spatial statistics in practice

Lab #3:- Jacobian estimation

- estimating autoregressive models

- estimating spatial filter models- estimating random effects

models - mapping spatial filters

Spatial statistics in practiceCenter for Tropical Ecology and Biodiversity,

Tunghai University & Fushan Botanical Garden

SAS code for the Jacobian approximationFILENAME EIGEN 'D:\JYU-SUMMERSCHOOL2006\LAB#3\PR-EIG.TXT';TITLE 'MATRIX C OR W JACOBIAN APPROXIMATION, IRREGULAR LATTICE';************************************************ THE APPROPRIATE EIGENVALES MUST BE SELECTED ************************************************;DATA STEP0; INFILE EIGEN; INPUT ID LAMBDAC LAMBDAW;LAMBDA=LAMBDAW;X0=1;RUN;PROC UNIVARIATE NOPRINT; VAR LAMBDA; OUTPUT OUT=EXTREMEL MAX=LMAX MIN=LMIN;RUN; PROC PRINT DATA=EXTREMEL; VAR LMAX LMIN; RUN;PROC MEANS DATA=STEP0 NOPRINT; VAR X0; OUTPUT OUT=STEP0A SUM=N; RUN;DATA EXTREMEL(REPLACE=YES); SET EXTREMEL; SET STEP0A;FINISH = 0.999/LMAX;START = 0.999/LMIN;INC = (FINISH - START)/201;RUN;DATA STEP1; IF _N_=1 THEN SET EXTREMEL; SET STEP0;ARRAY JACOB{202} JAC1-JAC202;RHO = START;DO I = 1 TO 202; JACOB{I} = -LOG(1 - RHO*LAMBDA); RHO = RHO + INC;END;RUN;PROC MEANS NOPRINT; VAR JAC1-JAC202; OUTPUT OUT=JACOB1 MEAN=;RUN;

PROC TRANSPOSE OUT=JACOB2; VAR JAC1-JAC202;RUN;DATA STEP2 (REPLACE=YES); SET EXTREMEL; DO RHO = START TO FINISH BY INC; OUTPUT; END;DROP START FINISH INC;RUN;DATA STEP2 (REPLACE=YES); SET JACOB2; SET STEP2;J = COL1;DROP COL1;RUN;

PROC GPLOT; PLOT J*RHO;RUN;PROC NLIN DATA=STEP2 NOITPRINT MAXITER=15000 METHOD=MARQUARDT; PARMS A1=0.15 A2=0.15 D1=1.1 D2=1.1; BOUNDS D1<2 , D2<2 ;

MODEL J = A1*(LOG(1 - RHO*LMIN)/(RHO*LMIN) + 1 - D1*LOG(1 - RHO*LMIN))+ A2*(LOG(1 - RHO*LMAX)/(RHO*LMAX) + 1 - D2*LOG(1 - RHO*LMAX));

OUTPUT OUT=TEMP3 ESS=ESS2 P=JHAT PARMS=A1 A2 D1 D2;RUN;PROC MEANS; VAR A1 A2 D1 D2; RUN;PROC GPLOT; PLOT J*RHO JHAT*RHO/OVERLAY; RUN;PROC UNIVARIATE DATA=STEP2 NOPRINT; VAR J; OUTPUT OUT=TOTALSS CSS=TSS;RUN;DATA TEMP3(REPLACE=YES); SET TOTALSS; SET TEMP3;RESS = ESS2/TSS;RUN;PROC PRINT; VAR ESS2 TSS RESS; RUN;

