Alternatives to Shephard interpolation?

Question

I am a chemist, so I have little experience in the field of math.

My program is that I have a set of points (approx. 20000) in some larger dimensional space (like 10-20 dimensions), and I want to be able to interpolate between these discrete points. I have found, after some research, the method of Shephard, which seems to be suitable for my needs.However, no matter how hard I look, I can not find alternatives. It looks like Shephard is the only common way to go.

Since my project should have as few fitting errors as possible, I would like to implement some additional methods, and compare the results. Are there more interpolation methods like Shephard's?

p.s. my function goes to zero at infinity of any coordinate, and it goes to infinity if some given coordinates have approximately the same value. Can you tell me anything about Shephard's performance in cases like this?

Answer 1

There is a vast literature on this problem. For a start have a look at the wikpedia article on "multivariate interpolation". The big challenge is not to get lost in the dazzling variety of approaches. There is no single approach which is best in all cases. The following criteria might be helpful in distinguishing between the various ways to approach the problem:

1. Is your data defined on a grid (gridded) or randomly dispersed (scattered)?
2. What do you know about the target function in addition to the behaviour you mention: Is it differentiable, merely continous or of special kind such as piecewise linear?
3. What kind of optimality do you try to achieve? Possible definitions of optimality are worst case optimal or optimal on average.
4. Are the observed function values you try to interpolate really exact or noisy?

For scattered data an approach called "radial basis function interpolation" (or Kriging or Gaussian Process Regression, all different names for basically the same thing) is very flexible in respect to criteria (2) to (4). Textbooks discussing this approach are:

• Wendland, H., "Scattered Data Approximation". Cambridge University Press, 2005 (Very thorough treatment, definitely heavy on the math though)
• "Gaussian Processes for Machine Learning", Carl Edward Rasmussen and Chris Williams, the MIT Press, 2006 (an online version of the book is available here)
• Jean-Paul Chiles, Pierre Delfiner "Geostatistics: Modeling Spatial Uncertainty", Wiley 2012