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Lab #1: - spatial autocorrelation games - constructing Moran scatterplots - constructing semivariogram plots

Lab #1: - spatial autocorrelation games - constructing Moran scatterplots - constructing semivariogram plots. Spatial statistics in practice Center for Tropical Ecology and Biodiversity, Tunghai University & Fushan Botanical Garden. SASIM. SASIM game: web & R versions .

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Lab #1: - spatial autocorrelation games - constructing Moran scatterplots - constructing semivariogram plots

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  1. Lab #1:- spatial autocorrelation games- constructing Moran scatterplots- constructing semivariogram plots Spatial statistics in practice Center for Tropical Ecology and Biodiversity, Tunghai University & Fushan Botanical Garden

  2. SASIM SASIM game: web & R versions ############################################# # THE AUTOCORRELATION GAME # ############################################# # to start: run the code below # ############################################# rm(list = ls()) library(spdep) s<-function(i,j){ a<-dat[i] b<-dat[j] dat[i]<-b dat[j]<-a return(dat) } plot.new<-function(dat){ plot(0:1,0:1,,xaxt="n",yaxt="n",bty="n",xlab="",ylab="",type="n") segments(seq(0,1,by=0.25),rep(0,5),seq(0,1,by=0.25),rep(1,5)) segments(rep(0,5),seq(0,1,by=0.25),rep(1,5),seq(0,1,by=0.25)) text(x,y,round(dat,2),cex=2) text(x+0.10,y+0.10,1:16,col="grey",cex=0.8) title(main=paste("Moran I=",round(moran.test(dat,listw)$estimate[1],3))) } x<-rep(seq(0.25/2,0.75+0.25/2,by=.25),4) y<-rep(seq(0.25/2,0.75+0.25/2,by=.25),each=4) dat<-runif(16,0,10) plot(0:1,0:1,,xaxt="n",yaxt="n",bty="n",xlab="",ylab="",type="n") segments(seq(0,1,by=0.25),rep(0,5),seq(0,1,by=0.25),rep(1,5)) segments(rep(0,5),seq(0,1,by=0.25),rep(1,5),seq(0,1,by=0.25)) text(x,y,round(dat,2),cex=2) text(x+0.10,y+0.10,1:16,col="grey",cex=0.8) neigh<-cell2nb(4, 4, type="rook", torus=FALSE) listw<-nb2listw(neigh, style="B") title(main=paste("Moran I=",round(moran.test(dat,listw)$estimate[1],3))) ###################################################### # THE AUTOCORRELATION GAME # ###################################################### # how to play: # # 1. to start: run the code above a 4-by-4 lattice # # will appear the values in black are simulate # # from a uniform distribution with range (0,10) # # and rounded up to 1/100 cells are labelled by # # the grey number on the top-right corner # # Moran I gives the value of the autocorrelation # # index for the current data on the lattice # # 2. to switch the values in cells i and j: # # type "dat<-s(i,j)" # # 3. to see the result: type "plot.new(dat)" # # 4. have fun # ###################################################### dat<-s(2,15) plot.new(dat) http://www.nku.edu/~longa/cgi-bin/cgi-tcl-examples/generic/SA/SA.cgi

  3. ArcView extension: MapStat3 • Generate neighbors matrix • Rook’s adjacency • Queen’s adjacency • Calculate MC

  4. Basic changes when customizing any of the SAS files • file paths • INPUT statements (e.g., list of variables) • n, n-1 (Puerto Rico: 73, 72) • variables in KEEP= statements • variables in outfile (DATA _NULL_) PUT statements (e.g., list of variables) • TRY definition

