ITERATIVE ESTIMATION PROCEDURE FOR GEOSTATISTICAL REGRESSION AND GEOSTATISTICAL KRIGING
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1 ESE 5 ITERATIVE ESTIMATION PROCEDURE FOR GEOSTATISTICAL REGRESSION AND GEOSTATISTICAL KRIGING Gven a geostatstcal regresson odel: k Y () s x () s () s x () s () s, s R wth () unknown () E[ ( s)], s R () cov[ ( s), ( v)] C sv ;, s, v R (v) ( rsa,, ) unknown (v) For each fnte set, { s,, s n } R the resdual vector [ ( s),, ( s n )] s dstrbuted ultvarate noral together wth observed data, y ( y :,, n) and X ( x,, xn), at spatal locatons { s :,, n} R, the task s to estate the paraeters, (, ), and to test hypotheses about the paraeters Ths s accoplshed by a varaton of the Unversal Krgng estaton procedure n whch one seeks to deterne a utually consstent set of paraeter estates, (, ), as specfed below [Ths procedure s also dscussed n Baley and Gatrell (p89), and s the sae as the Iteratvely Reweghted GLS procedure n Gotway and Waller (p337)] Iteratve Estaton Procedure [] Frst construct an OLS estate, () ( X X) Xy of wth correspondng resduals, () y X [] Use these resduals to estate an eprcal varogra, ( h), at soe set of selected dstance values, ( h :,, )
2 ESE 5 [3] Next use ths data (, h),,, to ft (by nonlnear least squares) a sphercal varogra, ( h ; ), wth paraeter vector, (,, ) (3) r s a [4] Then use the dentty, covarogra, ( ) ( h) Ch, to construct the correspondng sphercal ( ) ( ;,, ) (4) C h s h r s a for all dstances h [5] If the dstance between each par of data ponts, s and s s denoted by h, then the covarance, cov(, ), between the resduals at s and s s estated by C( h ) [where by defnton, s ], and the resultng estate of the covarance atrx, C cov( ), between resduals at all data ponts,, n s gven by (5) C n n [6] Usng ths covarance atrx, now apply GLS to obtan a new estate of : (6) ( X C X) XC y wth correspondng resduals, (7) y X [7] Then replace by and apply steps [] and [3] to obtan a new sphercal varogra, ( h; ), wth paraeter vector, (,, ) (8) r s a [8] At ths pont, one can check to see f there are any sgnfcant dfferences between the ntal paraeter estates, (, ), and the new estates, (, ) Here there are any crtera to check for dfferences If one s prarly nterested n the
3 ESE 5 paraeters (as s typcal n regresson), the splest approach s to focus on fractonal changes n these estates by lettng (9) ax :,,, k One ay then choose an approprate threshold value, (say ) and defne a sgnfcant change to be If one s also nterested n the varogra paraeters, ( rsa,, ), then one ay replace (9) by the broader set of fractonal changes r r s s a a ax,,, () r s a [9] If there s no sgnfcant change, e, f (or ), then stop the teratve estaton procedure and set the fnal paraeter estates to be () (, ) (, ) [] On the other hand, f (or ), then contnue the teratve estaton procedure by replacng wth n steps [4] through [7] to obtan a new estate, () ( X C X) XC y [based on the new covarance atrx, Ĉ, constructed fro ], and new varogra paraeter estates (,, ) (3) r s a [based on the new resduals, y X ] [] Wth these new paraeters, defne (or ) as n step [8] If (or ) then stop the procedure and set the fnal paraeter estates to (4) (, ) (, ) [] On the other hand, f (or ), then contnue the teratve estaton procedure by replacng (, ) by (, ) n steps [4] through [7]
4 ESE 5 [3] Contnue n the sae way untl a set of paraeters (, ) s found for whch (or ) Then stop the procedure and set the fnal estates to (5) (, ) (, ) These fnal paraeter estates are sad to be utually consstent n the sense that the covarance atrx, Ĉ, resultng fro (approxately) reproduces as, (6) ( ) X C X X C y and slarly, that the resduals, y X, yeld an eprcal varogra, ( h), (approxately) reproducng (,, rsa ) as the ftted paraeters of the sphercal varogra yeldng Ĉ Applcaton to Geostatstcal Regresson Gven the regresson estates,, one can use the paraeter estates, (,, rsa ), to construct the fnal covarogra as follows: (7) Ch ( ) s ( h; rsa,, ) Ths covarogra s n turn used to obtan a fnal estate, (8) C n n of the resdual covarance atrx, C cov( ) [entoned n (6) above] The covarance atrx, V, of the GLS estates,, can then be obtaned n ters of C as follows: (9) ( XC X) XC Y ( XC X) XC ( X ) ( XC X) ( XC X) ( XC X) XC ( XC X) XC V cov( ) cov ( X C X) X C
5 ESE 5 ( X C X) XC cov( ) C X ( XC X) ( X C X) XC CC X ( XC X) ( X C X) ( XC X)( XC X) ( X C X) Hence (8) yelds the followng estate of V, () v k V ( X C X) v k v v kk whch n turn yelds standard error estates () s v for each beta paraeter estate,,,,, k These standard errors can then be used to construct p-values for sgnfcance tests of these coeffcents based on the t-ratos: () t / s,,,, k 3 Applcaton to Geostatstcal Krgng Recall that Unversal Krgng used a pror estate of the varogra paraeters ( rsa,, ) based on OLS resduals But one can prove ths procedure by usng the utually consstent estates [( rsa,, ), ] obtaned above In partcular, usng the beta estates,, and covarance estates, Ĉ, [fro (7) and (8) above] for the estates n steps (5) and (6) of the Unversal Krgng notes, one can proceed to step (7) and nterpolate unobserved values, Ys, ( ) at any other locatons, s R Thus, the essental dfference between that procedure and the present one, whch we desgnate as Geostatstcal Krgng, s that rather than stoppng after step [5] above, one contnues untl a utually consstent set of paraeter values s obtaned Whle the present procedure s ore coputatonally ntensve, t should generally yeld ore stable paraeter estates and krged predctons
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