Uncertainty as the Overlap of Alternate Conditional Distributions

Transcription

1 Uncertanty as the Overlap of Alternate Condtonal Dstrbutons Olena Babak and Clayton V. Deutsch Centre for Computatonal Geostatstcs Department of Cvl & Envronmental Engneerng Unversty of Alberta An mportant task n modern geostatstcs s the assessment and quantfcaton of resource and reserve uncertanty. Ths uncertanty s valuable decson support nformaton for many management decsons. Uncertanty at specfc locatons and uncertanty n the global resource s of nterest. There are many dfferent methods to buld models of uncertanty ncludng Smple Krgng, Inverse Dstance, Smple Cokrgng, and so on. Each method leads to dfferent results. We propose a method for combnng local uncertantes predcted by dfferent models to obtan a combned measure of uncertanty that, deally, combnes the good features of each alternatve. The new estmator s obtaned as an overlap of alternate condtonal dstrbutons. Uncertanty as the Overlap of Alternate Condtonal Dstrbutons Let us consder n data on spatal random varable Y (u) at locatons u, = 1,, n n the doman of nterest A. We consder the problem of estmatng the value of the varable of nterest Y at an unsampled locaton u* and quantfcaton of the local uncertanty at ths estmaton locaton. Consder two dfferent estmaton technques (e.g., Smple Krgng and Smple Cokrgng) avalable to obtan local uncertanty at the estmaton locaton u*. Fgure 2 shows a schematc representaton of the results for the local uncertanty at locaton u*. Ideally, we would only have one dstrbuton of uncertanty. In presence of many alternatves, we should smply choose the best one. Best s defned as the dstrbuton wth the most theoretcally vald approach, the greatest fdelty wth the data, the smplest to apply and so forth. In practce, however, dfferent technques are best n dfferent senses. There may be a need to reconcle dstrbutons from dfferent sources that all have some legtmacy. In the case consdered above, t s known that both technques provde theoretcally vald model of uncertanty and estmate the same varable at the same estmaton locaton; however, the results are dfferent. Because both technques are theoretcally vald, there s a zero probablty that varable of nterest Y (u) wll be equal to some value for whch one of the estmators ndcates zero probablty. In general, the local uncertanty of Y (u) at an unsampled locaton u* could be consdered to be the mnmum or overlap of the local uncertantes obtaned by two dfferent estmaton technques (scaled to 1). The uncertanty model obtaned as an overlap of the two modeled local uncertantes s often narrow and appears as a reasonable result. The probablty s the hghest that the value at the unsampled locaton s n the nterval common to both estmators. Ths estmator wll be referred to as a overlap estmator. Fgure 1 shows schematc representaton of the overlap of the two the local uncertanty models at locaton u*. Ths approach can easly be extended to the case of several dfferent estmators and models. In partcular, Smple Krgng, Ordnary Krgng (wth bas correcton), Inverse Dstance, f secondary nformaton s avalable then also Smple Cokrgng, Bayesan Updatng, etc. all can be consdered. Then the overlap estmator can take on arbtrary shape, see schematc examples n Fgure 2. Example wth Inverse Dstance and Smple Krgng To explan the dea of the overlap uncertanty estmator, we wll lmt ourselves n ths secton to consderng standard normal spatal random varable and only two technques for ts estmaton, that s, Inverse Dstance and Smple Krgng. A short descrpton of each of the two technques follows

2 Inverse Dstance Interpolator An Inverse Dstance (ID) weghted estmate of the varable of nterest Y at an unsampled locaton u* s a spatally weghted average of the sample values wthn a search neghborhood (Shepard, 1968; Franke, 1982; Dadato and Ceccarell, 2005). It s calculated as Y ID n *( u) = β Y ( u ), (1) where β, = 1,, n, are the weghts assgned to each sample. The weghts are determned as = 1 1 p d β =, ( = 1,, n), (2) n 1 p 1 d = where d are the Eucldan dstances between estmaton locaton and sample ponts, and exponent p s the power or dstance exponent value. The mean and varance of the Inverse Dstance estmator Y ID * ( u) at estmaton locaton u * under assumpton of statonarty are gven by E( Y ID * ( u)) = m; (3) n n 1 1 Cov( Y ( u ), ( )) Y u p p = 1 = 1 d ( * ( )) d Var Y = ID u ; (4) 2 n 1 p k = 1 d k where m s statonary doman mean (for standard normal random varable Y t s zero), d are the Eucldan dstances between estmaton locaton and sample ponts, exponent p s the power or dstance exponent value used n nverse dstance nterpolaton and Cov( u,u ),, = 1,, n, denotes data-to-data covarance functon calculated though the semvarogram model 2γ ( h). Smple Krgng Smple Krgng s a well establshed lnear estmaton technque that provdes an estmate of the unsampled value Y(u) as a lnear combnaton of neghborng observatons Y ( u ), = 1,, n, accountng for the spatal contnuty of the varable under study (Deutsch, 2002). The Smple Krgng estmator predcts the value of the varable of nterest as n n Z * SK (u) Y (u ) = λ + 1 λ m, (5) = 1 = 1 where m denotes the statonary doman mean (for standard normal random varable Y t s zero), T λ = ( λ 1,, λn ) denotes the vector of the Smple Krgng weghts calculated from the normal system of equatons for the estmaton locaton u*, 408-2