one of threepossible

approximationequations

SAS code for simultaneous autoregressive (SAR) modeling

FILENAME ATTRIBUT 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-DEM&QUAD-DATA.TXT';FILENAME CONN 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-CON.TXT';FILENAME EIGEN 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-EIG.TXT';TITLE 'SAR FOR PUERTO RICO DEM';DATA STEP1; INFILE ATTRIBUT; 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 OUT=STEP1(REPLACE=YES); BY IDDEM; RUN;PROC STANDARD MEAN=0 STD=1 OUT=STEP1(REPLACE=YES); VAR Y; RUN;DATA STEP1 (REPLACE=YES); SET STEP1; INFILE CONN; INPUT MUNUM2 C1-C73; ARRAY CONY{73} CY1-CY73; ARRAY CON{73} C1-C73; CSUM = 0; DO I=1 TO 73; CSUM = CSUM + CON{I}; CONY{I} = Y*CON{I}; END;RUN;PROC MEANS SUM; VAR CSUM; RUN;PROC PRINT; VAR NAME IDDEM MUNUM2; RUN;PROC MEANS DATA=STEP1 NOPRINT; VAR CY1-CY73; OUTPUT OUT=CYOUT1 SUM=CY1-CY73; RUN;PROC TRANSPOSE DATA=CYOUT1 PREFIX=WY OUT=CYOUT2; VAR CY1-CY73;RUN;DATA STEP1 (REPLACE=YES); SET STEP1; SET CYOUT2;WY = WY1/CSUM;RUN;PROC REG; MODEL Y=WY; RUN;

DATA STEP2; INFILE EIGEN; INPUT IDE LAMBDAC LAMBDAW; LAMBDA=LAMBDAW; RUN;PROC TRANSPOSE DATA=STEP2 PREFIX=TLAM OUT=EOUT1; VAR LAMBDAW; RUN;DATA STEP1; SET STEP1; IF _N_ = 1 THEN SET EOUT1; RUN;

PROC NLIN DATA=STEP1 NOITPRINT METHOD=MARQUARDT; PARMS RHO=0.5 B0=0; BOUNDS -1.5<RHO<1; ARRAY LAMBDAJ{73} TLAM1-TLAM73; JACOB = 0; DERJ = 0; DO I=1 TO 73; JACOB = JACOB + LOG(1 - RHO*LAMBDAJ{I}); DERJ = DERJ + -LAMBDAJ{I}/(1 - RHO*LAMBDAJ{I}); END; J=EXP(JACOB/73); DERJ = -DERJ/73;

ZY = Y/J;

MODEL ZY = (RHO*WY + B0*(1 - RHO))/J; OUTPUT OUT=TEMP1 PRED=YHAT R=YRESID PARMS=RHO B0;

DER.RHO = ((RHO*WY + B0*(1 - RHO) - Y)*DERJ + WY - B0)/J;RUN;PROC REG; MODEL Y=YHAT; RUN;PROC UNIVARIATE NORMAL; VAR YRESID; RUN;

eigenvalues forthe exact Jacobian

SAS calculusnot completely

correct

*************************** ** JACOBIAN APPROXIMATION ** ***************************;PROC MEANS NOPRINT DATA=STEP2; VAR LAMBDA; OUTPUT OUT=EXTREMES MIN=LMIN MAX=LMAX; RUN;DATA STEP1(REPLACE=YES); SET STEP1; IF _N_=1 THEN SET EXTREMES(KEEP=LMIN LMAX); RUN;PROC NLIN DATA=STEP1 NOITPRINT METHOD=MARQUARDT MAXITER=500; PARMS RHO=0.5 B0=0; BOUNDS -1<RHO<1;

A1 = 0.4276827; A2 = 0.2998183; D1 = 0.9853629; D2 = 1.0664149;IF RHO=0 THEN J=1; ELSE J = EXP(A1*(LOG(1 - RHO*LMIN)/(RHO*LMIN) + 1 - D1*LOG(1 - RHO*LMIN))+ A2*(LOG(1 - RHO*LMAX)/(RHO*LMAX) + 1 - D2*LOG(1 - RHO*LMAX)) );IF RHO=0 THEN DERJ=0; ELSE DERJ = A1*(-LOG(1 - RHO*LMIN)/(LMIN*RHO**2) + (RHO*D1*LMIN - 1)/(RHO*(1 - RHO*LMIN)) ) + A2*(-LOG(1 - RHO*LMAX)/(LMAX*RHO**2) + (RHO*D2*LMAX - 1)/(RHO*(1 - RHO*LMAX)) ) ;

ZY = Y*J;

