Lab #1:- spatial autocorrelation

games- constructing Moran

scatterplots- constructing

semivariogram plots

Spatial statistics in practiceCenter for Tropical Ecology and Biodiversity,

Tunghai University & Fushan Botanical Garden

SASIMSASIMSASIM game: web & R versions http://www.nku.edu/~longa/cgi-bin/cgi-tcl-examples/generic/SA/SA.cgi #############################################

# 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)

ArcView extension: MapStat3

• Generate neighbors matrix

– Rook’s adjacency

– Queen’s adjacency

• Calculate MC

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

Calculating eigenfunctions: MCmaxFILENAME 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;

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 tooutput files

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;

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 approximationsto output files

writing tooutput file for SAR

MC as a linear regression solutionFrom 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

YTY

1Y

TYnumerator )( Czzzzb

C1111b T1Trdenominato )(

GR from MC

MCn

1n

z

cz2

2

1nGR n

1i

2iy,

n

1i

n

1iij

2iy,

T

C11

row sums of matrix Csquared

deviations

The GR contains all of the locational information captured by the MC, but emphasizes any marked deviations (e.g., outliers).

SAS code to calculate MC & GRFILENAME 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;

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;

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;

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;

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;

SAS code for semivariogram trend lineFILENAME 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;

coordinatesare labeled u, v

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;

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;

******************************************** ** 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*/

ArcGIS 9.1: semivariograms & MC

Semivariogram

– Open ArcMap

– Add (+) *.shp file

– TOOLS -> EXTENSIONS -> Geostatistical analyst

– VIEW -> TOOLBARS -> Geostatistical analyst

– Geostatistical wizard

• Select variable (e.g., STD_E)

select “kriging” option

select “prediction map” option

select “model”

• Inspect “semivariogram”

inspect “anisotropy”

ArcGIS 9.1: semivariograms & MC

MC

– TOOLS -> SPATIAL STATISTICS TOOLS -> ANALYZING PATTERN -> Spatial Autocorrelation

• Select “Display Output Graphically”

• Select “inverse distance squared”

What you learned in today’s lab:

1. Change path names, etc. (Slide #4—print for a ready reference)

2. Compute eigenfunctions

3. Compute MC and GR

4. Construct Moran scatterplots and semi-variogram plots

5. Fit trend lines to Moran scatterplots and semi-variogram plots

6. Implement kriging in ArcGIS