  5. Calculating eigenfunctions: MCmax FILENAME CONN 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-CON.TXT'; FILENAME OUTFILE1 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-EIGENVECTORS.TXT'; FILENAME OUTFILE2 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-MC-EIG.TXT'; FILENAME OUTFILE3 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-EIG.TXT'; ******************************************* * CHANGES NEED TO BE MADE FOR N AND (N-1) * ******************************************; TITLE 'EIGENFUNCTIONS FOR THE PR DATA'; ***************************************************************************************************************** * READ IN THE N-BY-N GEOGRAPHIC CONNECTIVITY MATRIX; CODE THE DIAGONAL WITH 1S, MAKE SURE NOT TO MISREAD THE ID * *****************************************************************************************************************; DATA STEP1; INFILE CONN; INPUT IDC C1-C73; DROP IDC; RUN; DATA STEP2; ARRAY M{73} M1-M73; DO I=1 TO 73; DO J=1 TO 73; M{J} = -1/73; IF I=J THEN M{J}=1-1/73; END; OUTPUT; END; DROP I J; RUN; PROC IML; USE STEP1; READ ALL INTO C; USE STEP2; READ ALL INTO M; IDEN=I(73); CM=C*M; MCM=M*CM; ONE=J(73,1); CSUM=C*ONE; CALL EIGEN(EVALS,EVECS,MCM); CREATE STEP3 FROM EVALS; APPEND FROM EVALS; CREATE STEP4 FROM EVECS; APPEND FROM EVECS; CREATE STEP5 FROM CSUM; APPEND FROM CSUM; QUIT;

  6. DATA STEP3(REPLACE=YES); SET STEP3; EVALS=COL1; DROP COL1; RUN; DATA STEP4(REPLACE=YES); SET STEP4; ARRAY EVECS{73} E1-E73; ARRAY IMLOUT{73} COL1-COL73; DO I=1 TO 73; EVECS{I} = IMLOUT{I}; END; DROP COL1-COL73 I; X0=1; RUN; DATA STEP5(REPLACE=YES); SET STEP5; CSUM=COL1; DROP COL1; RUN; DATA COMBINE; SET STEP3; SET STEP4; SET STEP5; RUN; PROC MEANS DATA=COMBINE NOPRINT; VAR CSUM X0; OUTPUT OUT=CSUM SUM=CSUM N; RUN; PROC MEANS DATA=COMBINE NOPRINT; VAR EVALS; OUTPUT OUT=LAM1 MAX=LAM1; RUN; DATA MAXMC; SET CSUM(KEEP=CSUM N); SET LAM1(KEEP=LAM1); MCMAX = N*LAM1/CSUM; RUN; PROC PRINT; RUN; DATA _NULL_; SET STEP4; FILE OUTFILE1 LRECL=2048; ID=_N_; PUT ID E1-E73; RUN; DATA _NULL_; SET STEP3; IF _N_=1 THEN SET CSUM; IF _N_=1 THEN SET LAM1; FILE OUTFILE2; ID=_N_; MCO=N*EVALS/CSUM; MCADJ=EVALS/LAM1; PUT ID EVALS MCO MCADJ; RUN; writing to output files

  7. DATA STEP1(REPLACE=YES); SET STEP1; ARRAY CONN{73} C1-C73; CSUM=0; DO I=1 TO 73; IF _N_=I THEN CONN{I} = 1; CSUM=CSUM + CONN{I}; END; IDC=_N_; X0=1; CSUM=CSUM-1; DROP I; RUN; PROC MEANS DATA=STEP1 NOPRINT; VAR X0 CSUM; OUTPUT OUT=APP SUM=N CSUM; RUN; PROC PRINCOMP DATA=STEP1(TYPE=CORR) OUTSTAT=EVECS NOPRINT; VAR C1-C73; RUN; DATA EVECSTMP; SET EVECS; IF _N_=1 THEN SET APP(KEEP=CSUM N); IF _TYPE_='EIGENVAL' THEN IKEEP=1; ELSE IKEEP=0; IF IKEEP=0 THEN DELETE; EVAL1=C2-1; MCMAXAPP = N*EVAL1/CSUM; DROP _TYPE_ _NAME_ IKEEP C1-C73; RUN; PROC PRINT; RUN; DATA EVALS; SET EVECS; IF _TYPE_="EIGENVAL" THEN IKEEP=1; ELSE IKEEP=0; IF IKEEP=0 THEN DELETE; RUN; PROC TRANSPOSE DATA=EVALS OUT=APPEVALS PREFIX=EVALS; VAR C1-C73; RUN; DATA EIGEN; SET APPEVALS(KEEP=EVALS1); LAMBDAC=EVALS1-1; DROP EVALS1; RUN; DATA APPEVALS(REPLACE=YES); SET APPEVALS; EVALS=EVALS1-1; IF _NAME_="C1" THEN DELETE; DROP _NAME_ EVALS1; RUN; PROC TRANSPOSE DATA=STEP1 OUT=TCSUM PREFIX=TCSUM; VAR CSUM; RUN; DATA MATRIXW; SET STEP1; IF _N_=1 THEN SET TCSUM; _TYPE_="CORR"; ARRAY TCSUM{73} TCSUM1-TCSUM73; ARRAY CONN{73} C1-C73; DO I=1 TO 73; CONN{I}=CONN{I}/(SQRT(TCSUM{I})*SQRT(CSUM)); IF _N_=I THEN CONN{I} = 1; END; IDC=_N_; DROP TCSUM1-TCSUM73 I CSUM _NAME_; RUN;