MODEL ZY = (RHO*WY + B0*(1 - RHO))*J; OUTPUT OUT=TEMP1 PRED=YHAT R=YRESID;

DER.RHO = ( ( (RHO*WY + B0*(1 - RHO) ) - Y)*DERJ + WY - B0)*J;RUN;

from theJacobian

approximation

The AR-SAR model specification

2/n-ARSAR

2

3SARARSAR

2ARSARARSAR

3

ARARSARSAR

2ARAR

2AR

1

)]ρ)det(ρ[det(J

ρ-)ρ-)(1ρ-μ(1

ρρ)ρ(ρ

]

)ρ-μ(1)[ρρ()ρ(

)ρ-μ(1ρ

)μ)(ρ(

μ

WIWI

εWXβXβ1

YWWYY

εXβ

1WYWIYWI

εXβ1WYY

εXβ1YWI

εXβ1Y

44thth order effect order effect

SAS code for spatial filter: Gaussian RV

FILENAME ATTRIBUT 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-DEM&QUAD-DATA.TXT';FILENAME MC 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-MC-EIG.TXT';FILENAME CONN 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-CON.TXT';FILENAME EVEC 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-EIGENVECTORS.TXT';FILENAME OUTFILE 'D:\JYU-SUMMERSCHOOL2006\LAB#3\GAUSSIAN-SF.TXT';

OPTIONS LINESIZE=72;TITLE 'SPATIAL FILTER MODEL FOR PUERTO RICO: GAUSSIAN RV';

DATA STEP1; INFILE ATTRIBUT; INPUT IDDEM MELEV SELEV U V QUAD NAME$;* Y=LOG(MELEV+17.5); Y=(SELEV-25)**0.5;Y0=Y;RUN;PROC SORT DATA=STEP1 OUT=STEP1(REPLACE=YES); BY NAME; RUN;PROC UNIVARIATE NORMAL; VAR Y; RUN;PROC STANDARD MEAN=0 STD=1 OUT=STEP1(REPLACE=YES); VAR Y0; RUN;

DATA STEP1 (REPLACE=YES); SET STEP1; INFILE EVEC LRECL=2048; INPUT IDE E1-E73;RUN;****************************************************************** ** THE SET OF CANDIDATE EIGENVECTORS IS DETERMINED BY MCADJ >= 0 ** ******************************************************************;PROC REG OUTEST=COEF; MODEL Y = E1-E18/SELECTION=STEPWISE SLE=0.10;

OUTPUT OUT=TEMP P=YHAT R=YRESID;RUN;PROC UNIVARIATE NORMAL; VAR YRESID; RUN;

stepwise regression

may wish to makea Bonferroni

adjustment here

PROC TRANSPOSE DATA=COEF PREFIX=B OUT=COEF2; VAR E1-E18; RUN;DATA COEF(REPLACE=YES); INFILE MC;

INPUT ID LAM_MCM MC MCADJ;IF MCADJ<0.25 THEN DELETE;RUN;DATA COEF(REPLACE=YES); SET COEF;

SET COEF2;IF B1='.' THEN DELETE; IF B1=0 THEN DELETE;BSQ=B1**2;EMC=BSQ*MC;P=1;RUN;PROC PRINT; RUN;PROC MEANS SUM NOPRINT; VAR LAM_MCM P EMC BSQ; OUTPUT OUT=EMC SUM=SUML P SPANUM SPADEN; RUN;PROC STANDARD DATA=TEMP MEAN=0 STD=1 OUT=TEMP(REPLACE=YES); VAR YRESID; RUN;

DATA STEP2(REPLACE=YES); INFILE CONN LRECL=1024;

INPUT ID C1-C73;RUN;DATA STEP2(REPLACE=YES); SET STEP2;

SET TEMP(KEEP=YRESID); SET STEP1(KEEP=Y0);

ARRAY CONN{73} C1-C73; ARRAY ZC{73} ZC1-ZC73; ARRAY Y0C{73} Y0C1-Y0C73; CSUM=0; DO I=1 TO 73; CSUM = CSUM + CONN{I}; ZC{I} = YRESID*CONN{I};

