Foster, WH

English

Brunel University Research Archive

                      TR/09/90 September 1990  METRIC STATISTICS ON THE SIMPLEX  by  W. H. FOSTER                                      w919901x     BRUNEL UNIVERSITY SEP 1991 LIBRARY 1. Introuduction  Given a finite set of points S = { } ,x,....,x 1nk1 −∆⊆  the standardn-1 simplex in  we define : ,RI n 1. ( ) { } ni,km,j:xxmaxSd mijii ∈∈∀−=  Where xj = { } kj,x,....,x jnji ∈ . Note that the distributions for di (S)are the same n∈∀ i .  2. d(S) = max{di (S): i∈n}  It is an easy exercise in Crofton's Theorem (2.2 of [2] or C h a p t e r  5  o f  [ 3 ] )  t o  o b t a i n  e x p r e s s i o n s  f o r  t h e  p r o b a b i l i t y  distribution of di(S). We present these in section 2 below.  The main purpose of this paper is to obtain an expression for the cumula t ive  p robab i l i ty d i s t r ibu t ion  o f d(S)  fo r a  g iven  n  ≥  2  and  k≥2  ( see   3 .4 ) .  This is achieved in section 3 by a crucial use of the agreement or c lus te r ing measure  c (S)  ,  s ee  [1 ]  ,  fo r  a  f in i t e  se t  S  .  1−∆⊆ nIn  3 .5  we provide calculat ions of  the  above probabi l i t ies  for  some values  of  n  for  k  = 2.  The ra the r messy express ions  fo r  the  p robab i l i t i e s  con t ras t s  wi th  the  same  measure  d (S)  for  S  =  k  po in t s  in  [0 ,1 ] n ⊆ nRI .  I t  i s  easy to  show,  using the independence of  the coordinates ,  probabi l i ty  d(S)≤λ  =  λ n ( k - 1 ) (k-(k-1)λ )nNotationThe  on ly non-s tandard  no ta t ion we use i s to deno te for n  ,  NI∈{ } { }nnx1 :NI  x ,..,2,1=≤≤∈=n   2.Distribution of id (S)Suppose  tha t  we  have  k  po in t s  S  =  { } .x,....,x 1nki −∆⊆  Choose  a  coord ina t e ,  say  the  i th .  L e t  ( ) { } ni,kmj, : xx maxSd mijii ∈∈∀−=  where  x j  =  ( ) kj,x,....,x jnji ∈   1 We can  app ly  Crof ton ' s  Theorem to  ob ta in  the  fo l l owing  expres s ion  fo r  the  cumula t ive  p robab i l i ty   o f  dλn.kp i (S ) .  [ ]{ } )1n(kr1knrnkr)1n(1k rnr1o1kork,n dxx)x()1()1n(kP−+−−−−−−=λ αλ+−λ−−= ∫∑  Where [ ][ ]1knk rnr1kr1nr1k1r)1(−−−−+−=−=α ∑   I f  bo th  n  and  k  a re  odd  then  0=α .Th i s i s l e f t as a p leasan t  Exerc i se  fo r  the  reader .  Examples 1.   432λ3,3 3λ8λ6λP +−=2. [ ][ ] ⎭⎬⎫⎩⎨⎧−−= −−−−−−−−=∑ 1221221112, )1(21rnnrnrnnrnnPλλλ  3 .  The p .d . f . fo r   i n the case k  =  2 can be ca lcu la ted  )S(diDi rec t ly  and  we  ob ta in :   dzzzxn nnxn2221922 )1()1(2 −−− −− ∫   o r  by d i f fe ren t i a t ing  2 .  