Testing for the Multivariate Gaussian Distribution of Spatially Correlated Data

Size: px

Start display at page:

Download "Testing for the Multivariate Gaussian Distribution of Spatially Correlated Data"

Noreen Ward
5 years ago
Views:

1 estg for the Multvarate Gaussa Dstrbuto of Spatally Correlated Data Olea Babak ad Clayto V. Deutsch Most geostatstcal smulato s based o a assumpto that the varable s multvarate Gaussa after a uvarate ormal scores trasform. It s terestg to test how far the data depart from the multvarate Gaussa dstrbuto; the decso of statoarty could be recosdered or a dfferet multvarate dstrbuto cosdered. ests for multvarate Gaussaty, however, requre data depedece, whch s rarely the case geostatstcal modelg. Dfferet techques are revewed ad a ew testg methodology s developed. hs test detfes the correct umber of falures a multvarate Gaussa settg ad provdes a measure of how far real data depart from beg multvarate Gaussa. Itroducto Geostatstcal smulato s a useful tool for modelg varables that caot be descrbed determstcally due to ther heret complexty. he most commo smulato approach s Gaussa smulato. Each varable s trasformed to a Gaussa dstrbuto. hs esures a uvarate Gaussa dstrbuto of each varable; the, a assumpto of multvarate Gaussa dstrbuto s made. Real multvarate dstrbutos are ot lkely multvarate Gaussa ad show such o-gaussa features as o-learty ad heteroscedastcty. I ths case, Gaussa smulato may ot reproduce mportat aspects of the spatal varablty of the pheomeo uder study. hs could result based predctos. herefore, a procedure for testg how far the data depart from a multvarate Gaussa dstrbuto s of great practcal terest. A umber of tests for departures from the multvarate Gaussa dstrbuto a spatal data cotext are revewed. he theory behd each test s developed ad practcal mplemetato s dscussed. A test s proposed that s based o a uvarate test after data orthogoalzato. hs test s show to be far, that s, the umber of falsely rejected tests matches perfectly the cofdece of the test. he performace of the tests s llustrated usg real petroleum reservor data. estg for a Multvarate Gaussa Dstrbuto Cosder data values Y ( u 1, Y ( u 2,, Y ( u at locatos u 1, u 2,, u that have bee ormal score trasformed. We would lke to test the assumpto of a multvarate Gaussa dstrbuto betwee these data. Specfcally, we would lke to determe f a by 1 vector of data Y = [ Y ( u 1 Y ( u 2 Y ( u ], where Y ( u, = 1,,, are ormal scores, s -varate Gaussa wth mea of zero ad by varacecovarace matrx C s gve below Var( Y ( u1 Cov( Y ( u1, Y ( u Cov( Y ( u1, Y ( u C = Cov( Y ( u, ( 1 ( ( ( (, ( Y u Var Y u Cov Y u Y u. Cov( Y ( u, ( 1 ( (, ( ( ( Y u Cov Y u Y u Var Y u Uder the assumpto of statoarty the varace-covarace matrx C s calculated through the ormal score trasformed data varogram model ad ca be rewrtte as follows C( u1, u1 C = C( u, u1 C( u, u1 C( u, u 1 C( u, u C( u, u C( u1, u C( u, u, C( u, u 15-1

