A Stochastic Trend Approach to Calculate Uncertainty in the Mean

Transcription

1 A Stchstic rend Apprch t Clculte Uncertinty in the Men Mrth E. Villlb nd Clytn V. Deutsch Gesttistics techniques ften led t less uncertinty fr lrger dmins even if the number f dt stys the sme. An lterntive t evlute uncertinty in these dmins culd be t use trend equtin nd define the men dependent n the lctin within the dmin. he trend mdel culd be used t predict the vlue t unsmpled lctins. A glbl men is evluted with the set f cefficients evluted t ll lctins within the dmin. he stchstic trend pprch prpses t rndmize these cefficients cnsidering the crreltin f the riginl fitted cefficients. Different cefficients prvide different men vlues tht re cmbined t distributin f uncertinty in the men. 1. Intrductin Estimtes re mde under uncertinty becuse few dt re present in the evlutins nd mny input prmeters re required in the estimtes. he uncertinty in the input prmeters must be trnsferred int the estimtes t prvide mre relistic ssessment f uncertinty. Decisin ming ften cnsiders the uncertinty. he chnce f success/filure, ptentil gin nd ptentil lss re sme cnsidertins in ris nlyses. hese cnsidertins rely n the estimtes nd their uncertinty (Rse, 001). We re mtivted t ccurtely evlute the uncertinty f ur estimtes. he cnditinl finite dmin requires the setting f mny prmeters tht ffect the distributin f uncertinty in nn-intuitive nd nn-trnsprent mnner. Reltively minr chnges in gesttisticl prmeters culd hve lrge effect n prbbilistic estimtes (Deutsch, eungthng, & Ortiz C., 006). Sensitivity nlysis shws tht n increse in the nugget effect, reductin f the rnge f crreltin r n increse in the size f the dmin leds t less uncertinty in lrge scle verges. his mes sttisticl sense becuse rndm vritins in the vrible verge ut. A dmin is n re r vlume where smples fllw nwn distributin (e.g., nrml r lgnrml distributin). he dmin includes smples tht re nt ften unifrmly spced; preferentil smpling is cmmn in plces f high grde. Gesttistics techniques ften led t less uncertinty fr lrger dmins even if the number f dt stys the sme. An lterntive t evlute uncertinty in these dmins culd be t relx the ssumptin f sttinrity. Sttinrity permits clcultin f the men, cvrince nd semi vrigrm by pling the dt within the chsen dmin (Gverts, 1997). he use f trend equtin relxes this ssumptin f sttinrity nd defines the men dependent n the smple lctin within the dmin, see Equtin (1). symblizes the number f drift r trend terms, l represents the unnwn cefficients nd the f l (u) terms re functinls tht represent the shpe f the trend. he functinl my be liner, qudrtic, sine/csine, drwn by hnd r specified rbitrrily. = l l l = 0 m( u) f ( u ) (1) Althugh the liner nd the qudrtic equtins re cnsidered in the implementtin here, different equtins culd be used if required. A FORRAN cde clled uncregcef.fr, which is mdifictin f the crrelte prgrm, ws develped t implement the stchstic trend technique. he trend mdel culd be used t predict the vlue t unsmpled lctins, tht is, ndes r lctins in the dmin re evluted with plynmil with cefficients cmputed by lest squre methd. A glbl men is then evluted with the set f cefficients evluted t ll lctins within the dmin. he stchstic trend (S) pprch develped belw prpses t rndmize these cefficients ting ccunt the crreltin f the riginl fitted cefficients. Mny sets f cefficients prvide different men vlues. hen, the uncertinty in the input prmeter is clculted frm the distributin f these mens. 11-1

