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<a name="Multi_002dparameter-regression"></a>
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Next: <a href="Regularized-regression.html#Regularized-regression" accesskey="n" rel="next">Regularized regression</a>, Previous: <a href="Linear-regression.html#Linear-regression" accesskey="p" rel="previous">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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<a name="Multi_002dparameter-regression-1"></a>
<h3 class="section">38.3 Multi-parameter regression</h3>
<a name="index-multi_002dparameter-regression"></a>
<a name="index-fits_002c-multi_002dparameter-linear"></a>
<p>This section describes routines which perform least squares fits
to a linear model by minimizing the cost function
</p><div class="example">
<pre class="example">\chi^2 = \sum_i w_i (y_i - \sum_j X_ij c_j)^2 = || y - Xc ||_W^2
</pre></div>
<p>where <em>y</em> is a vector of <em>n</em> observations, <em>X</em> is an
<em>n</em>-by-<em>p</em> matrix of predictor variables, <em>c</em>
is a vector of the <em>p</em> unknown best-fit parameters to be estimated,
and <em>||r||_W^2 = r^T W r</em>.
The matrix <em>W = </em> diag<em>(w_1,w_2,...,w_n)</em>
defines the weights or uncertainties of the observation vector.
</p>
<p>This formulation can be used for fits to any number of functions and/or
variables by preparing the <em>n</em>-by-<em>p</em> matrix <em>X</em>
appropriately.  For example, to fit to a <em>p</em>-th order polynomial in
<var>x</var>, use the following matrix,
</p>
<div class="example">
<pre class="example">X_{ij} = x_i^j
</pre></div>

<p>where the index <em>i</em> runs over the observations and the index
<em>j</em> runs from 0 to <em>p-1</em>.
</p>
<p>To fit to a set of <em>p</em> sinusoidal functions with fixed frequencies
<em>\omega_1</em>, <em>\omega_2</em>, &hellip;, <em>\omega_p</em>, use,
</p>
<div class="example">
<pre class="example">X_{ij} = sin(\omega_j x_i)
</pre></div>

<p>To fit to <em>p</em> independent variables <em>x_1</em>, <em>x_2</em>, &hellip;,
<em>x_p</em>, use,
</p>
<div class="example">
<pre class="example">X_{ij} = x_j(i)
</pre></div>

