Noise variance estimation

Suppose that you have a signal Y (Y can be a time series, a parametric surface or a volumetric data series) corrupted by a Gaussian noise with unknown variance. It is often of interest to know more about this variance. EVAR(Y) thus returns an estimated variance of the additive noise.

EVAR provides better results if the original function (i.e. the function without noise) is relatively smooth i.e. has continuous derivatives up to some order. Several tests, however, showed that EVAR works very well even with multiple discontinuities.

Note: EVAR only works with evenly-gridded data in one and higher dimensions.

Here are two examples:

%-- Let us estimate the noise variance from a corrupt signal --
% First create a time signal
t = linspace(0,100,1e6);
y = cos(t/10)+(t/50);
% Make this signal corrupted by a Gaussian noise of variance 0.02
var0 = 0.02; % noise variance
yn = y + sqrt(var0)*randn(size(y));
% Now estimate the variance with EVAR and compare with the "true" value
evar(yn)

%-- Now, let us estimate the noise variance from volumetric data --
% Create a volume array
[x,y,z] = meshgrid(-2:.2:2,-2:.2:2,-2:.2:2);
f = x.*exp(-x.^2-y.^2-z.^2);
% Make these data corrupted by a Gaussian noise of variance 0.5
var0 = 0.5; % noise variance
fn = f + sqrt(var0)*randn(size(f));
% Estimate the variance with EVAR and compare with the "true" value
evar(fn)

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Reference:
Garcia D. Robust smoothing of gridded data in one and higher dimensions with missing values.
Comput Statist Data Anal, 2010;54:1167-1178
http://www.biomecardio.com/publis/csda10.pdf
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Several examples are also given in:
http://www.biomecardio.com/matlab/evar_doc.html
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