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## SP TOPS STAT THRY STAT ANLYS MISS DATA

by: Helga Torp Sr.

28

0

13

# SP TOPS STAT THRY STAT ANLYS MISS DATA STA 695

Marketplace > University of Kentucky > Statistics > STA 695 > SP TOPS STAT THRY STAT ANLYS MISS DATA
Helga Torp Sr.
UK
GPA 3.87

Staff

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COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
13
WORDS
KARMA
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This 13 page Class Notes was uploaded by Helga Torp Sr. on Friday October 23, 2015. The Class Notes belongs to STA 695 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 28 views. For similar materials see /class/228275/sta-695-university-of-kentucky in Statistics at University of Kentucky.

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Date Created: 10/23/15
Anisotropic Variogram De nition Suppose Z is an intrinsically stationary process 0 Z is isotropic if the spatial dependence between Z s and Z s h is a function of the size or length but not the direction of h 0 Z is anisotropic if the spatial dependence between Z s and Z s h is a function of both the size or length and the direction of h Examples 0 Topography soil moisture 0 Wind directions air pollution 111 Anisotropic Variogram Types of anisotropy o Zonal anisotropy The sill changes with direction but the range remains the same for different directions This is not much discussed in the literature 0 Geometric anisotropy The range changes with direction but the sill remains the same for different directions We Will focus on geornetric anisotropy How to detect geornetric anisotropy 112 Anisotropic Variogram Geometric anisotropy 0 Let Z1s be a intrinsically stationary process with isotropic semiyariogram function MHhH mean u and variance 02 0 Let A be a real d X d matrix and consider the stochastic process Zs Z1As 0 Because Z1s is intrinsically stationary we have EZ u and VarZ 02 Furthermore 7h VarZs h Zs VarltZ1As h Z1As 71Ah 71llAhll Then 7h is geometrically anisotropic 0 To correct for geometric anisotropy this transformation of the coordinate system can be reversed o Zs ZBs is isotropic if B A4 113 Anisotropic Variogram Directional variogram 0 De ne a directional semivariogram as W m 2 ma as Where the sum is over all the pairs sz and sj that are h distance apart and that the direction from si to sj is some xed direction 0 Binnng ie tolerance region is based on not just the size but also the direction of h o For example directions could be 450i2250 or 450i11250 or 450 i 50 o For 6 i A6 the angle 8 is called the azimuth and A9 is called the tolerance azimuth R commands gt wellsvario4 lt variog4wells estimatortypequotmodulusquot maxdist07 gt plotwellsvario4 sameT typequotbquot gt plotwellsvario4 sameF typequotbquot 114 Anisotropic Variogram Example Directional semivariograms in one plot Lo 3 39 0 g 45 5 I 4 90 5 135 8 e L4 1 394 C E a T E I U Lo O I I I I I I I 00 01 02 03 04 05 06 distance 115 Anisotropic Variogram Example Directional semivariograms in separate plots sem i variance sem i variance 1O 15 5 1O 15 5 0 000 o 0 0 0 00 01 02 03 04 05 06 distance 9W 0 o 00 0 0 Oo 0 I I I I I I I 00 01 02 03 04 05 06 distance 450 a 539 0 6 o o oo t o E m39 o o 0 0 0 a 0 o 0 o o I I I I I I I 00 01 02 03 04 05 06 distance 13 a 539 2 g 9 0 0 0 0 0 m o In 0 0 E 0 D o I o I I I I I I I 00 01 02 03 04 05 06 distance 116 Anisotropic Variogram Additional example 1 l Z3m3y Qquot 0 00 00 09000 0 0 O 00 000 quotO D39 00 o 0 5 0 0 O 8 19 gt V 98 6 O 0 00 O 0 0 9 0 g 0 0 0o O O 000 0 X Coord 117 eg1z 2 O N 139 D b eg1x 00 00204060810 eg1y Anisotropic Variogram Additional example 1 0 45 2O 20 o a o o oooo o I I I I I I I I I I I I 00 02 04 06 08 10 00 02 04 06 08 10 senu vanance O I I sen va ance O I I 0 o o v o 0 distance distance 90 135 4O 4O 2O 2O oo ooo o semI varlance I O semI varlance I 02 o