an introduction to kriging using sas

19th New England Statistics Symposium

April 23rd, 2005

Prof. Roger Bilisoly

Department Mathematical Sciences

Central Connecticut State University

An Introduction to Kriging

Using SAS

1. Example of Spatial Data

• Magnetometer readings to

detect metal

• Goal here is to find

unexploded ordnance (UXO)

• In general, need spatial data

that is spatially correlated

2. In This Talk We Consider

Mining Data

Black = lowest value

White = highest value

(Think of burning embers)

The goal is to find high

concentrations of ore,

which are then mined.

Some Background on

Spatial Modeling

•Spatial model: Z(s), s = spatial location, and

Z(s) is a random variable for each s.

• If Z(s) has not been measured at a location s,

then its value is uncertain. We model this

uncertainty by a probability distribution.

. Z(s1) . Z(s2)

x Z(s)

. Z(s3)

Kriging = Spatial Interpolation

• Kriging is BLUE:

– B for Best meaning minimum variance

– L for Linear

– U for Unbiased

– E for Estimate

Form of

Kriging Estimate:

. Z(s1) . Z(s2)

x Z(s)

. Z(s3)

3

1

)()(i

ii sZsZ

Spatial Autocorrelation

All points uncorrelated here:

No hope for spatial prediction. Points close together tend

to be similar which allows

spatial prediction.

Variogram = Sill - Covariance

• Goal: We want to know the correlation between Z(s)

and Z(s+h) as a function of the distance |h|.

Sill

Range .

. . .

. .

Find pairs of data values

separated by the same distance

Empirical Variogram

Assume that the variogram is a function of s1 – s2 = h

and E(Z(s)) is constant.

γ(s1 – s2) = γ(h) = Var(Z(s1) – Z(s2))

2))()((

pairs#

1ji

hss

sZsZji

.

. . .

. .

Valid Models

• Any spatial prediction must have non-negative

variance:

N

i

N

j

jiji

N

i

ii ssCovsZVar1 11

0),())((

This restricts our choice of variogram models γ(h)

Example of Kriging with SAS

7. Goal: Strike it Rich!

Walker data set from

Isaaks and Srivastava(1989)

Sampling design: two stage

adaptive sampling

Measured:

Ore concentration

Goal: Find high concentrations of

ore so it can be profitably mined.

8. Map of Ore Concentration

Produced by

PROC G3GRID

PROC GCONTOUR

Using PATTERN

SAS Code for Map

axis1 order=(0 to 250 by 25) width=3 minor=(n=4) label=('Easting');

axis2 order=(0 to 300 by 25) width=3 minor=(n=4) label=('Northing');

symbol1 interpol=none line=1 value=plus color=black;

proc g3grid data=walker out=walker2;

grid y*x=v / naxis1=25 naxis2=30;

run;

pattern1 value=solid color=CX000000;

pattern2 value=solid color=CX180000;

pattern3 value=solid color=CX300000;

…

pattern31 value=solid color=CXFFFFFF;

proc gcontour data=walker2;

title 'Smoothed Data Plot';

plot y*x=v/haxis=axis1 vaxis=axis2 levels=0 to 1500 by 50 pattern nolegend;

run;

3. Read in the Data

data walker;

infile "M:\My Work\Spatial Stat Talks\walker data set.txt";

drop title nvar;

length title $ 50;

if _n_ = 1 then do;

input title $ &; put title;

input nvar;

input #(nvar+2);

end; else do;

input id x y v u t;

output;

end;

run;

x and y = spatial coordinates

v = variable analyzed here

4. Variogram Estimation

proc variogram data=walker outvar=outv;

compute lagdistance=5 maxlag=20;

coordinates xc=x yc=y;

var v; run;

This creates the empirical variogram,

which quantifies spatial autocorrelation.

Need spatial

spacing info.

4. PROC NLIN to Fit

a Parametric Variogram

proc nlin data=outv;

parms c0=20000 c1=80000 a1=80;

model variog = (c0 + c1*(1.5*(distance/a1) - 0.5*(distance/a1)**3))*(distance < a1) + (c0 + c1)*(distance >= a1);

der.c0 = 1;

der.c1 = 1.5*(distance/a1) - 0.5*(distance/a1)**3;

der.a1 = c1*(-1.5*(distance/a1**2) + 1.5*(distance**3/a1**4));

output out=fitted predicted=pvari

parms=c0 c1 a1;

run;

Need to

specify a

parametric

model

Need to give

derivative

information.

4. PROC KRIGE2D

Does the Kriging

proc krige2d data=walker outest=est;

predict var=v radius=200;

model form=spherical

range=&a1

scale=&c1

nugget=&c0;

coord xc=x yc=y;

grid x=0 to 250 by 2 y=0 to 300 by 2;

run;

This is a specific

parametric shape.

This specifies locations to make spatial predictions.

Isotropic Variogram

Model for h < a1:

c0 + c1*(1.5*h/a1 - 0.5*(h/a1)3)

Best fit by PROC NLIN:

c0 = 29474

c1 = 64313

a1 = 43.39

6. Kriging vs. Map

Note that the kriging predictions are smoother than the data.

7. Why Kriging?

•Both mapping and kriging interpolate

•Mapping is quicker

BUT

•Kriging comes with an error estimate

6. Kriging Estimate &

Error Estimate

5. Conclusions

1. Kriging is based on modeling, which makes

it harder to do, but it also comes with an

error estimate

2. SAS has kriging built into SAS/STAT via

PROC NLIN, PROC VARIOGRAM and

PROC KRIGE2D

3. SAS/GRAPH can be used for mapping

4. Note that hypothesis testing is not

prominent for these PROCs.