VARIABLE SECOND-ORDER INCLUSION PROBABILITIES AS A TOOL TO PREDICT THE SAMPLING VARIANCE

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1 VARIABLE SECOND-ORDER INCLUSION PROBABILITIES AS A TOOL TO PREDICT THE SAPLING VARIANCE Bataan Geelhoed; Delft Unverty of Tehnology, ekelweg 5, 69 JB Delft, The Netherland; b.geelhoed@tudelft.nl ABSTRACT A generalzaton of Gy theory for the varane of the fundamental amplng error revewed. Pratal tuaton where the generalzed model potentally lead to more aurate varane etmate are dentfed a: luterng of partle, dfferene n dente or ze of the partle or repulve nter-partle fore. Two general approahe for etmatng an nput parameter for the generalzed model are dued. The frt approah ont of modellng baed on phyal properte of partle uh a ze, denty and eletrotat fore between partle. The eond approah ue mage analy of atual ample. Further reearh nto both method propoed and a uggeton made to ue lne-nterept amplng ombned wth arkov Chan modellng n the eond approah. It onluded that although, at the moment, t too early for a routne applaton of the generalzed theory, the generalzaton ha the potental of provdng more aurate varane etmate than are poble n the theory of Gy. Therefore, further reearh nto the development and expanon of the generalzed theory worthwhle.

2 VARIABLE SECOND-ORDER INCLUSION PROBABILITIES 83 INTRODUCTION Durng the prevou World Conferene on Samplng and Blendng, WCSB-, a generalzaton of Gy' model for the fundamental amplng error (ee Gy 979, 983) wa propoed and an equaton to predt the varane of the fundamental amplng error wa derved (Geelhoed, 005A). It wa dued that applaton of th equaton ould, at leat n theory, lead to more aurate predton of the varane of the fundamental error than are urrently provded n the theory of Gy. In th ontrbuton, the generalzaton and the new theoretal development ne t wa propoed at WCSB- wll be revewed. Stuaton where t expeted that the generalzed theory wll beome relevant wll be dued. An mportant new nput parameter of the new model the parameter for the dependent eleton of partle. Therefore, two general approahe of etmatng th nput parameter wll be dued. The frt approah a modellng approah baed on phyal properte of partle uh a ze, denty and eletrotat fore between partle. The eond approah ue mage analy of atual ample. Further reearh nto both approahe wll be propoed. ETHODOLOGY Gy' theory baed on the underlyng aumpton that amplng orrepond to Poon amplng: every partle ndependently ubeted to a Bernoull experment, where the -th partle n the bath ha a probablty of q of beomng part of the ample and a probablty of q of not beomng part of the ample. If amplng orretne aumed, whh requre that all partle of the amplng target have an equal probablty of beomng part of the ample, all probablte q beome equal and an therefore be denoted by a ngle ymbol q. The parameter q generally quantfed (etmated) ung the rato of the ample ze and the bath ze from whh the ample wa taken, whh an be nterpreted a the frt-order nluon probablte of the partle. A generalzaton of the above model wa propoed (Geelhoed, 005A). The generalzaton onern the eond-order nluon probablte,.e. the probablte that a par of partle and partle beome part of the ample. Beaue the theory of Gy baed on Poon amplng, the eondorder nluon probablte are gven n Gy' theory by q q. In the generalzed model, the eond-order nluon probablty, denoted a π, gven by: π = q q ( C ) (Eq. ) where C the "parameter for the dependent eleton of partle", whh form a ymmetr matrx (C = C ). It noted that the value for q, q and C are generally nfluened by the hoe of the amplng trategy, the ample ma and the partle properte, o the eond-order nluon probablte (and ultmately the amplng varane) wll alo depend on thee rumtane. An equaton for the varane of the fundamental amplng error wa derved ung () and the followng fve general and pratally reaonable ondton (Geelhoed, 005A):. The partle an be lafed ung a fnte number of lae.. The ze of the bath from whh the ample drawn nfnte, o that amplng an effetvely be regarded a "amplng wth replaement".. The frt-order nluon probablte of partle may vary between lae, but do not vary for partle wthn a la. v. The eond-order nluon probablte may depend on the lae of both partle, but there no varaton n eond-order nluon probablte of dfferent partle par but wth eah member belongng to the ame la a the orrepondng member n the other partle par.