  8. PROC PRINCOMP DATA=MATRIXW(TYPE=CORR) OUTSTAT=WEVECS NOPRINT; VAR C1-C73; RUN; DATA WEVALSTMP; SET WEVECS; IF _N_=1 THEN SET APP(KEEP=CSUM N); IF _TYPE_='EIGENVAL' THEN IKEEP=1; ELSE IKEEP=0; IF IKEEP=0 THEN DELETE; DROP _TYPE_ _NAME_ IKEEP; RUN; PROC TRANSPOSE DATA=WEVALSTMP OUT=WEVALS PREFIX=WEVALS; VAR C1-C73; RUN; DATA EIGEN(REPLACE=YES); SET EIGEN; SET WEVALS(KEEP=WEVALS1); LAMBDAW=WEVALS1-1; DROP WEVALS1; RUN; PROC MEANS; VAR LAMBDAW; RUN; DATA _NULL_; SET EIGEN; FILE OUTFILE3; ID=_N_; PUT ID LAMBDAC LAMBDAW; RUN; ****************************************** * ADJUSTING THE APPROXIMATE EIGENVECTORS * ******************************************; DATA EVECS(REPLACE=YES); SET EVECS; IF _TYPE_='SCORE' THEN IKEEP=1; ELSE IKEEP=0; IF IKEEP=0 THEN DELETE; RUN; PROC TRANSPOSE OUT=STEP6 PREFIX=C; VAR C1-C73; RUN; DATA STEP6(REPLACE=YES); SET STEP6; SET STEP1(KEEP=IDC); RUN; PROC FACTOR DATA=STEP6 N=72 ROTATE=QUARTIMAX OUT=FINAL OUTSTAT=FINALE NOPRINT; VAR C2-C73; RUN; /* DATA _NULL_; SET FINAL; FILE OUTFILE1 LRECL=2048; ID=_N_; PUT ID FACTOR1-FACTOR72; RUN; DATA _NULL_; SET APPEVALS; IF _N_=1 THEN SET CSUM; IF _N_=1 THEN SET EVECSTMP; FILE OUTFILE2; ID=_N_; MCO=N*EVALS/CSUM; MCADJ=EVALS/EVAL1; PUT ID EVALS MCO MCADJ; RUN; */ writing to output file for SAR writing approximations to output files

  9. MC as a linear regression solution From OLS theory: b = (XTX)-1XTY • Convert the attribute variable in question to z-scores • Let X = zY and Y = CzY • Regress CzY on zY, with a no-intercept option • Let X = 1 and Y = C1 • Regress C1 on 1, with a no-intercept option • MC = bnumerator/bdenominator

  10. GR from MC row sums of matrix C squared deviations The GR contains all of the locational information captured by the MC, but emphasizes any marked deviations (e.g., outliers).

  11. SAS code to calculate MC & GR FILENAME INDATA ‘D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-AREAS-COMPETITION.TXT'; FILENAME CONN 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-CON.TXT'; TITLE 'MC AND GR FOR THE PR AREA DATA'; ************************************************************** * READ IN THE N-BY-N GEOGRAPHIC CONNECTIVITY MATRIX; * * CODE THE DIAGONAL WITH 1S, MAKE SURE NOT TO MISREAD THE ID * **************************************************************; DATA STEP1; INFILE CONN; INPUT IDC C1-C73; ARRAY CONN{73} C1-C73; CSUM = 0; DO I=1 TO 73; CSUM = CSUM + CONN{I}; END; RUN; PROC MEANS DATA=STEP1 NOPRINT; VAR CSUM; OUTPUT OUT=CSUM SUM=SUMCSUM; RUN; ************************************************ * READ IN GEOREFERENCED DATA; * * THEN CENTER THE SELEDTED ATTRIBUTE VARIABLE * ************************************************; DATA STEP2; INFILE INDATA LRECL=1024; INPUT ID AREAM AREACT MCT_RATIO TRMCT_RATIO AREAPT MPT_RATIO TRMPT_RATIO NAME$; Y = LOG(AREAM + 0.001); * Y = TRMPT_RATIO; X0=1; RUN; PROC STANDARD MEAN=0 STD=1 OUT=STEP2(REPLACE=YES); VAR Y; RUN;

