Fits models to sets of variograms and cross-variograms.
Options
PRINT = string tokens |
Controls printed output from the fit (model, summary, estimates, fittedvalues, monitoring); default mode, summ, esti |
|---|---|
WEIGHTING = string token |
Method to be used for weighting (counts, equal); default coun |
MAXLAG = scalar |
Maximum lag distance of points to be included in the modelling |
MINCOUNT = scalar |
Minimum number of points required at a particular lag point for a pair of variables for this to be used to model their cross-variogram; default 30 for equal weighting and 10 for counts |
MAXCYCLE = scalar |
Maximum number of iterations for model fitting; default 30 |
TOLERANCES = variate |
Tolerances for model fitting; default * i.e. appropriate default values |
COORDSYSTEM = string token |
Coordinate system used for the geometry for discretizing the lag (mathematical, geographical); default math |
COVARIOGRAM = pointers |
Experimental variograms, cross-variograms and associated information defining the data for fitting the model |
Parameters
MODELTYPE = string tokens |
Defines the model structures to be fitted (nugget, power, boundedlinear, circular, spherical, pentaspherical, cubic, stable, besselk1, cardinalsine, dampenedcosine); no default i.e. must be specified |
|---|---|
INITIAL = scalars or variates |
Scalar defining the initial distance parameter for fitting an isotropic model structure or a variate defining initial values for an anisotropic ellipse or ellipsoid for fitting an geometrical anisotropic model |
ISOTROPY = string tokens |
Specifies the zonal anisotropy to be used for model structure (isotropic, x, y, z, xy, xz, yz); default isot |
ESTIMATES = pointers |
Structures to store the estimated nonlinear parameters and sill values |
LOWER = scalars |
Lower bound for each nonlinear distance parameter |
UPPER = scalars |
Upper bound for each nonlinear distance parameter |
STEPLENGTH = scalars |
Initial step length for each nonlinear distance parameter |
SMOOTHNESS = scalars |
Value of exponent parameter for the power and stable models, or theta parameter for the dampened-cosine model |
Description
The MCOVARIOGRAM directive fits models to sets of auto- and cross-variograms. You can specify a combination of basic variogram functions to model the variograms, for example, nugget plus spherical. MCOVARIOGRAM uses the algorithms from the directives FIT and FITNONLINEAR to estimate the model parameters for the combination of basic variogram functions. It then fits a linear model of coregionalization using the Goulard & Voltz (1992) algorithm, where each step of the solution is checked for conditional semi-definiteness. The two-step process is iterated until convergence.
The MODELTYPE parameter selects the combination of model structures to be used in the model:
nugget |
c0 |
|---|---|
boundlinear |
ch/a for h ≤ a, otherwise 0 |
circular |
c {1 – (2/π)arccos(h/a) + (2h/(πa))√(1-h2/a2)} |
| for h ≤ a, otherwise 0 | |
spherical |
c {1.5h/a – 0.5(h/a)3 } |
| for h ≤ a, otherwise 0 | |
pentaspherical |
c {1.875h/a – 1.25(h/a)3 + 0.375(h/a)5} |
| for h ≤ a, otherwise 0 | |
cubic |
c {7(h/a)2 – 8.75(h/a)3 + 3.5(h/a)5 – 0.75(h/a)7} |
stable |
c {1 – exp(-(h/a)b))} |
| for 0 ≤ b ≤ 2 | |
besselk1 |
c {1 – h/a k1(h/a) } |
cardinalsine |
c {1 – a/h sin(h/a)} |
dampenedcosine |
c {1 – exp(-h/(as)) cos(h/a) } |
power |
ghα |
Initial values for the model structures should be supplied using the INITIAL parameter. For an isotropic model the initial value should be specified as a scalar. You can specify a geometrically anisotropic model by supplying the values within a variate. In two dimensions the variate should contain three values that define an anisotropy ellipse. The first value should define the first axis direction. This is the angle for the main direction of continuity (least change with separating distance) measured in degrees, counter-clockwise from East if option COORDSYSTEM is set to mathematical or clockwise from North if COORDSYSTEM is set to geographical. The second value should contain the initial value for the distance parameter of the first axis, and the last value of the variate should be the anisotropy ratio between the distance parameters along the first axis (principal direction of continuity) and the second axis.