we  ob ta in  the  equ iva len t  express ion  fo r  the  p .d . f .  [ ] ⎭⎬⎫⎩⎨⎧− −− −−−−=− ∑ rrnnnornxnxn 42221)!32()1()!1(2  4 .  The  mth  moment  abou t  ze ro  fo r  in  t he  case  k=2  i s  )S(di )!1()!22()!2(!)1(2 2−+−+−−=nmnmnmnmµ              23.Distribution of d(S)  F i r s t  we  ou t l ine  some  bas ic  r e sul t s  needed  in  th i s  sec t ion .  3.1 greement Measure Background  ALet  be a  random { } k∈=∆⊆= − j,)x,...,x(xwherex,...,x S jnj1j1nk1se t of  k  no t  necessa r i ly d i s t inc t  po in t s .  Let { }kx ∈∀= j:min  a jii . I n [1 ] we  de f in e d (  a n d  i t  w a s  s h o w n  t h e r e  t h a t  iaS) ∑=−=⊂ni 11⊂S)  was d i s t r i bu t ed a s a  b e t a d i s t r i b u t i o n w i t h p a r a me t e r s ( n - 1 ) ( k - 1 ) ,  n .  F u r t h e r m o r e ,  g i v e n  S  w e  o b t a i n  a n  o r d e r e d  k - p a r t i t i o n  o f  n ,  A i , … , A k  ,  w h e r e  A j  = { i }: jii xa =∈ n . ( w e h a v e t o  a s s u m e  t h a t  S  i s  i n  g e n e r a l  p o s i t i o n - s e e  2 . 5  i n  [ 1 ]  -  a l s o  a t  l e a s t  t w o o f t h e  s e t s  A 1 , … A k  a r e  n o n  e m p t y . )   I f  ,,...,1, kjAr jj ==  t h e n  t h e  p r o b a b i l i t y  t h a t  S  h a s  t h e  o r d e r e d  k - p a r t i t i o n  { }k1 AA ,...,=Π  a s s o c i a t e d  t o  i t  i s  )!knk(!)1rn!....()1rn(!)1rn())!1n((!)knnk(  )P(k21 −−−−−−−−−−=Πk (see theorem 20 in [1])  3 . 2  S u b s e t s  o f  a n d  t h e i r  m e a s u r e s .  1n−∆O u r  a p p r o a c h  i n  t h i s  s e c t i o n  i s  t o  f i n d  p r o b a b i l i t i e s  b y  c o n s i d e r i n g  t h e  m e a s u r e s  o f  s u b j e c t s  o f  1n−∆ .  Let ,⎭⎬⎫⎩⎨⎧ ∈∀≤≤≤= ∑=nixxxxB iin1in1n 1, 0  1,:),...,(t h e n  g i v e n   w e  d e f i n e  t h e  u n i q u e  r e g u l a r  ),...,(, n1n xxxBx =∈s u b s i m p l e x  o f  s p a n n e d  b y  )(x∆ 1n−∆ niyyx(i) n1 ∈= ),,...,(  where  ,, ijxy jj ≠=  iji xx  y +−= ∑=n1j1  see 2.4 in [1]).  3 o r  g i v e n ,  ,, RIan ∈∈L e t  λay,)y,...,y(yRI y(RH(1, Ln1n ≥=∈=We  t h e n  h a v e  f o r   nn1 Bxxx ∈= ),...,()xH(n)xH(1,x n111n ∩∩∩∆=∆ − ...)(  o r  { } )1..(..........,),...,(:)( iin11n xyyyyyx ≥=∆∈=∆ − .  A l s o  w e  h a v e  t h a t  t h e  m e a s u r e  o r  v o l u m e  o f   i s  )(X∆1nin1in x1)!(nyXV −=∑−−=∆ )1())(( ,  r e g a r d e d  a s  a  s u b s e t  o f  .  1n-RID e f i n i t i o n  L e t   w h e r e  nRI  b , a ∈ ),...,(),..., 1 nn1 bbb a(aa ==  We  d e f i n e   w h e r e  c),...,( n1 ccba =⊕ i  =  ma x  { a i  ,  b i } .  