fields (version 1.2)

Tps: Thin plate spline regression

Description

Fits a thin plate spline surface to irregularly spaced data. The smoothing parameter is chosen by generalized cross-validation. The assumed model is additive Y = f(X) +e where f(X) is a d dimensional surface. This is a special case of the spatial process estimate.

Usage

Tps
(x, Y, m = NULL, p = NULL, decomp = "WBW", scale.type = "range", ...)

Arguments

x

Matrix of independent variables. Each row is a location.

Y

Vector of dependent variables.

m

A polynomial function of degree (m-1) will be included in the model as the drift (or spatial trend) component. Default is the value such that 2m-d is greater than zero where d is the dimension of x.

p

Exponent for radial basis functions. Default is 2m-d.

decomp

Type of matrix decompositions used to compute the solution. Default is the more numerically stable "WBW" Wendelberger-Bates-Wahba. This is the strategy used in GCV pack. An alternative is "DR" Demmler-Reinsch. This must be used if one wants a red

scale.type

The independent variables and knots are scaled to the specified scale.type. By default the scale type is "range", whereby the locations are transformed to the interval (0,1) by forming (x-min(x))/range(x) for each x. Scale type of "user" allows spe

...

Any argument that is valid for the Krig function. Some of the main ones are listed below.

lambda

Smoothing parameter that is the ratio of the error variance (sigma**2) to the scale parameter of the covariance function. If omitted this is estimated by GCV.

cost

Cost value used in GCV criterion. Corresponds to a penalty for increased number of parameters.

knots

Subset of data used in the fit.

weights

Weights are proportional to the reciprocal variance of the measurement error. The default is no weighting i.e. vector of unit weights.

return.matrices

Matrices from the decompositions are returned. The default is T.

nstep.cv

Number of grid points for minimum GCV search.

x.center

Centering values are subtracted from each column of the x matrix. Must have scale.type="user".

x.scale

Scale values that divided into each column after centering. Must have scale.type="user".

rho

Scale factor for covariance.

sigma2

Variance of errors or if weights are not equal to 1 the variance is sigma**2/weight.

method

Character string specifiying the method for estimating the "smoothing" parameter. The default is 'GCV' -- generalized corss-validation.

verbose

If true will print out all kinds of intermediate stuff.

cond.number

maximum size of condition number to allow when using DR decomposition.

mean.obj

Object to predict the mean of the spatial process.

sd.obj

Object to predict the marginal standard deviation of the spatial process.

yname

Name of y variable

return.X

If true returns the big X matrix used for the estimate.

null.function

An S function that creates the matrices for the null space model. The default is make.tmatrix, an S function that creates polynomial null spaces.

offset

The offset to be used in the GCV criterion. Default is 0. This would be used when Krig/Tps is part of a backfitting algorithm and the offset has to be included to reflect other model degrees of freedom.

Value

• A list of class Krig. This includes the predicted surface of fitted.values and the residuals. The results of the grid search minimizing the generalized cross validation function is returned in gcv.grid. Please see the documentation on Krig for details of the returned arguments.

References

See "Nonparametric Regression and Generalized Linear Models" by Green and Silverman. See "Additive Models" by Hastie and Tibshirani.

Details

A thin plate spline is result of minimizing the residual sum of squares subject to a constraint that the function have a certain level of smoothness (or roughness penalty). Roughness is quantified by the integral of squared m-th order derivatives. For one dimension and m=2 the roughness penalty is the integrated square of the second derivative of the function. For two dimensions the roughness penalty is the integral of

(Dxx(f))**22 + 2(Dxy(f))**2 + (Dyy(f))**22

(where Duv denotes the second partial derivative with respect to u and v.) Besides controlling the order of the derivatives, the value of m also determines the base polynomial that is fit to the data. The degree of this polynomial is (m-1).

The smoothing parameter controls the amount that the data is smoothed. In the usual form this is denoted by lambda, the Lagrange multiplier of the minimization problem. Although this is an awkward scale, lambda =0 corresponds to no smoothness constraints and the data is interpolated. lambda=infinity corresponds to just fitting the polynomial base model by ordinary least squares.

This estimator is implemented simply by feeding the right generalized covariance function based on radial basis functions to the more general function Krig. This is a different approach than the older version in FUNFITS (tps) and provides simpler coding. One advantage of both the FUNFITS and FIELDS implementations is that once a Tps/Krig object is created the estimator can be found rapidly other data and smoothing parameters provided the locations remain unchanged. This makes simulation within S efficient ( see example below).

See Also

Krig, summary.Krig, predict.Krig, predict.se.Krig, plot.Krig, surface.Krig, sreg

Examples

#2-d example 

fit<- Tps(ozone$x, ozone$y)  # fits a surface to ozone measurements. 
plot(fit) # diagnostic plots of  fit and residuals. 
summary(fit)

# predict onto a grid that matches the ranges of the data.  

out.p<-predict.surface( fit)
image( out.p) 
surface(out.p) # perspective and contour plots of GCV spline fit 
# predict at different effective 
# number of parameters 
out.p<-predict.surface( fit,df=10)

#1-d example 
out<-Tps( rat.diet$t, rat.diet$trt) # lambda found by GCV 
plot( out$x, out$y)
lines( out$x, out$fitted.values)

# 
# compare to the ( much faster) one spline algorithm 
#  sreg(rat.diet$t, rat.diet$trt) 
# 
#
# simulation reusing
fit<- Tps( rat.diet$t, rat.diet$trt)
true<- fit$fitted.values
N<-  length( fit$y)
temp<- matrix(  NA, ncol=50, nrow=N)
sigma<- fit$shat.GCV
for (  k in 1:50){
ysim<- true + sigma* rnorm(N) 
temp[,k]<- predict(fit, y= ysim)
}
matplot( fit$x, temp, type="l")


# 
#4-d example 
fit<- Tps(BD[,1:4],BD$lnya,scale.type="range") 
surface(fit)   
# plots fitted surface and contours 
#2-d example using a reduced set of basis functions 
r1 <- range(flame$x[,1]) 
r2 <-range( flame$x[,2]) 
g.list <- list(seq(r1[1], r1[2],6), seq(r2[1], r2[2], 6)) 
knots<- make.surface.grid(g.list) 
# these knots are a 6X6 grid over 
# the ranges of the two flame variables 
out<-Tps(flame$x, flame$y, knots=knots, m=3)   
surface( out, type="I")
points( knots)