Krig: Kriging surface estimate

Description

Fits a surface to irregularly spaced data. The Kriging model assumes that the unknown function is a realization of a Gaussian random spatial processes. The assumed model is additive Y = P(x) + Z(X) + e, where P is a low order polynomial and Z is a mean zero, Gaussian stochastic process with a covariance that is unknown up to a scale constant. The main advantages of this function are the flexibility in specifying the covariance as an R language function and also the supporting functions plot, predict, predict.se, surface for subsequent analysis. Krig also supports a correlation model where the mean and marginal variances are supplied.

Usage

Krig(x, Y, cov.function=exp.cov, lambda=NA, df = NA,
cost=1, knots, weights=rep(1,length(Y)), 
m=2, return.matrices=TRUE, nstep.cv=80, scale.type="user", 
x.center=rep(0, ncol(x)), x.scale=rep(1, ncol(x)), rho=NA, sigma2=NA, 
method="GCV", decomp="DR", verbose=FALSE, cond.number=10^8, mean.obj=NULL, 
sd.obj=NULL, yname=NULL, return.X=TRUE, null.function=make.tmatrix, offset=0, 
 outputcall = NULL, cov.args=NULL,na.rm=FALSE,...)

Arguments

Value

A object of class Krig. This includes the predicted values in fitted.values and the residuals in residuals. The results of the grid search to minimize the generalized cross validation function are returned in gcv.grid.callCall to the functionyVector of dependent variables.xMatrix of independent variables.weightsVector of weights.knotsLocations used to define the basis functions.transformList of components used in centering and scaling data.npTotal number of parameters in the model.ntNumber of parameters in the null space.matricesList of matrices from the decompositions (D, G, u, X, qr.T).gcv.gridMatrix of values from the GCV grid search. The first column is the grid of lambda values used in the search, the second column is the trace of the A matrix, the third column is the GCV values and the fourth column is the estimated value of sigma conditional on the vlaue of lambda.lambda.estA table of estimated smoothing parameters with corresponding degrees of freedom and estimates of sigma found by different methods.costCost value used in GCV criterion.mOrder of the polynomial space: highest degree polynomial is (m-1). This is a fixed part of the surface often referred to as the drift or spatial trend.eff.dfEffective degrees of freedom of the model.fitted.valuesPredicted values from the fit.residualsResiduals from the fit.lambdaValue of the smoothing parameter used in the fit.ynameName of the response.cov.functionCovariance function of the model.betaEstimated coefficients in the ridge regression formatdEstimated coefficients for the polynomial basis functions that span the null spacefitted.values.nullFitted values for just the polynomial part of the estimatetraceEffective number of parameters in model.cEstimated coefficients for the basis functions derived from the covariance.coefficientsSame as the beta vector.just.solveLogical describing if the data has been interpolated using the basis functions.shatEstimated standard deviation of the measurement error (nugget effect).sigma2Estimated variance of the measurement error (shat**2).rhoScale factor for covariance. COV(h(x),h(x)) = rho*cov.function(x,x) If the covariance is actually a correlation function then rho is also the "sill".mean.varNormalization of the covariance function used to find rho.best.modelVector containing the value of lambda, the estimated variance of the measurement error and the scale factor for covariance used in the fit.

References

See "Additive Models" by Hastie and Tibshirani, "Spatial Statistics" by Cressie and the FIELDS manual.

Details

This function produces a object of class Krig. With this object it is easy to subsequently predict with this fitted surface, find standard errors, alter the y data ( but not x), etc.

The Kriging model is: Y(x)= P(x) + Z(x) + e

where Y is the dependent variable observed at location x, P is a low order polynomial, Z is a mean zero, Gaussian field with covariance function K and e is assumed to be independent normal errors. The estimated surface is the best linear unbiased estimate (BLUE) of P(x) + Z(x) given the observed data. For this estimate K, is taken to be rho*cov.function and the errors have variance sigma**2. In more conventional geostatistical terms rho is the "sill" if the covariance function is actually a correlation function and sigma**2 is the nugget variance or measure error variance (the two are confounded in this model.)

If these parameters rho and sigma2 are omitted in the call, then they are estimated in the following way. If lambda is given, then sigma2 is estimated from the residual sum of squares divided by the degrees of freedom associated with the residuals. Rho is found as the difference between the sums of squares of the predicted values having subtracted off the polynomial part and sigma2.

A useful extension of a stationary correlation to a nonstationary covariance is what we term a correlation model. If mean and marginal standard deviation objects are included in the call. Then the observed data is standardized based on these functions. The spatial process is then estimated with respect to the standardized scale. However for predictions and standard errors the mean and standard deviation surfaces are used to produce results in the original scale of the observations.

