For example I have the variables x, y (or higher dimensional data in general) and a probability density distribution p(x,y). I want to approximate p(x,y) as a linear function, a plane in this case, at least somewhere in the domain. However I only have samples from the distribution. In case of big amount of data the easy way is to collect them into bins, and fit a plane onto the estimated density function. But I dont want to compress data, I would like to extract the maximum information and use the points itself. Are there any method to do this? Once more: I do not have (x,y,z) triples, I only have (x,y) samples from the p(x,y) density.
I recalled that the "Graphics Gems" series of books had a routine that did this, so I did a Google search for "fit a plane to data graphics gems".
One of the hits was this: "Graphics Gems, by Category" at http://tog.acm.org/resources/GraphicsGems/category.html.
This was what I was looking for there:
Tampieri, Filippo, Newell's Method for the Plane Equation of a Polygon, Graphics Gems III, p. 231-232, code: p. 517-518, newell.c.
The code is here: http://tog.acm.org/resources/GraphicsGems/gemsiii/newell.c
The routine is quite short, less than 30 lines long.
A little more searching turned up this:
This is a very good resource - highly recommended.