best p for inverse distance weighting ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:53:11Z http://mathoverflow.net/feeds/question/33657 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33657/best-p-for-inverse-distance-weighting best p for inverse distance weighting ? denis 2010-07-28T14:12:19Z 2010-07-28T14:12:19Z <p><a href="http://en.wikipedia.org/wiki/Inverse_distance_weighting" rel="nofollow">Inverse distance weighting</a> is a common way of interpolating values z<sub>j</sub> at scattered data points X<sub>j</sub> in Rn:</p> <p>&nbsp;&nbsp;&nbsp;&nbsp;idw(P) = &Sigma; w<sub>j</sub> z<sub>j</sub> / &Sigma; w<sub>j</sub><br> &nbsp;&nbsp;&nbsp;&nbsp;w<sub>j</sub> = f( |P - X<sub>j</sub>| )<br> &nbsp;&nbsp;&nbsp;&nbsp;f(d) = 1 / d<sup>p</sup></p> <p>Is there a "best" p, for say X<sub>j</sub> uniformly distributed in the unit cube and z(X) = cos( c . X ) + normal noise ?<br> (For that matter, is there a rationale for 1/d at all -- why not say Gaussian ?)</p> <p>The Wikipedia article say that IDW minimizes a &phi;(x,u) which looks like least squares minimization with variance ~ distance<sup>p</sup>: maybe a connection to least squares, maybe not.</p> <p>(Please add tag "interpolation", thanks.)</p>