Probabilities of hitting a convex hull

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1 Pobabilities of hittig a covex hull Zhexia Liu ad Xiagfeg Yag Liköpig Uivesity Post Pit N.B.: Whe citig this wok, cite the oigial aticle. Oigial Publicatio: Zhexia Liu ad Xiagfeg Yag, Pobabilities of hittig a covex hull, 4, Comptes edus. Mathematique, (35,, Copyight: Elsevie Masso Postpit available at: Liköpig Uivesity Electoic Pess

2 Pobabilities of hittig a covex hull Pobabilités d atteite d ue evelopple covexe Zhexia Liu Xiagfeg Yag August 7, 4 Abstact I this ote, we coside the o-egative least squae method with a adom matix. This poblem has coectios with the pobability that the oigi is ot i the covex hull of may adom poits. As elated poblems, suitable estimates ae obtaied as well o the pobability that a small ball does ot hit the covex hull. Abstact Das cette Note ous appliquos la méthode des moides caés o-égatifs avec ue matice aléatoie. Ce poblème est coecté à la pobabilité que l eveloppe covexe de poits aléatoies e cotiee pas l oigie. E elatio avec ce poblème ous obteos aussi des estimatios de la pobabilité qu ue petite boule e ecote pas ue eveloppe covexe. Keywods ad phases: Covex hull, uifom distibutio, o-egative least squae method AMS subject classificatios: 6D5, 5A Itoductio Let ad m be two positive iteges with m. Suppose that A is a m matix ad b is a vecto i R. I mathematical optimizatio ad othe eseach fields, it is fequet to coside the o-egative least squae solutio to a liea system AX = b with X = (x, x,..., x m T R m ude the costait mi i m x i. The o-egativity costaits occu atually i vaious models ivolvig o-egative data; see [], [3] ad [7]. Moe geeally fo o-egative adom desigs, the matix A is assumed to be adom; see [4] ad efeeces theei fo this aspect. The fist topic of this ote is to ivestigate the pobability P {AX = b, mi i m x i } whe A is a adom matix with suitable estictios; see Theoem.. The idea of the poof is to chage this pobability to the oe ivolvig the evet that the oigi is ot i the covex hull of may adom poits, ad the apply a well-kow esult by Wedel []. Howeve, istead of applyig Wedel s esult diectly, we popose a ew pobabilistic poof of it. This pobabilistic poof allows us to wok o a moe geeal pobability of hittig a covex hull by a small ball (istead of the oigi i R ; see Theoem 4.. zhexia.liu@hotmail.com, Blåeldsväge B, Stuefos, Swede xiagfeg.yag@liu.se, Depatmet of Mathematics, Liköpig Uivesity, SE Liköpig, Swede

3 The study o adom covex hulls dates back to 96s fom vaious pespectives. Fo istace, i [] ad [] the expected peimete of a adom covex hull was deived. The expected umbe of edges of a adom covex hull was obtaied i [8]. Fo expected aea o volume of a adom covex hull, we efe to [5]. As metioed ealie, i [] the pobability that the oigi does ot belog to a adom covex hull was pefectly established. I Sectio 3, we deive a explicit fom fo the pobability that a ball with a small adius i R does ot belog to the covex hull of may i.i.d. adom poits; see Theoem 3.. This type of pobability was cosideed i [6] togethe with cicle coveage poblems. Because of additio assumptios thee, ufotuately the esults (Coollay 4. ad Example 4. i [6] caot ecove ou esult Theoem 3. i this ote. A moe detailed suvey o adom covex hulls is icluded i [9]. A liea system with a adom matix Sice the oe-dimesio = is tivial, we coside highe dimesios. I the poof of the ext esult, a coectio is established betwee the pobabilities of hittig a covex hull ad the o-egative solutios to a liea system. Theoem.. Let A be a m, m, matix such that the eties ae idepedet oegative cotiuous adom vaiables. Suppose that these adom vaiables have the same mea µ, ad ae symmetic about the mea. The the liea system AX = (,,..., T has a o-egative solutio with pobability ( m+ m. k Whe m =, it simplifies to +. k= Poof. We set the eties of A as {a ij }, the m j= a ijx j = fo i. Summig o i, we obtai m j= ( i= a ijx j =. Let c j = i= a ij, the m j= c jx j =. Thus, we ca ewite the liea system m j= a ijx j = as m j= (a ij c j x j =. Let a,..., a m be the colum vectos of A, ad v = (,,..., T. If we deote w j = a j c j v, The the liea system m j= a ijx j = fo i has a o-egative solutio if ad oly if thee exist x, x,..., x m with x + x x m > such that m j= x jw j =. I othe wods, the oigi belogs to the covex hull of {w, w,..., w m }. We show that {w j } ae symmetic. Ideed, P{w j > (t, t,..., t T } { } = P a ij a kj > t i, i k= { } = P (a ij a kj > t i, i k= { } = P [(µ a ij (µ a kj ] > t i, i k= { } = P (a ij a kj > t i, i = P { w j > (t, t,..., t T }. k=

