircleSp - 0),theabsolute aluelg,(9o)Is simplyhe ength f rc(6,p),and thesignofgo(s) being- or + depends n whetherhe directionngoingfrom to f is counterclockwiser clockwise.H,9( is used to representheresulting istributionn thetangent ine Lo with ts entire robability asson (-7r, 7). Let SD,(-) be the simplicialdepthon thetangent ine Lo w.r.t. thedistribution 0(-). The depthfunctions SDQ and SD,(-) are connected sfollows.

PROPOSITION3.4 (On thecircle). If H(-) is continuous,hen or ll 0 onthe

circle(3.4) ASD( 0) + ASD- 0) = SD6(O).

PROOF. Let {W1,W2)be a random amplefromH(-). Exceptfor null setthefollowinghree vents re equivalent:

(o E g0(Wg60(W2)},

{W1andW2 re on twodifferentidesof hediagonaloining and - 01

and

(3.5) (6 E arc( W1,W2)} u (-60 E arc(W1,W2)}

Thepropositionollows rom he fact hat he ntersectionfthetwo ventsn(3.5) has probability. E

We now turn to the two-dimensionalphere.Similarlywe let H(-) be adistributionnd 0 a fixed ointon the unitsphere.Let Po be the tangentplaneto thesphere t 0. For a pointSp 'p # 0, - 0), consider hegreat irclewhichpassesthrough and p. The planeof this circle uts Po alonga line

L0 which s just thetangentineto the circle t 0. This means that fwerestrict ur attentiono thiscircle nd ineLo,. weareexactlynthesituationofthecirclediscussedn theprevious aragraph.n particular,heconstruc-tiondescribed here pplies nd pcan bemappednto pointg(yp)on the ineLoo between-7r, 7r).As 'pmoves ideways n thesphere,t is evident hatthe line L0,pwillrotate on the plane Po. The spherewithout 0 willbemapped ygo into disc n P0 with enter (whichs theorigin fPo now).As before,weuse Ho(-) todenote heresulting istributionn Po,whichhasits totalprobability ass on the disccentered t theorigin withradius7r.

Theanalogof 3.4) for hesphere an nowbe stated s follows.PROPOSITION .5 (On thesphere). Assume hatH(-) is absolutelyontinu-

ous. Then

(3.6) ASD(0) + ASD(-0) = SD0(O),

where D0( ) is the implicial epth n theplanecorrespondingoHo(-).




PROOF. Fix W1 and W2. The great circles oining {W1, 1 and {W2, 1,respectively,plit hesphere ntofourpieces.Let S(W1,W2, ) be thesmallerpiecewhichdoes notcontainW1 and W2. Thisexistswithprobability.) For

anyW3,0 E A(g0(W1),g9(W2),g0(W3)) ifandonlyif W3E S(W1,W2, ).

The eventW3 E S(W1, W2,0) can be divided nto the followingwomutuallyexclusive vents:

(I) Both 0 and W3 lie on one of the twohemispheres eterminedy thegreat ircleoiningW1 nd W2.

(II) Both - 0 andW3 lie on oneof the twohemisphereseterminedy thegreat ircle oiningW1 ndW2.

Note that I) is equivalent o the eventthat 0 EIA,(W1,W2,W3) and (II) to-0 E A,(W1,W2,W3).This proves he assertion.R

We nowderive he followingimple haracterizationfantipodallyymmet-ricdistributionsn the sphere.

PROPOSITION 3.6 (On the sphere). Assume that h( ) is continuous. ThenASD(O) = 1/8 forall 0 ifand onlyif h(H) = h(- 0) forall 0.

PROOF. (<=) If H( ) is antipodallyymmetrici.e., h(0) = h(-0)], thenASD(O) = ASD( - 0). It is clearly obecausefor n observationWfromuchanH( ) the random variablesW and - W have the same distribution. heinduceddistribution o(-) is symmetricboutthe origin0 on the tangentplanePo.As a result seeTheorem of Liu (1990)],

SD>(O) = 4

and theresultfollows.(=) SupposeASD(0) = 1/8 throughout.henbyProposition .5 we have

SD0(O) = 1/4 for ll 0. However,twas shownnLiu (1990) that ftheSD(-)is equal to 1/4 on the plane at some point, hen the distribution 0(-) issymmetricround hatpoint. ince this s truefor ll 0,the distribution (-)assignsprobability/2 toeachhemisphere.herefore (O) = h(- 0) for ll 0.