  12. PROC MEANS NOPRINT; VAR X0; OUTPUT OUT=STEP0 SUM=N; RUN; DATA STEP0(REPLACE=YES); SET STEP0; SET CSUM; APPSEMC = SQRT(2/SUMCSUM); RUN; DATA STEP2(REPLACE=YES); SET STEP2; SET STEP1; ARRAY CONN{73} C1-C73; ARRAY YCONN{73} CY1-CY73; DO I=1 TO 73; YCONN{I} = Y*CONN{I}; END; Y2CSUM=(Y**2)*CSUM; RUN; PROC MEANS NOPRINT; VAR CY1-CY73; OUTPUT OUT=CYOUT1 SUM=CY1-CY73; RUN; PROC TRANSPOSE DATA=CYOUT1 PREFIX=CY OUT=CYOUT2; VAR CY1-CY73; RUN; DATA STEP3; SET STEP2(KEEP=X0 Y2CSUM Y CSUM); SET CYOUT2(KEEP=CY1); CZY=CY1; ZY=Y; DROP CY1; X0GR=X0; RUN; PROC REG OUTEST=OUTNMCB; MODEL CZY=ZY/NOINT; OUTPUT OUT=MCPRED P=CZYHAT; RUN;

  13. AND construct a Moran scatterplot ********************************* * * * CONSTRUCTING A MC SCATTERPLOT * * * *********************************; PROC GPLOT; PLOT CZY*ZY CZYHAT*ZY/OVERLAY; RUN; PROC REG DATA=STEP3 OUTEST=OUTDMCB; MODEL CSUM=X0/NOINT; RUN; PROC REG DATA=STEP3 OUTEST=OUTNGRB NOPRINT; MODEL Y2CSUM=X0GR/NOINT; RUN; DATA STEP0(REPLACE=YES); SET STEP0; SET OUTNMCB(KEEP=ZY); SET OUTDMCB(KEEP=X0); SET OUTNGRB(KEEP=X0GR); MC = ZY/X0; EMC = -1/(N-1); GR = X0GR/X0 - ((N-1)/N)*MC; IF MC> 0 THEN PROBMC=1 - PROBNORM((MC-EMC)/APPSEMC); ELSE PROBMC= PROBNORM((MC-EMC)/APPSEMC); DROP ZY X0; RUN; PROC PRINT; VAR N GR MC EMC APPSEMC PROBMC; RUN;

  14. SAS code for constructing a semivariogram plot FILENAME INDATA1 ‘D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-AREAS-COMPETITION.TXT'; FILENAME INDATA2 ‘D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-DEM&QUAD-DATA.TXT'; TITLE 'SEMIVARIOGRAM CONSTRUCTION FOR THE PR DATA'; ************************************************ * READ IN GEOREFERENCED DATA; * * THEN CENTER THE SELEDTED ATTRIBUTE VARIABLE * ************************************************; DATA STEP1; INFILE INDATA1 LRECL=1024; INPUT ID AREAM AREACT MCT_RATIO TRMCT_RATIO AREAPT MPT_RATIO TRMPT_RATIO NAME$; * Y = LOG(AREAM + 0.001); * Y = TRMPT_RATIO; RUN; PROC SORT DATA=STEP1 OUT=STEP1(REPLACE=YES); BY NAME; RUN; DATA STEP2; INFILE INDATA2 LRECL=1024; INPUT IDDEM MELEV SELEV U V QUAD NAME$; U=U/1000; V=V/1000; * Y=LOG(MELEV+17.5); Y=(SELEV-25)**0.5; RUN; PROC SORT DATA=STEP2 OUT=STEP2(REPLACE=YES); BY NAME; RUN; DATA STEP3; MERGE STEP1 STEP2; BY NAME; RUN;