4 Clealy, {w j } ae adom vectos i R that lie o the hypeplae L = {(y, y,..., y R : y + y y = }. Let p(k, m be the pobability that does ot belog to the covex hull of m symmetic adom vectos i R that lie o a k-dimesioal subspace of R. We ow compute the pobability p(, m. The method below is a pobability vesio of a geometic agumet of Wedel []. Let h be the idicato fuctio of the evet / cov(w, w,..., w m. That is, h(w, w,..., w m = if thee exists a o-zeo vecto b such that w i, b fo all i m, ad h(w, w,..., w m = othewise. The, p(, m = P { / cov(w, w,..., w m } = E w h(w, w,..., w m. Because {w i } ae symmetic, if we let {ε i } be i.i.d. Beoulli adom vaiables, the p(, m = E ε E w h(ε w, ε w,..., ε m w m. Noticig that coditioig o ε = (ε, ε,..., ε m, we have whee p(, m = E ε E w E εm h(ε w, ε w,..., ε m w m = E ε E we εm h(ε w, ε w,..., ε m w m + E ε E w[e εm h(ε w, ε w,..., ε m w m h(ε w, ε w,..., ε m w m ] = p(, m + E ε E wr R :=h(ε w, ε w,..., w m + h(ε w, ε w,..., w m h(ε w, ε w,..., ε m w m. We see that R {, }, ad R = if ad oly if h(ε w, ε w,..., w m = h(ε w, ε w,..., w m =. That is, thee exists vectos b, b such that ε i w i, b, ε i w i, b fo i m ad w m, b, w m, b. Thus we ca fid α, β > such that fo c = αb + βb, we have ε i w i, c fo i m, ad w m, c =. O the othe had, if we ca fid such a vecto c, the of couse h(ε w, ε w,..., w m = h(ε w, ε w,..., w m =. Theefoe, R = if ad oly if thee exists a vecto c such that c w m such that ε i w i, c fo i m. If we let u i be the othogoal pojectio of w i o to w m fo i m, the R = if ad oly if h(ε u, ε u,..., ε m u m =. Fom the fact that {u i } ae vectos i R that lies o the ( dimesioal subspace w m L, it follows that Hece, we obtai the idetity E ε E w R = E ε E u h(ε u, ε u,..., ε m u m = p(, m. p(, m = p(, m + p(, m fo all m. Note that p(, k = k+ ad p(k, = fo k. By usig iductio ad the combiatoial idetity ( ( ( m m m + =, k k + k + 3

5 it is staightfowad to check that fo all m. ( p(, m = m+ m k k= 3 Pobability of avoidig a small disk i R Let adom vectos {X i } i=,,...,m be idepedetly ad uifomly distibuted i the uit ball of R. The esult i Sectio states that the pobability that the oigi is ot i the covex hull of {X i } i=,,...,m is p(, m = m m+. I this sectio, ou goal is to fid a moe geeal esult, amely, the pobability that a ball with a small adius i R does ot belog to the covex hull of {X i } i=,,...,m. We will pove the followig esult. Theoem 3.. Suppose that {X i } i=,,...,m ae idepedetly ad uifomly distibuted adom vectos i the uit ball of R. Let p (, m deote the pobability that a ball with a small adius i R does ot belog to the covex hull of {X i } i=,,...,m. The p (, m = [ m m m ( (] si. (3. Poof. Thee ae two diffeet cases: the closest poit o the covex hull of {X i } i=,,...,m to the oigi is a vetex (see Case, ad the closest poit o the covex hull of {X i } i=,,...,m to the oigi is ot a vetex but a poit o a edge (see Case. Fo each case, we compute the pobability espectively. Case Case Step. Let P ad Q be two idepedetly ad uifomly distibuted adom poits i the uit ball. We calculate the pobability of the evet E( that the distace betwee the oigi ad the lie segmet P Q is less tha o equal to, ad the closest poit to the oigi is ot P o Q. Let (λ, θ be the pola coodiates of P. Let L be the lie passig though P ad beig pepedicula to OP. The the lie L divides the uit disk ito two pats, say R ad R, whee R is the lage 4