The corollarieselowfollow rom ropositions.4 and3.5,andthey rovideupperboundsforASD( ) on the circle nd on thesphere, espectively.

COROLLARY.2 (On the circle). Under the conditionsof Proposition 3.4,ASD(O) < 1/2 forevery0 on the circle. The equalityholds at a point 00 ifandonly ifthe entireprobabilitymass is on a semicircleand H(00) = 1/2.

The claim is based on the factthat SDG(sP) 1/2, whichholds sinceSD,((p) = 2H6(P)[1 - H6(P)].




COROLLARY.3 (On thesphere). Under heconditionsf Proposition .5,ASD(O) < 1/4 for ny0 on the phere.Theequality olds at a point00 iftheentiredistributions concentratedn a hemisphereontaining 0 and the

induceddistributiono -) is symmetricroundtheorigin.

Corollary .3 is obtained ycombiningroposition.5 withTheorem ofLiu (1990).

Statistical pplications fPropositions.3 and 3.6. Theresult fProposi-tion3.3 (Proposition.6) givesriseto a simple estofwhether given ircular(spherical) istributions antipodallyymmetricboutthecenter fthe circle

(sphere).Wemayuse(3.7) sup ASDn(o) - 4

on thecircle nd

(3.8) sup ASDn(0) - 810

onthesphere s test tatistics.argevaluesof 3.7) and 3.8) indicatehat hedistributionsunlikelyobeantipodallyymmetric.eedless osay, heactualimplementationfthesetestingdeas wouldrequireknowledgefthe exactorapproximateampling istributionsfthese est tatistics. erhaps n approx-imation fthe sampling istributionan be obtained rom omeresamplingprocedures, or xample, hebootstrapmethod.A different ethod as beensuggested yFisher 1989) for btaininghesampling istributionnder henullhypothesisfantipodal ymmetry:se randomreflectionthroughheorigin) ftheoriginal ata set toproduce n newsampleseachof size n) andcompute n values fortest statistic3.7) [or 3.8)]. The histogramased onthese2n values is an approximationfthe desired ampling istribution.detailed tudy f theproposed ests 3.7) and (3.8) and their omparison ithAjne'stest Ajne 1968)]shall be reportedlsewhere.

4. Properties of angular Tukey's depth. It may be instructive oconsider irsthe case where heunderlyingistribution is supportednlyon a semicirclehemisphere).n this case we can easilyrelateATDQ toTukey'sdepth n the ine plane).

PROPOSITION4.1. IfH(-) is a distributiononcentratedn a semicircle,ayfrom to r, thenATDH( ) assumes he amiliar ormfTukey's epth ntheline, namely, or0 < 0 < 2r,

(4.1) ATDH(M) = min{H(O), 1 - H(6)).




ClearlyATDH( ) vanishes outside theinterval 0, r]. Furthermore,

ATDJO) = min{HJ(O), 1 - Hn(0 -)

Consider he center-outwardanking fdata pointsbased on decreasingATDn(*)values. Proposition.1 implies hattheaboverankingoincideswiththecenter-outwardanking asedupontheusual order tatisticsn the ine, fthedistributions supported n a semicircle.

In thespherical aseweassumethatH( ) is supportedn a hemisphere0.Given nypoint0 on SO,,weconsiderhestereographicrojection, ith oleat- 0,from hesphere o thetangent laneat 0. The distribution ( ) on thesphere hen nduces distributionn thetangent lanewhichwe denoteby

Ho(-).Now, given nyhemisphere containing in its interior e can find

anotherhemisphere,ay S', containing in itsboundaryatisfying(S') <P(S). Wecan visualizeS' as follows. et L bethe ineof ntersectionetweenthe planes supportinghe boundariesof S and that of So. Rotate theboundary f S aroundL as axis until t passes through . One of the twohemispheres husobtainedwill haveprobabilityess than or equal to P(S),and this s theone we take as S'. This mplies hefollowingroposition.

PROPOSITION 4.2. Let H(-) be a distribution upportedon thehemisphere

So.Then the

followinghold: (a) ATDH(0) = 0

forall 0 0

So.(b) For

any0 in

So, ATDH(0) agrees with Tukey's depth (2.5) taken w.r.t. the distributionHo(-) induced on thetangentplane at 0 bystereographic rojectionfrom - 0.