2 where C( u, u j s the statoary covarace betwee u, u j, j = 1,, (Goovaerts, Now let us ote that Y = [ Y ( u1 Y ( u2 ~ N(, C, f ad oly f ~ 1 Y = L [ Y( u1 Y ( u2 ~ N(, I, ~ ~ ~ ~ 1 where Y = [ Y ( u 1 Y ( u2 ; I deotes the detty matrx of sze by ; L stads for verse of the lower tragular matrx L the Cholesky decomposto (Golub ad Va Loa, 1996 of the varace-covarace matrx C, that s, C = LL. hus, to test f Y = [ Y ( u1 Y ( u2 ~ N(, C, we eed to test f varables correspodg to dfferet elemets Y ~ are (1 stadard ormally dstrbuted ad (2 depedet (Lehma, Ufortuately, t s mpossble to test f each of the elemets vector Y ~ s stadard ormal because we have oly oe multvarate observato (spatally correlated sample. We ca, however, test f there s strog departure from the multvarate Gaussa dstrbuto. Specfcally, f Y = [ Y ( u1 Y ( u2 ~ N(, C, ~ ~ ~ ~ the vector Y = [ Y ( u 1 Y ( u2 s uvarate stadard ormally dstrbuted. hs follows drectly from the fact that f Y = [ Y ( u1 Y ( u2 ~ N(, C, the each elemet Y ~ s stadard ormally dstrbuted. herefore, to test f the data departs strogly from the multvarate Gaussa we ca apply Komogorov- ~ ~ ~ ~ Smrov test (Berry ad Ldgre, 199 to test f Y = [ Y ( u Y ( u Y ( ] s uvarate stadard 1 2 u ormal. If the trasformed data Y ~ fals the Komogorov-Smrov test, the we coclude that our data s strogly o multvarate Gaussa ( I departure from a multvarate Gaussa dstrbuto. However, f the ull hypothess of the stadard ormalty of the Y ~ s ot rejected, the we caot coclude that the data s multvarate Gaussa. It oly meas that there s o strog departure from the multvarate Gaussa dstrbuto at the partcular of sgfcace. I partcular, f our data fal to reject the uvarate test for multvarate Gaussaty, we ca devse a test for a bvarate Gaussa dstrbuto. We kow that f Y = [ Y ( u1 Y ( u2 ~ N(, C, the matrx Y ( u Y ( u Y M = Y ( u Y ( u Y ( u 4 1 Y ( u s bvarate stadard ormal. A test for a bvarate Gaussa dstrbuto amouts to a Komogorov-Smrov test f Y M s bvarate stadard ormal. he f the data fals ths test, the t s ot multvarate Gaussa ormally dstrbuted ( II departure from a multvarate Gaussa dstrbuto; however f the ull hypothess s ot rejected t does ot mea mea that our data s multvarate Gaussa, t oly meas that addtoal testg s requred. Hgher order tests ca be desged smlar fasho. Depedg o the sze of the spatally correlated data set at had, multvarate tests of dfferet orders ca be coducted to detfy evdet departures from a multvarate Gaussa dstrbuto. I geeral, however, because real spatal data are ot multvarate Gaussa ature, coductg a uvarate test for a multvarate Gaussa dstrbuto s lkely suffcet. A logcal cocluso from the statemet that real 15-2

3 data are ot multvarate Gaussa s that testg for a multvarate Gaussa dstrbuto s amless. However, test results are terestg because they provde a measure of how far the data depart from a multvarate Gaussa assumpto. hs measure could be used to detfy problematc data. Data that depart a great deal from a assumpto of multvarate Gaussa dstrbuto could be vestgated further. here may be problem data, there may be treds or other o-statoary features, ad there may be other techques better suted to the data at had. hs applcato wll be further developed Secto 5. Uvarate est of a Multvarate Gaussa Dstrbuto (UMG he test proposed above requres kowledge of the statoary varogram model ; however, the true uderlyg varogram s ot kow. o accout for the fact that the varogram/covarace of the ormal score trasformed data s ukow, the followg modfcato s cosdered. Istead of testg f ~ 1 Y = L [ Y( u1 Y( u2 Y( u] s stadard ormal we wll test ~ Y ˆ 1 = L [ Y ( u1 Y ( u2 s ormally dstrbuted 1 where L ˆ stads for verse of the lower tragular matrx Lˆ the Cholesky decomposto of the varace-covarace matrx Ĉ obtaed based o modeled expermetal varogram. No restrcto s made o the mea ad varace of the ormal dstrbuto. he test s (, 1967 used rather tha the covetoal Kolmogorov-Smrov test because t accouts for the fact that the uderlyg statstcal parameters are ot kow. he recommeded test ths paper wll be deoted UMG for a Uvarate est for a Multvarate Gaussa dstrbuto. he test accouts for spatal correlato ad the fact that the varogram s ukow. I summary, (1 ormal score trasform the data, (2 calculate ad ft a varogram, (3 orthogoalze the data accordg to the covarace matrx, ad (4 perform the test o the result. A umber of small studes are documeted to show that the proposed correctos are ecessary. Cosder 5 samples of 5 data spaced a ut dstace apart from a multvarate Gaussa dstrbuto to wth spatal correlato. he followg sotropc sphercal varograms were used: γ 1( h = Spha= 5 ( h; γ Sph ( ; 2 ( h = a= 2 h 3( h = Sph a = 1 ( h +.2Spha= 2 ( h 4 ( h = Sph a = 5 ( h +.2Spha= 1 ( h γ γ Because the data are already multvarate Gaussa, o ormal score trasformato was doe. ests were coducted at four dfferet s of sgfcace α, α =.2,.1,. 5 ad.1. he umber of Komogorov-Smrov tests that rejected the ull hypothess for each value of α was recorded. he same procedure based o Komogorov-Smrov tests but wth varace-covarace matrx C calculated usg modeled expermetal varograms was also calculated for comparso. Note that expermetal varogram for each of 5 sampled was calculated ad modeled separately based o gamv from GSLIB (Deutsch ad Jourel, 1998 usg two sphercal varogram structures. Results of the test for each cases are gve ables 1-4 for all four dfferet varogram models. ables 1-4 also show respectve results of the corrected UMG (correcto s made for the ukow covarace. he results of the corrected uvarate test of multvarate Gaussa dstrbuto show ables 1-4 were obtaed usg test of ormalty (, 1967 coducted usg modeled expermetal varograms. Note that the case of a kow varogram model, the Komogorov-Smrov test results are very close to that predcted by theory; however, whe the varogram model s ukow, the results of Komogorov- Smrov test are usually very dfferet from the theoretcal expectato. Results of the test, o the other had, are very close to the theoretcal expectato for the ukow varogram model. he results ;. 15-3