N o t e  t h a t  i t  i s  p o s s i b l e  f o r  a , b  b u t  .  nB∈ nBba ∉⊕C l e a r l y ,  i f  a  , b  , … , w   t h e n  nRI∈ wba ⊕⊕⊕ ..  i s  w e l l  d e f i n e d .  L e m ma  1  L e t  a  , b  , … , w ,  nB∈1 ,  nBw..baifw)..b∆(a  ∆(w)..∆(b)∆(a) ∈⊕⊕⊕⊕⊕⊕=III2 .  nB w .. b  a if  ∆(w)..∆(b)∆(a) ∉⊕⊕⊕= φIIIP r o o f  F o l l o w s  d i r e c t l y  f r o m ( 1 )  a b o v e .  E . O . P .  A s  a n  a p p l i c a t i o n  o f  l e m m a  1  w e  c o n s i d e r  t h e  p r o b l e m  o f  f i n d i ng  t h e  v o l u me  o f  t h e  u n i o n  o f  s  s u b s e t s   ,  i n  t h e si , ASi ∈⊆s p e c i a l  c a s e  w h e r e  e a c h  o f  t h e  S i  i s  a  p a r t i c u l a r  r e g u l a r  s u b s i mp l e x  o f  A = ,  1n−∆I f  T  w e  l e t  Vs⊆ T  b e  t h e  v o l u me  o f  I ,  w h e r e  a n d  o ( T )  b e  TiiS∈,oV =φt h e  n u mb e r  o f  e l e me n t s  o r  o r d e r  o f  T .  L e t  V  b e  t h e  v o l u me  o f  s1 S...S UU .  B y  t h e  c o mb i n a t o r i a l  p r i n c i p l e  o f  i n c l u s i o n - e x c l u s i o n ,  C h a p t e r  5  O f  [ 4 ] ,  w e  h a v e  TT VV 1)()1( +⊆−= ∑ osT.  N o w  l e t  ni    i ∈∀=∈ ,)0,.,0,1,..,0,0(  ,  b e t h e s t a n d a r d  u n i t  v e c t o r  w i t h  i t h  c o o r d i n a t e  = 1  a n d  t h e  r e s t  0 .  F i x   a n d  l e t  ,10, ≤≤ λλ .,..,,)( n1  i   λe  S ii =∆=  N o t  t h a t  t h e  v o l u me  o f  1ni λ11)(nnλe∆ −−−= )(!)(  4 I t  i s  easy  to  see  tha t  { }λ:x∆)x,...,(x  S i1nn1i ≥∈= −  Then ,  in  th i s  case ,  we  have  { }Un1ij1nn1i  λxthatsuch j :∆)x,...,(x    S=− ≥∃∈=  { }{ } ..(2).......... λx:∆)x,...,(x  jmaxnj1nn1 ≥∈= ∈−  Now V T  =  volume of  I = volume of  I  TiiS∈ Tiiλe∈∆ )(Note  tha t  ∑∈∈≤≤∈=⊕TiniiTi λ1o(T)i.e1λo(T)iffBeλλe  Hence  V T  =  volume of  I volume of   Tii)∆(λe∈= ∑∈Tii)e∆(λ = oo(T)λ1 if o(T)(1 1)(nn 1n ≥−−−−   =  0  oo(T)λ1i ≤−+  Thus  [ ]∑ ∑⊆ =−++ −−−=−= nTm1r1nnr1rT1o(T) rλλ(11)(1)!(nnV1)(V  Where :  NI m ∈  i s  such  tha t  m = min{n,  b}  and  i s  such  tha t  NI  b ∈ 1bλ1b +<≤ .  We wr i te  ⎥⎦⎤⎢⎣⎡=λ1b  The  fo l lowing Lemma is  wel l  known,  ( see  2 .14  of [2] )  however the proof we present here is, we believe, new and is elementary depending only upon Lemma 1 and the inclusion-exclusion principle of combinatorics           5 Lemma 2  Le t  be  a  r andom po in t  i n  )x,...,x(x n1= 1n−∆ .  Then  the  p robab i l i t y  t ha t  max  { } λ≤n1 x,...,x  i s   [ ] }λ1,{nminm,rλλ(11)(mor1nnrr ⎥⎦⎤⎢⎣⎡=−−∑=− .  