Constraining the Sum of Multivariate Estimates. Behrang Koushavand and Clayton V. Deutsch

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1 Constranng the Sm of Mltarate Estmates Behrang Koshaand and Clayton V. Detsch Geostatstcans are ncreasngly beng faced wth compostonal data arsng from fll geochemcal samplng or some other sorce. Logratos are sed at tmes to ensre the resltng estmates sm to nty; howeer, the logarthm transform can ntrodce a bas. We may want to cokrge the arables drectly whle ensrng that the estmates sm to nty. A lnear constrant s added to the Cokrgng system. The large cokrgng system estmates all arables at the same tme sch that the sm of all arables to be a specfed constant ale. The methodology and practcal concerns are dscssed. Introdcton Compostonal data are represented as ector arables, wth nddal ector components rangng between zero and a poste maxmm ale representng a constant sm constrant, sally nty (or 00%. The earth scences commonly enconter spatal dstrbtons of compostonal data, sch as concentratons of consttents n natral waters (e.g., mole, mass, or olme fractons, mneral percentages, ore grades, or proportons of mtally exclse categores (e.g., a water ol rock system. Tradtonal geostatstcal methods, sch as krgng or co-krgng, wll not respect the compostonal natre of data f appled drectly becase these methods calclate estmates that are ndependently optmal. A constant sm constrant ntrodces negate correlatons. The constant sm constrant also prodces cokrgng systems of eqatons that are ntractable for conentonal lnear eqaton solers (Carle, 004. Most of the methods amed at compostonal data are based on data re-expresson or transformaton. The data mst be transformed to new arables to elmnate the constrants. Then geostatstcal models of new arables hae been created, ths model shold be back transformed to orgnal nt data, whch most of these methods do not hae a lnear back transformaton and ntrodce seros rsk of bas. The approach presented here s a lnear estmaton that mnmzes estmaton arance. In practce, the proposal s the conentonal cokrgng wth an addtonal constrant. Methodology Consder the staton where the composte data { z,,..,n,,..., } = = at any, possbly dfferent, locatons. The lnear estmator for each of arables s (Gooaerts, 997: n ( k Zk( mk( = λ ( Z m k,..., = = k λ s the weght assgn to the datm Z ( to estmate Zk where random arable (RV k mnmze error arance ( Z s denoted m σ, that s (Gooaerts, 997: k k( Var{ Zk( Zk( } n ( nj( k k..c j j β = j= = β j= n ( k Ckk ( 0 (.C k = = σ = k = (. The expected ale of the. All co-krgng estmator shold be nbased and = λ λ + λ k =,..., ( where C j s cross coarance fncton between arables and j, C k s coarance fncton of arable k. The estmaton arance s mnmzed nder the constant that the expected error s zero: 308-

2 { k k} E Z Z = 0 ; k =,..., (3 It has been shown that smple co-krgng (SCK s nbased snce statonary mean of each RVs s constant oer the entre doman: mk( = m k, =..n ; k =,..., (4 Smple Co-krgng system s wrtten as: n ( nj( n ( k k λ. λβ.cj β = Ck =,...,, k =,..., j j = = β j= = The matrx notaton for fll Co-krgng system s shown below: A sck. Γ sck = bsck (5 where, C ( C ( β β Asck = C( β C ( β λ β λ ( β Γ sck = ( λβ λ ( β C ( C bsck = C( C ( There are lnear systems that shold be soled separately. There s no garantee for sch lnear systems that sm of estmated arables to be a constant ale whch s ery mportant n compostonal data. A compostonal Co-krgng (CCK system has been defned here, soles a ery bg lnear system sch that sm of all estmated arables hae to be a constant ale. The compostonal smple co-krgng (CSCK system s defned as below: A CSCK. Γ CSCK = bcsck (6 T C ( β C Z Z( β [ 0] [ 0 ] C C Z ( Z A CSCK ( β ( β [ 0] [ 0] T C ( C β ( β = Z( Z( [ 0] [ 0] C( β C Z Z β T T Z( Z( Z Z [ 0] Z ( Z Z Z 308-

