Comparison of Simple Indicator Kriging, DMPE, Full MV Approach for Categorical Random Variable Simulation

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1 Papr 17, CCG Aual Rport 11, 29 ( 29) Compariso of Simpl Idicator rigig, DMPE, Full MV Approach for Catgorical Radom Variabl Simulatio Yupg Li ad Clayto V. Dutsch Ifrc of coditioal probabilitis at usampld locatios is a critical problm i gostatistics. I th cotxt of catgorical variabls, th probabilitis of ach catgory could b calculatd by idicator rigig or multipl poit statistics - two matur coditioal probability stimatio approachs. Ths tchiqus ar compard with th wly dvlopd Dirct Multivariat Probability Estimatio tchiqu. Th compariso is basd o th iformatio providd by th diffrt tchiqus. Th iformativ strgth fuctio is usd as a quatitativ ucrtaity assssmt to th stimatio rsults. Th iformativ strgth fuctio masurs how our ucrtaity is rducd ad how th usampld locatio gais iformatio from rlatd ighbourig locatios. Itroductio Catgorical variabls such as facis or roc typs usually rflct th origi of a roc uit. Although, th tru facis for a spcific usampld locatio is uiqu, it is usually viwd as a radom variabl wh th tru facis o som locatio is iaccssibl. Th paradigm of gostatistics is to charactriz ay facis or roc typs for usampld locatio as a catgorical radom variabls which ar dfid as th st of possibl valus or stats that ca ta ovr th study ara or at ay particular locatio. Usually, th radom variabls ar locatio dpdt ad doatd as Z ( u ), th uppr-cas lttr such as Z ( u) will rfr to a radom variabl at locatio u. Th st of possibl outcoms of th radom variabl is dotd as{, = 1, 2,..., }. Th, th probability of o catgory to xist at locatio uis xprssd as a probability: p( u; ) = prob{ Z( u ) = } (1) Th aim is to giv a mor accuracy ad prcis stimatio for p( u ; ). For xampl, giv catgory umbr with o othr prior iformatio, w ca us th uiform distributio (last iformatio probability) as th stimatio, which will b: p( u ; ) = 1. Mor practically, w hav som global ad dotd as: iformatio for th probability of th catgory to prvail at locatio u, will b p( u; ) p( u ; s ) = p( ). If mor iformatio ar giv, such as th outcoms of th rlatd surroudig ighborig locatios, dotd as { Z( u ) = z, = 1,..., } or {( )} α α α, th probability distributio fuctio as xprssd i (1) will b updatd to a postrior probability, which is calld coditioal probability ad ca b writt as: p( u u, u,... u ) = prob{ Z( u ) = z Z( u ) = z, = 1,..., } (2) 1 2 α α α { z, z,..., z } ar th outcoms of th radom variabls I quatio(2), th lowr cas 1 { Z( u ) = z, =,1,..., }, which ar z [1,2,... ], = 1,...,. α α α α Basd o th coditioal probability dfiitio, th probability P( u u1, u2,..., u) ca b calculatd from th multivariat probability as: P( u, u1, u2,..., u ) P( u u1, u2,..., u ) = (3) P( u1,..., u ) I gostatistics, charactrizatio of ucrtaity about a spatially distributd phomo o th usampld locatio is do through coditioal simulatio o th uivariat cumulativ distributio fuctio as i quatio(2). Mor ovr from th viw of iformatio gaiig, w ar mor itrstd i stimatig th coditioal probability basd o th ighborig data as xprssd i quatio (2) bcaus it will provid mor iformativ stimatio for spcific locatio aftr obtaiig som surroudig α 17-1