We discuss now some moregeneral propertiesofATD(*).

PROPOSITION4.3. On the circle as well as on thesphere,ATD( ) is boundedaboveby 1/2. The value 1/2 is achieuedat a point 6 on a circle 0 on a sphere)if and only if each semicircle (hemisphere) containing 6 (0) has probabilitygreater than or equal to 1/2.

In particular,he bound1/2 is achieved t themodeofany member fthevonMisesclassofdistributions,ndeverywheref he distributions uniform.

PROPOSITION4.4. On the circle(sphere)ATD(-) has theconstantvalue 1/2throughoutfand only if any semicircle hemisphere) has probability1/2.

Since the property hat each semicircle hemisphere)has probability1/2canbeviewed s an alternative efinitionf n antipodallyymmetricistribu-tion,Proposition .4 is a characterizationfantipodally ymmetricistribu-tions.

These observationsmay suggestthat a distribution ith constantATDwouldhave to be antipodallyymmetricnd that the maximum TD for nydistributions always1/2. Neither tatement s true,as is shown n thefollowingxample.



ORDERING DIRECTIONAL DATA 1479

EXAMPLE .1. Let E = 1/28, and let H( ) be a distributionon the unitcircle with the densityfunctionh(-) defined s

(1 + 3E)(w7T/2) forO 6< -w

4 ~~~~26(77/4) , for _< 0 < 37

(4- 2E)(7/4)', for4 <0 <r,

(h) - E)(T/2) for r < 0 < 2

(4 - 2E)(wr/4)1, for 1T < 6 < 4-,

E(77-/4) for wT < H< 27.

For this distributionATDH(6) = 1/2 - 1/14 for all 6, 0 < 0 < 2w. To checkthis we note that each one ofthese threesemicircles:fromw/2 to (3/2)7, w to2wr nd (7/4)rr to (3/4)wrhas probability1/2 - 2E. This also turns out to bethe minimumprobability ver all semicircles.

This example contrasts with the following act relatedto ASD: The ASD isconstanton a circle fand only fthe distribution s antipodally ymmetricndthe value of the constant is always 1/4 (cf. Proposition3.3).

We now state the key monotonicity roperty fATD( ).

PROPOSITION.5. Let 00 be a point on the sphere. We introduceforeachpoint 0 theEuler angle X, 0 < 4 < 7r,which is the angle between000 and00, in otherwordsthe atitudeof 0. Ifwefix meridian,theposition of 0 willbe characterizedby band its longitude-q,0 < -q< 2wr,nd a densityh on thesphereis ust a function f (0, -q).Assume we have a distributionwithdensityh(G, ') which decreases in 0 foreach rq, nd satisfiesh(b, -q)= h(0, -q+ 7r).Then

ATDH(O) = ATDH((X, i))

is a monotonicallynonincreasing functionof 0 foreach -q. In particular, itattains its maximum 1/2 at 00.

PROOF. The argument relies upon the followingobservation: If a hemi-sphere contains 00 then it has probabilitygreaterthan or equal to 1/2; if itdoes not then it has probability ess than or equal to 1/2.

Fix a longitude q. Consider two latitudes 01 and 02 such that 0 < +p <

02 < 7. We claimthat

ATDH((Ol, 77)) < ATDH((02, )

To show thiswe considerany hemisphereS+ such that it contains k1, -q)butnot (02, -q). Let S- represent the complement which obviously contains(02, -q).EvidentlyS+ also contains the mode 00. In viewof the above observa-tion we obtain P(S+) P(S-). The propositionfollows. O




We omitthediscussion fthe similarmonotonicityropertyn thecirclecase.

Next, we point out a simplebut peculiarproperty f ATD, which also

contrastswith he general trictmonotonicityfASD.PROPOSITION4.6. For anydistributionn the phere circle), here xists t

leastone hemispheresemicircle)where heATD(Q) s constantnd thevalueoftheconstants its minimum alue.

In fact f we let S be a hemispherewiththe smallestprobability,henATD(O) willbe equal to P(S) for nypoint0 in S. This observation ill beusefulfor hecomparisonsnSection .

Evidently,roposition.6 applies o the empirical ersion fATD also. ThispropertyfATD suggests natural rimmingfangulardata,namely,o trimoff ll the datapointswith heminimum TD.