2 . Methdlgy he cefficients f the trend re clculted bsed n the well-nwn liner regressin thery. he sum f the squred difference between the estimtes nd the vilble dt re s smll s pssible. his difference is nt zer becuse the estimted vlue fluctutes but its expected vlue, tht is, the methd f lest squre selects the regressin cefficients with the criteri f minimizing the sum squre f these fluctutins (Crl Friedrich Guss, 1794), (Jhnsn & Wichern, 007) he cefficients f the trend nd their individul vrinces re clculted by the thery f multiple regressin mdels. he liner regressin mdel is defined by the equtin mtrix (). he dependent vrible is dented s Z nd the independent vribles X. he dependent vrible in S is the dt vlues nd the independent vribles re the crdintes f the dt. he mdel culd be written in mtrix nttin s: Z = X + ε () he dependent vrible Z represents the (n 1) vectr f the dt, where n is the number f dt vilble. X is the (n ) mtrix f the levels f the independent vribles, where is the number f prmeters, explntry vribles r regressr predictr tht define the fitted equtin f the mdel equtin. Fr instnce, if the dt hve crdintes t Est, Nrth nd Elevtin, the (n, ), (n, 3) nd (n, 4) f mtrix X my crrespnd t the crdinte dt. hen, (n, 5 - ) clumns re interctin f the crdintes t better define the shpe f the trend with mre prmeters. crrespnds t the ( 1) vectr f the cefficients r vectr f regressin prmeters. he lst element ε regrds t n errr (n 1) vectr, this errr vectr is ssumed s rndm prt f the mdel equtin tht hs distributin f men zer nd unnwn vrince s. he next equtin shws the explined vribles: z1 1 x11 x1 x1 0 ε1 z 1 x1 x x 1 ε = + M M M M O M M M zn 1 xn 1 xn K xn ε n Z = Χ ε he lctin f the vilble dt is trnslted t the X mtrix bsed n the specified functinl plynmils r trend mdel. he methd f lest squres is cmmnly used t estimte the regressin cefficients in multiple liner regressin mdels, the term liner becuse the mdel is liner functin f the unnwn prmeters 0, 1,...,. he mdel describes hyper-plne in -dimensinl spce f the regressr vribles {x l } (Mntgmery, 000). he cefficients selected by the lest squre re clled lest squred estimtes f the regressin prmeter. hey re dented by â becuse they re estimtes f. ˆ ( ) 1 = X X X Z (4) hen, the fitted regressin equtin is given in mtrix nttin. his fitted mdel predicts the vlue t unsmpled lctins s functin f its crdintes: ˆ ˆ ( ) 1 Z = X = X X X X Z (5) he residul describes the errr in the fitting f the trend mdel t the n smples z i, i = 1,...,n. he vectr f residuls is the difference between the riginl vlue smples nd the nes clculted by the fitted equtin t nwn smple lctins. he vrince f the residuls (7) is the sum f the squred residuls divided by the (n - (+1)) degree f freedm, where (+1) is the number f prmeters r cefficients. Ntice tht the number f dt must be greter thn number f cefficients. εˆ = Z Z ˆ (6) ˆ ˆ s = ε ε n + { } E s ( 1) (3) (7) = (8) 11-

3 he methd f lest squres prduces n unbised estimtr f the prmeters in the multivrite liner regressin mdel. Prperties f the â estimtrs re defined belw: { ˆ} E = (9) Cv( ˆ) ( ) 1 = X X (10) he next mtrix shws the cvrince between the regressin cefficients. hen, the vrinces f the cefficients re cquired frm the dignl f the symmetric mtrix becuse C( l, l ) = l c c 00 c 01 0 c 00 c ( ˆ c c c c c Cv ) = = M M O M M M O M c c 0 c 1 c c 0 1 Nw ech term f the cvrince is divided by their respective cmbined stndrd devitins t btin the crreltin mtrix between the cefficients. 1/ / c 00 c / / ( ˆ c c ρ ) = 0 0 O M M O M O / c / c ρ ρ ( ˆ ρ ρ ρ ) = M M O M ρ ρ Mny sets f cefficients culd be generted by Mnte Crl simultin t evlute uncertinty in the fitted trend. he crreltin between the regressin cefficients will be preserved. Ech cefficient â l, l = 0, 1,..., hs distributin defined by its men nd its vrince. hen, mny sets f cefficients re drwn frm their distributins. hse sets te ccunt the crreltin f the riginl regressin cefficients. Otherwise, different techniques lie sptil btstrp use the cvrince mtrix f the smple lctins (C 11 ) t cnditinl the smpling nd evlute uncertinty. Bth mtrices re symmetric nd psitive definite. he steps f the prpsed stchstic trend (S) pprch re s fllws: Define the number f prmeters r terms fr the trend equtin, the mdel culd be liner r qudrtic. Cmpute the regressin cefficients f the trend by lest squre methd â l, l = 0, 1,..., where the best fit minimizes the sum f squred residuls. Define the cvrince mtrix nd deduce the vrince f the cefficients l, l = 0, 1,...,. Define the crreltin mtrix fr the regressin cefficients. Perfrm Chlesy decmpsitin f the crreltin mtrix ρ = U Smple independent nrml Gussin scre vlues w = G -1 (p) nd crrelte them by the use f the lwer decmpsed crreltin mtrix, Y = w. Nn-stndrdize these Y vlues by the use f their respective fitted regressin cefficient l nd their respective stndrd devitin l. (11) = ˆ + y, l = 0, 1,..., (1) l l l l Use the set f cefficients in the plynmil t evlute the fitted men Z(u) t ll lctins within the dmin. he men f the vlues t these lctins is the men f the first reliztin m, = 1,...,K (where K is the number f times the wrflw is repeted) 11-3