4 of the Komorov-Smrov test ad test were smlar whe modeled expermetal varogram were close to true varogram models used smulato, see ables 3 ad 7. ables 5-8 show results of the smulato study for 5 samples of 2 data each. he same varogram models as before were used smulato. he coclusos of the aalyss of ables 5-8 are the same. able 1: Number of rejected tests for 5 samples of sze 5 data each from multvarate Gaussa dstrbuto. Results were obtaed based o 1( h Cofdece heoretcal Komogorov-Smrov Komogorov-Smrov test test wth kow test wth ukow (wth ukow 8% % % % able 2: Number of rejected tests for 5 samples of sze 5 data each from multvarate Gaussa dstrbuto. Results were obtaed based o 2 ( Cofdece heoretcal Komogorov-Smrov Komogorov-Smrov test test wth kow test wth ukow (wth ukow 8% % % % able 3: Number of rejected tests for 5 samples of sze 5 data each from multvarate Gaussa dstrbuto. Results were obtaed based o 3 ( Cofdece heoretcal Komogorov-Smrov Komogorov-Smrov test test wth kow test wth ukow (wth ukow 8% % % % able 4: Number of rejected tests for 5 samples of sze 5 data each from multvarate Gaussa dstrbuto. Results were obtaed based o 4 ( Cofdece heoretcal Komogorov-Smrov Komogorov-Smrov test test wth kow test wth ukow (wth ukow 8% % % %

5 able 5: Number of rejected tests for 5 samples of sze 2 data each from multvarate Gaussa dstrbuto. Results were obtaed based o 1( h Cofdece heoretcal Komogorov-Smrov test wth kow Komogorov-Smrov test wth ukow test (wth ukow 8% % % % able 6: Number of rejected tests for 5 samples of sze 2 data each from multvarate Gaussa dstrbuto. Results were obtaed based o 2 ( Cofdece heoretcal Komogorov-Smrov Komogorov-Smrov test test wth kow test wth ukow (wth ukow 8% % % % able 7: Number of rejected tests for 5 samples of sze 2 data each from multvarate Gaussa dstrbuto. Results were obtaed based o 3 ( Cofdece heoretcal Komogorov-Smrov Komogorov-Smrov test test wth kow test wth ukow (wth ukow 8% % % % able 8: Number of rejected tests for 5 samples of sze 2 data each from multvarate Gaussa dstrbuto. Results were obtaed based o 4 ( Cofdece heoretcal Komogorov-Smrov Komogorov-Smrov test test wth kow test wth ukow (wth ukow 8% % % % he Multvarate Aspect of the UMG he data beg tested are uvarate Gaussa by desg (Deutsch, 22. he uvarate test of a multvarate Gaussa dstrbuto (UMG truly tests the multvarate dstrbuto. Cosder a small example to llustrate ths fact. A strg of 613 data s avalable for aalyss. he data the strg are equally sampled wth a dstace of.1 meters betwee the samples. he hstogram of the data s show Fgure 1. he uvarate dstrbuto of the data s o-gaussa. Fgure 1 also shows the hstogram of the ormal score trasformed data; the dstrbuto of the ormal score trasformed data s perfectly Gaussa (p-value of the 15-5

