Modelng Local Uncertanty accontng for Uncertanty n the Data Olena Babak and Clayton V Detsch Consder the problem of estmaton at an nsampled locaton sng srrondng samples The standard approach to ths problem s krgng Krgng ses the spatal correlatons provded by the varogram to calclate the weghts of the sample vales srrondng an nsampled locaton The weghts obtaned from the krgng eqatons mnmze the estmaton varance and accont for the spatal correlaton between the srrondng samples and the estmaton locaton (that s, closeness to the estmaton locaton) and between sample themselves (that s, data redndancy) Krgng reslts n optmal estmaton (n the case of a known varogram model) and provdes a model for local condtonal dstrbtons; n the Gassan framework, the krgng estmate and krgng estmaton varance are exactly the mean and varance of the local condtonal Gassan dstrbtons Oftentmes, however, the exact sample data are not known de to measrement errors In ths case smple krgng can not be drectly appled to nfer the local condtonal dstrbtons A theoretcal framework for ncorporatng data ncertanty nto calclaton of the local ncertanty dstrbtons s reqred Smple Krgng The smple krgng estmator predcts the vale of the varable of nterest at the estmaton locaton as a lnear combnaton of nearby observatons ), =,, n (, (Jornel and Hbregts, 978): ) z * ( ) ( ) = λ + λ m, () = = T where m denotes the statonary mean, λ = ( λ,, λn ) denotes the vector of the smple krgng weghts calclated from the normal system of eqatons = λ Cov( ), )) = Cov(, )), =,,, () where Cov( ), )),, =,,, denotes the data-to-data covarance vales and Cov(, )), =,,, s the data-to-estmaton pont covarance vales The covarance fncton s calclated nder statonarty throgh the semvarogram model γ (h) Smple krgng s the best lnear nbased estmator, that s, t provdes estmates wth mnmm error varance n the least sqare sense gven by = C(0) λ Cov(, )), () where C (0) s the statonary varance = In the Gassan framework the local condtonal dstrbtons are derved by smple krgng as follows Uncertanty at the estmaton locaton s z *, ) () Accontng for the Uncertanty n Data Let s assme that each of the observatons ), =,, n (, avalable for analyss was measred wth some measrement error Frther assme that the measrement errors are dstrbted accordng to 7-
Gassan (normal) dstrbton; ths, ncertanty n each observaton (random varable) ), =,, n ( can be expressed as follows: ), ), =,, n (, (5) where and denote the mean and varance of the ncertanty dstrbton n -th data For now let s assme that the observatons ), =,, n (, represent ndependent random varables, eg, each data locaton was measred sng dfferent measrement tool When the observatons are no longer assmed to be known, the mean of the local condtonal dstrbtons s a random varable The varance of the local condtonal dstrbtons gven n () s not a random varable becase the smple krgng varance s homoscedastc (see ) Becase the mean of the local condtonal dstrbton s a random varable, the ncertanty at the nsampled locaton s descrbed the followng herarchcal model, ), ], ]), (6) where ) ( ) ( ) = λ + λ m (7) = = Note that dstrbton of s Gassan becase t s a lnear combnaton of Gassan random varables Frthermore, de to (7), the mean and varance of the dstrbton for can be calclated as follows: ] E λ = + λ m = = (8) λ )] λ m λ = + = + λ m = ; = = = = + ] = Var λ ) λ m = = (9) = Var λ ) = λ )] = λ = = = = Ths, t follows that the local condtonal dstrbtons n the case of data ncertanty can be expressed sng the followng herarchcal model:, ), and ), ), (0) where are gven n (8)-(9) Moreover, note that the mean and the varance of local condtonal dstrbtons are gven by: E ] = ]] = ] = ; () [ ] = ]] + = ] + ] = + Z The shape of the local ncertanty n Z ( s Gassan * ]] () 7-
It worth notng that when the observatons ), =,, n (, do not represent ndependent random varables, bt are correlated wth a prescrbed correlaton strctre, the mean and varance of the local condtonal dstrbtons can be calclated followng the same steps as before expect varance of needs to be calclated as ] = Var = λ + = λ m = Var λ ) = λ λ Cov[ ), )] = = = = Moreover note that the above dervatons heavly rely on the assmpton that the varogram model for the stdy doman s known; ncertanty n the data does not mpact the assmpton of the statonary varogram model Small Examples Example : Consder the data confgraton shown n Fgre In total, there are condtonng data avalable for nference of the local condtonal dstrbton at the nsampled locaton All condtonng data are known sbect to measrement errors; the dstrbtons of the condtonng data are Gassan wth dfferent means and varances, =,,, () see Table below Stdy doman of sze 0 by 0 nts s assmed to be statonary; statonary mean and varance are 0 and, respectvely The varogram of the data s a sngle strctred sphercal wth ngget effect of 0 and range of correlaton of 0 nts Table : Data locatons and