o I I I I I I I I I I I I 00 02 04 06 08 00 02 04 06 08 10 12 00 o oo O O 0 o distance distance 118 Anisotropic Variogram R commands gt gt VVV VVVVVVV VVVV librarygeoR libraryscatterplot3d wells lt readtablequotwellstxtquot headerT wellscoords lt cbindwellsx wellsy wellsdata lt wellsz egl lt NULL eg1x lt wellsx eg1y lt wellsy eg1z lt grf59 wells nsim1 covmodelquotexponentialquotcovparsc9 04 nugget0data 3wellsx 3wellsy eg1data lt eg1z eg1coords lt cbindeg1x eg1y eg1vario4 lt variog4eg1 estimatortypequotmodulusquot parmfrowc12 pointsgeodataeg1 scatterplot3deg1x yeg1y zeg1z typequothquot ploteg1vario4 sameF typequotbquot 119 Anisotropic Variogram Rose diagram After drawing directional variograms we t variogram models and draw a rose diagram The idea of a rose diagram is to t smooth curves to directional variograms at xed sill and look at the ranges Suppose the tted ranges for the directional variograms are 3 2 1 and 2 for 00 450 900 1350 A rose diagram plots the ranges corresponding to given angles We de ne i major axis ratio minor ax1s For geometrically anisotropic data the rose diagram is an ellipse If the rose diagram is not an ellipse either there is no geometric anisotropy or the estimate of the ellipse is too noisy In general anisotropy is dif cult to detect using rose diagram due to low power 120 Anisotropic Variogram Example WLS tted directional semivariograms 15 15 sem IvarIance I sem ivariance I o o 0 L0 0000 I I I I I I I I I I I I I I 00 01 02 03 04 05 06 00 01 02 03 04 05 06 distance distance 90 135 15 sem IvarIance I sem ivariance I o I I I I I I I I I I I I I I 00 01 02 03 04 05 06 00 01 02 03 04 05 06 distance distance 121 Anisotropic Variogram R commands gt wellsvario4 lt variog4wells estimatortypequotmodulusquot maxdist07 gt wellswlsO lt variofitwellsvario4quot0quot inicovparsc904 covmodelquotexponentialquotweightsquotcressiequot gt cwellswlsOcovpars wellswlsOnugget 1 64612760 01373669 00000000 gt wellswls45 lt variofitwellsvario4quot45quot inicovparsc904 covmodelquotexponentialquotweightsquotcressiequot gt cwellswls45covpars wellswls45nugget 1 79992850 01489105 00000000 gt wellsw1590 lt variofitwellsvario4quot90quot inicovparsc904 gt cwellsw1590covpars covmodelquotexponentialquotweightsquotcressiequot wellsw1590nugget 1 122884710 03732955 00000000 gt gt wellswlsl35 lt variofitwellsvario4quot135quot inicovparsc904 covmodelquotexponentialquotweightsquotcressiequot cwellswlsl35covpars wellswlsl35nugget 1 125730848 02163537 00000000 VVVVVVVVVVVVV parmfrowc22 plotwellsvario4quot0quot lineswellswlsO titlequot0quot plotwellsvario4quot45quot ylimc020 lineswellswls45 titlequot45quot plotwellsvario4quot90quot ylimc020 lineswellsw1590 titlequot90quot plotwellsvario4quot135quot ylimc020 lineswellswlsl35 titlequot135quot ylimc020 122 Anisotropic Variogram R commands gt wellsml4 lt likfitwells inicovparsc65 05 covmodelquotexponentialquot likmethodquotMLquot fixpsiAF psiApi4 fixpsiRF psiR2 gt summarywellsml4 Summary of the parameter estimation Estimation method maximum likelihood Parameters of the mean component trend beta 67412 Parameters of the spatial component correlation function exponential estimated variance parameter sigmasq partial sill 451 estimated cor fct parameter phi range parameter 01385 anisotropy parameters estimated anisotropy angle 1416 81 degrees estimated anisotropy ratio 1665 Parameter of the error component estimated nugget 0 Transformation parameter fixed BoxCox parameter 1 no transformation Practical Range with cor005 for asymptotic range 04147844 Maximised Likelihood logL nparams AIC BIC n1105n H6quot quot233quot quot2455quot non spatial model logL nparams AIC BIC quot 1377quot quot2quot quot2794quot quot2836quot Call likfitgeodata wells inicovpars c65 05 fixpsiA F psiA pi4 fixpsiR F psiR 2 covmodel quotexponentialquot likmethod quotMLquot 123

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