6 egio that cotais the oigi. Futhe, we let D be the disk with OP as its diamete. The it is obvious that D R. If Q R, the P is the closest poit to the oigi. If Q D, the Q is the closest poit to the oigi. If Q R \ D, the the closest poit of the lie segmet P Q to the oigi is ot P o Q. If λ, the fo all Q R \ D, the distace betwee the oigi ad the lie segmet P Q is less tha o equal to ; if λ >, the to esue that the distace betwee the oigi ad the lie segmet P Q is less tha o equal to, the poit Q must lad i the egio S which is betwee the two taget lies fom P to the cicle cetes at the oigi with adius. I coclusio, we have the followig: if λ, the Q R \ D; if λ >, the Q S (R \ D. The set R has aea / + λ x dx, ad D has aea λ /4. Thus R \ D has aea / + λ x dx λ /4. To calculate the aea F := S (R \ D, we let T ad T be the two taget poits. The agle betwee the two taget lies is si (/λ. We daw two lies though the oigi which ae paallel to the two taget lies. The egio G that lies betwee these two lies ad iside F has aea si (/λ. To calculate the aea of the egio F \ G, we coect O with T ad T. Let A be the aea betwee the lie segmet OT ad the small ac OT o D. The the aea of F \ G is x dx A. To calculate A, we let M be the cete of D. The OMT = si (/λ, the fa OMT has aea (λ/ si (/λ, ad the OMT has aea λ /4. Hece, the aea of F is si (/λ + x dx λ si (/λ + λ. Theefoe, give P at (λ, θ, if λ, the the evet E( occus with pobability + λ If λ >, the the evet E( occus with pobability x dx λ /4. si (/λ + x dx λ si (/λ + λ. Thus, the evet E( occus with pobability P {E(} = ( + + λ ( si (/λ + x dx λ /4 λdλ x dx λ si (/λ + λ λdλ. 5

7 This implies that d P {E(} ( d = + ( x dx /4 + ( + λ + ( = λ + λdλ = 4 ( 3/. λ λ + x dx /4 λ λ λdλ We ote that hee a paticula case is P {E(} = 4 ( 3/ d = 3 4, which is the pobability that the closest poit is ot eached at a vetex poit. Step. Now we calculate the pobability P ( that the distace betwee the oigi ad the covex hull is at least. If the closest poit is a vetex of the covex hull, the it could be ay of the m poits. Thus we eed to fist choose a poit, say P (, θ, ad we have m diffeet choices. Let L be the lie passig though P which is pepedicula to OP. The all the othe poits must lad o the oute side of the lie L. The aea of that egio is x dx. Thus, the coespodig pobability is P {} = m [ x dx] m d. I paticula, if =, the we have P ( = /. I othe wods, with pobability /4, the closest poit is a vetex. If the closest poit is ot a vetex, the it is o the lie segmet betwee two vetices. Sice ay two vetices ae equally likely, we have m(m / diffeet choices. The pobability i this case is P {} = Hece, the total pobability is P ( = m m(m [ m(m + [ = m( [ ] m 4 x dx ( 3/ d. x dx] m d = m m ( ] m 4 x dx [ m x dx] [ whee the secod equality is fom itegatio by pats. ( 3/ d si ] m, 6