  15. PROC VARIOGRAM DATA=STEP3 OUTP=PAIRS; COMPUTE NOVARIOGRAM; COORDINATES XC=U YC=V; VAR Y; RUN; DATA PAIRS(REPLACE=YES); SET PAIRS; GAMMA=(V1-V2)**2; DROP X1 Y1 X2 Y2 COS VARNAME; RUN; PROC MEANS MAX; VAR DISTANCE; RUN; PROC VARIOGRAM DATA=STEP3 OUTVAR=GAMMA; COMPUTE LAGD=3 MAXLAGS=50; COORDINATES XC=U YC=V; VAR Y; RUN; DATA GAMMA(REPLACE=YES); SET GAMMA; IF LAG =0 THEN DELETE; IF LAG=-1 THEN LAG=0; IF LAG=0 THEN DISTANCE=0; IF LAG=0 THEN VARIOG=0; RUN; PROC PRINT; RUN; PROC GPLOT; PLOT VARIOG*DISTANCE; RUN;

  16. SAS code for semivariogram trend line FILENAME INDATA1 ‘D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-AREAS-COMPETITION.TXT'; FILENAME INDATA2 ‘D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-DEM&QUAD-DATA.TXT'; TITLE 'CONSTRUCT A SEMIVARIOGRAM FOR THE PR AREA DATA'; DATA STEP0; MAXDIST=80; RUN; ************************************************ * READ IN GEOREFERENCED DATA; * * THEN CENTER THE SELEDTED ATTRIBUTE VARIABLE * ************************************************; DATA STEP1A; INFILE INDATA1 LRECL=1024; INPUT ID AREAM AREACT MCT_RATIO TRMCT_RATIO AREAPT MPT_RATIO TRMPT_RATIO NAME$; * Y = LOG(AREAM + 0.001); * Y = TRMPT_RATIO; X0=1; RUN; PROC SORT DATA=STEP1A OUT=STEP1A(REPLACE=YES); BY NAME; RUN; PROC MEANS NOPRINT; VAR X0; OUTPUT OUT=OUTN SUM=N; RUN; DATA STEP1B; INFILE INDATA2 LRECL=1024; INPUT IDDEM MELEV SELEV U V QUAD NAME$; U=U/1000; V=V/1000; * Y=LOG(MELEV+17.5); Y=(SELEV-25)**0.5; RUN; PROC SORT DATA=STEP1B OUT=STEP1B(REPLACE=YES); BY NAME; RUN; DATA STEP1; MERGE STEP1A STEP1B; BY NAME; RUN; coordinates are labeled u, v

  17. PROC VARIOGRAM DATA=STEP1 OUTP=PAIRS; COMPUTE NOVARIOGRAM; COORDINATES XC=U YC=V; VAR Y; RUN; DATA PAIRS(REPLACE=YES); SET PAIRS; GAMMA=(V1-V2)**2; DROP X1 Y1 X2 Y2 COS VARNAME; STDDIST=DISTANCE; DIST=DISTANCE; COUNT=1; RUN; PROC MEANS SUM; VAR COUNT; RUN; PROC MEANS MAX; VAR DISTANCE; RUN; PROC STANDARD OUT=STEP1(REPLACE=YES) MEAN=0 STD=1; /* CREATES STANDARDIZED DISTANCE INDEX */ VAR STDDIST; RUN; DATA PAIRS(REPLACE=YES); SET PAIRS; STDDIST=ROUND(STDDIST,.001); /* ROUNDS STANDARDIZED DISTANCE INDEX */ RUN; PROC SORT DATA=PAIRS(REPLACE=YES); BY STDDIST; RUN; PROC MEANS MEAN NOPRINT; BY STDDIST; VAR DIST GAMMA; /* INITIAL CLUSTERING USING STANDARDIZED DISTANCES INDEX */ OUTPUT OUT=TMEAN MEAN=MDIST MGAMMA; RUN; DATA STEP2; SET TMEAN; SEQID=_N_; DIST=MDIST; GAMMA=MGAMMA; DROP MDIST MGAMMA; RUN; PROC SORT OUT=STEP2(REPLACE=YES); BY DIST; RUN; DATA STEP2(REPLACE=YES); SET STEP2; SEQID = _N_; MERGEVAL=1; RUN;