Finally,we state without he proofthe connection etweenATD andmedian xis in the nextproposition,hich anbe seenas thecounterpartfProposition.2 forATD.

PROPOSITION.7. Proposition .2 holdswithASD replaced yATD.

5. Aspects of robustness. The notion of breakdowndescribed nHampel,Ronchetti, ousseeuw nd Stahel 1986) is an intuitivemeasureofrobustnessn RP.The followingefinitions a natural daption fbreakdownfordirectional unctions. nderthisdefinitionhebreakdowns nonzero orall threemedians.

DEFINITION. Let H be a distribution n Sd with00 as a median.We definethe breakdown f thismedian s the infimumf E such thatthemedianofHE = (1 - e)H + EG is - 00 for omecontaminatingistribution.

Under this definition he following roposition stablishes ome lowerboundsofthe breakdown or he threemedians.

PROPOSITION.1. On a circle,for ny distributions and G, wehave

(i) |ASDH-() ASDH(O) ?<2?,

(ii) ATDH (0) - ATDH(0) ?<

and

(iii) ADDH'(60) - ADDHM( ?EI',

whereHE = (1- )H + EG.

Theinequalityi) implies hefollowing:f00 s an ASM on thecircle nd thedepthof 00 is strictly igher han thatof -00, then the breakdownfthis



ORDERING DIRECTIONALDATA 1481

median s nonzero.The same statement ppliesto ATM and ADM. In fact,Proposition.1 implies

breakdown fASM ? ASD(6Q) - ASD( - 00)4

ATD(6 0) - ATD(-00)breakdownfATM ? 2

and

breakdownofADM > ADD(00) - ADD( -00)2w

PROOF. (i) Let {W1,W2} e a random amplefromH; {Z1,Z2}from ; andhhl} from Bernoulli istributionithP(Gi = 1) - 1 - E and PG7i= O) -

r. Weassume hat heWi's, i's and <js areall independentandom ariables.Define

=Zi if 77= O.

We obtain

ASDH(0) = P(6 c arc(W,*,W2W))- P(6 E are(W*W2f) n (m 1, 12 1))

+ P(0 E arc(WE*t*, ) n (ij?l 1 q =-1)c)

- (1 _e) ASDH(0) + R,

where ? R < 2e. The result ollows rom hefact hat1 - (1 - 0)2 < 2e.(ii) The inequalityn ATD is easilydeduced rom he observationhatfor

any semicircle , lPH(S) -PH(S)I

< r.Theproof f iii) is straightforwardnd is thusomitted.

REMARK .1. In the case of a sphere, he boundforATD and ADD inProposition.1 remains hesame,and it becomes e forASD.

Besidesthe notionofbreakdown,he influence unctions another om-monly sed toolfor hestudy frobustness.ee Hampel,Ronchetti,ousseeuwand Stahel 1986) for hedescriptionf the influenceunctionf a statistic.Sincethe nfluence unctionsfmost ircularocation stimatorsrebounded,Ko andGuttorp1988) proposed o divide he nfluence unctiony a measureofscale andthen ake thesupremumver he circle nd a reasonable lass ofdistributions.f thesupremumsbounded, hen he estimators consideredobe standardized-biasobustSB-robust). o andGuttorp1988) show hat hedirectional ean s notSB-robust; owever,nthevonMises classofdistribu-tions,Mardia'smedian or ADD) is SB-robust. or a symmetricnimodal





ORDERING DIRECTIONALDATA 1483

decreases p tohalfwayn each direction nd becomes constant fterwards.The value ofATD at themodeof unimodal istributionquals 1/2, whereasthatforASD at themode on a circle) anges rom /4to 1/2.Thevalue1/2

forASD at the mode occurs fand only fthe entiredistributions concen-trated n a semicircle.huswe notethatthe maximum alue ofATD at themode is the same as its constantvalue in thecase oftheuniformdistribution(= 1/2), while the notionof ASD makes a clear distinctionbetweenthe twosituations.

REMARK6.3. For symmetricnimodaldistributionse.g., the von Misesclass),ADM,ASM andATM all coincidewiththemode.The examplegivenbelowfurther hows thatASD, ATD and ADD mayhelpone pickout the

centralmost oint n the presence fmultiplemedian axes and multiplemodes.