P roo f  Now we  have  f rom (2 )  t ha t  g iven  a  r andom po in t  ,  t he  1nn1 ∆)x(x −∈,...,probab i l i t y  t ha t  a t  l e a s t  one  o f  i t  coo rd ina t e s  i s   i s  λ≥[ ] }λ1 n, {min m , rλλ-(11)(q(λ( 1-nrnm1r1r ⎥⎦⎥⎢⎣⎢=−= ∑=+ .  Hence  t he  p robab i l i t y  t ha t  none  o f  i t s  coo rd ina t e s  i s   i s  λ≥[ ]∑=− ⎥⎦⎤⎢⎣⎡=−−=−=mor1nnrr }λ1,{nminm,rλλ(11)(q(λ(1p(λ( .  E .O.P 3 .3 Simplexes  and  Barycentr i c  Coord ina te s  Le t  be  an  a f f i ne  i ndependen t  s e t  o f  vec to r s  i n .  }V,...,V{A q1= pRIThen  they  span  a  s imp lex  ∆  o f  d i mens ion  q -1  g iven  by  (A)∑=∈∀≤≤=++=q1iiiqq11 q}i,1λ0,1λ:vλv{λ  ∆(a) ...  I f  qq11 vλ...vλ  xand∆(A)  x ++=∈  t hen   a r e ca l l ed the  ba rycen t r i c  coo rd ina t e s  o f  x .  )λ,..., (λ q1U s i n g  t h e  b a r y c e n t r i c  c o o r d i n a t e s  g i v e s  a n  a f f i n e  l i n e a r  i s o m o r p h i s m  f r o m  t o  t h e  s t a n d a r d  s i m p l e x  .  )A(∆ 1q∆ −We then  have   Lemma  3  (No ta t i on  a s  above )  Le t  x  be  a  r andom po in t  i n  )A(∆ wi th  ba rycen t r i c  coo rd ina t e s  .  Then  t he  p robab i l i t y  t ha t  max  )λ(λ q 1 ,..., λ}λ{λ q 1 ≤,...,  i s  [ ]∑=− ⎥⎦⎤⎢⎣⎡=−−m0r1qqrr }λ1,{qminm,rλλ(11)( .        6 3 . 4  C a l c u l a t i o n s  f o r  D ( S )   L e t  S  =  1nk1 ∆}x{x −⊆,...,  w h e r e  kj,)x(xx jnj1,...,j ∈=  a n d  f u r t h e r mo r e  we  a s s u me  t h a t  c ( S ) ∑−= ia1c  w h e r e  k}j:{xmina jii ∈∀= .  A l s o  w e  a s s u m e  t h a t  t h e  a s s o c i a t e d  k - p a r t i t i o n  Π  }A {A k1 ,...,=w h e r e  jj Ar = .  N o t e  t h a t  a t  l e a s t  t w o  o f  t h e  s e t s  }A{A k 1 ,...,  a r e  n o n  e m p t y .  I f  w e  l e t  a ( p ) =  n1P,)a,...,aa-,1a(a npi1p 1 ,..,,..., =+∑−  T h e n  t h e  s e t  A  =  { a ( l ) , a ( 2 ) , . . , a ( n ) }  i s  a n  a f f i n e  i n d e p e n d e n t  s e t  s p a n n i n g  t h e  s i mp l e x   w h i c h  i s  t h e   s ma l l e s t  r e g u l a r  s u b s i mp l e x  )A(∆o f   c o n t a i n i n g  S .  1n−∆L e t  ,  a n d  w e  t h e n  h a v e :  k∈∀>∈<= jAi:{a(i)σ cjj } 1 .   I n t  ∈jx )(σj2 .  S i n c e  jcj σ,2A ≥  i s  a  r e g u l a r  s i m p l e x  o f  e d g e  l e n g t h  √ 2 c ,  ( s e e  2 . 4  o f  [ l ] )  3 .  jiin1j Ai,ay)y(yy,σy ∈∀=⇒=∈ ,.., .   F i x  .  k∈jL e t  }kS1,ni1:xx{maxd sijii ≤≤≤≤−=  A l s o ,  s i n c e   w e  h a v e :  )∆(Aσx cjjj =∈a(i)λx iAijcj∑∈=  w h e r e  t h e   a r e  t h e  b a r y c e n t r i c  c o o r d i n a t e s  o f  iλjx  i n   a n d    1λi.e.),∆(AcjAijcj =∑∈cji Ai,1λ0 ∈∀≤≤          7 L e m ma  4 ( No t a t i o n  a s  a b o v e )  }Ai:maxcd jijε∈=  λ{P r o o f  We  h a v e  ∑∈=cjAiij (i)aλx .  