  18. PROC CLUSTER METHOD=WARD NOPRINT OUTTREE=TREE; ID SEQID; VAR DIST; COPY GAMMA; FREQ _FREQ_; RUN; ************************************************** * THE NUMBER OF GROUPS (N = ???) NEEDS TO BE SET * **************************************************; PROC TREE DATA=TREE NOPRINT N=75 OUT=TEMP1; ID SEQID; RUN; PROC SORT DATA=TEMP1 OUT=TEMP1(REPLACE=YES); BY SEQID; RUN; DATA STEP2 (REPLACE=YES); MERGE STEP2(DROP=MERGEVAL) TEMP1; BY SEQID; RUN; DATA OUTN(REPLACE=YES); SET OUTN; DIST=0.0001; STDDIST=DIST; GAMMA=0; _TYPE_=0; _FREQ_=N; SEQID=0; CLUSTER=0; CLUSNAME='CL000'; RUN; DATA STEP2(REPLACE=YES); SET STEP2 OUTN; RUN; PROC SORT DATA=STEP2 OUT=STEP1(REPLACE=YES); BY CLUSTER; RUN; PROC MEANS DATA=STEP1 NOPRINT; BY CLUSTER; VAR DIST GAMMA; FREQ _FREQ_; OUTPUT OUT=RESULTS MIN=DISTMIN GAMMAMIN MAX=DISTMAX GAMMAMAX MEAN=DISTM GAMMAM; RUN; PROC SORT OUT=STEP1(REPLACE=YES); BY DISTM; RUN; PROC PRINT; VAR GAMMAM DISTM _FREQ_; RUN; PROC MEANS SUM; VAR _FREQ_; RUN; PROC GPLOT DATA=RESULTS; PLOT GAMMAM*DISTM; RUN; DATA RESULTS(REPLACE=YES); SET RESULTS; IF _N_=1 THEN SET STEP0; IF DISTM > MAXDIST THEN DELETE; RUN;

  19. ******************************************* * * * TREND LINES FOR THE SEMIVARIOGRAM PLOTS * * * *******************************************; /* SPHERICAL */ TITLE 'SPHERICAL SEMIVARIOGRAM MODEL'; PROC NLIN DATA=RESULTS NOITPRINT MAXITER=500 METHOD=MARQUARDT; PARMS C0=0.01 C1=1 R=10; BOUNDS C0>0, C1>0, O<R; IF DISTM<=R THEN DO; MODEL GAMMAM = C0 + C1*((3/2)*(DISTM/R) - (1/2)*(DISTM/R)**3); _WEIGHT_ = _FREQ_; END; ELSE DO; MODEL GAMMAM = C0 + C1; _WEIGHT_ = _FREQ_; END; OUTPUT OUT=TEMP1 P=GAMMAMHAT; RUN; PROC GPLOT; PLOT GAMMAM*DISTM GAMMAMHAT*DISTM/OVERLAY; RUN; /* GAUSSIAN */ . . . /* BESSEL */

  20. ArcGIS 9.1: semivariograms & MC Semivariogram • Open ArcMap • Add (+) *.shp file • TOOLS -> EXTENSIONS -> Geostatistical analyst • VIEW -> TOOLBARS -> Geostatistical analyst • Geostatistical wizard

  21. Select variable (e.g., STD_E) select “kriging” option select “prediction map” option select “model” • Inspect “semivariogram” inspect “anisotropy”

  22. ArcGIS 9.1: semivariograms & MC MC • TOOLS -> SPATIAL STATISTICS TOOLS -> ANALYZING PATTERN -> Spatial Autocorrelation • Select “Display Output Graphically” • Select “inverse distance squared”

  23. What you learned in today’s lab: • Change path names, etc. (Slide #4—print for a ready reference) • Compute eigenfunctions • Compute MC and GR • Construct Moran scatterplots and semi-variogram plots • Fit trend lines to Moran scatterplots and semi-variogram plots • Implement kriging in ArcGIS

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