Y0C{I} = Y0*CONN{I}; END;

Z=YRESID;X0=1;RUN;PROC MEANS SUM NOPRINT; VAR CSUM X0; OUTPUT OUT=CSUM SUM=CSUM N; RUN;PROC PRINT; VAR CSUM; RUN;

PROC MEANS DATA=STEP2 NOPRINT; VAR Y0C1-Y0C73; OUTPUT OUT=Y0COUT1 SUM=Y0C1-Y0C73; RUN;PROC TRANSPOSE DATA=Y0COUT1 PREFIX=Y0C OUT=Y0COUT2; VAR Y0C1-Y0C73; RUN;PROC MEANS DATA=STEP2 NOPRINT; VAR ZC1-ZC73; OUTPUT OUT=ZCOUT1 SUM=ZC1-ZC73; RUN;PROC TRANSPOSE DATA=ZCOUT1 PREFIX=ZC OUT=ZCOUT2; VAR ZC1-ZC73; RUN;DATA STEP2(REPLACE=YES); SET STEP2(KEEP=X0 Z CSUM Y0); SET ZCOUT2(KEEP=ZC1);

SET Y0COUT2(KEEP=Y0C1);Y0C=Y0C1;ZC=ZC1;DROP ZC1;RUN;PROC REG DATA=STEP2 OUTEST=DEN NOPRINT; MODEL CSUM=X0/NOINT; RUN;PROC REG DATA=STEP2 OUTEST=NUM0 NOPRINT; MODEL Y0C=Y0/NOINT; RUN;PROC REG DATA=STEP2 OUTEST=NUM NOPRINT; MODEL ZC=Z/NOINT; RUN;DATA STEP3; SET DEN; SET NUM0; SET NUM; SET CSUM(KEEP=CSUM N); SET EMC(KEEP=SUML SPANUM SPADEN P);MC0=Y0/X0;MC=Z/X0;EMC = -(1+(N/CSUM)*SUML)/(N-P-1);ZMC = (MC-EMC)/SQRT(2/CSUM);MCSPA=SPANUM/SPADEN;RUN;PROC PRINT; VAR MC0 MC EMC ZMC MCSPA; RUN;

PROC STANDARD DATA=TEMP OUT=TEMP(REPLACE=YES) MEAN=0; VAR YHAT; RUN;DATA _NULL_; SET TEMP;

FILE OUTFILE;PUT IDDEM Y YHAT;

RUN;

YHAT is the spatial filter;output for mapping purposes

Residual diagnostics;MC distribution theory known

for this case

SAS code for spatial filter: binomial RVFILENAME INDATA 'D:\JYU-SUMMERSCHOOL2006\LAB#3\PR-POP-1899&2000.TXT';FILENAME EVEC 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-EIGENVECTORS.TXT';FILENAME OUTFILE 'D:\JYU-SUMMERSCHOOL2006\LAB#3\BINOMIAL-SF.TXT';TITLE 'SPATIAL FILTER MODEL FOR PUERTO RICO: BINOMIAL RV';

DATA STEP1; INFILE INDATA; INPUT ID P1899 U1988 P2000 U2000 QUAD NAME$;Y=U2000/P2000;RUN;PROC SORT DATA=STEP1 OUT=STEP1(REPLACE=YES); BY NAME; RUN;DATA STEP1 (REPLACE=YES); SET STEP1; INFILE EVEC LRECL=2048; INPUT IDE E1-E73;RUN;********************************************************************* THE SET OF CANDIDATE EIGENVECTORS IS DETERMINED BY MCADJ >= 0.25 *********************************************************************;PROC LOGISTIC; MODEL U2000/P2000 = E1-E18/SELECTION=STEPWISE SLE=0.10 SCALE=WILLIAMS;

OUTPUT OUT=TEMP P=YHAT;RUN;PROC REG; MODEL Y=YHAT; RUN;