2 Papr 17, CCG Aual Rport 11, 29 ( 29) locatios iformatio. Thus, th ifrc of this coditioal probability distributio is th ctral problm of gostatistics. Thr ar svral stablishd ad w dvlopmt approachs to ifr th coditioal probability dirctly or idirctly, such as idicator rigig or multipl poit gostatistics. I idicator rigig approach (Goovarts, 1994; Jourl, 1983), th coditioal probability multivariat probability is calculatd as a liar combiatio of idicator data: P( u = z u = z,..., u = z ) = λ P( u = z u = z ) (4) 1 1 i i i i= 1 Th wights charactriz th dpdc btw th sampld locatios uiad th iformatio from sampld locatios to usampld locatio. Th wights λi ar stimatd by idicator rigig (I) basd o th idicator covariac. Thortically, th coditioal probability ca b calculatd from quatio(3), th umrator is a +1 variats multivariat probability, whil th domiator is th -variats margial probability of this -variats multivariat probability. I th multipoit statistics approach, th rquird +1 variat multivariat probability is costructd from a traiig imag. Th, th rquird coditioal probability i xprssio (3) ca b calculatd by just coutig th rlativly stats umbr occurrd i th traiig imag: Cout( u = z, u1 = z1,..., u = z ) P( u = z u1 = z1,..., u = z} = (5) Cout( u = z,..., u = z ) 1 1 Th w dvlopd DMPE approach stimat th coditioal probability by stimat th multivariat probability, which is th umrator i quatio(3), from th bivariat margial. I this approach, i this algorithm, th bivariat margial probabilitis ar imposd to a iitial multivariat probability as th costraits ad satisfid by itrativly modifyig th multivariat probability. Ad also i DMPE, th lowr margial probabilitis ar usd as a tool to charactriz th iformatio ad th rdudacy btw th sampld locatios ad th usampld locatio. Th thory of all ths thr mthods is ot th mai poit of this papr. Th dtails of idicator rigig approach wr writt i dtails i papr (Dutsch, 26; Goovarts, 1994; Jourl, 1983). Th multipl-poit approach ca b foud i(liu, 26; Ortiz ad Dutsch, 24; Strbll, 22; Wag, 1996). Th dtails of th w dvlopd DMPE approach was giv i th papr i this rport. I this papr, th stimatio rsults wr compard from th iformativ poit btw thos two matur approachs ad th wly dvlopd approach DMPE (Dirct Multivariat Probability Estimatio). Spatial statistics for thos thr mthods I this rsarch, th mai poit is giv a objctiv compariso with th aim of algorithm improvmt. To ma a vry good stimatio, thr ar may practical dtails for ayo of thm. I ordr to xclud th practical issu such as variogram modl costructio, all th spatial statistics ar ifrrd from th sam multivariat probability that is obtaid by scaig th traiig imag with crtai data cofiguratio. Multivariat probability from traiig imag Giv catgorical radom variabls, ad assumig ach of thm may hav catgoris { = 1, 2,..., } outcoms, ths radom variabls will form a multivariat probability P ( u, u,..., u ) which is dfid as: mv 1 2 P ( u,..., u ) = prob( Z( u ),..., Z( u )); α = 1,..., (6) mv 1 1 Latr, th otatio of Pmv ( u1, u2,..., u ) will rfr to th multivariat probability mass fuctio, whil p ( z, z,..., z ) will rfr to o probability stat, which rprsts th probability of a spcific mv 1 2 cofiguratio of catgory zα = with ( = 1,...,, α = 1,..., ) xistig at locatios u1, u2,..., u. Totally, th probability stats umbr will b. I this rsarch, th origial multivariat probability is obtaid 17-2