N o w  w e  h a v e  j8j8 AS,ax ∈∀=   … … … … … … ( 3 )    ∑∑=∈−+⎥⎥⎦⎤⎢⎢⎣⎡=n1tt8Ai8ij8 )a(1λaλxcj  … … … … … … … … … ( 4 )  cj88 As,cλa ∈∀+=S i n c e  a 8  =  m i n  nS,}kt1:x{ t8 ∈≤≤  We  c l e a r l y  h a v e  k}S1,ni1:xx{maxd 8ijij ≤≤≤≤−=    }{ ni1:ax max iji ≤≤−=  ( f r o m ( 3 ) , ( 4 ) a b o v e )  }{ cj8 Asc:λ max ∈=  }{ cjS As:λ max c ∈= H e n c e  r e s u l t .  E . O . P   L e m m a  5  L e t   b e  t h e  p r o b a b i l i t y  t h a t  cλ,jP λdj ≤ . T h e n  w e  h a v e :  [ ]∑=− ⎥⎦⎥⎢⎣⎢=−−=mor1qqrrcλ,j }λc,q{minm,)cλr(11)(P .  Wh e r e  .  jrnq −= P r o o f   We  h a v e  jcj rnA −= ,  h e n c e   i s  a n  jσ 1rn j −−  s i m p l e x .  T h e  r e s u l t  t h e n  f o l l o w s  f r o m  L e m m a s  3 , 4 .  E . O . P        8  N . B .  A l l  o f  t h e  a b o v e  p r o b a b i l i t i e s  a r e  c o n d i t i o n a l  i n  t h a t  w e  a s s u m e  f i x e d  c  a n d  p a r t i t i o n  Π .  W e  l e t   b e  t h e  p r o b a b i l i t y  t h a t  d ( S )  ( )λp c,Π λ≤ g i v e n  c  =  c ( S )  a n d  t h e  p a r t i t i o n  Π .   L e m m a  6   ∏=Π =k1jcλ,jc pλp )(,   P r o o f  T h e  o n l y  c o n d i t i o n  w e  n e e d  t o  c h e c k  i s  i n d e p e n d e n c e ,  a s  i t  i s  c l e a r  t h a t  kj,λdλd(S) j ∈∀≤⇔≤ .  B u t  e a c h   i s  r a n d o m  p o i n t  i n   a n d  s o  c a n  t a k e  a n y  b a r y c e n t r i c  c o o r d i n a t e s  i n d e p e n d e n t l y  o f  t h e  o t h e r  p o i n t s .  jx jσE . O . P .  T h e  p . d . f .  f o r  c  i s  1)!(nk)!n(nkc)(1ck)!(nk 1nknnk−−−−− −−−  a s  c  i s  d i s t r i b u t e d  a s  a  b e t a  d i s t r i b u t i o n  w i t h  p a r a m e t e r s  ( n - 1 ) ( k - 1 ) ,  n  .  H e n c e  , g i v e n  t h e  p a r t i t i o n  Π ,  l e t   b e  t h e  p r o b a b i l i t y  t h a t  d ( s ) .  We  t h e n  h a v e  )(λpΠλ≤( ) dc1)!(nk)!n(nkc)(1ck)!(nkλpλP1nknnk1oc,−−−−−=−−−ΠΠ ∫)(  dcc)(λλ)(p1)!(nk)!n(nkk)!(nk  1n1ocΠ, −−−−−−= ∫  Wh e r e   a n d  ∏=Π =k1jcλ,jp   λp )(,c[ ]∑=−−−=m0r1qqrrcλ,j  rλcp ,)()1(  m  =  m i n  jrnqλcq −=⎥⎦⎥⎢⎣⎢ ,},{          9 T h e  p r o b a b i l i t y  t h a t  w e  h a v e  t h e  p a r t i t i o n  Π  i s  k)!(nk1)!r1)!....(nr(n1)!r(n)1)!