PROC GENMOD; MODEL U2000/P2000 = E1 E3 E4 E10 E14 E18/DIST=BINOMIAL SCALE=DEVIANCE; OUTPUT OUT=SF XBETA=XBETA P=YHAT; RUN;DATA STEP1(REPLACE=YES); SET STEP1;Y=(U2000-1280)/(P2000-1057);TRY=LOG(Y/(1-Y));W=1/((P2000-1057)*Y*(1-Y));RUN;PROC REG; MODEL TRY=E1-E18/NOINT SELECTION=STEPWISE SLE=0.10; WEIGHT W; RUN;

PROC STANDARD DATA=SF OUT=SF(REPLACE=YES) MEAN=0; VAR XBETA; RUN;DATA _NULL_; SET SF; FILE OUTFILE;YRESID=Y-YHAT;

PUT ID Y XBETA YRESID;RUN;

quasi-likelihood

XBETA is the spatial filter;output for mapping purposes

stepwiseselection

normal approximation

SAS code for spatial filter: Poisson RVFILENAME INDATA1 'D:\JYU-SUMMERSCHOOL2006\LAB#3\PR-POP-1899&2000.TXT';FILENAME INDATA2 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-AREAS-COMPETITION.TXT';FILENAME EVEC 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-EIGENVECTORS.TXT';FILENAME OUTFILE 'D:\JYU-SUMMERSCHOOL2006\LAB#3\POISSON-SF.TXT';

OPTIONS LINESIZE=72;TITLE 'SPATIAL FILTER MODEL FOR PUERTO RICO: POISSON RV';DATA STEP1A; INFILE INDATA1; INPUT ID1 P1899 U1988 P2000 U2000 QUAD NAME$;DENOM=100000000000;RUN;PROC SORT DATA=STEP1A OUT=STEP1A(REPLACE=YES); BY NAME; RUN;DATA STEP1B; INFILE INDATA2; INPUT ID2 AREAM AREACT MCT_RATIO TRMCT_RATIO AREAPT MPT_RATIO TRMPT_RATIO NAME$; LNAREAM=LOG(AREAM);RUN;PROC SORT DATA=STEP1B OUT=STEP1B(REPLACE=YES); BY NAME; RUN;DATA STEP1; MERGE STEP1A STEP1B; BY NAME; RUN;

DATA STEP1(REPLACE=YES); SET STEP1; INFILE EVEC LRECL=2048; INPUT IDE E1-E73;TRY=LOG(P2000/AREAM - 441830);RUN;********************************************************************* ** THE SET OF CANDIDATE EIGENVECTORS IS DETERMINED BY MCADJ >= 0.25 ** *********************************************************************;PROC GENMOD; MODEL P2000 = E1-E18/DIST=POISSON OFFSET=LNAREAM; RUN;

no stepwisePoisson regression

option in SAS

offset variablemust be converted

to its log form

denominatorfor binomial

approximation

****************************************************************************** ** ORDER OF REMOVAL (BACKWARD ELIMINATION): 13, 5, 17, 14, 11, 15, 16, 9, 18 ** ******************************************************************************;PROC GENMOD; MODEL P2000 = E1-E4 E6-E8 E10 E12/DIST=POISSON OFFSET=LNAREAM

SCALE=DEVIANCE; RUN;

********************************************************************** ** ORDER OF REMOVAL (BACKWARD ELIMINATION): 17, 14, 11, 13, 15, 5, 9 ** **********************************************************************;PROC GENMOD; MODEL P2000 = E1-E4 E6-E8 E10 E12 E16 E18/DIST=NB OFFSET=LNAREAM; OUTPUT

OUT=TEMP XBETA=XBETA P=YHAT; RUN;DATA TEMP(REPLACE=YES); SET TEMP; Y=P2000/AREAM; YHAT=YHAT/AREAM;RUN;PROC REG; MODEL Y=YHAT; RUN;

PROC LOGISTIC; MODEL P2000/DENOM = E1-E18/SELECTION=BACKWARD OFFSET=LNAREAM SCALE=WILLIAMS SLS=0.10; RUN;

PROC REG; MODEL TRY=E1-E18/SELECTION=BACKWARD SLS=0.10; RUN;