EXAMPLE6.1. Let h( ) be a density unction n the unit circledefined sfollows:

H 1 7r

/2-, forO?< 2'

T-0 1 v

h(0) for-' for <0<7

7r/2 2-

12-, for Tr < 0 < 2w.

In otherwords, hedistributions triangularn[0,TrIand uniformn 7, 2X).It is easyto check hatthere re twoperpendicular edian xes along he twoaxes. On the otherhand, here s a uniquemaximum oint fASD( ), namely,the point 7r/2.This seems moresensiblebecause among the fourmediancandidates , 7w/2, and 37/2 suggested ymedian xes,7r/2 tandsout as

thepointwith hehighest robabilityoncentrationround it. The claimofuniquemaximizationt 1r/2 anbe verifiedy usingProposition.1 and thefollowingact:

AO Co 0 for E (? 2 ) u (I,27w)

and

7r 31rAO-Co<0 for0E- 2' 2.

Thus, ASD and ADD decrease monotonicallyn the ranges7r/2to 3v/2clockwise s well as 7/2 to 37r/2 ounterclockwise.imilarly, TD decreasesmonotonicallyn both idesof r/2with trictmonotonicityetweenr/2and7r/2 w/4. Beyond hisrange,ATD stays onstant ndassumes ts minimumvalue.




Now, in this example new mode can actuallybe created t w or 0 byaltering he densityocally, huscreating nothermodekeeping he maximumpoint fASD( ),ATD0,) ADD(-) and thetwo median xes unaffected.

REMARK .4. In comparing he rankings erivedfrom hreedepths,wenote that the ranking ased on ADD is not even consistentwiththe linearrankingwhen he distributions on a half-circlecf.Example5.1).As forASDand ATD, even though heybothsatisfy his consistencyroperty,TD isunableto distinguishll points ying n the hemispherewith the smallestprobabilitycf.Proposition .6 and Remark .2). In conclusion, SD seems oprovidea finerranking f data points n the orderof centrality, hich sparticularlyseful n detectingutliersseeCollett1980) for discussion noutliers n circular

data].On the otherhand,ATD may be expected o be

superior n terms of the robustness f the associated center. Anotheradvantage fASD is that, ngeneral, SD seemseasiertocomputehanATD,especially or d, d > 2.

REFERENCES

AJNE,B. (1968). A simple estforuniformityf a circular istribution.iometrika5 343-354.BROWN, B. M. (1983). Statistical ses of the spatial median.J. Roy. Statist.Soc. Ser. B 45

25-30.COLLETT, D. (1980). Outliersncircular ata. J. Roy. Statist. oc. Ser. C 29 50-57.

EFRON, B. (1979). Bootstrapmethods: notherook t the ackknife. nn.Statist. 1-26.FISHER, N. I. (1985). Sphericalmedians.J. Roy. Statist. oc. Ser. B 47 342-348.FISHER, N. I. (1989). Personal ommunication.FISHER, N. I., LEWIS, T. and EMBLETON, B. J. J. 1987). Statistical nalysis f SphericalData.

Cambridge niv.Press.GoWER,J. C. (1974). The median entre.J. Roy.Statist. oc. Ser. C 23 466-470.GRoss, S. and LIU,R. (1989). Classificationulesbasedon concepts f datadepth.Unpublished

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Ko, D. andGUTTORP, P. (1988). Robustness f estimators ordirectional ata. Ann. Statist. 6609-618.

Liu, R. (1988). On a notion f implicial epth. roc.Nat. Acad. Sci. U.S.A. 85 1732-1734.Liu, R. (1990). On a notion fdata depth ased on random implices. nn. Statist. 8 405-414.MARDIA,K. V. (1972). Statistics fDirectional ata. Academic, ewYork.SMALL, C. G. (1987). Measures fcentralityormultivariatend directional istribution.anad.

J. Statist. 31-39.TuKEY, . W. 1975). Mathematicsnd the picturingfdata. In Proceedingsfthe nternational

Congress fMathematicians,ancouver, 974 R. D. James, d.)2 523-531.CanadianMathematicalongress.

WATSON,G. S. (1983). Statistics n Spheres.Wiley, ewYork.WEHRLY, T. E. and SHINE, E. P. (1981). Influence urvesof estimators ordirectional ata.

Biometrika 8 334-335.

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Documents

Liu R. and Singh K. (1992) Ordering Directional Data Concepts of Data Depth on Circles and Spheres