3 n = 1 λ Cov( Y (u ), Y (u )) = C( Y ( u*), Y (u )), = 1,, n, (6) where Cov( Y (u ), Y (u )),, = 1,, n, denotes data-to-data covarance functon and Cov( Y (u*), Y (u )), = 1,, n, s data-to-estmaton pont covarance functon calculated though the semvarogram model 2γ ( h). Smple Krgng s the best lnear unbased estmator, that s, t provdes estmates wth mnmum error varance σ n the least square sense gven by 2 SK n 2 2 σ SK = σ λcov( Y (u*), Y (u )), (7) = 1 2 where σ s the statonary doman varance. The mean and varance of the Smple Krgng estmator *( u) where Var Y SK at estmaton locaton * u are gven by E( Y SK *( u)) = m; (8) n n 2 2 ( YID * ( u)) = λλ Cov( Y ( u ), Y ( u )) = σ σ SK = 1 = 1 2 σ SK s Smple Krgng varance defned n (7)., (9) Overlap Uncertanty Let us consder 10 smulated data n the doman 20 by 20 unts shown n Fgure 4. The data were generated usng zero nugget sphercal varogram model wth range of correlaton equal to the sze of the doman. Data dstrbuton s also shown n Fgure 3. Let us now determne the local uncertanty at the two estmaton locatons (8,12) and (9,9) usng nverse dstance nterpolaton and smple krgng, then subsequently calculate the overlap uncertanty estmator as the overlap of the two alternate condtonal dstrbutons. Fgure 4 shows results obtaned for estmaton locaton (8,12). Lookng at Fgure 4 we can clearly note that the local uncertanty predcted by the nverse dstance nterpolaton and smple krgng are qute dfferent. As we combne both technques to obtan the overlap uncertanty estmator, we observe that the local uncertanty by ths approach s sgnfcantly narrower than predcted by other of the two estmaton approaches. To quantfy the change n the local uncertanty at the estmaton locaton (8,12), we wll use the nterval (P10,P90). For the nverse dstance estmator ths nterval s (-2.55,-0.92); for the smple krgng estmator t s (-1.93,-0.09) and for the overlap uncertanty estmator t s (-2.14,-0.74). So, the change s sgnfcant. Fgure 5 shows results obtaned for estmaton locaton (9,9). The local uncertanty predcted by the nverse dstance nterpolaton and smple krgng are qute smlar As we combne both technques to obtan the overlap uncertanty, we observe that the local uncertanty by ths approach s smlar to local uncertanty predcted by two dfferent estmaton approaches. To quantfy the change n the local uncertanty at the estmaton locaton (9,9), we wll use agan the nterval (P10,P90). For the nverse dstance estmator ths nterval s (-2.66,-1.14); for the smple krgng estmator t s (-2.40,-1.15) and for the overlap uncertanty estmator t s (-2.44,-1.13). Thus, we can see that results n ths case of all consdered approaches are smlar. Case Study I: Cluster.dat Example To llustrate the performance of the overlap uncertanty estmator a well known GSLIB (Deutsch and Journel, 1998) data set cluster.dat s selected. The data conssts of about 100 data that are sampled on a 408-3