PROC STANDARD DATA=TEMP OUT=TEMP(REPLACE=YES) MEAN=0; VAR XBETA; RUN;DATA _NULL_; SET TEMP; FILE OUTFILE;XBETA=XBETA-LNAREAM;YRESID=Y-YHAT;

PUT ID1 Y XBETA YRESID;RUN;

binomialapproximation

normalapproximation

quasi-likelihood

negativebinomial

XBETA is the spatial filter;output for mapping purposes

ArcView mapping of spatial filters• The SAS output files need to be converted

to *.dbf files with column headers• MCs can be computed with MapStat3 for

the binomial and Poisson regressions residuals

Gaussianbinomial Poisson

RV MCSF MCresid

Gaussian 0.675 -0.048

binomial 0.859 -0.142

Poisson 0.834 -0.147

Dark red: very highLight red: highGray: mediumLight green: lowDark green: very low

SAS code for spatially structured random effects linear modeling

FILENAME INDATA1 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-AREAS-COMPETITION.TXT';FILENAME INDATA2 'D:\JYU-SUMMERSCHOOL2006\LAB#1\PR-DEM&QUAD-DATA.TXT';TITLE 'MIXED MODEL 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;X0=1;RUN;PROC SORT DATA=STEP2 OUT=STEP2(REPLACE=YES); BY NAME; RUN;DATA STEP3; MERGE STEP1 STEP2; BY NAME; RUN;PROC REG; MODEL Y=/; RUN;PROC MIXED METHOD=ML MAXITER=500 COVTEST SCORING=10; MODEL Y=X0/NOINT S; REPEATED/SUB=INTERCEPT; RUN;

PROC MIXED METHOD=ML MAXITER=500 COVTEST SCORING=10; MODEL Y=X0/NOINT S; PARMS (13, 10); REPEATED/SUB=INTERCEPT TYPE=SP(EXP) (U V); RUN;

PROC MIXED METHOD=ML MAXITER=500 COVTEST SCORING=10; MODEL Y=X0/NOINT S; PARMS (13, 10); REPEATED/SUB=INTERCEPT TYPE=SP(GAU) (U V); RUN;

PROC MIXED METHOD=ML MAXITER=500 COVTEST SCORING=10; MODEL Y=X0/NOINT S; PARMS (13, 10); REPEATED/SUB=INTERCEPT TYPE=SP(SPH) (U V); RUN;

PROC MIXED METHOD=ML MAXITER=500 COVTEST SCORING=10; MODEL Y=X0/NOINT S; PARMS (0.1, 13); REPEATED/SUB=INTERCEPT TYPE=SP(POW) (U V); RUN;

PROC MIXED METHOD=ML MAXITER=500 COVTEST SCORING=10; MODEL Y=X0/NOINT S; PARMS (13, 10, 1); REPEATED/SUB=INTERCEPT TYPE=SP(MATERN) (U V); RUN;

semivariogrammodel

randomeffect

initialparameter

values

geocodings

maximum likelihood estimation

Linear mixed model (Gaussian assumption) without nugget

SA model a saSA random

intercept variance

nugget Residual variance

None 9.4364 0.4121 0 0 12.3968

exponential 8.5340 1.2523 17.5256 0 14.4643

Gaussian 9.4233 0.5950 9.3222 0 12.0672

Spherical 9.0118 0.8007 28.2559 0 13.5297

Power 8.5344 1.2522 0.9445 0 14.4634

Bessel 8.5925 1.1900 15.2664 0 14.3009

OLS:12.5690

0.5ν

ν

0.5584ν̂

Linear mixed model (Gaussian assumption) with nugget

SA model a saSA random

intercept variance

nugget Residual variance

None 9.4364 0.4121 0 0 12.3968

exponential 8.5271 1.2593 17.6185 14.4184 0.0671

Gaussian 8.9213 0.8591 16.2443 9.8902 3.4054

Spherical 9.0237 0.7869 28.6201 12.6564 0.4515

Power failed to converge

Bessel 8.6029 1.1559 11.9560 12.5227 1.8082

OLS:12.5690

0.5ν

ν

0.9156ν̂

SAS code for spatially structured random effects generalized linear modeling

FILENAME INDATA 'D:\JYU-SUMMERSCHOOL2006\LAB#3\PR-URBAN-DATA.TXT';

