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Is there a good, modern treatment covering the various methods of multivariate interpolation, including which methodologies are typically best for particular types of problems? I'm interested in a solid statistical treatment including error estimates under various model assumptions.

An example:

Shepard's method

Say we're sampling from a multivariate normal distribution with unknown parameters. What can we say about the standard error of the interpolated estimates?

I was hoping for a pointer to a general survey addressing similar questions for the various types of multivariate interpolations in common use.

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Can you be a little more specific? – mbq♦ Jul 22 '10 at 16:26
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Interpolation involves 3 things: 1) a class of functions to interpolate, e.g. sound, pictures, terrain; 2) grids of known / unknown points, 4 cases regular <-> scattered; and 3) a model of noise added to 1). There are many many interpolation methods for various cases, most ad hoc, not "in common use"; even IDW has variants. Can you describe what you're interpolating ? – Denis Jul 28 '10 at 16:11

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Sorry, no quick answer. There are thick books dedicated to answering this question. Here's a 600-page long example: Harrell's Regression Modeling Strategies

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