3 ( λ β C ( λβ ( C( Γ = CSCK λ β ( b = CSCK C ( λβ ( C ( 0 Const The estmaton arance for each of arables s larger than SCK estmator. Ordnary and tradtonal ordnary co-krgng systems also ntrodce lnear constrants for each arable, whch shold consder at lnear system (6. Cokt3D_ (Detsch and Jornel 997 program has been modfed to sole a fll cokrgng whch estmated ales hae a constrant sm to a constant. Dscsson The constraned cokrgng system ensres that the estmates sm to nty, bt there s no garantee to get non-negate estmates. To sole ths problem, we need to mnmze the estmaton arance wth other algorthms. Ths leads to excesse CPU reqrements. The second and more reasonable method s to replace negate ales wth 0 and remoe them form system of eqatons and resole for the optmal estmates. Ths method also forces algorthm to sole cokrgng two tmes for each negate estmate, bt t s more effcent and faster then solng krgng system wth another algorthm. Ths algorthm s referred to as constraned poste Cokrgng. The second sse s bas n the constraned co-krgng system. All krgng estmates are nbased becase the expected ale of dfference between estmate and tre ale s 0. Ths means that these estmates dose not hae any systematc error. E{ Zk( Zk( } = 0 So, E λ( Z( Z( = 0 (7 = Howeer, n constraned co-krgng, the system of eqatons s soled sch that estmates are forced to sm to nty. These means that krgng weght λ ( are not ndependent from arables Z( and may case based estmate ale. Ths sse needs more stdy. Small Example Consder two data wth three arables: Z = 0.0 Z = 0.50 Z = Z =? Z =? Z =? Z = 0.0 Z = 0.70 Z = m where the arogram of ths data set s: 6m 308-3

4 h h γ Z ( h = 0.5.Sph Sph 5 0 h h γ Z ( h = 0..Sph 0.8.Sph h h γ Z ( h = 0.8.Sph 0..Sph h h γ ZZ ( h =0..Sph 0.3.Sph 5 0 h h γ ZZ ( h =0.3.Sph 0..Sph h h γ ZZ3( h = 0. 3.Sph 0.3.Sph Tables below show the Left hand sde matrx A and rght hand sde ector CSCK b. CSCK The table below shows the solton of lnear system of CSCK, estmate and estmaton arance ales. Z W W W3 ZW ZW ZW Large Data Set Z = Z = Z = σ csck = σ csck = σ csck = = Z = There s three arable n ths data set that shold be sm to at each sample or estmated locaton. Fgre shows the locaton map and fgre shows hstograms and Scatter plot of arables. Cokt3D and Cokt3D_ wth and wthot forced to poste estmate were sed to an ordnary fll co-krgng estmaton to compare the reslts. All three methods were sed wth the same LMC model sed n small data set aboe. Pxel plot of estmate and estmaton arance ales are presented at fgre 3 and 4 respectely. The blank blocks on rght colmn maps, whch appear on the second arable, are zero estmates replaced wth negate estmate ales on the mddle colmn. Hstograms of estmated ales and scatter plot of estmate arance ales are shown at fgre 5, 6 and 7 respectely. There s no sch a bg dfference at estmate arance ales between three methods, bt t s clear that addng sm to constrant and forcng to get poste estmates ncrease estmaton arance

5 There are 36 negate estmates wth Cokrgng and 35 estmates wth constraned Cokrgng and the sm of all negate ales of second arable for frst and second method respectely are and -3.. Addng sm to constrant to Cokrgng system decrease nmber of negate estmates for second arable bt we stll hae negate estmates, whch are replaced wth zero by constraned poste Cokrgng; Sgsm (Detsch and Jornel 997 was sed to generate 00 realzatons of ncondtonal smlaton for each of arables n Gassan space and then all realzatons were back transferred to orgnal nts and standardzed to get sm of arables n eery locaton to. The same sample locatons were extracted from each realzaton. Cokrgng and constraned poste Cokrgng were sed to estmate n-sampled locatons that we already hae the tre ales. Fgre 8 shows the hstograms of Error (Estmate-Tre wth these two methods. As we expected there s a ery small based on estmaton of V and V, bt for ths case stdy constraned poste Cokrgng dose not nject sgnfcant based to the estmaton. Conclson Geostatstcans commonly enconter compostonal data. on lnear transformaton of these arables wth technqes lke logratos s problematc becase of sgnfcant bas. Krgng n orgnal nts prodes a best lnear nbased estmator. Tradtonal cokrgng does not constran the sm of estmates. The approach presented here leads to a large cokrgng system that has a constrant to force estmated ales to sm to a constant. egate estmated arables was set to 0 and remoed form system of eqatons and a new system of eqatons s soled. Ths method reqres more CPU tme than tradtonal Cokrgng becase algorthm soles a bg lnear system rather than (nmber of arables smaller lnear systems n Cokrgng. References Carle, S. F., 004, Geostatstcal Analyss of Compostonal Data : Oxford Unersty Press, ew York Detsch, C. V., and Jornel, A. G., 998, GSLIB: Geostatstcal software lbrary and sers gde, nd ed.: Oxford Unersty Press, ew York, 369 p. Gooaerts, P., 997, Geostatstcs for atral Resorces Ealaton: Oxford Unersty Press, ew York Woronow, A. 995, A Psedo-Genetc Algorthm Sted to Compostonal Data: Math Geology,. 7, no., p Locatons of V 6 Locatons of V 6 Locatons of V Fgre. Locaton map of samples and three arables content 308-5