((nk)!n(nk)p(k21k−−−−−−−−−−=Π  T h e  n u m b e r  o f  s u c h  p a r t i o n s  i s  .!r..!r!rn!k21 L e t   b e  t h e  p r o b a b i l i t y  t h a t  ( )λp λ  d(s) ≤  T H E O R E M  ( N o t a t i o n  a s  a b o v e )  ( ) ( ) ( )∑ ⎟⎟⎠⎞⎜⎜⎝⎛ Π= Πn k21λPrrrn!Pλp!!..!   ( )( )∑ ∫ ⎟⎟⎠⎞⎜⎜⎝⎛ −⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛= −Π−−nnn101c,k1n11ndcc1λP... rrS u m  o v e r  a l l  o r d e r e d  p a r t i t i o n s  Π  =  { r 1  , . . , r k }  o f  n  s u c h  t h a t  .k1,1nr2.nr1.lk1ll∈∀−≤=∑=  P r o o f  T h e  s e t  o f  a l l  s u c h  p a r t i t i o n s  a r e  e x c l u s i v e  a n d  e x h a u s t i v e .  T h e  R e s u l t s  f o l l o w s  a f r e r  a  s m a l l  a mo u n t  o f  a l g e b r a i c  m a n i p u l a t i o n .  E . O . P  3 . 5  E x a m p l e s  W e  c o n s i d e r  t h e  c a s e  k = 2  a n d  c o m p u t e  p ( λ ) f o r  s o m e  v a l u r s  o f  n .  L e t  e  =  ⎥⎦⎥⎢⎣⎢2n  We  h a v e  f o r  .em1,m1λ1m1 <≤≤≤+  ( ) ( )( )∑ ∫−=−∏− ⎟⎟⎠⎞⎜⎜⎝⎛ −⎟⎟⎠⎞⎜⎜⎝⎛−−⎟⎠⎞⎜⎝⎛=1n1r101nc.1n dcc1λPrn1nrnλp  Wh e r e     10 ( )( ) ( )( ) ( )( )( ) ( ) ( ) { }emwithabovetheusewee1λForo.m,1r1,rnminw,sλcsr1λpanddcc1λpλpdcc1λp1rt0asct,r1ncrnυ0tλ1tλtct,r101nc.=≤≤−−−=−⎥⎦⎤⎢⎣⎡−=−=−−=−−=+−∏∑∑∫∫ We  u s e d  R E D U C E  r u n n i n g  o n  a n  I B M  P S 2  t o  o b t a i n  t h e  c u m u l a t i ve  p r o b a b i l i t y  f u n c t i o n  p (λ )  f o r  n = 3  t o  n  =  1 O  ( a l l  f o r  k  =  2  i . e .  2  p o i n t s  i n  ) .  1n−∆S o m e  s a m p l e  r e s u l t s  a r e  s h o w n  b e l o w .   n  =  3  [  0  ,  1  ]  x 2 ( 3 x 2 - 8 X + 6 )         n  =  4  [ O ,  1 / 2 ]  -52 x 3 ( 8 8 x 3 - 2 9  x 2 + 2 2 5 X - 8 0 )  [  1 / 2 ,  1 ]  51  ( 1 6 X 6 - 1 O 8 X 5 + 2 7 O X 4 - 9 2 O X 3 + 1 8 O X 2 - 3 6 x + 3 )   4  A p p l i c a t i o n  O n e  a p p l i c a t i o n  o f  d ( S ) , w h e r e  k  =  2  , i s  g i v e n  b y  d e c i s i o n  m a k i n g  t e c h n i q u e s  u s i n g  p a i r w i s e  c o m p a r i s o n s  b e t w e e n  n  o p t i o n s .  T h e  r e s u l t i n g  d e c i s i o n  i s  t h e n  u s u a l l y  e x p r e s s e d  i n  n o r m a l i z e d  f o r m  a s  s c o r e s  ( x 1 , . . , x n )  w h e r e   .e.ini,0x,1x in1ii ∈∀≥=∑=a s  a  p o i n t  i n  ∆ n - 1  .  I f  t h e  n u m b e r  o f  o p t i o n s  i s  l a r g e  t h e n  s a m p l e s  o f  t h e  p a i r w i s e  c o mpa r i s o n s  a r e  us u a l l y  t a ke n .  T h e  r e su l t i ng  no rma l i s ed  s co re s  f r om t h e  s a m p l e  c a n  b e  c o n s i d e r e d  a s  a p p r o x i m a t i o n s  t o  t h e  s c o r e  i d e a l l y  o b t a i n e d  f r o m  u s i n g  a l l  t h e  c o m p a r i s o n s .  