3 84 B. GEELHOED v. Varaton n ample ma reman mall, o that the ample onentraton an effetvely be lnearzed a a funton of the ample ma. Under thee ondton, the equaton for the varane of the fundamental amplng error equvalent to the followng equaton for the varane of the onentraton n the ample, V( ample ): V( ample ) Nm ( ) C NN mm ( )( ) (Eq. ) Where the parameter m and denote repetvely the ma of and onentraton n a partle belongng to the -th partle la, ', N' and ' are repetvely the expeted value of the ample ma, the expeted value of the number of partle belongng to the -th la n the ample and the expeted value of the onentraton of the property of nteret n the ample. In () and ubequent part of th artle, C repreent the parameter for the dependent eleton of partle of a par ontng of a partle belongng to the -th la and a partle belongng to the -th la. The frt term on the rght-hand de of () orrepond to the equaton derved n the theory of Gy (Geelhoed, 005A), where the varane nverely proportonal to the expeted value of the ample ma when the expeted value N' are proportonal to the expeted value of the ample ma. It an thu be een that Gy method to derve a model equaton for the fundamental amplng error rele on the mplt aumpton that frt-order nluon probablte play a domnant role n explanng the ample-to-ample varaton and that the hgher eond-order nluon probablte do not gnfantly nfluene the reult,.e. the eond term, lnear n C, neglgble. The generalzaton therefore offer the potental of a more aurate derpton of the varane of the fundamental amplng error. The eond term on the rght-hand de an thu be ondered to be a orreton term, derbng the effet of non-zero eond-order nluon probablte. It noted that th orreton term may alo depend on the expeted value of the ample ma. Therefore, an eay determnaton of the orreton term by onderng a ere of ample wth large ample mae, uh that the frt term on the rght-hand de would beome neglgble ompared to the orreton term, not generally poble. The obervaton that the varane of the amplng error vare nverely proportonal wth the expeted value of the ample ma hould not be een a an ndaton that the orreton term (and hene the effet of non-zero eond-order nluon probablte on the amplng varane) doe not gnfantly nfluene the amplng varane. Under ertan aumpton about the amplng proe, Lyman (998) derved an expreon for the amplng varane for amplng from a bath of partle n a totally egregated tate and aumed that th value repreent a maxmum poble value for the amplng varane. It would be nteretng to nvetgate whether () an gve a theoretal ba for th aumpton. Further duon of th ubet however outde the ope of th artle. If the expeted value ', N' and ' are not known (a generally the ae n prate), the varane may be etmated by replang thee expeted value n () by the orrepondng value n a ample. The reultng varane etmator then denoted a Var( ample ) (ntead of V( ample )) and the orrepondng ample value of ', N' and ' are repetvely denoted a, N and. The thu obtaned equaton :

4 VARIABLE SECOND-ORDER INCLUSION PROBABILITIES 85 Var( ample ) N m ( ) C N N m m ( )( (Eq. 3) ) where, N and repreent repetvely the ma of the ample, the number of partle belongng to the -th la n the ample and the onentraton n the ample. Another way of etmatng the varane baed on the ue of the Horvtz-Thompon etmator (ee e.g. Särndal et al, 99), whh lead to the followng varane etmator (Geelhoed, 006), denoted a V HT ( ample ): N m C N N m m V HT (ample) (Eq. 4) - C - C A dervaton preented n the Appendx. The equaton baed on the addtonal aumpton that the amplng proe ued to draw the ample lead to ample of a ontant ma and that the frtorder nluon probablty of the partle gven by the rato of the ample ma and the ma of the bath from whh the ample wa drawn (.e. amplng orret). (4) therefore not applable when C =0 for all poble value of and (Gy model), beaue n that ae the ample ma ould not be ontant, but would vary between zero and the ma of the bath from whh the ample wa drawn. If the ample value, N and are not known, thee may be replaed by ther expeted value ', N' and ' repetvely, f avalable, n order to arrve at an etmated value for V HT. It hould be noted that the ue of (4) an lead to dfferent etmate for the varane than the ue of (3), o future pratal ue of thee equaton need to take nto aount the trength and weaknee of both opton. A full