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In C h a p te r 3. a n d S e c tio n 4.5. we e sta b lis h th e fo u n d a tio n for d e riv in g p ro p e rty - b ased v a ria te p q -m e tric s th ro u g h a se t-th e o re tic a p p ro a c h . T h e c o m p le te pro cess is illus­ tr a te d in F ig u re 4.5. N o te t h a t d o m ain know ledge is o n ly re q u ire d for th e very first ste p of th e p ro cess - id e n tify in g th e in itial set o f o b se rv a b le p ro p e rtie s . In th is section, th e d e riv a tio n process is d e m o n s tr a te d in d e ta il th ro u g h a sim p le e x a m p le b a se d u p o n the six ste p s d e sc rib e d in F ig u re 4.5.

STEP 1 STEP 2 STEP 3

D o m a i n K n o w l e d g e 1 Differentiation: | define O Metric define Derivation: ; d from 0 j Metric Evaluation: compare with o

STEP 4 STEP S STEP 6

j Property Topologisation: j construct o* Metric define Derivation: j d' from o* j Metric Evaluation: compare Tj, with o*

F ig u re 4.5: T h e P rocess o f V a riate M e tric D e riv a tio n

C o n sid e r a v a r ia te X — { 9 . 1 1 .1 2 . 1 6 . 2 0 . 2 5 . 3 0 . 4 9 } . A lth o u g h th e re is a n a tu ra l o rd e r a n d m e tric on in te g e rs, let us ignore th a t a n d tr e a t X as u n o rd c re d a n d n onnietric. In fact, we d eriv e a p q -m e tric on X w hich is co m p letely d iffe re n t from th e n a tu ra l one. E ach p o sitiv e in te g e r o th e r th a n 1 can be deco m p o sed in to a p r o d u c t o f p r im e num bers. In

''U n fo rtu n a te ly , th e r e is n o precise definition for one se t to b e “sim ilar" to a n o th e r . H ere, we loosely sav two s e ts a re sim ila r if th e ir in te rs e c tio n is relatively large c o m p a re d to th e ir sizes.

(i.S

tLiis ex a m p le , we ta k e th e view th a t th e ex iste n c e o f a specific p rim e nu m b er in a n in te g e r’s p ro d u c t e x p a n s io n is a n observable p ro p e rty . T h e p rim e fa c to riz a tio n of each elem en t in .V is p re se n te d below :

9 = 32 11 = 1 1 ' 12 = 2~ 3 1 16 = 2 '

2 0 = 22 5 1 25 = 52 3 0 = 2 l 3 l 5 I 49 = 72

S T E P 1 A lth o u g h th e re are five p rim e n u m b e rs - 2. 3. 5. 7. a n d 11 th a t ex ist in th e e x p a n sio n s o f th e integers in .V. let us a ssu m e t h a t o n ly th e ex isten ce o f 2. 3. a n d 5 a re o b serv ab le o r o f in te re st. L et P>. P i- a n d P-, d e n o te th e su b se ts o f integers in X c o n ta in in g p rim e n u m b e rs 2. 3. a n d 5 respectively. N e ith e r 11 o r 4 9 has 2. 3. or 5 as p rim e facto rs. T h e fact itse lf c a n be defined as a n a d d itio n a l o b se rv a b le p ro p e rty . P),). We have

P , = { 1 2 . 1 6 .2 0 . 3 0 } . P } = { 9 .1 2 .3 0 } . P , = { 2 0 .2 5 .3 0 } . = { 1 1 .4 9 } .

L et O - { P i . P:\-Pr>. P\a) rep resen t a set o f o b se rv a b le p ro p e rtie s . N ote th a t a lth o u g h in teg ers a re u se d in A”, th e y are a c tu a lly tr e a te d like n o m in al categ o rical d a ta . O n ce we have O d efin ed , th e elem e n ts in X a r e on ly o b se rv a b le th ro u g h O . For in sta n c e , w ith

O = { Pi . P:\. Pn. Pi/)}, th e fact th a t 16 h as four 2 's in it is no lo n g er observable. From O . all

we know is t h a t 16 has a t least one 2 facto r.