vales Data Data Data Data Unsampled Locaton X poston 5 9 5 Y poston 7 8 5 Vale N (, ) N (, ) N (, ) N (, )? We wll vary the means and varances of the condtonal data dstrbtons to assess the mpact of data ncertanty on the resltng local ncertanty dstrbton nferred from smple krgng Frst, let s fx s as follows = 08; = 0; = 0; = 0; and examne the effect of s Table show reslts for fve dfferent scenaros for s Note that Table shows only reslts for the varance of the local dstrbton of ncertanty accontng for data ncertanty, that s, Var [ ] ; ths s becase the mean of the local condtonal dstrbton s ndependent of s and eqal to 0088 Table : Effect of s on the local ncertanty dstrbton Case Case Case Case Case 5 0 0 0 05 08 0 0 0 06 09 0 0 0 0 06 0 0 0 0 07 Var [ ] 009 ( ) 0507 0587 0657 0795 7-
It can be clearly noted from Table that wth an ncrease n the data ncertanty (that s, ncrease n the varance of the condtonal data dstrbtons), the varance of the local condtonal dstrbton at the nsampled locaton ncreases Moreover, when there s no ncertanty n the condtonng data; the varance of the local condtonal dstrbton at the nsampled locaton s eqal to smple krgng varance On the other hand, f we fx s as: = 08; = 0; = 0; = 0; we can observe that wth ncrease n the mean of the condtonal data dstrbtons, the mean of the local condtonal dstrbton at the nsampled locaton ncreases, see Table Table : Effect of s on the local ncertanty dstrbton Case Case Case Case -08-0 0-0 0 0-0 0 0-0 0 0 E [ ] -09 065 0765 09780 Note that Table shows only reslts for the mean of the local dstrbton of ncertanty accontng for data ncertanty, that s, E [ ] ; ths s becase the mean of the local condtonal dstrbton s ndependent of s and eqal to0576 It s worth notng that the reslts shown n Tables - were theoretcally calclated from Eqatons ()- () There s, however, another mch more comptatonally ntensve approach based on Monte Carlo smlaton to obtan the same reslt Specfcally, n order to calclate the mean and varance of the local ncertanty dstrbton accontng for parameter ncertanty va Monte Carlo smlaton approach the followng steps need to be ndertaken: At each of the condtonng data locatons draw a vale from the condtonng data dstrbton sng Monte Carlo smlaton approach; Apply smple krgng to calclate the mean and varance of the local condtonal dstrbton sng the condtonal data generated n ; Draw a vale from the local condtonal dstrbton obtaned n Add to the database; Repeat steps - many tmes, say 0000 To show the eqvalence of the theoretcally derved local condtonal dstrbtons of ncertanty and the ones obtaned sng Monte Carlo smlaton, let s repeat analyss of Table Reslts are shown n Tables Table : Theoretcally-derved approach vs Monte-Carlo smlaton: Varance of the local ncertanty dstrbton Var [ ] Case Case Case Case Case 5 Theory 009 0507 0587 0657 0795 Smlaton 007 05078 0588 0655 0797 The reslts of theoretcally-derved approach vs Monte-Carlo smlaton approach match perfectly; the dfference between reslts of both approaches cold have been even smaller f nstead of 0000 drawngs n Monte-Carlo approach 00000 or more were sed Example : To frther nderstand the nflence of the data ncertanty on the local condtonal dstrbtons at the nsampled locatons, let s asses the change n the varance of the local condtonal dstrbtons 7-
(accontng for data ncertanty) over the stdy doman Let s consder the same data confgraton as before; set the means of the condtonng data dstrbtons at: = 8; = 0; = 0; = 0; 0 and consder three dfferent cases, that s, case, case and case 5, for s, see Table In present stdy let s also consder two dfferent varogram models, both sngle strctred sphercal wth ngget effect of 0, bt one wth range of correlaton eqal to 0 nts and the other one wth a range of 5 nts and let s compare reslts Fgre shows reslts obtaned n each case It can be clearly noted from Fgre that wth ncrease n the range of contnty, the varance of the local condtonal dstrbtons decreases The varance of the local condtonal dstrbtons s sally les n the nterval from 0 to However, t can be also hgher than, see Table 5 Table 5: Maxmm varance f the local condtonal dstrbtons over the stdy doman Maxmm Var [ ] Case Case Case 5 Range 5 00 Range 0 09967 0999 06 Conclsons In ths paper a new nterestng framework for ncorporaton of the data ncertanty nto geostatstcal estmaton s presented The theory behnd the methodology was developed n detal; theoretcal reslts were compared wth practcal reslts obtaned va drect Monte Carlo smlaton Two small examples llstratng the change n the local ncertanty when ncorporatng data ncertanty were presented Fgre : Data confgraton for Examples 7-5
Fgre : Varance of the local condtonal dstrbtons accontng for the ncertanty n the data obtaned based on a sngle strctred sphercal varogram wth ngget effect of 0 and range of contnty 5 (left) and 0 (rght) : case (top), case (mddle) and case 5 (bottom) 7-6