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<title>GNU Scientific Library – Reference Manual: Linear regression</title>
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<a name="Linear-regression"></a>
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<p>
Next: <a href="Linear-fitting-without-a-constant-term.html#Linear-fitting-without-a-constant-term" accesskey="n" rel="next">Linear fitting without a constant term</a>, Previous: <a href="Fitting-Overview.html#Fitting-Overview" accesskey="p" rel="previous">Fitting Overview</a>, Up: <a href="Least_002dSquares-Fitting.html#Least_002dSquares-Fitting" accesskey="u" rel="up">Least-Squares Fitting</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Linear-regression-1"></a>
<h3 class="section">37.2 Linear regression</h3>
<a name="index-linear-regression"></a>
<p>The functions described in this section can be used to perform
least-squares fits to a straight line model, <em>Y(c,x) = c_0 + c_1 x</em>.
</p>
<a name="index-covariance-matrix_002c-from-linear-regression"></a>
<dl>
<dt><a name="index-gsl_005ffit_005flinear"></a>Function: <em>int</em> <strong>gsl_fit_linear</strong> <em>(const double * <var>x</var>, const size_t <var>xstride</var>, const double * <var>y</var>, const size_t <var>ystride</var>, size_t <var>n</var>, double * <var>c0</var>, double * <var>c1</var>, double * <var>cov00</var>, double * <var>cov01</var>, double * <var>cov11</var>, double * <var>sumsq</var>)</em></dt>
<dd><p>This function computes the best-fit linear regression coefficients
(<var>c0</var>,<var>c1</var>) of the model <em>Y = c_0 + c_1 X</em> for the dataset
(<var>x</var>, <var>y</var>), two vectors of length <var>n</var> with strides
<var>xstride</var> and <var>ystride</var>. The errors on <var>y</var> are assumed unknown so
the variance-covariance matrix for the
parameters (<var>c0</var>, <var>c1</var>) is estimated from the scatter of the
points around the best-fit line and returned via the parameters
(<var>cov00</var>, <var>cov01</var>, <var>cov11</var>).
The sum of squares of the residuals from the best-fit line is returned
in <var>sumsq</var>. Note: the correlation coefficient of the data can be computed using <code>gsl_stats_correlation</code> (see <a href="Correlation.html#Correlation">Correlation</a>), it does not depend on the fit.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005ffit_005fwlinear"></a>Function: <em>int</em> <strong>gsl_fit_wlinear</strong> <em>(const double * <var>x</var>, const size_t <var>xstride</var>, const double * <var>w</var>, const size_t <var>wstride</var>, const double * <var>y</var>, const size_t <var>ystride</var>, size_t <var>n</var>, double * <var>c0</var>, double * <var>c1</var>, double * <var>cov00</var>, double * <var>cov01</var>, double * <var>cov11</var>, double * <var>chisq</var>)</em></dt>
<dd><p>This function computes the best-fit linear regression coefficients
(<var>c0</var>,<var>c1</var>) of the model <em>Y = c_0 + c_1 X</em> for the weighted
dataset (<var>x</var>, <var>y</var>), two vectors of length <var>n</var> with strides
<var>xstride</var> and <var>ystride</var>. The vector <var>w</var>, of length <var>n</var>
and stride <var>wstride</var>, specifies the weight of each datapoint. The
weight is the reciprocal of the variance for each datapoint in <var>y</var>.
</p>
<p>The covariance matrix for the parameters (<var>c0</var>, <var>c1</var>) is
computed using the weights and returned via the parameters
(<var>cov00</var>, <var>cov01</var>, <var>cov11</var>). The weighted sum of squares
of the residuals from the best-fit line, <em>\chi^2</em>, is returned in
<var>chisq</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005ffit_005flinear_005fest"></a>Function: <em>int</em> <strong>gsl_fit_linear_est</strong> <em>(double <var>x</var>, double <var>c0</var>, double <var>c1</var>, double <var>cov00</var>, double <var>cov01</var>, double <var>cov11</var>, double * <var>y</var>, double * <var>y_err</var>)</em></dt>
<dd><p>This function uses the best-fit linear regression coefficients
<var>c0</var>, <var>c1</var> and their covariance
<var>cov00</var>, <var>cov01</var>, <var>cov11</var> to compute the fitted function
<var>y</var> and its standard deviation <var>y_err</var> for the model <em>Y =
c_0 + c_1 X</em> at the point <var>x</var>.
</p></dd></dl>
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<p>
Next: <a href="Linear-fitting-without-a-constant-term.html#Linear-fitting-without-a-constant-term" accesskey="n" rel="next">Linear fitting without a constant term</a>, Previous: <a href="Fitting-Overview.html#Fitting-Overview" accesskey="p" rel="previous">Fitting Overview</a>, Up: <a href="Least_002dSquares-Fitting.html#Least_002dSquares-Fitting" accesskey="u" rel="up">Least-Squares Fitting</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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