R help - Apr 2003 - polynomial fitting

I'm trying to find a way to fit a polynomial of degree n in x  and y to
a set of x, y, and z data that I have and obtain the coefficients for
the terms of the fitted polynomial.  However, when I try to use the
surf.ls function I'm getting odd results.  
> x <- seq(0, 10, length=50)
> y <- x
> f <- function (x, y) {x^2 + y}  
> library(spatial)
> test <- data.frame(x=x, y=y, z=f(x, y))
> test.kr <- surf.ls(2, test)
> test.kr$beta
[1] 0 0 0 0 0 0

When I try the example from the help I get:
>      library(MASS)
>      data(topo, package="MASS")
>      topo.kr <- surf.ls(2, topo)
> topo.kr$beta
[1] 801.217617 -11.018887  68.229148 -73.992968   3.343573   8.342717

Why is my test data causing problems?  Also it seems that the beta
attribute from the object returned the surf.ls correspond with the terms
of the fitted polynomial.  If this is correct, in what order are the
coefficients for the fitted polynomial given?  Finally, the R source
code for the spatial library indicate that both surf.ls and surf.gls are
limited to polynomials of degree 6 or below.  Is there another function
that will work with higher order polynomials.  

I'm working with R 1.7.0 using the binary rpm for Redhat 7.3 from CRAN. 
> version
         _                
platform i686-pc-linux-gnu
arch     i686             
os       linux-gnu        
system   i686, linux-gnu  
status                    
major    1                
minor    7.0              
year     2003             
month    04               
day      16               
language R

On 29 Apr 2003, Suchandra Thapa wrote:
> I'm trying to find a way to fit a polynomial of degree n in x  and y to
> a set of x, y, and z data that I have and obtain the coefficients for
> the terms of the fitted polynomial.  However, when I try to use the
> surf.ls function I'm getting odd results.  
> 
> > x <- seq(0, 10, length=50)
> > y <- x
> > f <- function (x, y) {x^2 + y}  
> > library(spatial)
> > test <- data.frame(x=x, y=y, z=f(x, y))
> > test.kr <- surf.ls(2, test)
> > test.kr$beta
> [1] 0 0 0 0 0 0
> 
> When I try the example from the help I get:
> 
> >      library(MASS)
> >      data(topo, package="MASS")
> >      topo.kr <- surf.ls(2, topo)
> > topo.kr$beta
> [1] 801.217617 -11.018887  68.229148 -73.992968   3.343573   8.342717
> 
> Why is my test data causing problems?  
They lie on a line in the x-y plane, so there is collinearity in the fit
and you cannot fit a 2-dimensional trend surface uniquely.
> Also it seems that the beta
> attribute from the object returned the surf.ls correspond with the terms
> of the fitted polynomial.  If this is correct, in what order are the
> coefficients for the fitted polynomial given?  
It is not fully correct, as there is scaling of the data going on before 
fitting the polynomial.
> Finally, the R source
> code for the spatial library indicate that both surf.ls and surf.gls are
> limited to polynomials of degree 6 or below.  Is there another function
> that will work with higher order polynomials.  
You can use the R function poly() to fit orthogonal polynomials using lm
(provided you have genuinely two-dimensional data).
> lm(z ~ poly(x,y, degree= 2), data=topo)
Coefficients:
              (Intercept)  poly(x, y, degree = 2)1.0  
                   827.07                     -20.54  
poly(x, y, degree = 2)2.0  poly(x, y, degree = 2)0.1  
                   167.27                    -337.40  
poly(x, y, degree = 2)1.1  poly(x, y, degree = 2)0.2  
                    67.26                      20.13  


Reminder: spatial is part of the VR bundle, whose DESCRIPTION file says it
is software to support a book.  It is not intended to be comprehensive,
just sufficient for the techniques covered in that book.  I have never
seen a real application of trend surfaces of degree more than 6.

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595