8 4 Pobability of avoidig a small ball i R ( 3 Let i.i.d. adom vectos {X i } i=,,...,m be uifomly distibuted i the uit ball of R, 3. I this sectio we study the pobability that a ball with a small adius i R does ot belog to the covex hull of {X i } i=,,...,m. If we use a simila method as i Sectio 3, the ew difficulties aise o takig ito accout too may diffeet cases, ad computig seveal complicated volumes, multiple itegals, etc. Istead of computig the exact value of the pobability, we give o-tivial uppe estimates of it i this sectio based o the idea used i Sectio. To this ed, let p (k, m be the pobability that the ball i R with adius does ot belog to the covex hull of {X i } i=,,...,m which lie o a k-dimesioal subspace of R. Theoem 4.. Let {X i } i=,,...,m be idepedetly ad uifomly distibuted adom vectos i the uit ball of R, 3, ad p (, m be the pobability that a ball with a small adius i R does ot belog to the covex hull of {X i } i=,,...,m. It holds that p (, m p (, m whee p (, m solves { p (, m = p (, m + p (, m, (4. p (k, = k ad p (, k = ( k, fo k. k I paticula, ( + m p (, m + m, fo m ; (4. p (, + [ ( + + ]. (4.3 Remak 4.. We ecall the pobabilities p(, m that the oigi does ot belog to the covex hull of m adom vectos i R discussed i Sectio. The pobabilities ae ( p(, m = m+ m fo < m, k k= ad p(, m = fo m. It is the obvious that a tivial uppe boud of p (, m is p(, m, that is p (, m p(, m. This gives p (, m fo m, ad p (, +. Thus the uppe bouds i (4. ad (4.3 ae slightly bette tha these. Poof of Theoem 4.. Followig the idea i the poof of Theoem. i Sectio, we will show p (, m p (, m + p (, m. (4.4 To this ed, let h be the idicato fuctio of the evet that the ball with adius is ot i the covex hull of {X i } i=,,...,m, ad {ε i } be i.i.d. Beoulli adom vaiables. The by the same easoig i Sectio, we have, with ε = (ε,..., ε m, p (, m = E ε E X E εm h(ε X,..., ε m X m = E ε E XE εm h(ε X,..., ε m X m + E ε E X [E εm h(ε X,..., ε m X m h(ε X,..., ε m X m ] = p (, m + E ε E XR 7

9 whee the adom vaiable R {, } is R = h(ε X,..., X m + h(ε X,..., X m h(ε X,..., ε m X m. I Sectio, a equivalet statemet of the evet R = was foud. But hee we ca oly show {R = } {h(ε X,..., ε m X m = } (4.5 To see (4.5, we otice that whe h(ε X,..., X m = h(ε X,..., X m =, the the othogoal pojectio u i of X i oto X m, i m, should satisfy h(ε u,..., ε m u m =. This is (4.5. Thus the pobabilities p (, m satisfy (4.4 with kow bouday values p (k, = k ad p (, k = ( k k fo k. Now we solve the coespodig diffeece equatio (4.. Obviously p (, m p (, m fom compaisos. What is moe, the equatio (4. ca be solved as ( + m p (, m = + m, fo m ; p (, + = [ ( + + ]. (4.6 Thus (4. ad (4.3 ae diectly fom (4.6. By iductios, it is also feasible to fid geeal p (, m, which have moe complicated expessios. Ackowledgmet. We ae gateful to Pofesso Fak Gao fo stimulatig discussios ad useful suggestios, to a aoymous efeee fo helpful commets. Refeeces [] J. Badsley, J. Nagy, Covaiace-pecoditioed iteative methods fo oegatively costaied astoomical imagig, SIAM Joual o Matix Aalysis ad Applicatios, 7, 4, 84-97, (6 [] G. Baxte, A combiatoial lemma fo complex umbes, The Aals of Mathematical Statistics, 3, 3, 9-94, (96 [3] D. Dooho, et al., Maximum etopy ad the ealy black object, Joual of the Royal Statistical Society, 54,, 4-8, (99 [4] D. Dooho, J. Tae. Coutig the faces of adomly-pojected hypecubes ad othats, with applicatios, Discete & Computatioal Geomety, 43, 3, 5-54, ( [5] B. Efo, The covex hull of a adom set of poits, Biometika, 5, , (965 [6] N. Jewell, J. Romao, Coveage poblems ad adom covex hulls, Joual of Applied Pobability, 9, , (98 [7] L. Li, T. Speed, Paametic decovolutio of positive spike tais, The Aals of Statistics, 8, 5, 79-3, ( [8] A. Réyi, R. Sulake, Übe die kovexe Hülle vo zufällig gewählte Pukte, Pobability Theoy ad Related Fields,,, 75-84, (963 [9] S. Majumda, A. Comtet, J. Rado-Fulig, Radom covex hulls ad exteme value statistics, Joual of Statistical Physics, 38, 6, 955-9, ( [] F. Spitze, H. Widom, The cicumfeece of a covex polygo, Poceedigs of the Ameica Mathematical Society,, 3, 56-59, (96 [] J. Wedel, A poblem i geometic pobability, Mathematica Scadiavica,, 9-, (96 8