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copied!<p>According to the 1D example, we are given a sequence of values <code>(xi, yi)</code>, where the <code>xi</code> values are vectors of dimension <code>n</code> (<code>n = 1</code> or <code>n = 2</code>), and the <code>yi</code> values are scalars. We wish to find mean <code>mu</code> and covariance matrix <code>sigma</code> for a Gaussian function <code>f</code> of dimension <code>n</code> such that the <code>f(xi)</code> values are close to the <code>yi</code> values.</p> <p>To solve this problem we have to define what <em>close</em> means. We can choose for example to minimize the sum of the squares <code>(yi - f(xi))^2</code>. The resulting non-linear optimization problem in <code>mu</code> and <code>sigma</code> can be solved with an iterative method such as <a href="http://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm" rel="nofollow">Levenberg–Marquardt</a>.</p> <hr> <p>If MatLab is used perhaps see their <a href="http://www.mathworks.se/support/tech-notes/1500/1508.html" rel="nofollow">Curve Fitting Guide</a>, specifically the <a href="http://www.mathworks.se/support/tech-notes/1500/1508.html#section2bb" rel="nofollow">Nonlinear Curve Fitting</a> section and the <code>lsqnonlin</code> and <code>lsqcurvefit</code> examples. If Python is used, you can find library bindings to Levenberg–Marquardt implementations in C/C++ or Fortran.</p> <p>Whatever approach is used, I recommend trying out the solver on the 1D example first.</p> <p>Another relevant resource is <a href="http://www.scipy.org/Cookbook/FittingData" rel="nofollow">SciPy's data fitting cookbook</a>, which includes a section on <a href="http://www.scipy.org/Cookbook/FittingData#head-11870c917b101bb0d4b34900a0da1b7deb613bf7" rel="nofollow">fitting a 2D gaussian</a>.</p>

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