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<title>Fitting Overview - GNU Scientific Library -- Reference Manual</title>
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<a name="Fitting-Overview"></a>
Next: <a rel="next" accesskey="n" href="Linear-regression.html">Linear regression</a>,
Up: <a rel="up" accesskey="u" href="Least_002dSquares-Fitting.html">Least-Squares Fitting</a>
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<h3 class="section">36.1 Overview</h3>
<p>Least-squares fits are found by minimizing \chi^2
(chi-squared), the weighted sum of squared residuals over n
experimental datapoints (x_i, y_i) for the model Y(c,x),
<pre class="example"> \chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2
</pre>
<p class="noindent">The p parameters of the model are <!-- {$c = \{c_0, c_1, \dots\}$} -->
c = {c_0, c_1, <small class="dots">...</small>}. The
weight factors w_i are given by w_i = 1/\sigma_i^2,
where \sigma_i is the experimental error on the data-point
y_i. The errors are assumed to be
gaussian and uncorrelated.
For unweighted data the chi-squared sum is computed without any weight factors.
<p>The fitting routines return the best-fit parameters c and their
p \times p covariance matrix. The covariance matrix measures the
statistical errors on the best-fit parameters resulting from the
errors on the data, \sigma_i, and is defined
<a name="index-covariance-matrix_002c-linear-fits-2371"></a>as <!-- {$C_{ab} = \langle \delta c_a \delta c_b \rangle$} -->
C_{ab} = <\delta c_a \delta c_b> where <!-- {$\langle \, \rangle$} -->
< > denotes an average over the gaussian error distributions of the underlying datapoints.
<p>The covariance matrix is calculated by error propagation from the data
errors \sigma_i. The change in a fitted parameter \delta
c_a caused by a small change in the data \delta y_i is given
by
<pre class="example"> \delta c_a = \sum_i (dc_a/dy_i) \delta y_i
</pre>
<p class="noindent">allowing the covariance matrix to be written in terms of the errors on the data,
<pre class="example"> C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>
</pre>
<p class="noindent">For uncorrelated data the fluctuations of the underlying datapoints satisfy
<!-- {$\langle \delta y_i \delta y_j \rangle = \sigma_i^2 \delta_{ij}$} -->
<\delta y_i \delta y_j> = \sigma_i^2 \delta_{ij}, giving a
corresponding parameter covariance matrix of
<pre class="example"> C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i)
</pre>
<p class="noindent">When computing the covariance matrix for unweighted data, i.e. data with unknown errors,
the weight factors w_i in this sum are replaced by the single estimate w =
1/\sigma^2, where \sigma^2 is the computed variance of the
residuals about the best-fit model, \sigma^2 = \sum (y_i - Y(c,x_i))^2 / (n-p).
This is referred to as the <dfn>variance-covariance matrix</dfn>.
<a name="index-variance_002dcovariance-matrix_002c-linear-fits-2372"></a>
The standard deviations of the best-fit parameters are given by the
square root of the corresponding diagonal elements of
the covariance matrix, <!-- {$\sigma_{c_a} = \sqrt{C_{aa}}$} -->
\sigma_{c_a} = \sqrt{C_{aa}}.
The correlation coefficient of the fit parameters c_a and c_b
is given by <!-- {$\rho_{ab} = C_{ab} / \sqrt{C_{aa} C_{bb}}$} -->
\rho_{ab} = C_{ab} / \sqrt{C_{aa} C_{bb}}.
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