4 random stratfed grd and 40 data that are clustered n hgh valued areas. We dscard the clustered data. The 2-D area of nterest s 50 by 50 dstance unts. The dstrbuton of data s approxmately lognormal wth a mean of 2.5 and a standard devaton of 5.0. The spatal contnuty of the data n the normal space s descrbed by sotropc Sphercal varogram model wth range of correlaton 12 and nugget effect of 0.3. Fgure 6 shows the locaton map of the 100 data and ther dstrbuton. Fgure 6 also shows the same analyss but for the normal score transformed 100 data. The am of our analyss s to establsh local condtonal dstrbutons for all 100 data ponts n crossvaldaton usng Inverse Dstance nterpolaton approach, Smple Krgng and Overlap Uncertanty method and check results. Specfcally, the farness and accuracy of the local uncertantes s checked as well as drect results of crossvaldaton for the errors n estmaton. All analyss s conducted n normal space. Results Fgure 7 shows accuracy plots obtaned for Inverse Dstance, Smple Krgng and Overlap Uncertanty estmator predcton n crossvaldaton of 100 nscore transformed data of Fgure 6. From Fgure 7 we can clearly see that Overlap Uncertanty estmator s both accurate and precse, thus far, estmator of uncertanty. Both Smple Krgng and Inverse Dstance are accurate, but not precse estmators of uncertanty. Fgure 8 shows the crossplots between p10, p50 and p90 of Smple Krgng and Overlap Uncertanty estmator for 100 nscore transformed data. Analogous results for Inverse Dstance and Overlap Uncertanty Estmator are shown n Fgure 9. Fgure 10 shows the p10-p90 local uncertanty ntervals for the frst 10 data. One can clearly note from Fgures 8-10 that takng uncertanty as the overlap of the local condtonal dstrbutons from Smple Krgng and Inverse Dstance (for example) can result n sgnfcant reducton of the probablty ntervals and, moreover, more far local uncertanty (see Fgure 7). To assess how much narrower are local condtonal dstrbutons predcted by Overlap Uncertanty versus Smple Krgng and Inverse Dstance, crossplots between the varance of the local condtonal dstrbutons (smoothng effect) of the Smple Krgng, Inverse Dstance and Overlap Uncertanty estmators for the 100 data of the fle cluster.dat are prepared. Fgure 11 shows the results. We can observe that the average varance of the local condtonal dstrbutons obtaned by the Smple Krgng s 0.706, by the Inverse Dstance s and by the overlap of the local condtonal dstrbutons s Thus, we see that the average varance of the local condtonal dstrbutons s ndeed sgnfcantly smaller than that of Smple Krgng (more than 6% smaller) and that of Inverse Dstance (more than 16% smaller). Fgure 12 shows results of estmates crossvaldaton for Inverse Dstance, Smple Krgng and Overlap Uncertanty estmator for 100 nscore transformed cluster.dat data. Table below summarzes results shown n Fgure 12. Statstcs Inverse Dstance Smple Krgng Overlap Uncertanty Estmator Bas Correlaton From the table above, we can observe that all three estmators are vrtually unbased, The correlaton between true values and estmates are the hghest for the Smple Krgng approach, and only slghtly lower s for the Overlap Uncertanty method. Case Study II: Mne650.dat Example The second case study s based on the data set mne650.dat. The data conssts of 310 data. The 2-D area of nterest s 3000 by 5000 dstance unts. The spatal contnuty of the data s descrbed by sotropc Sphercal varogram model wth range of correlaton 1450 and zero nugget effect. Fgure 13 shows the locatons of the 310 data and ther representatve (declustered) dstrbuton. Fgure 13 also shows the same analyss but for the normal score transformed mne650.dat data. The am of our analyss s the same as before, that s, to establsh the farness and wdth of local condtonal dstrbutons for all data ponts n crossvaldaton usng Inverse Dstance nterpolaton approach, Smple Krgng and Overlap Uncertanty method and check results. All analyss s conducted n normal space

5 Results Fgure 14 shows accuracy plots obtaned for Inverse Dstance, Smple Krgng and Overlap Uncertanty estmator predcton n crossvaldaton of the nscore transformed mne650.dat data of Fgure 13. From Fgure 14 we can clearly see that all consdered uncertanty estmators are accurate, but not precse estmators of uncertanty. The queston s now how wde are local dstrbutons of uncertanty predcted by each of the approaches. Fgure 15 shows the crossplots between p10, p50 and p90 of Smple Krgng and Overlap Uncertanty estmator. One can see that local dstrbutons of uncertanty are vrtually the same, there are only a few data cases when the local uncertanty dstrbutons obtaned based on the Overlap Uncertanty estmator are narrower. The crossplots between p10, p50 and p90 for Inverse Dstance and Overlap Uncertanty estmator are shown n Fgure 16. Note that there s a huge dfference n the wdth of the local uncertanty dstrbutons. Local uncertanty dstrbutons obtaned by the Overlap Uncertanty estmaton are sgnfcantly narrower. To summarze, we can say that bascally only Smple Krgng has an mpact on the local condtonal dstrbutons obtaned by the Overlap Uncertanty approach. Ths s manly because the respectve means of the local dstrbutons modeled by Smple Krgng and Inverse Dstance are very smlar, however local dstrbutons obtaned n Inverse Dstance are much wder than obtaned n Smple Krgng. Concluson A flexble approach for combnng alternate local condtonal dstrbutons to create Overlap Uncertanty estmator was proposed. Both smulated and real case studes (one wth 100 data from fle cluster.dat and the other one wth 310 data from fle mne650.dat ) were consdered. It was shown that Overlap Uncertanty estmator can result n sgnfcantly narrower ntervals for the local uncertanty. Good results are not guaranteed. One should always check whether the local dstrbutons obtaned by the approaches are accurate. References Deutsch, C.V., Geostatstcal Reservor Modelng, Deutsch, CV and Journel, A.G.: GSLIB: Geostatstcal Software Lbrary and Users Gude, Oxford Unversty Press, New York, second edton, Dadato, N. and Ceccarell: Interpolaton Processes usng Multvarate Geostatstcs for Mappng of Clmatologcal Precptaton Mean n the Sanno Mountans (southern Italy), Earth Surface Processes and Landforms, Franke, R., Scattered Data Interpolaton: Tests of Some Methods, Mathematcs of Computaton, Shepard, D.: A Two-Dmensonal Interpolaton Functon for Irregularly Spaced Data, Proc. 23 rd Nat. Conf. ACM,

