Tps {fields} | R Documentation |
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.
Tps(x, Y, m = NULL, p = NULL, decomp = "WBW", scale.type = "range", ...)
To be helpful, a more complete list of arguments are described that are the same as those for the Krig function.
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 reduced set of basis functions (specifying knots). |
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 specification of an x.center and x.scale by the user. The default for "user" is mean 0 and standard deviation 1. Scale type of "unscaled" does not scale the data. |
... |
Any argument that is valid for the Krig function. Some of the main ones
are listed below.
|
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 this implementation is that once a Tps/Krig object is created the estimator can be found rapidly for other data and smoothing parameters provided the locations remain unchanged. This makes simulation within R efficient (see example below).
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.
See "Nonparametric Regression and Generalized Linear Models" by Green and Silverman. See "Additive Models" by Hastie and Tibshirani.
Krig, summary.Krig, predict.Krig, predict.se.Krig, plot.Krig,
surface.Krig
,
sreg
#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)