Spectral covariance estimates

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Spectral covariance estimates

The following figures are generated with cov_est_1dwave.m in the osimorph repository.

Covariance model

We will sample a two variable model of the following form:

<math>X_j=h\exp{(-((x_j-c)/w)^2)}</math>

<math>Y_j=.3X_j+.7W_j</math>

  • <math>n=128</math>
  • <math>j=1,\cdots,n</math>
  • <math>x_j=(j-1)/(n-1)</math>
  • <math>h=1+.1\omega</math>
  • <math>c=.3+.1\omega</math>
  • <math>w=.025+.0025\omega</math>
  • <math>\omega\sim\mathcal{N}(0,1)</math> i.i.d.
  • <math>W</math> is a smooth random field independent of <math>X</math>

A sample of size 5 looks as follows:

The purpose of this choice of variable is to create <math>X</math> with a spatially varying density and a correlation of <math>.3</math> between <math>X</math> and <math>Y</math>. In the following figures, a large sample is used to estimate the true covariance of our model. Compare this to covariance estimation using a sample of size 10 for the different methods. The color scale is the same between all of the images.

Using the usual sample covariance, the estimation is very bad. Estimation by FFT results in a distribution that is homogeneous in space, smearing the distribution across the domain. Wavelet estimation keeps the spatial structure of the distribution, while filtering out long distance correlations.

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