4 he distributin f m mens cn be ssembled t mdel the uncertinty in the men. 3. Criteri in the Implementtin All techniques t clculte the uncertinty in the men require implementtin chices. he stchstic trend pprch des nt require s mny input prmeters s mst techniques. his pprch nly requires the frm f the trend mdel. One shuld chse the plynmil representtin tht ppers resnble given the dt. he uncertinty in the men will be sensitive t the number f terms. he uncertinty in the men using the stchstic trend pprch des nt require vrigrm; the vrigrm is required fr ther techniques lie the sptil btstrp nd the cnditinl finite dmin. he fitted regressin equtin r trend mdel is used t predict the men t ll lctins. he liner reltinship my nt be necessrily vlid fr extrpltin purpses (Mntgmery & Runger, 006). Uncertinty in the regressin mdel is used specificlly t clculte the uncertinty in the men. he genertin f the stchstic trends is explined with simple exmple. A synthetic dt set f 11 vlues is lcted in ne dimensin. he men f the vlues is.409 nd the vrince ε.4 = M Z = Χ he trend f the dt is mdel with liner trend tht cnsiders nly tw cefficients 0 nd 1. he lctin f the dt is trnsferred t mtrix X(11 ) ccrding the liner trend mdel, where the Est crdinte f the dt crrespnd t the clumn tw. he vectr Z with dimensin (11, 1) represent the vlues f the dt set. he lest squres methd gives the estimtes â 0 nd â 1 f the prmeters 0 nd 1 thrugh the simplified pertins f mtrices. hen, the first cefficient is nd secnd cefficient is ˆ ( ) 1 = X X X Z = ε ε 11 he trend mdel is defined by the equtin tht evlutes Z(u) s functin f lctin in ne dimensin. Z( u ) = x he dt lctin is replced in the regressin line t clculte the ε residul r devitin f the dt frm the estimted regressin mdel. hen, the vrince f the residuls is s fllw: = εˆ εˆ n p = 11 = he cvrince f the cefficients is defined nd the vrince f the regressin cefficients re extrcted frm the dignl f this Cv(â) E 0 6.E 04 Cv( ˆ ) = ( XX ) = 6.E E 05 ε 11-4

5 hen, ech term f the cvrince mtrix is divided by their respective stndrd devitins t btin the crreltin mtrix. C C ρ ρ ρ = C C = = ρ 11 ρ he crreltin mtrix is symmetric nd is psitive-definite, then, the Chlesy decmpsitin is pssible. he lwer tringulr mtrix helps t crrelte the independent nrml vlues w 0 nd w 1. y ρ w = y ρ 1 w = ( 0.837) ρ wer mtrix he reliztin f nd cefficients re in Gussin units. hese re nn-stndrdized t get nd in riginl units. = + y = = + y = ( 0.493) he vlues f the first reliztin = 1 crrespnd t the set f cefficients = nd = hen, ne hundred times = 1,...,100 re smpled frm the distributin f regressin cefficients with men â 0 nd â 1 nd stndrd devitin 0 nd 1. he riginl crreltin f the cefficients is , nd then the crreltin between the tw cefficients fter ne hundred reliztins is preserved. he sctter plt f the first stchstic trend cefficients is illustrted in red dt nd the next nes re illustrted in blc smll dts see Figure 1. Figure 1: Verifictin whether the crreltin f the 100 reliztins f â0 nd â1 reprduce the input crreltin between the regressin cefficients, the red dt crrespnd t the first reliztin develped befre. One hundred sets f cefficients re used in the trend mdel. hen, ne hundred stchstic trends re defined. A trend mdel is used t evlute the vlue t ll lctins nd the crrespnding glbl men. he fluctutins f the trend prvide ne hundred mens. he fluctutins f the trend re illustrted in Figure. 11-5