3 Papr 17, CCG Aual Rport 11, 29 ( 29) from scaig a traiig imag with crtai data cofiguratio. Th traiig imag ca b ay ids of catgorical variabls distributio map which ca rprst th htrogitis charactrs i th spatial spac that th gologist xpct to s i th rsarch ara. It is usd most oft i multipl poit statistics. From this traiig imag, udr th statioary assumptio, th poit statistics will b rtrivd by scaig th data vt from th traiig imag. Cout( u1 = z1, u2 = z2,..., u = z ) P( u1 = z1, u 2 = z2,..., u = z } = (7) Cout( u1, u 2,..., u ) Th xprssio(7) charactrizs th joit ucrtaity about th actual valus z 1,... z. It should satisfy th costrait of: pmv ( z1, z2 Trasitio Probability from multivariat probability owig th multivariat probability, th rlatd ay lowr ordr margial probability ca b calculatd through th multivariat probability margializatio. For xampl, dot th variat multivariat probability as P( u l = z l, l = 1,...., ), th scod ordr margial or bivariat margial probability P( u, u ; z, z ), i, j = 1, 2,...,, i j ca b calculatd from th multivariat probability as: P( u, u ; z, z ) = P( = z ; l = 1,...,, z = 1,..., ) i j i j,..., z ) [,1] ad i j i j Alll with ui = zi & u j = z j ; i j pmv z1 z2 z = 1 u l l (,,..., ) 1 Th bivariat margial probability satisfy P( ui, u j; zi, z j ) 1ad P( ui, u j; zi, z j ) = 1. For ay two radom variabls (uu i, u j; i, j = 1,...,, i j), this bivariat probability will b a matrix which is also calld trasitio probability matrix. For xampl, if thr ar 3 locatios with possibl 3 catgoris i all th locatios, thr will b 27 possibl multivariat stats 3 trasitio probability matrics as show i Figur 1.. l zi = 1 z j = 1., ) (8) Figur 1 Diagram of multivariat probability ad th rlatd bivariat margial probability (All th multivariat probability stats ar show as a gr poit i th spac. Th bivariat probabilitis btw radom variabl u2 ad u3 ar show as a rd ad thos btw u1 ad u3 ar show as blu circl, th bivariat btw u1 ad u2 is ot show i this figur). P ( u, u,..., u ), thr ar totally For th multivariat probability distributio MV 1 2 possibl valus. 1 Assumig thr facis ca b xit at 1 grid locatios, th probability spatial spac will b 3, which is a hug umbr. Thus, it is a big challg to sav ad rtriv it i th stimatio procss. Whil, th lowr ordr margial distributio has a lowr statioary rquirmt ad ca b asily obtaid. I DMPE 17-3

4 Papr 17, CCG Aual Rport 11, 29 ( 29) approach, th trasitio probability is usd to rcostruct th origial multivariat probability. Thus, th coditioal probability ca b calculatd dirctly from quatio(3). Covariac from Trasitio probability I idicator rigig approach, idicator covariac, which is a bivariat statistics, is usd to charactriz th spatial rlatioship. Whil i DMPE, th trasitio probability is usd. I this part, it will show that th covariac ca b ifrrd from th trasitio probability with som simplicity assumptio. This poit will sur that th diffrcs of coditioal probability stimatio for usampld locatio oly com from th algorithm itslf. I idicator rigig approach, a catgorical radom variabl is always trasformd to a biary idicator variabl. For a catgorical variabl Z(u), th idicator variabl for catgory is dfid as: 1, if xist at locatio u I ( u ; ) = = 1,..., (9), othrwis Th idicator covariac modl is usually giv by idicator variogram modl, which is a masur of th spatial corrlatio btw vry two locatios. I statioary assumptio, th idicator variogram I ( ; ) γ h for a idicator variabl at two locatios with dpartur distac of h is dfid as: { 2 } ( ) [ ] 2 γ I h; = E I( u; ) I( u + h ; ), = 1,..., (1) From idicator variogram calculatio as i quatio(1), th