O n e  m e t h o d  b y  w h i c h  t he  s a m p l e  s c o r e s  c a n  b e  a s s e s s e d  i s  t o  m e a s u r e  t h e  c o n s i s t e n c y  o f  t h e  c o m p a r i s o n s  u s i n g  w e i g h t e d  t r a n s i t i v i t y .  I f  t h e  c o n s i s t e n c y  i s  g o o d  t h e n  i t  i s  r e a s o n a b l e  t o  i n f e r  t h a t  a l l  t h e  c o m p a r i s o n s  f o l l o w  f r o m  t h e  s a m p l e  a n d  t h a t     11  t h e  n o r m a l i s e d  s c o r e s  f r o m t h e  s a m p l e  a r e  r e l i a b l e .  H o w e v e r ,  i t  i s  c l e a r  t h a t  a  n o t i o n  o f  d i s t a n c e  i s  n e e d e d  w h i c h  a l l o w s  s t a t e m e n t s  t o  b e  m a d e  c o n c e r n i n g  t h e  l i n k  b e t w e e n  c o n s i s t e n c y  o f  t h e  s a m p l e  a n d  t h e  l i k e l y  c o n s e q u e n t  e r r o r .  A l s o  d i f f e r e n t  s a m p l i n g  s c h e m e s  n e e d  t o  b e  a s s e s s e d .  d ( S )  g i v e s  a  c l e a r  i d e a  o f  t h e  e r r o r  i n  t h a t  i f  S  =  { x , y }  ,  t h e n  d ( S )≤ . 0 5  m e a n s  t h a t  n o  t w o  c o m p o n e n t s  d i f f e r  m o r e  t h a n  . 0 5  o r  5 % .  S e e [ 5 ]  f o r  m o r e  d e t a i l s . .                                       12 REFERENCES  [ 1 ]  F o s t e r  ,  W. H .  a n d  C , E .  T r i p p  ( 1 9 9 0 )  A  C l u s t e r i n g  S t a t i s t i c  F o r  S e t s  o f  P o i n t s  o n  t h e  S i m p l e x .  B r u n e l  T e c h n i c a l  R e p o r t ,  M a t h e m a t i c s  D e p t . ,  B r u n e l  U n i v e r s i t y .  [2] Kendall ,  M.G. and Moran ,  P.A.P. (1963) Geometrical Probability  Griffin ,London. [ 3 ]  S o l o m o n ,  H . ( 1 9 7 8 )  G e o m e t r i c  P r o b a b i l i t y .  S o c i e t y  f o r  I n d u s t r i a l  a n d  A p p l i e d  M a t h e m a t i c s  ,  P h i l a d e l p h i a .  [ 4 ]  A n d e r s o n  ,  I .  ( 1 9 8 9 )  A  F i r s t  C o u r s e  i n  C o m b i n a t o r i a l  M a t h e m a t i c s  ,  O x f o r d  A p p l i e d  M a t he m a t i c s  a n d  C o m p u t i n g  S e r i e s  ,  O x f o r d .  [ 5 ]  F o s t e r  ,  W. H .  ( 1 9 9 0 )  S a m p l i n g  P a i r w i s e  C o m p a r i s o n s  .  I n  p r e p a r a t i o n  .                            13                    NOT TO BE REMOVED FROM THE LIBRARY            XB 2321442 2  

Metric statistics on the simplex

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