6 Figure : Stchstic trend mdel, where the blc pints re the dt; the slid blc line is the liner trend mdel with tw prmeters; nd the grey lines re the reliztins. One hundred stchstic trend mdels prvide n uncertinty f his vlue represent 45 % f the uncertinty ssuming independence f the dt / n = Applictin nd Chllenges he stchstic trend is pplied t 3-D dt set. Als, the influence f vrible dmins n the uncertinty nd the cmplexity f the mdel in the uncertinty re evluted. he gld vlues re lcted in dmin f 00 meters by 00 meters in the hrizntl directin nd 0 meters in the verticl directin. he men f the gld vlues is 1.10 g/t nd the stndrd devitin is 1.4. he evlutins cnsider three different dmins: Pessimistic criteri, the dmin which limits re less thn hlf distnce between smples. Nrml criteri, the resnble dmin which limits re rund the hlf f the medin distnce between smples. Optimistic criteri, the dmin which limits re excessively fr frm the smple, it mens lmst twice r mre thn the medin distnce between smples. he uncertinty f the gld vlues is clculted by qudrtic trend f 10 cefficients nd liner trend f 4 cefficients. Z( u) = + x + y + z + x + y + z + xy + xz + yz Z( u) = + x + y + z he vribles x, y nd z re the crdintes Est, Nrth nd Elevtin f the vectr lctin u. hese vlues re trnsferred t the mtrix X t btin the regressin cefficients, â = (X X) -1 X Z. he vectr Z cntins the 49 gld vlues. he regressin cefficients re cmputed fr bth trend mdels. Z( u) x y z x = y z xy xz yz Z( u) = x y 0.153z Ech trend mdel requires cvrince mtrix nd crreltin mtrix f the regressin cefficients. One hundred reliztins f the cefficients re simulted with the U methdlgy described bve. Sets f the cefficients â re replced in their trend mdels t estimte the gld vlue t ech nde f the dmin. he uncertinty with the qudrtic trend results greter thn the liner trend fr the three dmins. 11-6

7 () (b) < 0.5 g/t 0.50 g/t g/t 0.8 g/t - 1 g/t > 1.0 g/t (c) Figure 3: he dt re illustrted with pints, the first grphic () shws the pessimistic criteri used t mdel the dmin, the secnd grphic (b) shws the resnble criteri nd the third grphic (c) shws the ptimistic criteri. Frty-nine dt f gld vlues lcted in pessimistic dmin give n uncertinty f 0.7 (std) using liner trend nd (std) using qudrtic trend. hse vlues re greter thn 1.4/ 49 = 0.0 (std) with CB tht ssumes independence. As expected, the uncertinty increses s the dmin increses. he ptimistic (lrge) dmin prvides n uncertinty f 0.34 using liner trend nd 0.71 using qudrtic trend. he sudden increse f uncertinty s the dmin increses withut supprt f dt is n dvntge f the S pprch becuse trditinl gesttistics techniques struggle with less uncertinty in this scenri. he technique is simple t pply becuse it des nt require cvrince mdel f the dt nd des nt need t be set mny prmeters during the implementtin. his technique cnsiders the size f the dmin nd ssume tht the men dependent f the lctin dt; hwever, the specific frm f the trend is n imprtnt chice. Mrever, the mdel prmeters re ssumed t be multivrite Gussin. 5. Cnclusins Gesttistics techniques ften led t less uncertinty fr lrger dmins even if the number f dt stys the sme. An lterntive t evlute uncertinty in these dmins culd be t use trend equtin nd define the men dependent n the lctin within the dmin. he trend mdel culd be used t predict the vlue t unsmpled lctins. A glbl men is evluted with the set f cefficients evluted t ll lctins within the 11-7