idicator variogram of catgory will oly cout thos catgorical trasitios that catgory chags to othr catgoris from th o locatio to th othr locatio. From prvious sssio, th bivariat trasitio probability btw ay two catgoris is dfid as: p h ;, ' ;, ' = 1,..., ( ) Thus, for ach idicator variogram, it summarizs ths 2(-1) diffrt bivariat trasitio probabilitis (Dutsch, 25). ( ) = p ( ) + p ( ) 2 γ h ; h ;, ' h ; ', (11) ' = 1 ' = 1 ' ' For xampl, for thr catgoris ( = 1, 2,3), th idicator variogram ad th trasitio probability hav th rlatioship as: 2 γ ( h;1) = p( h;1, 2) + p( h;1,3) + p( h;2,1) + p( h;3,1) 2 γ ( h;2) = p( h;2,1) + p( h;2,3) + p( h;1, 2) + p( h;3, 2) (12) 2 γ ( h;3) = p( h;3,1) + p( h;3, 2) + p( h;1,3) + p( h;2,3) With th asymmtric assumptio ( p( h;1, 2) = p( h ;2,1) ad p( h;1,3) = p( h ;3,1) ) ad th quality of trasitio probability matrix ( p( h;1,1) + p( h;1,2) + p( h ;1,3) = p1 ): γ ( h;1) = p( h;1,2) + p( h;1,3) = p(1) p( h;1,1) γ ( h;2) = p( h;2,1) + p( h;2,3) = p(2) p( h;2,2) (13) γ ( h;3) = p( h;1,3) + p( h;2,3) = p(3) p( h;3,3) Whr p1 is th uivariat probability for catgory o. Basd o th rlatioship of C1 ( h) = C1 () γ1( h ), from quatio(12) ad(13), th idicator covariac for catgory 1 ca b calculatd from th trasitio probability as: C1 ( h) = C1() p1 + p11 ( h) = p (1 p ) p + p ( h) (14) = p11 ( h) p1i p1 It is th sam for catgory two ad thr. Th idicator covariac ca b calculatd from quatio(14) giv a trasitio probability matrix, ad usd i th simpl idicator rigig dirctly. 17-4

5 Papr 17, CCG Aual Rport 11, 29 ( 29) Th compariso critria Th coditioal probability p( u u1, u2,... u) for a giv data vt ca b calculatd from full multivariat probability, DMPE or from idicator rigig. For compariso purposs, it is hlpful to hav a quatitativ masurmt to say how iformativ of a stimatio is giv th coditioig data. I othr words, it is cssary to quatitativly valuat how much ucrtaity has b rducd rgardig of this stimatio for th usampld locatio? Th iformatio cott is rlatd to th coditioal probability p( u; u1, u2,... u ). Say, if p( ;,,... ) = 1 u u1 u2 u, it is crtai that at locatio surroudig locatios 1 2 u is catgory giv th catgoris at th ( u, u,... u ), thrfor this stimatio is vry iformativ o usampld locatio. Similarly for th cas wh 1 2 p( u ; u, u,... u ) =. it is crtai that is ot goig to happ giv th situatio at th surroudig locatios, hc 1 2 stimatio o usampld locatio. Covrsly, if 1 2 p( u ; u, u,... u ) = is also vry iformativ p( u ; u, u,... u ) 1 = it is ot crtai which catgory is goig to happ o usampld locatio, hc th iformatio cott of this stimatio rachs th miimum. I mor practical cas, w hav som prior iformatio about catgory, say, its global proportio p( u; s ) = p( ) p( u ; u, u,... u ) = p( ) (1). ω [,1] ad ω wh p 1 2 th th lowst iformativ stimatio should b shiftd th poit whr 1 2 Basd o this udrstadig, w ca ma a gralizd dfiitio of iformativ strgth fuctio ω for th stimatio satisfyig th followig coditios(liu, 25): ω is a fuctio of stimatioω = f { p( u; u1, u2,... u )}; ( u ; u, u,... u) is most iformativ ( p( u; u1, u2,... u ) 1or ); ω wh B is ot iformativ ( p( u 1 ; u1, u2,... u ) or p( ) ); ω dcras withi [, 1 ] or [ p( )] ad icras withi[ 1,1] or [ p( ),1] (2). 1 (3). (4). moroically; Liar iformativ strgth dfiitio Giv sampld aroud locatios, th postrior probability (th stimatd coditioal probability distributio for usampld locatio) will b p( u; u1, u2,... u ), which will brig mor iformatio rgardig to th prior probability p( )(Global proportio). Th iformativ strgth ω ca b dfid as a liar fuctio (Liu, 25): p( u; u1, u2,... u) p( ) if p( u; u1, u2,... u) p( ) 1 p( ) ω( u; u1, u2,... u) = (15) p( ) p( u; u1, u2,... u d) if p( u; u1, u2,... u) < p( ) p( ) Figur 2 is a xampl of iformativ strgth fuctio giv diffrt prior probabilitis. 17-5