duon of th however outde the ope of th paper. However, a bref and prelmnary duon wll be gven. A general advantage of V HT ( ample ) that, beaue t baed on the Horvtz-Thompon etmator, whh unbaed, t expeted that the ba n V HT ( ample ) generally maller than the ba n Var( ample ). Another advantage of the ue of V HT ( ample ) that, n the mple rumtane where there only one partle la k ontng of partle wth a non-zero value for the onentraton k, whle all other lae ( k) have =0, the equaton mplfed o that t doe not expltly depend on the mae of the other partle: V HT ( ample ) = (/ )( ( N k C kk ) k m k /( C kk )). If a relable empral etmate for the amplng varane, denoted here a V e avalable, e.g. determned ung the analy of a ere of ample of ontant ma, th etmate an be ubttuted nto the above equaton, from whh the parameter C kk an then be olved: C kk =(V e k m k / )/(V e N k k m k / ). Hene, C kk zero when V e = k m k /, whh wll here be denoted a V GY, beaue n Gy amplng model C kk would be zero by defnton. Table how the thu obtaned etmate for C kk for everal value of the rato of V e and V GY and the number N k of partle wth non-zero onentraton n the ample (or t expeted value N' k ).

5 86 B. GEELHOED Table - Value of the etmated value of C kk for ome value of the rato of the empral varane etmate V e and the varane etmate V GY and (the expeted value of) the number of partle wth non-zero onentraton n the ample for the mple ae of only one partle la (la k) wth non-zero onentraton. N k, N k V e /V GY A general drawbak of the ue of V HT ( ample ) that t trtly only applable when the ample ma ontant. Whle t tehnally poble to draw ample of approxmately ontant ma, remanng varaton n ample ma ould lead to a ba n the etmate alulated ung V HT ( ample ). Therefore, further tudy requred nto pratally aeptable value for the ba n varane etmate and aeptable level of varaton n ample ma for whh V HT ( ample) an tll be ued wthout ba orreton. Three general advantage of the ue of Var( ample ) wll be dued. The frt advantage of the ue of Var( ample ) that t an ealy be een that the value of Var( ample ) wll be zero when all onentraton and are equal. Th derable, beaue f all partle have the ame onentraton, the varane of the ample onentraton wll be zero. A eond advantage that the value of Var( ample ) not nfluened by ontant ytemat error n the determnaton of the parameter and. Fnally, the thrd advantage that Var( ample ) depend n a lnear way on the parameter C, makng t le entve to error n the determnaton of C than V HT ( ample ), for value of C loe to one. A further duon of the dfferene between the etmator expreed n (3) and (4) and the potental pratal onequene outde the ope of th artle. A prerequte of applaton of the equaton of the generalzed theory knowledge of the parameter for the dependent eleton of partle, C. It therefore ueful to dentfy the patal dtrbuton of partle n the bath before the ample drawn a a domnant underlyng oure of dependent eleton of partle (Geelhoed, 005B). Two type of partle that tend to be more n the vnty of eah other than would be expeted on the ba of ompletely ndependently randomly patal dtrbuton of partle wll have an nreaed eond-order nluon probablty and wll therefore have a negatve value of C. On the other hand, two type of partle that tend to be further away from eah other than would be expeted on the ba of a ompletely random and ndependent patal dtrbuton of partle wll have a dereaed eond-order nluon probablty and wll therefore have a potve value of C. In vew of the above remark, fator ontrbutng to a potental dereae of the value of C are dentfed a: luterng of partle aued by attratve nter-partle fore, uh a attratve eletrotat fore or moture that make partle wet and tk to eah other. Fator leadng to a potental nreae n the value of C are dentfed a: denty dfferene ombned wth the nfluene of gravty durng tranport or repulve nter-partle fore. Wth all of the effet mentoned, the ze and hape of the partle alo nfluene the magntude of the effet. Sze alo n telf a fator leadng to a potental nreae of the value of C. Th an be demontrated ung a mple example of two pheral partle wth dameter D and D. The dtane between (the entre of mae of) the partle annot be maller than D / + D /. Hene, larger partle wll on average be further away from eah other than maller partle, leadng to a potental nreae of C.