8 dmin. he stchstic trend pprch prpses t rndmize these cefficients cnsidering the crreltin f the riginl fitted cefficients. Different cefficients prvide different men vlues tht re cmbined t distributin f uncertinty in the men. 6. References Chilès, J.-P., & Delfinier, P. (1999). Gesttistics: Mdeling Sptil Uncertinty. New Yr: Jhn Wiley & Sns. Deutsch, C. V. (000). Specil pics in Gesttistics. Albert: University f Albert. Deutsch, C. V. (00). Gesttisticl Reservir Mdeling. New Yr: Oxfrd University Press, Inc. Deutsch, C. V. (004). A sttisticl Resmpling Prgrm fr Crrelted Dt, Sptil Bstrp. Six Annul Reprt f the Centre fr Cmputtinl Gesttistics. Edmntn: Deprtment f Civil & Envirnmentl Engineering University f Albert. Deutsch, C. V., & Jurnel, A. (1998). GSIB: Gesttisticl Sftre ibrry nd User's Guide (nd Editin ed.). New Yr: Oxfrd University Press. Deutsch, C. V., eungthng, O., & Ortiz C., J. (006). A Cse fr Gemetric Criteri in Resurces nd Reserves Clssifictin. Eight Annul Reprt f the Centre fr Cmputtinl Gesttistics (p. 1). Edmntn: Deprtment f Civil & Envirnmentl Engineering-University f Albert; Deprtment f Mining Engineering-University f Chile. Dminy, S. C., Nppé, M. A., & Annels, A. E. (00, Octber). Errrs nd Uncertinty in Minerl Resurce nd Ore Reserve Estimtin: he Imprtnce f Getting it Right. Explrtin nd Mining Gelgy, 11, Efrn, B. (1979). Btstrp Methds: Anther t the Jcnife. he Annls f Sttistics, 7, 1-6. Emery, X. (008). Uncertinty mdeling nd sptil predictin by multi-gussin riging: ccunting fr n unnwn men vlue. Cmputer & Gesciences 34, Gverts, P. (1997). Gesttistics fr Nturl Resurces Evlutin. New Yr: Oxfrd University Press. Jhnsn, R. A., & Wichern, D. W. (007). Applied Multivrite Sttisticsl Anlysis. rnt: Persn Eductin, Inc. Jurnel, A. G. (1994). Resmpling frm Stchstic Simultin. Envirnmentl nd Eclgicl Sttistics, 1, eungthng, O., Khn, K. D., & Deutsch, C. V. (008). Slved Prblem in Gesttistics. Hben, New Jersey.: Jhn Wiley & Sns, Inc. eungthng, O., Mcennn, J., & Deutsch, C. V. (005). Acceptble Ergdic Fluctutins nd Simultin f Sewed Distributins. Edmntn: Centre fr Cmputtinl Gesttistics, Deprtment f Civil&Envirnmentl Engineering University f Albert. eungthng, O., Mcennn, J., & Deutsch, C. V. (005). Acceptble Ergdic Fluctutins nd Simultin f Sewed Distributins. Seventh Annul Reprt f the Centre fr Cmputtinl Gesttistics. Edmntn: Deprtment f Civil&Envirnmentl Engineering-University f Albert. Mntgmery, D. C. (000). Design nd Anlysis f Experiments Fifth Editin. New Yr: Jhn Wiley & Sns, Inc. Mntgmery, D. C., & Runger, G. C. (006). Applied Sttistics nd Prbbility fr Engineers. Arizn: Jhn Wiley & Sns, Inc. Neufeld, C.., Ortiz, J. M., & Deutsch, C. V. (005). A Nn Sttinry Crrectin f the Prbbility Field Cvrince Bis. Seventh Annul Reprt f the Centre fr Cmputtinl Gesttistics. Edmntn: Deprtment f Civil & Envirnmentl Engineering-University f Albert; Deprtment f Mining Engineering-University f Chile. Pyrcz, M., & Deutsch, C. (00). w rtifcts f prbbility field simultin. Mthemticl Gelgy, 33, Rse, P. R. (001). Ris Anlysis nd Mngement f Petrleum Explrtin Ventures. uls, Olhm: he Americn Asscitin f Petrleum Gelgist. Strun Jr., W., & White, E. (009). he Elements f Style (Fiftieth Anniversry ed.). New Yr, United Sttes: Persn Eductin, Inc. 11-8