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<a name="Fitting-Overview"></a>
<div class="header">
<p>
Next: <a href="Linear-regression.html#Linear-regression" accesskey="n" rel="next">Linear regression</a>, Up: <a href="Least_002dSquares-Fitting.html#Least_002dSquares-Fitting" accesskey="u" rel="up">Least-Squares Fitting</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Overview-4"></a>
<h3 class="section">38.1 Overview</h3>

<p>Least-squares fits are found by minimizing <em>\chi^2</em>
(chi-squared), the weighted sum of squared residuals over <em>n</em>
experimental datapoints <em>(x_i, y_i)</em> for the model <em>Y(c,x)</em>,
</p>
<div class="example">
<pre class="example">\chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2
</pre></div>

<p>The <em>p</em> parameters of the model are <em>c = {c_0, c_1, &hellip;}</em>.  The
weight factors <em>w_i</em> are given by <em>w_i = 1/\sigma_i^2</em>,
where <em>\sigma_i</em> is the experimental error on the data-point
<em>y_i</em>.  The errors are assumed to be
Gaussian and uncorrelated. 
For unweighted data the chi-squared sum is computed without any weight factors. 
</p>
<p>The fitting routines return the best-fit parameters <em>c</em> and their
<em>p \times p</em> covariance matrix.  The covariance matrix measures the
statistical errors on the best-fit parameters resulting from the 
errors on the data, <em>\sigma_i</em>, and is defined
<a name="index-covariance-matrix_002c-linear-fits"></a>
as <em>C_{ab} = &lt;\delta c_a \delta c_b&gt;</em> where <em>&lt; &gt;</em> denotes an average over the Gaussian error distributions of the underlying datapoints.
</p>
<p>The covariance matrix is calculated by error propagation from the data
errors <em>\sigma_i</em>.  The change in a fitted parameter <em>\delta
c_a</em> caused by a small change in the data <em>\delta y_i</em> is given
by
</p>
<div class="example">
<pre class="example">\delta c_a = \sum_i (dc_a/dy_i) \delta y_i
</pre></div>

<p>allowing the covariance matrix to be written in terms of the errors on the data,
</p>
<div class="example">
<pre class="example">C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) &lt;\delta y_i \delta y_j&gt;
</pre></div>

<p>For uncorrelated data the fluctuations of the underlying datapoints satisfy
<em>&lt;\delta y_i \delta y_j&gt; = \sigma_i^2 \delta_{ij}</em>, giving a 
corresponding parameter covariance matrix of
</p>
<div class="example">
<pre class="example">C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i) 
</pre></div>

<p>When computing the covariance matrix for unweighted data, i.e. data with unknown errors, 
the weight factors <em>w_i</em> in this sum are replaced by the single estimate <em>w =
1/\sigma^2</em>, where <em>\sigma^2</em> is the computed variance of the
residuals about the best-fit model, <em>\sigma^2 = \sum (y_i - Y(c,x_i))^2 / (n-p)</em>.  
This is referred to as the <em>variance-covariance matrix</em>.
<a name="index-variance_002dcovariance-matrix_002c-linear-fits"></a>
</p>
<p>The standard deviations of the best-fit parameters are given by the
square root of the corresponding diagonal elements of
the covariance matrix, <em>\sigma_{c_a} = \sqrt{C_{aa}}</em>.
The correlation coefficient of the fit parameters <em>c_a</em> and <em>c_b</em>
is given by <em>\rho_{ab} = C_{ab} / \sqrt{C_{aa} C_{bb}}</em>.
</p>
<hr>
<div class="header">
<p>
Next: <a href="Linear-regression.html#Linear-regression" accesskey="n" rel="next">Linear regression</a>, Up: <a href="Least_002dSquares-Fitting.html#Least_002dSquares-Fitting" accesskey="u" rel="up">Least-Squares Fitting</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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