6 Fgure 1: A schematc representaton of the example results for two dstrbutons at locaton u*. Sold and dashed lnes represent local uncertanty obtaned by two dfferent approaches; red area s the overlap. Fgure 2: A schematc representaton of the two example results for the four local uncertanty at locaton u*.sold dashed, doted and dash-dot lnes represent local uncertanty obtaned by four dfferent estmaton approaches; red area s an overlap. Fgure 3: Locaton map of 10 data (left) and ther dstrbuton

7 Fgure 4: Locaton map of 10 data (crcles) together wth estmaton locaton 1 (8,12) (astersk) (top left), result of the Inverse Dstance and Smple Krgng for the local uncertanty at the estmaton locaton 1 (top rght), not scaled Overlap Uncertanty estmator together wth Inverse Dstance and Smple Krgng local uncertanty models (bottom left), and (scaled). Fgure 5: Locaton map of 10 data (crcles) together wth estmaton locaton 2 (9,9) (astersk) (top left), result of the Inverse Dstance and Smple Krgng for the local uncertanty

8 Fgure 6: Locaton map of 100 data from fle cluster.dat (top left) and ther dstrbuton (top rght); Locaton map of the 100 normal score transformed data from fle cluster.dat (bottom left) and ther dstrbuton n normal space (bottom rght). Fgure 7: Accuracy plots for Inverse Dstance, Smple Krgng and Overlap Uncertanty estmator predcton n crossvaldaton of 100 nscore transformed prmary data from fle cluster.dat

9 Fgure 8: Crossplot between p10 (top left), p50 (top rght) and p90 (bottom) of Smple Krgng and Overlap Uncertanty Estmator for 100 nscore transformed prmary data from fle cluster.dat. Fgure 9: Crossplot between p10 (top left), p50 (top rght) and p90 (bottom) of Inverse Dstance and Overlap Uncertanty Estmator for 100 nscore transformed prmary data from fle cluster.dat

10 Fgure 10: p10-p90 probablty ntervals obtaned for the frst 10 data n the cluster.dat data set based on Smple Krgng (dash-dot lnes), Inverse Dstance nterpolaton (dashed lnes) and Overlap Uncertanty Estmator (sold lnes). Medans (p50) for each of the three consdered approaches are shown by dots. Fgure 11: Crossplots of between the varance of the local condtonal dstrbutons (smoothng effect) of the Smple Krgng, Inverse Dstance and Overlap Uncertanty estmators obtaned for the 100 data of the fle cluster.dat

11 Fgure 12: Crossvaldaton result for Inverse Dstance (top left), Smple Krgng (top rght) and Overlap Uncertanty estmator (bottom) for 100 nscore transformed prmary data from fle cluster.dat. Fgure 13: Locaton map of the data from fle Mne650.dat (top left) and ther representatve dstrbuton (top rght); Locaton map of the normal score transformed data (bottom left) and ther dstrbuton n normal space (bottom rght). Fgure 14: Accuracy plots for Inverse Dstance (top left), Smple Krgng (top rght) and Overlap Uncertanty estmator (bottom) predcton n crossvaldaton of nscore transformed data from fle Mne650.dat

12 Fgure 15: Crossplot between p10 (top left), p50 (top rght) and p90 (bottom) of Smple Krgng and Overlap Uncertanty Estmator for nscore transformed data from fle Mne650.dat. Fgure 16: Crossplot between p10 (top left), p50 (top rght) and p90 (bottom) of Inverse Dstance and Overlap Uncertanty Estmator for nscore transformed data from fle Mne650.dat