FILENAME EVECS 'D:\JYU-SUMMERSCHOOL2006\LAB#3\PR-EXPANDED-EVECS.TXT';

OPTIONS LINESIZE=72;

TITLE 'GENERALIZED LINEAR MIXED MODEL FOR THE PR DATA';

*******************************************************************

* READ IN GEOREFERENCED DATA; THEN CENTER THE ELEVATION VARIABLE *

*******************************************************************;

DATA STEP1; INFILE INDATA;

INPUT ID Y1899 Y1910 Y1920 Y1930 Y1935 Y1940

Y1950 Y1960 Y1970 Y1980 Y1990 Y2000 MELEV SELEV U V AAR NAME$;

NUM=Y1899;

U=U/1000; V=V/1000;

ELEV=LOG(MELEV/SELEV-0.6);

UTELEV=MELEV/SELEV-0.6;

IF AAR=1 THEN ISJ=1; ELSE ISJ=0;

IF AAR=2 THEN IA =1; ELSE IA =0;

IF AAR=3 THEN IM =1; ELSE IM =0;

IF AAR=4 THEN IP =1; ELSE IP =0;

IF AAR=5 THEN IC =1; ELSE IC =0;

IA=IA-ISJ;

IM=IM-ISJ;

IP=IP-ISJ;

IC=IC-ISJ;

DENOM=100;

Y=NUM/DENOM;

P=Y;

RUN;

PROC STANDARD MEAN=0 STD=1 OUT=STEP1(REPLACE=YES); VAR ELEV; RUN;

DATA STEP2; INFILE EVECS LRECL=1024; INPUT IDE E1-E45; RUN;DATA STEP2(REPLACE=YES); SET STEP2; SET STEP1; RUN;

DATA STEP4; SET STEP2(KEEP=ID DENOM ELEV UTELEV IA IP IM IC Y1899 E1-E45); NUM=Y1899; TIME=1899; DROP Y1899; RUN;

DATA STEP3; SET STEP2(KEEP=ID DENOM ELEV UTELEV IA IP IM IC Y1910 E1-E45); NUM=Y1910; TIME=1910; DROP Y1910; RUN;

DATA STEP4(REPLACE=YES); SET STEP4 STEP3; RUN;DATA STEP3(REPLACE=YES); SET STEP2(KEEP=ID DENOM ELEV UTELEV IP IM IC Y1920 E1-E45); NUM=Y1920; TIME=1920; DROP

Y1920; RUN;DATA STEP4(REPLACE=YES); SET STEP4 STEP3; RUN;DATA STEP3(REPLACE=YES); SET STEP2(KEEP=ID DENOM ELEV UTELEV IP IM IC Y1930 E1-E45); NUM=Y1930; TIME=1930; DROP









Y2000; RUN;DATA STEP4(REPLACE=YES); SET STEP4 STEP3; RUN;PROC SORT OUT=STEP5; BY ID DESCENDING TIME; RUN;

PROC NLMIXED DATA=STEP5; PARMS B0=0 BE=0 BIM=0 BIP=0 BIC=0 S2U=1

BE1 =0 BE2 =0 BE3 =0 BE4 =0 BE5 =0 BE6 =0 BE7 =0 BE9 =0 BE10=0 BE11=0 BE16=0 BE17=0 BE18=0 BE19=0 BE20=0 BE21=0 BE23=0 BE26=0 BE27=0 BE33=0 BE35=0; ETA = B0 + ALPHA + BE*ELEV + BIM*IM + BIP*IP + BIC*IC + BE1*E1+BE2*E2+BE3*E3+BE4*E4+BE5*E5+BE6*E6+BE7*E7+BE9*E9+BE10*E10+BE11*E11+BE16*E16+ BE17*E17+BE18*E18+BE19*E19+BE20*E20+BE21*E21+BE23*E23+BE26*E26+BE27*E27+BE33*E33+BE35*E35; P = EXP(ETA)/(1+EXP(ETA)); MODEL NUM ~ BINOMIAL(100,P); RANDOM ALPHA ~ NORMAL(0,S2U) SUBJECT=ID; PREDICT P OUT=POUT;RUN;