T h e in tro d u c tio n o f Pg is to m ak e s u re t h a t O is a cover o f .V (i.e.. A” C ( J p - e 1

a co n v en ien t fe a tu re w hich allows e a sie r m e tric fo rm u la tio n . If P„j = 0. it ca n be safely rem oved from O . S im ilarly, if any o f th e p ro p e rtie s o f in te re st re su lts in a n e m p ty se t. it

c a n also be rem o v e d from O .

S T E P 2 A triv ia l m e tric d\ can b e defined as

d i i p . q ) = 0. i f p = r/:

d i ( p . q ) = \ 0 \ - Z o ( p . q ) . o th erw ise .

w h ere Z o ( p . q ) is th e n u m b e r of o b serv ab le p ro p e rtie s s h a re d by p .q . i.e.. Z o ( p . q ) = |{ P €

0 \ { p . q } C P \ \ . ' i p . q 6 X . It is easy to verify t h a t d \ is in d eed a m etric a c c o rd in g to

D efin itio n 3.2.4. T h e in tu itio n o f d i is t h a t th e m o re p ro p e rtie s p .q £ X sh a re , th e closer th e y are. F or in sta n c e . d i ( 1 2 .3 0 ) = 2 < d [ ( 1 2 .1 6 ) = 3 < d t ( 1 2 . 2 5 ) = 4. In T a b le 4.1. all values o f d\ o v er X a re listed.

S T E P 3 S in ce m e tric d\ satisfies M 4. T,ix = V ( X ) (see E x am ple 3.4.8). We have

dx 9 11 12 16 20 2 5 3 0 4 9 9 0 4 3 4 4 4 3 4 11 4 0 4 4 4 4 4 3 12 3 4 0 3 3 4 2 4 1 6 4 4 3 0 3 4 3 4 2 0 4 4 3 3 0 3 2 4 2 5 4 4 4 4 3 0 3 4 3 0 3 4 2 3 2 3 0 4 4 9 4 3 4 4 4 4 4 0

T a b le 4.1: D ista n c e s S pecified by d\ over X

is g en eral e n o u g h t h a t for a n a r b itr a r y se t X a n d O C V ( X ) su c h th a t P$ G O. we have

77/, = V { X ) a n d 77/, h as e v e ry p ro p e rty as a n op en . It c a n b e seen from th e fo rm u la tio n that. d[ is p o p u la tio n -in d e p e n d e n t. T h e o b je c tiv e o f su b se q u e n t ste p s is to d eriv e a p o p u la tio n - d e p e n d e n t m etric.

S T E P 4 S ince (P> U P \ U P.x) D P$ = 0. th e to p o lo g iz a tio n o f O ca n b e m a d e e a sie r In- d iv id in g O in to 0 \ = {Ptf} a n d Oo = { P>• P.\-Pn\- a n d to p o lo g iz in g s e p a ra te ly 0 \ a n d 0>- A fter 0 \ a n d O ' a re d e riv e d , from th e d e fin itio n o f to p o lo g y we have 0 ' = {.4[ U -4-j|.4i G

0 \ . A ± G O ' } . W e have O* = {0. { 1 1 .4 9 } } a n d 0 * c o n sistin g o f th e follow ing 14 o p e n sets:

0

{30} { 1 2 .3 0 } { 2 0 .3 0 } { 9 . 1 2 .3 0 } { 1 2 .2 0 .3 0 } { 2 0 . 2 5 .3 0 } { 9 . 1 2 . 2 0 . 3 0 } { 1 2 . 1 6 ,2 0 . 3 0 } { 1 2 . 2 0 .2 5 . 3 0 } { 1 2 . 1 6 . 2 0 . 2 5 . 3 0 } { 9 . 1 2 . 2 0 . 2 5 . 3 0 } { 9 . 1 2 . 1 6 . 2 0 . 3 0 } { 9 . 1 2 . 1 6 . 2 0 . 2 5 . 3 0 }