6 Papr 17, CCG Aual Rport 11, 29 ( 29) Figur 2 liar iformativ strgth fuctio giv thr diffrt uivariat margial probabilitis o-liar iformativ strgth dfiitio I th iformatio thory(covr ad Thomas, 26), th ucrtaity of owldg is masurd by tropy H ( X ). Th tropy of a radom variabl X with a probability mass fuctio p( x ) is dfid as: H ( X ) = p( x) log( p( x)) (16) x Th mor uprdictabl, th highr it s tropy. Th tropy rachs its maximum valu wh X is uiformly distributd, corrspodig to miimum iformativ stimatio. It rachs th miimum valu wh thr is o ucrtaity about X, i.. X happs with probability 1 or. Cosidr th catgorical radom variabl uis dfid o a spatial domai such that all possibl outcoms of ( ; = 1,..., ) is th qual probability p( u 1 ; u1, u2,... u ) =. Th tropy H ( ) u will b: 1 1 H( u) = l( ) = 1 1 = l( ) (17) = l With ths uiform probabilitis, H ( u) is th uppr boud for th avrag tropy with lowst p( u ; u, u,... u ) o th usampld locatio. Thus, th uiform probability iformativ stimatio 1 2 is calld last iformativ probability. Practically, o mor iformatio rsourc is th global proportio of ach catgory p( ), = 1,...,. Basd o this th avrag tropy would b: H ( u) p( ) i l p( ) (18) = Aftr som coditioig data ar obtaid, th stimatio for usampld locatio is updatd from th global proportio to a postrior probability distributio dotd as 1 2 coditioal probability distributio for o usampld locatio, th local tropy would b: = 1 p( u; u, u,... u ), for th H ( u u, u,... u ) p( u; u, u,... u ) i l( p( u; u, u,... u ) (19) = = 1 This w tropy H( u u1, u2,... u ) will always b lss tha H( u ), says that owig mor iformativ ca oly rduc th ucrtaity. That is: Iformatio ca t hurt (Covr ad Thomas, 26). But it is still d to say how much of ucrtaity is rducd aftr gaiig th w iformatio rsourc. I Baysia statistics th L divrgc is usd as a masur of th iformatio gai i movig from a prior distributio to a postrior distributio. L divrgc also calld rlativ tropy or ullbac-liblr distac (L distac). For a radom variabl X, with th probability mass fuctios p( x) ad q( x ), L distac is dfid as: 17-6

7 Papr 17, CCG Aual Rport 11, 29 ( 29) p( x) D( p q) = p( x)l (2) x q ( x ) I th abov dfiitio, i l( ) =, il( ) = ad p( x ) i l( p( x) ) =. q( x) It is show that D( p q) is always ogativ ad is zro if ad oly if p( x) = q( x). I iformatio thory, L distac is usd to say how may xpctd umbr of bits would hav addd to th mssag lgth by usig th origial cod basd o q( x ) istad of usig a w cods basd o th p( x )(Covr ad Thomas, 26). If th p( x) i quatio(2) is th coditioal probability p( u; u1, u2,... u) ad q( x) is th prior probability p( ), this thrfor rprsts th amout of usful iformatio, or iformatio gaiig, about locatio u aftr upgradig th stimatio from th global proportio to coditioig probability. Th L distac will b: p( u ; u,..., u ) 1 ( ( u; u1,..., u ) (1 )) = ( u; u1,..., u)l{ } = 1 (1 ) D p p = { p( u; u1,..., u )l{ p( u; u1,..., u )} p( u; u1,..., u )l(1 )} (21) = p( u ; u,..., u )l( ) + { p( u ; u,..., u )l( p( u ; u,..., u ))} = l( ) + { p( u ; u,..., u )l( p( u ; u,..., u ))} = H ( u) H ( u u,..., u ) Th largr th distac is, th bttr th iformativ, th gratr our ucrtaity rducd o th usampld locatio. So th L distac D ca b usd as a masur to show how th iformativ th stimatio is aftr upgradig from a prior probability to postrior probability. Th lowr boud will b, wh usig th last iformativ probability as th stimatio. Th maximum will b 1 wh thr is o ucrtaity for ach catgory. It is itrstig to show that th distac to th last iformativ probability distributios for biary radom variabl. At this cas, lt th logarithms to bas 2, th quatio(21) will b: D( p( u ; u... u ) (1 )) = log (2) H ( u u,..., u ) = 1 + p( u; u1,..., u)log 2( p( u; u1,..., u)) (22) + (1 p( u; u1,..., u))log 2(1 p( u; u1,..., u)) Th abov distac satisfis th prvious iformativ strgth fuctioω rquirmts. Wh th probability distributio p( u; u1,..., u ) approachs to th uiform distributio, th iformativ strgth is dcrasig, ad it icrass wh p( u; u1,..., u) is farthr away from th uiform distributio. O both sids, th iformativ strgth ω mootoically icrass or dcrass. For biary variabl, wh th prior global probability p( ), = 1,..., is giv, th lowst poit i th iformativ strgth fuctio should b p( ), = 1,...,, th quatio (22) will b modifid as: D( p '( u; u1,..., u ) p( )) = 1 + p '( u; u1,..., u )log 2( p '( u; u1,..., u)) (23) + (1 p '( u ; u,..., u ))log (1 p '( u ; u,..., u )) With p( u; u1,..., u) if p '( u; u1,..., u ) ( ) 2 ( ) p p p '( u; u1,..., u) = ( p( u; u1,..., u) 2 p( ) + 1) if p( u; u1,..., u ) ( ) 2(1 ( )) > p p Figur 3 ar som o-liar iformativ strgth fuctio curvs. 17-7

8 Papr 17, CCG Aual Rport 11, 29 ( 29) Figur 3 o-liar iformativ strgth curvs giv diffrt global margial probabilitis (Lft: th biary variabl without owig th uivariat probability; Right: aftr obtaiig th uivariat probability, th lowst iformativ strgth should shift to th uivariat margial) Thr ar also som othr stablishd approach approachs to compar modls, such as cross validatio ad jacif. I cross validatio, data ar lft out o at a tim ad r-stimatd from th surroudig data. I th jacif, a sparat st of validatio data ar hld bac from th vry bgiig ad oly usd at th d for chcig. Mor dtails ca b foud i (Dutsch, 1999; Goovarts, 1997). Compariso mthods ad compariso rsults Bcaus th data cofiguratio will hav a svr ffct o th stimatio rsults, all fforts should b ta to ma sur our compariso is rprstativ. Basd o th prvious sssio th followig compariso worflow as show i Figur 4 is adoptd. Multivariat probability distributio Trasitio probability Matrix Idicator covariac Traiig imag Estimatd Multivariat probability(dmpe) Idicator rigig Estimatd Coditioal probability p ( MV u... ) u1 u5 Estimatd Coditioal probability p ( DMPE u... ) u1 u5 Estimatd Coditioal probability p ( I u... ) u1 u5 Data cofiguratio Figur 4 thr algorithm compariso worflow Th data cofiguratio as show i Figur 4 is composd by 6 locatios. Sampld data locatios ar u, 1 u, 2 u 3, u4adu 5. Th locatio u is usampld ad movig i th data cofiguratio widow (1 by 1 grids). Totally, thr will b 95 data cofiguratios compariso. Th first stp is multivariat probability costructio which is basd o quatio(7). Alog th rd arrow, th trasitio probability is calculatd accordig quatio(8). For idicator rigig approach, th idicator covariac is comig from trasitio probability basd o(11). Th first compariso could b usig th traditioal cross validat approach. Usig 5 locatios i th traiig imag as th crosschc locatio, th accuracy plots with thos thr approachs ar plottd i Figur 5. It shows that th DMPE stimatio is bttr tha S approach for dpictig th htrogity of this traiig imag. 17-8