6 VARIABLE SECOND-ORDER INCLUSION PROBABILITIES 87 RESULTS A generalzaton of the theory of Gy wa revewed, whh an potentally lead to more aurate etmate for the varane of the fundamental amplng error. Beaue a prerequte of applaton of the equaton of the generalzed theory knowledge of the parameter for the dependent eleton of partle, an mportant reult the dentfaton of the patal dtrbuton of partle n the bath before the ample drawn a a domnant underlyng oure of dependent eleton of partle. Th then reult n the dentfaton of luterng of partle, dfferene n dente or ze of the partle and repulve nter-partle fore a fator nfluenng the magntude of the parameter for the dependent eleton of partle. DISCUSSION Beaue the generalzaton ha the potental of provdng more aurate predton for the varane of the fundamental error, poble approahe to evaluate the eental new nput parameter for the generalzed theory, the parameter for the dependent eleton of partle, are dued below. A frt approah would be to fnd a model equaton for predtng the value of C (and ultmately alo of the varane) baed on the phyal properte of the partle uh a ze, denty, hape and eletrotat properte. Th a dffult and omplex tak, whh, f poble, may be ueful for ertan applaton, but wll not be further dued here. A eond approah would be the applaton of mage analy of ample, whh allow obervng dretly the patal dtrbuton of partle. Although t generally eay to obtan an mage, there are many potental way of extratng nformaton about the value of the parameter C from the mage. Below, one poble way wll be dued. A poble way of evaluatng the parameter for the dependent eleton of partle would be to ue lne-nterept amplng (alo known a lne-tranet amplng ) of the mage, a method of amplng partle n a regon whereby, roughly, a partle ampled f a hoen lne egment, alled a tranet, nteret the partle (Kaer, 983). Th produe a one-dmenonal han of partle from whh nformaton about the patal dtrbuton, pefally C an be derved. Th done by frt ountng the number of tranton between the dfferent partle type. The number of tranton gong from a partle of type to a partle of type denoted a N. Intutvely t lear that there wll be a relatonhp between C and N : a large negatve value of C wll lead to a hgh value of N, beaue partle type that have a hgh eond-order nluon probablty tend to have more tranton between eah other. arkov Chan modellng (ee e.g. Freedman, 97) ould be appled to quantfy th relatonhp. However, th wll not be further developed here. A potental problem wth the above-derbed method that the lne-nterept ample ould be baed toward larger partle, beaue of the nreaed probablty of nteretng a larger partle. Therefore, further tudy requred nto the effet of th potental ba on the etmate for C and t effet on the varane etmate. Alo further tudy nto way of overomng th ba requred for ae where the effet of th ba on the varane etmate non-neglgble. Some general obervaton onernng the eond approah an be made baed on the fat that th approah rele on an mage of the ample. Beaue two-dmenonal mage an be obtaned ung a varety of routne tehnque rangng from dgtal photography to Sannng Eletron roopy, further tudy nto the feablty of the eond approah requred for a range of materal and tehnque to obtan two-dmenonal mage. A ommon problem wth thee D tehnque,

