Multidimensional autoregression

LEVELED INVERSE INTERPOLATION

Eighteenth- and nineteenth- century mathematics literature gives us many methods of interpolating functions. These classical methods are generally based on polynomials. The user specifies some order of polynomial and the theory works out the coefficients. Today our interest is in both interpolating and extrapolating wavefields (which are solutions to low order differential equations) and we use methods that are much better behaved than polynomials when extrapolating data, methods which behave acceptably when faced with contradictory data values, and methods which also apply in two and three dimensions.

In Chapter , subroutine invint1() solved the problem of inverse linear interpolation, which is, given scattered data points, to find a function on a uniform mesh from which linear interpolation gives the scattered data points. To cope with regions having no data points, the subroutine requires an input roughening filter. This is a bit like specifying a differential equation to be satisfied between the data points. The question is, how should we choose a roughening filter? The importance of the roughening filter grows as the data gets sparser or as the mesh is refined.

Figures 22-25 suggest that the choice of the roughening filter need not be subjective, nor a priori, but that the prediction-error filter (PEF) is the ideal roughening filter. Spectrally, the PEF tends to the inverse of its input hence its output tends to be ``level''. Missing data that is interpolated with this ``leveler'' tends to have the spectrum of given data.

Multidimensional autoregression