In three dimensions the variate should contain six values that define an anisotropy ellipsoid. The first value defines the angle for the first axis (principal direction of continuity) which is measured in degrees, counter-clockwise from East if COORDSYSTEM is set to mathematical or clockwise from North if COORDSYSTEM is set to geographical. The second value defines the dip angle for the first axis (rotation angle around the y-axis) which is measured in degrees up from horizontal. The third value defines the rotation angle of the second and third axis around the first axis (defined by the two previous angles). The fourth value should contain the initial value for the distance parameter along the first axis. The fifth value defines the anisotropy ratio between distance parameters along the first and second axis of the ellipsoid. The last value of the variate defines the anisotropy ratio between the distance parameters along the second and third axis of the ellipsoid.
Another form of anisotropy can occur when the sill of a semi-variogram varies in different directions. This is known as zonal anisotropy and you can set a model structure to be zonal in particular directions using the ISOTROPY parameter. A model structure can be zonal and geometrically anisotropic.
For the power and stable models the SMOOTHNESS option controls the power parameter for the model. By default, the parameter is estimated, however, you can supply a value to fix the parameter for the model fitting.
The WEIGHTING option controls the weights that are used when fitting the model. The default setting counts uses the values supplied for the counts within the COVARIOGRAM option, and equal uses equal weights (of one).
The MAXLAG option can be used to specify the maximum lag distance of points to be included in the modelling. The MINCOUNT option specifies the minimum number of points to be used to model the variograms at a particular lag.
The TOLERANCES option controls the criterion for convergence of the nonlinear regression and Goulard & Voltz algorithm. The values should be supplied in a variate where the first value is the criterion for the nonlinear regression and the second value is the criterion for the Goulard & Voltz algorithm. The option MAXCYCLE can be used to change the maximum number of iterations performed by the nonlinear regression from the default of 30.
The COVARIOGRAM option allows you to specify a pointer containing the auto-variograms, cross-variograms and associated information. This structure can be saved from the FCOVARIOGRAM directive.
The geometry used for the directions supplied using the COVARIOGRAM option is given by the COORDSYSTEM option, where the setting mathematical specifies directions counter-clockwise from East, and geographical clockwise from North (for the first angle only in 3 dimensions).
The ESTIMATES parameter allows you to specify an identifier to save the estimated nonlinear parameters, sill values and associated information. This structure stores the information required for the DCOVARIOGRAM procedure or COKRIGE directive.
The PRINT option controls the output to be displayed, with settings:
model |
description of the models fitted, |
|---|---|
summary |
summary of analysis, |
estimates |
parameter estimates, |
fittedvalues |
fitted semi-variances, |
monitoring |
monitoring information at each iteration of the nonlinear regression. |
Options: PRINT, WEIGHTING, MAXLAG, MINCOUNT, MAXCYCLE, TOLERANCES, COORDSYSTEM, COVARIOGRAM.
Parameters: MODELTYPE, INITIAL, ISOTROPY, ESTIMATES, LOWER, UPPER, STEPLENGTH, SMOOTHNESS.
Reference
Goulard, M. & Voltz, M. (1992). Linear coregionalization model: tools for estimation and choice of cross-variogram matrix. Mathematical Geology, 24, 269-286.
See also
Directives: FCOVARIOGRAM, COKRIGE, FVARIOGRAM, KRIGE.
Procedures: DCOVARIOGRAM, KCROSSVALIDATION, MVARIOGRAM, DVARIOGRAM DHSCATTERGRAM.
Commands for: Spatial statistics.