6 Komogorov-Smrov test s 1. he varogram of the ormal score trasformed data s show Fgure 2. he modeled expermetal varogram s gve below: γ ( h = Sph a = 1( h +.2Spha= 2 ( h. Fgure 2 also shows the varogram of the Y ~ data obtaed by removg the spatal structure from the ormal score trasformed data. It ca be clearly see from Fgure 2 that observatos Y ~ are depedet; the varogram model of Y ~ s pure ugget. Fgure 3 shows the hstogram of Y ~. It s clear from Fgure 3 that these modfed values are ot uvarate Gaussa. he statstc for testg uvarate ormalty at sgfcace α =. 1 s.128 (crtcal value s.446, p-value s <<.1. hus, the data are strogly ot multvarate Gaussa; however, the uvarate dstrbuto of the ormal score trasformed data s perfectly Gaussa. Software Implemetato A program called testg_mg was specfcally prepared to test strog departures from multvarate Gaussaty usg UMG. he software mplemetato of ths program s cosstet wth all the FORRAN programs of GSLIB group (Deutsch ad Jourel, he parameter fle for program testg_mg s preseted below: A ru of the program testg_mg creates a output drectly the output wdow wth the followg formato: reject or do ot reject the ull hypothess; crtcal value of the test ad statstc. Note that the data putted to the program testg_mg must be ormal score trasformed. Selectg Data wth the UMG he UMG has oe very terestg applcato. hs applcato allows us to measure how real data depart from a multvarate Gaussa dstrbuto. We do ot expect ay of the data (eve after ormal score trasformato to be multvarate Gaussa; however, the UMG ca rak data accordg to the statstc. he data wth lowest value of the statstc ca be thought of most closely resemblg the multvarate Gaussa dstrbuto. Cosder a small case study. Fve strgs of 9 data each are avalable for aalyss. he data the strgs are equally sampled wth dstace of.1 meters betwee the samples. he hstograms of the data are show Fgure 4. Lookg at Fgure 4 we ca ote that all data sets exhbt dfferet o-gaussa features, therefore ormal score trasformato s employed to make each data set uvarate Gaussa. he expermetal varograms ad ther fts for the fve ormal score trasformed data sets are show Fgure 5. I order to rak the data wth respect to departure from a multvarate Gaussa dstrbuto, the ~ test of ormalty for Y ˆ 1 = L [ Y ( u1 Y ( u 2 Y ( u ] wll be employed for each ormal score trasformed data set. able 9 shows results of the test for all fve ormal score trasformed data sets. 15-6

7 able 9: Result of test for all 5 data sets cosdered the case study. statstc Rak Crtcal Value at α =. 2 Decso Crtcal Value at α =. 1 Decso Crtcal Value at α =. 1 Decso Reject H.268 Reject H.368 Reject H Reject H.268 Reject H.368 Reject H Reject H.268 Reject H.368 Reject H Reject H.268 Reject H.368 Reject H Reject H.268 Reject H.368 Reject H Note that all fve data sets are uvarate Gaussa after ormal score trasformato; however, oe of them are multvarate Gaussa. hey all depart strogly from a multvarate Gaussa dstrbuto. hs result holds true for ay commoly accepted of sgfcace α. Note also that each of the fve data sets has a dfferet value of the statstc. he data sets ca be raked wth respect to the value of the statstc. Data wth the smallest value of the test statstc wll receve rak 1 ad s cosdered to be closest to the multvarate Gaussa amog all data sets. I the partcular case cosdered above data set 4 s the closest data set to beg multvarate Gaussa, whle data set 3 s the furthest away from beg multvarate Gaussa. Subsets of dfferet szes were cosdered to see f the test would preserve the rak order departure from a multvarate Gaussa dstrbuto. able 1 shows results for the average test statstc calculated based o 5 data sets of szes 899 data pots, 5 data pots ad 2 data pots from the fve data sets. o calculate the test statstc each of the selected data sets was ormal score trasformed depedetly of other data. he varogram models for the selected data sets were ot recalculated. hs was doe maly to avod artfacts that could be observed automatc fttg of the varograms. Fgure 6 shows the varogram models obtaed for 9 ormal score data (all data of data set 3 ad to ormal score trasformed 899 frst data data sets 3. Note that despte the fact that expermetal varograms are vrtually detcal, there s a clear msmatch the varogram fts produced by automatc fttg program varft. he dfferece s especally sgfcat at short lag dstaces that has the most mpact o the testg procedure. able 1: Result of test for all 5 data sets cosdered the case study. statstc Rak Average statstc for 5 data sets of 899 data Rak based o average statstc Average statstc for 5 data sets of 5 data Rak based o average statstc Average statstc for 5 data sets of 2 data Rak based o average statstc Data Data Data Data Data he preservato of the rak posto able 1 allows us to coclude that the uvarate test of a multvarate Gaussa dstrbuto s robust. Fgure 7 shows the hstograms of the values of test statstc for 5 data sets of sze 5 data pots obtaed for each of the fve data sets. Lookg at Fgure 7 we ca ote that values of test statstc chage from data set to data set selected from the same data. he crtcal value of the test for data sets of sze 5 at sgfcace α =. 1 s.493. Noe of the data sets are 5-varate Gaussa eve at such low of sgfcace. Note, however, that as sze of the data sets decreases to 2, we ca observe that some of the chose data sets do ot cotue to exhbt strog departures from multvarate Gaussaty. I partcular, majorty of data sets selected from data set 1 (5.2% ad great majorty of data sets from data 4 (65% ca be cosdered 2-varate Gaussa at sgfcace 15-7