The GCV function has several alternative definitions when replicate observations are present or if one uses a reduced set knots. Here are the choices based on the method argument:

GCV: leave-one-out GCV. But if there are replicates it is leave one group out. (Wendy and Doug prefer this one.)

GCV.one: Really leave-one-out GCV even if there are replicate points. This what the old tps function used in FUNFITS. rmse: Match the estimate of sigma**2 to a external value ( called rmse)

pure error: Match the estimate of sigma**2 to the estimate based on replicated data (pure error estimate in ANOVA language).

GCV.model: Only considers the residual sums of squares explained by the basis functions.

WARNING: The covariance functions often have a nonlinear parameter that control the strength of the correlations as a function of separation, usually referred to as the range parameter. This parameter must be specified in the call to Krig and will not be estimated.

Examples

Run this code

#2-d example 
# fitting a surface to ozone  
# measurements. Range parameter is 10 (in miles) 

fit <- Krig(ozone$x, ozone$y, exp.cov, theta=10)  
 
summary( fit) # summary of fit 
plot(fit) # diagnostic plots of fit  
surface( fit, type="C") # look at the surface 
out.p<- predict.surface.se( fit) 
image(out.p)

# predict at data

predict( fit)

# predict on a grid ( grid chosen here by defaults)
out<- predict.surface( fit)
persp( out)

# predict at arbitrary points (10,-10) and (20, 15)
xnew<- rbind( c( 10, -10), c( 20, 15))
predict( fit, xnew)

# standard errors of prediction based on covariance model.  
predict.se( fit, xnew)

#
# Roll your own: using a user defined Gaussian covariance 
#
test.cov <- function(x1,x2,theta){exp(-(rdist(x1,x2)/theta)**2)} 
# use this and put in quadratic polynomial fixed function 
fit.flame<- Krig(flame$x, flame$y, test.cov, m=3, theta=.5)
#
# note how range parameter is passed to Krig.   
# BTW:  GCV indicates an interpolating model (nugget variance is zero) 
#
# take a look ...
surface(fit.flame, type="I") 

# 
# Thin plate spline fit to ozone data using the radial 
# basis function as a generalized covariance function 
#
# p=2 is the power in the radial basis function (with a log term added for 
# even dimensions)
# If m is the degree of derivative in penalty then p=2m-d 
# where d is the dimension of x. p must be greater than 0. 
#  In the example below p = 2*2 - 2 = 2  
#
# See also the Fields function Tps

out<- Krig( ozone$x, ozone$y, rad.cov, m=2,  p=2, 
scale.type="range",decomp="WBW") # these last two options make sure the
                                 # the results with Tps are identical
# A Knot example

data(ozone2)
y16<- ozone2$y[16,] # there are some missing values! 
good<- !is.na( y16)
y<- y16[good] # Krig does not remove missing values. 
x<- ozone2$lon.lat[ good,]

#
# the knots can be arbitrary but just for fun find them with a space 
# filling design. Here we select  50 from the full set of 147 points
#
xknots<- cover.design( x, 50)$design # select 50 knot points

out<- Krig( x, y, knots=xknots,  rad.cov, m=2,  p=2,scale.type="range")

# Zee plot 
surface( out, type="C")
US( add=TRUE)
points( x, col=2)
points( xknots, cex=2, pch="O")

summary( out)
#
# note that that trA found by GCV is around 15 so 50>15  knots may be a 
# reasonable approximation to the full estimator. 
#
surface( out, type='C')
# obs and knots
points( x)
points( xknots, col=2)


# Correlation model example

# fit krig surface using a mean and sd function to standardize 
# first get stats from 1987 summer Midwest O3 data set 
# Compare the function Tps to the call to Krig given above 
# fit tps surfaces to the mean and sd  points.  
# (a shortcut is being taken here just using the lon/lat coordinates) 
data(ozone2)
stats.o3<- stats( ozone2$y)
mean.o3<- Tps( ozone2$lon.lat, c( stats.o3[2,]))
sd.o3<- Tps(  ozone2$lon.lat, c( stats.o3[3,]))

# Now use these to fit particular day ( day 16) 

y16<- ozone2$y[16,] # there are some missing values! 
good<- !is.na( y16) 
 
fit<- Krig( ozone2$lon.lat[good,], y16[good],exp.earth.cov, theta=353, 
mean.obj=mean.o3, sd.obj=sd.o3)
#
# the finale
surface( fit, type="I")
US( add=TRUE)
title("Estimated ozone surface")

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