T h u s O ' has 28 o p e n s e ts in t o ta l w hich in clu d e all th e m e m b ers in O l a n d th e u n io n of { 1 1 .4 9 } w ith ea ch e le m e n t in O ',. E ach o p e n set c o rre sp o n d s to a n o b se rv a b le p ro p e rty . For in sta n c e . { 9 .1 2 . 2 0 . 3 0 } = P j U ( P‘> n P^ ) c o rre sp o n d s to th e p ro p e rty P:i V ( P , A A ,). S T E P 5 T h e re m ig h t o r m ig h t not b e a p q -m e tric (see S e c tio n 3.4) w hich c a n induce

O * . F or p .q G A'. let us d efin e a p q -m e tric d-> o n X as d-,(p .q ) = Z o - { p - p ) ~ Z o - {p. q).

w here Z o ' ( p - q ) is th e n u m b e r o f o p e n s in O* c o n ta in in g b o th p a n d q (th e sa m e fu n c tio n

Z used in th e d e fin itio n o f d \ ) . C learly. d-, (p. p) — 0. Vp G X . It sh o u ld b e o b v io u s th a t d.

is n o t s y m m e tric a l th o u g h Z o -{ p - q) = Z o * (q .p ).V p .< i 6 X . In T a b le s 4.2 a n d 4.3. we list a ll Z o - a n d do v a lu e s o v er A'. For p. q. r € X . we have

d o ( p . r ) + d o (r. q) - do {p. q) = Z o - ( r . r ) + Z o - ( p . q ) — Z o - ( p - r ) - Z 0 - { r .q ) > I).

a n d th u s do s a tisfie s th e tria n g le inequality. It sh o u ld b e n o te d th a t 6 0 fissures

d>(p. q) ^ 0 for p € Pin a n d q € X — P^. Z o - 9 11 12 16 20 25 3 0 4 9 9 10 5 10 4 8 4 10 5 11 5 14 10 4 10 5 13 14 12 10 10 20 8 16 8 20 10 1 6 4 4 8 8 8 4 8 4 2 0 8 10 16 8 20 10 20 10 2 5 4 5 8 4 10 10 10 5 3 0 10 13 20 8 20 10 26 13 4 9 5 14 10 4 10 5 13 14 T able 4.2: V alues o f Z o - do 9 11 12 16 20 25 3 0 4 9 9 0 5 0 6 2 6 0 5 11 9 0 4 10 4 9 1 0 12 10 10 0 12 4 12 0 10 1 6 4 4 0 0 0 4 0 4 2 0 12 10 4 12 0 10 0 10 2 5 6 5 2 6 0 0 0 5 3 0 16 13 6 18 6 16 0 13 4 9 9 0 4 10 4 9 1 0 Table 4.3: V alues o f d o

T h e p q - m e tr ic d o c a n b e in te rp re te d in several d iffe re n t w ays, d e p e n d in g on how wo choose to in te r p r e t t h e c o lle c tio n o f o bservable p ro p e rtie s O . S o m e possib le in te rp re ta tio n s in clu d e:

1. L et us look a t a ll th e p o in t p q -m etric s • • • • <^2 |49 - w h ich co rresp o n d to rows in th e ta b le o f d o : c le a rly 3 0 is th e m ost d is ta n t fro m a ll th e o th e rs. T h is is because? 3 0 has a ra r e q u a lity - b e in g th e o n ly n u m b e r h a v in g a ll th r e e p ro p e rtie s P>. P\ . a n d

Pr,. O n th e o t h e r e n d o f th e sp e c tru m . 16 is th e c lo se st to all th e o th e rs. T h is is

2. It is easy to see t h a t //-^g ~ ck \2 5 (he., t/^lg a n d d oI25 give the ex a ct sam e co llec tio n o f d is ta n c e a ssig n m e n ts, only in differen t p e r m u ta tio n s ). T h is tells us th a t th e se ts o f p ro p e rtie s p o ssessed by 9 a n d 25 have s im ila r o c c u rre n c e frequencies in th e v a ria te p o p u la tio n , a lth o u g h they m ight b e very d iffe ren t. T h e sam e th in g c a n b e sa id for d o

1

12 ~ d >120• T h e o b serv atio n . d o | 11 = d o | 4 9. in d ic a te s t h a t 11 a n d 4 9 a re in d is tin ­

g u ish a b le in O . F ro m a n o th e r p e rsp e c tiv e , p q -m e tric d o is oidy sym m etric: on p a irs ( 9 . 2 5 ) . ( 1 2 . 2 0 ) . a n d ( 1 1 .4 9 ) .