8 α =.1 (crtcal value of test for data sets of sze 2 at α =. 1 s.78. Note also that oe of the data sets from data 3 ad oly 4% of data sets from data 2 ad 8.6% of data sets at sgfcace α =. 1 ca be cosdered as ot exhbtg strog o mult-gaussa features. Further, to vestgate f the raks calculated based o test statstc are preserved for data selected wth the same cofgurato from 5 dfferet data sets aother small aalyss was performed. For each of 5 data sets of sze 5 we calculated how may tmes of the 5 data sets selected from data 1 was rak as umber 1, 2, 3, 4 ad 5. he same was doe for 5 data sets selected from data 2, data 3, data 4 ad data 5. Results of ths aalyss are gve able 11. It s apparet from able 11 that the rak of test statstc s preserved for majorty of data sets selected from each of the 5 data sets cosdered a case study. Moreover, lookg at the results preseted able 11, we cofrm that the uvarate test of a multvarate Gaussa dstrbuto s robust ad therefore applcable for rakg multple datasets wth respect to departures from a multvarate Gaussa dstrbuto. able 11: Results for rakg of 5 data sets of 5 data each selected for each of the 5 data sets cosdered the case study. Rak 1 Rak 2 Rak 3 Rak 4 Rak 5 Data Data Data 3 5 Data Data Coclusos he logstadg problem of testg for a multvarate Gaussa dstrbuto of spatally correlated data s cosdered. A umber of smple tests for testg strog departures from mult-gaussa dstrbuto have bee developed. he proposed UMG test s far; the umber of falsely rejected tests matches perfectly the cofdece of the test. Performace of the tests was llustrated usg real petroleum data. A terestg ew approach for rakg data accordg to closeess to a multvarate Gaussa dstrbuto was proposed. hs approach uses the statstc as a measure of departure from a multvarate Gaussa dstrbuto was show to be very robust a case study. Refereces Berry, D. A., ad Ldgre, B. W., 199, Statstcs: heory ad Methods. Brook/Cole, Pacfc Grove, CA. Deutsch, C.V., 22, Geostatstcal Reservor Modelg. Oxford Uversty Press, New York. Deutsch, C.V., Jourel, A.G., 1998, GSLIB: Geostatstcal Software Lbrary ad Users Gude. Oxford Uversty Press, New York Goovaerts, P., 1997, Geostatstcs for Natural Resources Evaluato. Oxford Uversty Press, New York. Golub, G.H. ad Va Loa, C.F., 1996, Matrx Computatos. Johs Hopks Uversty Press Lehma, E.L., 1999, Elemets of Large-Sample heory. Sprger-Verlag., H., 1967, O the Kolmogorov-Smrov test for ormalty wth mea ad varace ukow. Joural of the Amerca Statstcal Assocato, 62:

9 Fgure 1: Hstogram of the 613 data orgal uts (left ad ormal score uts (rght. Fgure 2: he expermetal varogram of the ormal score trasformed 613 data (dark dots ad ts varogram model (le. Expermetal varogram of the Y ~ data obtaed by removg the spatal structure from the ormal score trasformed data are show lght dots. Fgure 3: Hstogram of Y ~. 15-9

10 Fgure 4: Hstograms of the fve data sets of 9 data each used a case study. 15-1

11 Fgure 5: he expermetal ad model varograms for the fve ormal score data sets. Fgure 6: Varogram models ftted by varft to 9 ormal score data (all data of data set 3 (left ad to ormal score trasformed 899 frst data data sets 3 (rght

12 Fgure 7: Hstograms of the values of test statstc for 5 data sets of sze 5 data pots obtaed for each of the fve data sets cosdered a case study

Summary of the lecture in Biostatistics

Summary of the lecture in Biostatistics Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the