By ICM-1966, Moscow.
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The spectral density of the correlation function in the R2 Mat`ern class, with range λ > 0 and smoothness ν > 0, is given by f (Υ ) ∝ 1/(λ2 + Υ 2 )ν/2+1 . 2) where Kν is the modiﬁed Bessel function of the second kind of order ν (Abramowitz and Stegun, 1964). In the rest of this chapter, we focus the discussion and examples on processes in one dimension. However, the results are readily applied to higher dimensions as well. In practice, the need to model the background process tends to limit the application of the convolution approach to three or fewer dimensions as, with more dimensions than that, the curse of dimensionality starts to kick in and it is typically more eﬃcient to model the Gaussian process directly.
This relationship is based on the convolution theorem for Fourier transforms; more details can be found in Barry and Ver Hoef (1996). An example of a ﬂexible family of isotropic kernels is obtained from the Mat`ern class of isotropic correlations (see, for example, Stein, 1999), which is known to correspond to processes with widely diﬀerent degrees of smoothness. The spectral density of the correlation function in the R2 Mat`ern class, with range λ > 0 and smoothness ν > 0, is given by f (Υ ) ∝ 1/(λ2 + Υ 2 )ν/2+1 .
A) Doppler function. (b) Doppler function with noise. Signal-to-noise ratio equal to 5. On the contrary, the wavelet coeﬃcients at the higher resolution levels are confounded with noise. 90, respectively. These values for α and β assume that both the prior variance of nonzero wavelet coeﬃcients and the prior probability of a nonzero wavelet coeﬃcient decrease 50% from a given resolution level to the next higher resolution level. 4(c) present the wavelet coeﬃcients’ posterior mean and median, respectively.
Abstracts of reports on invitation by ICM-1966, Moscow.