PROC NLMIXED DATA=STEP5; PARMS B0=0 BE=0 BIM=0 BIC=0 S2U=1

BE2 =0 BE6 =0 BE10=0 BE11=0 BE19=0 BE35=0; ETA = B0 + ALPHA + BE*ELEV + BIM*IM + BIC*IC + BE2*E2+ BE6*E6+ BE10*E10+BE11*E11+ BE19*E19+ BE35*E35; P = EXP(ETA)/(1+EXP(ETA)); MODEL NUM ~ BINOMIAL(100,P); RANDOM ALPHA ~ NORMAL(0,S2U) SUBJECT=ID; PREDICT P OUT=POUT;RUN;

PROC SORT OUT=POUT(REPLACE=YES); BY TIME ID; RUN;DATA POUT(REPLACE=YES); SET POUT; SET STEP4(KEEP=NUM); P=NUM/100; RUN;PROC REG; MODEL P=PRED; BY TIME; RUN;PROC GPLOT; PLOT P*PRED; BY TIME; RUN;

PROC GENMOD DATA=STEP2; MODEL NUM/DENOM= ELEV E1 E4 E6 E7 E11 E17 E23 E26 E33/DIST=B SCALE=P; OUTPUT OUT=POUT P=PRED; RUN;

Comparative static and space-time resultsyear covariate vectors MC GR dev Pseudo-R2 MC GR

1899 *** *** -0.05 1.03 26.0 0 0.633 0.17 0.82

1910 Elev, IC 4, 6, 7;

11, 16, 20, 23, 26

-0.04 0.84 19.4 0.438 0.775 0.09 0.81

1920 Elev, IC 4, 6, 7;

11, 23, 26, 33-0.08 0.87 19.6 0.436 0.726 0.06 0.84

1930 Elev 4;

23-0.08 0.99 25.2 0.194 0.765 0.09 0.80

1935 Elev 4, 6;

23-0.10 0.97 21.8 0.286 0.731 0.08 0.79

1940 Elev 4;

21, 23-0.09 1.00 25.1 0.192 0.760 0.16 0.70

1950 Elev 4, 6;

21, 33-0.17 1.11 24.7 0.265 0.744 0.05 0.84

1960 Elev, IP 4, 6;

17, 23-0.12 0.98 24.1 0.386 0.733 0.05 0.82

1970 Elev 1, 4, 6, 7;

26-0.05 0.91 18.2 0.424 0.768 0.13 0.69

1980 Elev 1, 6, 7, 18 -0.04 0.95 22.1 0.472 0.640 0.30 0.59

1990 Elev, IP 1, 2, 6, 10, 18;

27, 35-0.09 0.99 23.5 0.550 0.490 0.32 0.62

2000 Elev 1, 3, 4, 5, 6, 9, 27;

19, 26

-0.10 1.15 4.7 0.778 0.184 0.15 0.85

Spatial filters structuring the random effects

MC = 0.274GR = 0.795

MC = 0.877GR = 0.227

MC = -0.459GR = 1.582

The random effect22 0.633σ̂ 0.001,μ̂

MC = -0.074, GR = 0.973

What you should have learned in today’s lab:

1. Calculate the J approximation for a given connectivity matrix (both C and W)

2. Normal approximation: estimate AR, SAR, AR-SAR and SF models

3. Estimate a SF GLM (Poisson or binomial)

4. Estimate a univariate LMM

5. Estimate a bivariate (time) SF-GLMM

6. Construct maps of the SFs

7. Construct maps of the random effects

spatial statistics in practice

Documents

set step1 set cyout2wy

var y rundata step1

set extremel set step0afinish

set jacob2 set step2j

sum var csum runproc

means noprint data

lambdaw runproc transpose

eout1 runproc nlin data