3. T ak in g d oI20 for ex a m p le , we have c/> (20.3 0 ) = 0. since for all th e o b serv ab le p ro p ­ e rtie s in O (also O ' ) . th e ex isten c e o f 2 0 alw a y s im plies 3 0 . In o th e r w ords. 3 0 h as all th e o b se rv a b le p ro p e rtie s 2 0 possesses. d o (2 0 . 1 2 ) = 4 tells us t h a t 2 0 a n d 12

• a re likely to c lu s te r to g eth er, since r7> (20.30) = d-.>(12.30) = 0. F u rth e r. 2 5 is a b it

I

{ closer to 2 0 t h a n 9 a n d 16. b ec au se d>(2 5 . 2 0 ) = 0.

T h e re ex ist m a n y o th e r in te rp re ta tio n s as well. T h e im p o r ta n t th in g is th a t do is p o p u la tio n - d e p e n d e n t. F or in sta n c e , a d d in g elem e n ts to X o r rem o v in g elem ents from A' m ay affect, th e d is ta n c e m e a s u re b etw e en two o rig in al e lem e n ts.

? ? 1

S T E P 6 N ow . we w a n t to ask: How good is do in m o d elin g (o r describing) O ' ? To a n sw e r th is q u e stio n , we have to co m p are 77/.,. th e to p o lo g y in d u c e d by do. w ith O '. L et S t>. Vp 6 X re p re se n t th e s m a lle s t o f all possible sp h eres S ({.,(p. s ) c e n te re d a t p (i.e.. {r/| d o (p .q ) < 5 } as

C — ► 0 ) .

So = { 9 .1 2 .3 0 } S n = { 1 1 .4 9 }

| S 12 = { 1 2 .3 0 } S ie = { 1 2 .1 6 .2 0 .3 0 }

j S 20 = { 2 0 .3 0 } S 2 5 = { 2 0 .2 5 .3 0 }

S 30 = {30} 5 49 = { 1 1 .4 9 }

I

U sing th e sp h e re s. 77/._, c a n b e easily d eriv ed . If d o is re a so n a b ly well defined. 77/. w ould n o t

b e fa r d iffe ren t from O ' . It is not h a rd to see t h a t 77/_, = O ' a n d . therefore, d o is a n o p tim a l p q -m e tric o f O ' .

In th e a b o v e ex a m p le , o th e r o b serv ab le p ro p e rtie s c a n be defined to describe: th e v a ria te values m o re precisely. For in stan ce , if we p u t a new value 8 into X . th e re is 110 w ay to d is tin g u is h 8 from 16 u sin g O - {P>. P ;{. Pt,. P«} o r O ' . W ith a m ore co m p reh en siv e O . we

c a n d eriv e p q -m e tric s b a se d on su b tle r c h a ra c te ris tic s , su c h as th e g re a te st co m m o n d iv iso r o f tw o e le m e n ts - th e g re a te r the gcd. th e s h o rte r th e d ista n c e . In fact, in th e m e tric -d riv e n know ledge d isco v ery pro cess, the fo rm u latio n o f th e set o f observable p ro p e rtie s m ight b e

p ro g ressiv ely refined b ase d on d om ain know ledge a n d th e v a lid ity o f th e m e tric derived (see F ig u re 4.5).

P ro p e r ty to p o lo g iz a tio n a p p e a rs to b e a v iab le a p p r o a c h to d eriv e p q -m e tric s from u n o rd e re d a n d n o n m e tric v a ria te dom ains, su c h as n o m in a l c a te g o ric a l d a ta . Even th o u g h d o m a in know ledge is re q u ire d for identifying th e o rig in a l o b se rv a b le p ro p e rtie s, th e to p o l­ o g iz a tio n p ro cess itse lf (d e riv in g O* from O ) c a n b e d o n e a u to m a tic a lly w ith o u t d o m a in know ledge a n d h u m a n in terv e n tio n .

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