Principal Directions with surf.ls

Thanks. I was hoping that either there were an existing
function or the components of the output of surf.ls were
sufficiently documented to allow one to do it oneself.

surf.ls seems to do all the real work in a C function
and I don't relish having to go through that but
since its just a regression I guess I can ditch surf.ls
and use lm in which case I can get the coefficients
explicitly and figure it out from there.

On 3/14/07, Roger Bivand <Roger.Bivand at nhh.no> wrote:

On Wed, 14 Mar 2007, Gabor Grothendieck wrote:

Suppose we plot a quadratic surface as in:

library(spatial)
example(surf.ls)

How can we plot lines on the graph to show the
principal directions/axes of the elliptical contours
and how do we get them?

In the example there are no closed ellipses, with an open, north-facing
depression with flow from the west, south-west, south, south-east and
east towards the north at the centre of the northern edge.

The contour lines can be retrieved with contourLines() on the same object
as contour() is used on. However, they are just a generalised
visualisation of the surface represented by the matrix. The underlying
question might then be: what is the aspect and slope of the surface, in
this case a smoothed surface, what are the plan and especially profile
curvatures, and what are their length attributes (ie. patches of similar
aspect along a profile curvature.

In GIS terms this would involve geomorphometrics, often computed using a 3
x 3 filter, and then flow analysis to find the directions (on a digital
elevation model, these would be similar to predicting drainage channel
networks. Landslip and avalanche studies often need slope length too,
which involves measuring patch characteristics in aspect and profile
curvature - that is extracting patches with interesting pattern
signatures, and checking whether the patch is "straight" in some sense,
and longer than a minimum accumulation threshold to feed mass movement.

These are general observations in the case that we don't have a smoothed
surface. But here we have a model, and I'd guess that the underlying model
could be used, rather than its predictions.

I couldn't see a package with a vector field plot, there was a suggestion
in:

http://finzi.psych.upenn.edu/R/Rhelp02a/archive/35933.html

but you'd still need the slope and aspect to get the length and direction
of the arrows (for display only). A classic reference is:

Horn, B. K. P. (1981). Hill Shading and the Reflectance Map, Proceedings
of the IEEE, 69(1): 14-47

and Horn's algorithm has stood the test of time, as has Zevenbergen and
Thorne:

Zevenbergen, L. W., Thorne, C. R. 1987. Quantitative analysis of land
surface topography. Earth Surface Processes and Landforms, 12 (1), 4756.

It could well be that the surface you are considering is something other
than elevation (temperature, etc.), but getting further would depend on
the specific characteristics of the study, and whether the surface is a
fitted trend surface or not. With a fitted trend surface model, it ought
to be possible to use the model. If thinking in terms of geomorphometrics
is unhelpful, please accept that geographers "see" things in those kinds
of ways!

Hope this helps,

Roger

Thanks.

--
Roger Bivand
Economic Geography Section, Department of Economics, Norwegian School of
Economics and Business Administration, Helleveien 30, N-5045 Bergen,
Norway. voice: +47 55 95 93 55; fax +47 55 95 95 43
e-mail: Roger.Bivand at nhh.no