7 88 B. GEELHOED however, that a urfae ample may be unrepreentatve for the whole ample. Th problem an be overome by ung a 3D magng tehnque (f avalable of oure!) or ung multple twodmenonal ro eton of the ame ample. However, for fat and heap reult, ung a dgtal mage of the top urfae would be deal. Therefore, further reearh alo propoed nto th applaton. A fnal pont of duon onernng the generalzed theory a a whole that although t ha the potental of provdng more aurate varane etmate than the theory of Gy, pratally relevant partle ytem mut be dentfed for whh t an be demontrated that th a gnfant mprovement. Further expermental work therefore requred before the new approah an beome a routnely appled and generally aepted method. CONCLUSIONS At the moment t too early for a routne applaton of the generalzed theory. However, the generalzaton ha the potental of provdng more aurate varane etmate than now poble n the theory of Gy. In vew of th, two approahe for evaluatng an eental nput parameter of the generalzed model are dued: a modellng approah baed on the partle properte and an approah baed on mage analy of an atual ample. Only one method, a method baed on lnenterept amplng and arkov Chan modellng, n the eond approah brefly dued n th artle. There are many other potental method of determnng the parameter for the dependent eleton of partle. Beaue the generalzed theory an potentally lead to more aurate etmate for the varane of the fundamental amplng error, further reearh nto the development of new method n both approahe propoed. APPENDIX A π-expanded etmator (ee e.g. Särndal et al, 99) for the onentraton n the bath gven by: N m π (Eq. 5) bath π bath where < bath > π the π-expanded etmator for the onentraton n the bath, N the number of partle n the ample belongng to the -th partle la, m and are repetvely the ma of and the onentraton n a partle belongng to the -th la, bath the ma of the bath and π the frt-order nluon probablty of a partle belongng to the -th la. A dervaton of the above equaton an be found n Geelhoed (004). If the ample ma ontant and the frt-order nluon probablty equal to the rato of the ample ma ( ) and the bath ma ( bath ), the π- expanded etmator beome equal to the ample onentraton,. Under thee aumpton (and ondton (), () and (v) tated n the man text of th artle), Geelhoed (004) derved the followng equaton for the varane of the ample onentraton, baed on the general Horvtz- Thompon etmator for the varane of the π-expanded etmator: mm m V HT (ample) N N ( ) N ( ) (Eq. 6) π π π π π bath bath

8 VARIABLE SECOND-ORDER INCLUSION PROBABILITIES 89 n whh π the eond-order nluon probablty of a partle par n whh the frt-partle belong to the -th la and the eond to the -th la. Subttuton of () for the eond-order nluon probablty and π = / bath, reult n: m m V HT (ample) N N ( ) ( )N (Eq. 7) C C bath Aumng that the bath from whh the ample wa drawn muh larger than the ample,.e. bath >> ample o that /( C ) / bath /( C ), the above reult an be rearranged to yeld (4) n the man text. m REFERENCES Geelhoed, B, 005A. A generalaton of Gy model for the fundamental amplng error. Seond World Conferene on Samplng and Blendng. The Autralaan Inttute of nng and etallurgy, Autrala, (ISBN ). pp 9-5. Geelhoed, B, 005B. A amplng tudy of ndutral mxture of partle, Appled Statt program and abtrat. Slovena, (ISBN ). pp Geelhoed, B, 006. Varable eond-order nluon probablte durng the amplng of ndutral mxture of partle, Appled Stohat odel n Bune and Indutry. Vol., pp Gy, P, 979 and 983. Samplng of partulate materal, Theory and prate. Elever: Amterdam. 43 p. Lyman, G, 998. The nfluene of egregaton of partulate on amplng varane. the queton of dtrbutonal heterogenety. Internatonal Journal of neral Proeng. pp 95-. Kaer, L, 983. Unbaed Etmaton n Lne-Interept Samplng, Bometr 39. pp Freedman, D, 97. arkov Chan. San Frano: Holden-day. ISBN/ISSN/CN p. Särndal, C E, Swenon, B, and Wretman, J. 99. odel Ated Survey Samplng. New York: Sprnger-Verlag.