Example
" Examples 2:8.3.6, 2:8.3.7, 2:8.3.8 & 2:8.3.9 "
" Data are measurements of concentrations of trace metals in the topsoil
of the Swiss Jura. Data analyzed are Cadmium, Nickel and Zinc taken
from Goovaerts prediction subset. See Goovaerts (1997) Geostatistics
for Natural Resources Evaluation."
FILEREAD [PRINT=summary; NAME=\
'%GENDIR%/Examples/GuidePart2/Goovaerts.dat']X1,X2,Cd,Ni,Zn
FCOVARIOGRAM [PRINT=statistics; MAXLAG=2.1; STEP=0.1; DIRECTIONS=0;\
SEGMENTS=180; MAXCONE=500; MINCOUNT=1;\
COVARIOGRAM=Save_cov] DATA=Cd,Ni,Zn; X1=X1; X2=X2
" Plot the variograms and covariograms."
GETATTRIBUTE [ATTRIBUTE=columns] Save_cov['semivar']; Lab
FRAME 11...16; YLOWER=2(0.66,0.33,0); YUPPER=2(0.98,0.65,0.32);\
XLOWER=(0,0.5)3; XUPPER=(0.5,1)3
TEXT scr; VALUE='clear'
PEN 1; SYMBOL='circle'
XAXIS 11...16; TITLE='Lag distance/km'; LOWER=0; LROTATION=45
YAXIS 11...16; TITLE='Semi-variance'; LOWER=0
FOR [INDEX=i; NTIMES=6]
DGRAPH [WINDOW=i+10; KEY=0; TITLE=Lab['columns']$[i]; SCREEN=#scr]\
Save_cov['semivar']$[*;i]; Save_cov['distances']$[*;i]
TEXT scr; VALUE='keep'
ENDFOR
" Model the coregionalization."
MCOVARIOGRAM [PRINT=summary,estimates; WEIGHTING=counts;\
MAXLAG=3; MINCOUNT=20; COVARIOGRAM=Save_cov]\
MODELTYPE=nugget,spherical,spherical; INITIAL=*,0.2,1.3;\
ESTIMATES=Save_est
DCOVARIOGRAM [ESTIMATES=Save_est] Save_cov
" Read the locations of the prediction points."
MATRIX [ROWS=1547; COLUMNS=2] Mpoints
OPEN '%GENDIR%/Examples/GuidePart2/Mpoints.dat'; CHANNEL=2
READ [CHANNEL=2] Mpoints
CLOSE 2; FILETYPE=input
" Produce predictions and variances for target variable Cadmium."
COKRIGE [PRINT=description; Y=Cd; POINTS=Mpoints; RADII=20;\
SEARCHNEIGHBOURHOOD=local] Cd; X1=X1; X2=X2;\
ESTIMATES=Save_est; PREDICTIONS=Predictions;\
VARIANCES=Variances
" Plot the predictions and variances."
VARIATE [NVALUES=NROWS(Mpoints)] Xpos,Ypos
EQUATE T(Mpoints); !p(Xpos,Ypos)
GROUPS [REDEFINE=yes] Xpos,Ypos; FACTOR=Xfac,Yfac; LEVELS=Xlevs,Ylevs
TABULATE [CLASSIFICATION=Yfac,Xfac] Predictions,Variances;\
MEANS=Zvals,Zvars
MATRIX [ROWS=!(#Ylevs); COLUMNS=!(#Xlevs)] Mpredictions; !(#Zvals)
MATRIX [ROWS=!(#Ylevs); COLUMNS=!(#Xlevs)] Mvariances; !(#Zvars)
XAXIS [RESET=yes] 1
YAXIS [RESET=yes] 1
PEN 2,3; COLOUR='azure','midnightblue'
DSHADE [TITLE='Cokriged estimates for cadmium in the Swiss Jura';\
YORIENTATION=normal; GRIDMETHOD=*] Mpredictions; PEN=!(2,3)
DSHADE [TITLE='Cokriging variances for cadmium in the Swiss Jura';\
YORIENTATION=normal; GRIDMETHOD=*] Mvariances; PEN=!(2,3)
PEN [RESET=yes] 1,2,3