Fitting a Gaussian to Data

When fitting a single Gaussian to data, one can take a log and fit a parabola. In matlab, this can be carried out as in the following example:

x = -1:0.1:1;
sigma = 0.01;
y = exp(-x.*x) + sigma*randn(size(x)); % test data:
[p,s] = polyfit(x,log(y),2); % fit parabola to log
yh = exp(polyval(p,x)); % data model
norm(y-yh) % ans =  1.9230e-16 when sigma=0
plot(abs([y',yh']));

In practice, it is good to avoid zeros in the data. For example, one can fit only to the middle third or so of a measured peak, restricting consideration to measured samples that are positive and ``look Gaussian'' to a reasonable extent.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.