6 mber of Data 67 mean 0.8 std. de coef. of ar maxmm 0.63 pper qartle 0.95 medan 0.4 lower qartle 3 mnmm 00 V mber of data 67 mber plotted 67 X Varable: mean 0.8 std. de mn. 00 max Y Varable: mean 39 std. de. 55 mn. 00 max correlaton 0.5 rank correlaton V V mber of Data 67 mean 39 std. de. 55 coef. of ar.409 maxmm 0.74 pper qartle 5 medan lower qartle 0 mnmm 00 V mber of data 67 mber plotted 67 X Varable: mean 0.8 std. de mn. 00 max Y Varable: mean std. de mn max..000 correlaton rank correlaton V V mber of Data 67 mean std. de coef. of ar 0.30 maxmm.000 pper qartle 0.96 medan lower qartle mnmm 0.85 V mber of data 67 mber plotted 67 X Varable: mean 39 std. de. 55 mn. 00 max Y Varable: mean std. de mn max..000 correlaton rank correlaton V3 Fgre. Hstograms and Scatter plot of arables V 308-6

7 600 V Co-Krggn Estmate V Compo. Co-Krggn Estmate 600 V Compo. Co-Krggn Estmate V Co-Krggn Estmate V Compo. Co-Krggn Estmate V3 Co-Krggn Estmate V3 Compo. Co-Krggn Estmate 300 V Compo. Co-Krggn Estmate V3 Compo. Co-Krggn Estmate Fgre 3. Co krgng Estmate map: wthot constrants(left, constrant sm to (mddle and constrant sm to whch forced to poste ales (rght 308-7

8 600 V Co-Krggn Estmate Var. V Compo. Co-Krggn Estmate Var. 600 V Compo. Co-Krggn Estmate Var V Co-Krggn Estmate Var. V Compo. Co-Krggn Estmate Var V3 Co-Krggn Estmate Var. V3 Compo. Co-Krggn Estmate Var. 300 V Compo. Co-Krggn Estmate Var V3 Compo. Co-Krggn Estmate Var Fgre 4. Co krgng Estmaton arance map: wthot constrants(left, constrant sm to (mddle and constrant sm to whch forced to poste ales (rght 308-8

9 mber of Data 800 mean 0.69 std. de. 0.0 coef. of ar maxmm 0.67 pper qartle 0.38 medan 0.73 lower qartle 0.97 mnmm mber of Data 800 mean std. de. 0.3 coef. of ar maxmm 0.68 pper qartle medan lower qartle 0.65 mnmm mber of Data 800 mean 0.35 std. de. 0. coef. of ar maxmm 0.68 pper qartle medan lower qartle 0.64 mnmm estmate V estmate V_C estmate V_C mber of Data 800 mean -33 std. de. 67 maxmm 0.33 pper qartle 05 medan -37 lower qartle -74 mnmm mber of Data 800 mean 47 std. de. 5 maxmm 0.8 pper qartle 70 medan 34 lower qartle mnmm mber of Data 567 nmber trmmed 33 mean 57 std. de. 5 coef. of ar maxmm 0.80 pper qartle 78 medan 4 lower qartle 0 mnmm estmate V estmate V_C estmate V_C mber of Data 800 mean std. de coef. of ar 0.65 maxmm pper qartle 0.68 medan lower qartle 0.48 mnmm mber of Data 800 mean std. de coef. of ar 0.4 maxmm pper qartle 0.73 medan 0.60 lower qartle 0.47 mnmm mber of Data 800 mean std. de coef. of ar 0.4 maxmm pper qartle 0.70 medan 0.60 lower qartle mnmm estmate V estmate V3_C estmate V3_C Fgre 5. hstogram of Co krgng Estmate: wthot constrants(left, constrant sm to (mddle and constrant sm to whch forced to poste ales (rght 308-9