9 Papr 17, CCG Aual Rport 11, 29 ( 29) Figur 5 th accplt with diffrt stimatio approach From th viw of ucrtaity, this improvmt ca b quatitivly masurd. Th liar fuctio of th iformativ strgthω for ach catgory, = 1,2 ad 3is plottd i Figur 6. As show i Figur 6, th iformativ strgth ordr for catgory 1 ad 3 is ω ( MV ) > ω ( DMPE) > ω ( I) ; for catgory 2 is ω ( MV ) > ω ( I) > ω ( DMPE). Figur 6 Li Iformativ Strgth comparisos for thr catgoris with diffrt stimatio approachs Th iformativ strgth masurd by th o-liar approach is show i Figur 7. Figur 7 th o-liar ucrtaity dcrasig usig diffrt stimatio approachs (Lft: tropy; Right: L distac) From th L distac, w ca calculat how th stimatio is improvd from th viw of ucrtaity rductio. Th maximum improvmt of iformativ strgth ω ( DMPE) ω ( I) is 1.6%, th 17-9

10 Papr 17, CCG Aual Rport 11, 29 ( 29) miimum of ω ( DMPE) ω ( I) is 4.5%; o avrag, it is 6.4%. From th iformativ strgth critria, th stimatio improvmt o th ucrtaity rductio from DMPE is largr tha I, which will courag a widly applicatio of this w algorithm i th futur. Discussio Although th multipl poit approach has th bst iformativ strgth, it is difficult to gt ough rplicatios for stabl highr ordr multivariat probabilitis bcaus of th hug stat spac. Th statioary assumptio i multipl poit gostatistics may also caus som difficulty. For I approach, th liar probability combiatio approach may ot rsult i modls that show th appropriat lvl of spatial dtail as th rsults show i this rsarch. It has th miimum iformativ strgth to th usampld locatio. Th advatag is its computatioal spd. Whil for DMPE, th lowr margial probabilitis i DMPE will hav miimal statioary rquirmt ad th oliar combiatio will b mor appropriat i may cass. It will giv bttr iformativ stimatio tha I approach. Th spatial statistics tool i DMPE is th trasitio probability which could b obtaid from a traiig imag or vrtical profil. Ifrc of thr dimsioal trasitio probabilitis is problmatic ad CPU rquirmts ar larg. Rfrc Covr, T.M., ad Thomas, J.A., 26, Elmt of Iformatio Thory, Joh Wily & Sos, Ic. Dutsch, C.V., 1999, A short ot o Cross VAlidatio of Facis Simulatio Mthods, i Dutsch, C.V., d., Ctr of Computatioal Gostatistics Aual Rport, Volum 1: Edmoto, Albrta., 26, A squtial idicator simulatio program for catgorical variabls with poit ad bloc data: BlocSIS: Computrs & Goscics, v. 32, p Goovarts, P., 1994, Comparativ prformac of idicator algorithms for modlig coditioal probability distributio fuctios: Mathmatical Gology, v. 26, p , 1997, Gostatistics for atural Rsourcs Evaluatio, Oxford Uivrsity Prss. Jourl, A., 1983, oparamtric stimatio of spatial distributios: Mathmatical Gology, v. 15, p Liu, Y., 25, A Iformatio Cott Masur Usig Multipl-poit Statistics, Gostatistics Baff 24, p , 26, Usig th Ssim program for multipl-poit statistical simulatio: Computrs & Goscics, v. 32, p Ortiz, J., ad Dutsch, C., 24, Idicator simulatio accoutig for multipl-poit statistics: Mathmatical Gology, v. 36, p Strbll, S., 22, Coditioal Simulatio of Complx Gological Structurs Usig Multipl-Poit Statistics: Mathmatical Gology, v. 34, p Wag, L., 1996, Modlig complx rsrvoir gomtris with multipl-poit statistics: Mathmatical Gology, v. 28, p