10 estmate V_C mber of data 800 mber plotted 800 X Varable: mean 0.69 std. de. 0.0 mn. 5 max Y Varable: mean std. de. 0.3 mn. 43 max correlaton 0.93 rank correlaton 0.93 estmaton arance V_C mber of data 800 mber plotted 800 X Varable: mean std. de. 0.7 mn. 9 max..050 Y Varable: mean std. de mn. 8 max..3 correlaton rank correlaton estmate V estmaton arance V 0.59 mber of data 800 mber plotted mber of data 800 mber plotted 800 estmate V_C X Varable: mean -33 std. de. 67 mn max Y Varable: mean 47 std. de. 5 mn. -4 max. 0.8 correlaton rank correlaton estmaton arance V_C X Varable: mean std. de. 0.0 mn. max..034 Y Varable: mean 0.47 std. de. 0.3 mn. max..090 correlaton rank correlaton estmate V estmaton arance V mber of data 800 mber plotted mber of data 800 mber plotted 800 estmate V3_C X Varable: mean std. de mn max Y Varable: mean std. de mn max correlaton rank correlaton estmaton arance V3_C X Varable: mean 0.79 std. de mn. 5 max..06 Y Varable: mean 0.79 std. de mn. 5 max..36 correlaton rank correlaton estmate V 3 estmaton arance V 3 Fgre 6. Scatterplots of estmated ales and estmaton arance ales: Cokrgng s. Constraned Cokrgng 308-0

11 estmate V_C mber of data 800 mber plotted 800 X Varable: mean 0.69 std. de. 0.0 mn. 5 max Y Varable: mean 0.35 std. de. 0. mn. 4 max correlaton rank correlaton 0.93 estmaton arance V_C mber of data 800 mber plotted 800 X Varable: mean std. de. 0.7 mn. 9 max..050 Y Varable: mean std. de mn. 8 max..59 correlaton rank correlaton estmate V estmaton arance V estmate V_C mber of data 567 mber plotted 567 mber trmmed 33 X Varable: mean - std. de. 6 mn max Y Varable: mean 57 std. de. 5 mn. 00 max correlaton 0.73 rank correlaton estmaton arance V_C mber of data 567 mber plotted 567 mber trmmed 33 X Varable: mean 0.47 std. de. 0.0 mn. max..034 Y Varable: mean 0.47 std. de mn. max..8 correlaton rank correlaton estmate V estmaton arance V estmate V3_C mber of data 800 mber plotted 800 X Varable: mean std. de mn max Y Varable: mean std. de mn max correlaton 0.98 rank correlaton estmaton arance V3_C mber of data 800 mber plotted 800 X Varable: mean 0.79 std. de mn. 5 max..06 Y Varable: mean 0.79 std. de mn. 5 max..84 correlaton rank correlaton estmate V 3 estmaton arance V 3 Fgre 7. Scatterplots of estmated ales and estmaton arance ales: Cokrgng s. Constraned Poste Cokrgng 308-

12 V: Tre-Est mber of Data nmber trmmed 45 mean -50 std. de maxmm pper qartle 89 medan -48 lower qartle -586 mnmm V: Tre-Est mber of Data nmber trmmed 457 mean -368 std. de maxmm pper qartle 53 medan lower qartle -853 mnmm V: Tre-Est _Tre-Est mber of Data nmber trmmed 45 mean -060 std. de. 943 maxmm pper qartle 366 medan 056 lower qartle -389 mnmm V: Tre-Est _Tre-Est mber of Data nmber trmmed 3 mean -95 std. de. 86 maxmm 0.93 pper qartle 056 medan -60 lower qartle -67 mnmm V3: Tre-Est _Tre-Est mber of Data nmber trmmed 45 mean 746 std. de maxmm pper qartle 0.30 medan 670 lower qartle 9 mnmm V3: Tre-Est _Tre-Est mber of Data nmber trmmed 454 mean 650 std. de maxmm pper qartle 0.05 medan 598 lower qartle 084 mnmm _Tre-Est _Tre-Est Fgre 8. Hstograms of Error: Cokrgng Estmate Tre Vale ; Left: Cokrgng (left and Rght: Constraned poste Cokrgng 308-

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