<p>where <em>x_j(i)</em> is the <em>i</em>-th value of the predictor variable
<em>x_j</em>.
</p>
<p>The solution of the general linear least-squares system requires an
additional working space for intermediate results, such as the singular
value decomposition of the matrix <em>X</em>.
</p>
<p>These functions are declared in the header file <samp>gsl_multifit.h</samp>.
</p>
<dl>
<dt><a name="index-gsl_005fmultifit_005flinear_005falloc"></a>Function: <em>gsl_multifit_linear_workspace *</em> <strong>gsl_multifit_linear_alloc</strong> <em>(const size_t <var>n</var>, const size_t <var>p</var>)</em></dt>
<dd><a name="index-gsl_005fmultifit_005flinear_005fworkspace"></a>
<p>This function allocates a workspace for fitting a model to a maximum of <var>n</var>
observations using a maximum of <var>p</var> parameters. The user may later supply
a smaller least squares system if desired. The size of the workspace is
<em>O(np + p^2)</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005flinear_005ffree"></a>Function: <em>void</em> <strong>gsl_multifit_linear_free</strong> <em>(gsl_multifit_linear_workspace * <var>work</var>)</em></dt>
<dd><p>This function frees the memory associated with the workspace <var>w</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005flinear_005fsvd"></a>Function: <em>int</em> <strong>gsl_multifit_linear_svd</strong> <em>(const gsl_matrix * <var>X</var>, gsl_multifit_linear_workspace * <var>work</var>)</em></dt>
<dd><p>This function performs a singular value decomposition of the
matrix <var>X</var> and stores the SVD factors internally in <var>work</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005flinear_005fbsvd"></a>Function: <em>int</em> <strong>gsl_multifit_linear_bsvd</strong> <em>(const gsl_matrix * <var>X</var>, gsl_multifit_linear_workspace * <var>work</var>)</em></dt>
<dd><p>This function performs a singular value decomposition of the
matrix <var>X</var> and stores the SVD factors internally in <var>work</var>.
The matrix <var>X</var> is first balanced by applying column scaling
factors to improve the accuracy of the singular values.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005flinear"></a>Function: <em>int</em> <strong>gsl_multifit_linear</strong> <em>(const gsl_matrix * <var>X</var>, const gsl_vector * <var>y</var>, gsl_vector * <var>c</var>, gsl_matrix * <var>cov</var>, double * <var>chisq</var>, gsl_multifit_linear_workspace * <var>work</var>)</em></dt>
<dd><p>This function computes the best-fit parameters <var>c</var> of the model
<em>y = X c</em> for the observations <var>y</var> and the matrix of
predictor variables <var>X</var>, using the preallocated workspace provided
in <var>work</var>.  The <em>p</em>-by-<em>p</em> variance-covariance matrix of the model parameters
<var>cov</var> is set to <em>\sigma^2 (X^T X)^{-1}</em>, where <em>\sigma</em> is
the standard deviation of the fit residuals.
The sum of squares of the residuals from the best-fit,
<em>\chi^2</em>, is returned in <var>chisq</var>. If the coefficient of
determination is desired, it can be computed from the expression
<em>R^2 = 1 - \chi^2 / TSS</em>, where the total sum of squares (TSS) of
the observations <var>y</var> may be computed from <code>gsl_stats_tss</code>.
</p>
<p>The best-fit is found by singular value decomposition of the matrix
<var>X</var> using the modified Golub-Reinsch SVD algorithm, with column
scaling to improve the accuracy of the singular values. Any components
which have zero singular value (to machine precision) are discarded
from the fit.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005flinear_005ftsvd"></a>Function: <em>int</em> <strong>gsl_multifit_linear_tsvd</strong> <em>(const gsl_matrix * <var>X</var>, const gsl_vector * <var>y</var>, const double <var>tol</var>, gsl_vector * <var>c</var>, gsl_matrix * <var>cov</var>, double * <var>chisq</var>, size_t * <var>rank</var>, gsl_multifit_linear_workspace * <var>work</var>)</em></dt>
<dd><p>This function computes the best-fit parameters <var>c</var> of the model
<em>y = X c</em> for the observations <var>y</var> and the matrix of
predictor variables <var>X</var>, using a truncated SVD expansion.
Singular values which satisfy <em>s_i \le tol \times s_0</em>
are discarded from the fit, where <em>s_0</em> is the largest singular value.
The <em>p</em>-by-<em>p</em> variance-covariance matrix of the model parameters
<var>cov</var> is set to <em>\sigma^2 (X^T X)^{-1}</em>, where <em>\sigma</em> is
the standard deviation of the fit residuals.
The sum of squares of the residuals from the best-fit,
<em>\chi^2</em>, is returned in <var>chisq</var>. The effective rank
(number of singular values used in solution) is returned in <var>rank</var>.
If the coefficient of
determination is desired, it can be computed from the expression
<em>R^2 = 1 - \chi^2 / TSS</em>, where the total sum of squares (TSS) of
the observations <var>y</var> may be computed from <code>gsl_stats_tss</code>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005fwlinear"></a>Function: <em>int</em> <strong>gsl_multifit_wlinear</strong> <em>(const gsl_matrix * <var>X</var>, const gsl_vector * <var>w</var>, const gsl_vector * <var>y</var>, gsl_vector * <var>c</var>, gsl_matrix * <var>cov</var>, double * <var>chisq</var>, gsl_multifit_linear_workspace * <var>work</var>)</em></dt>
<dd><p>This function computes the best-fit parameters <var>c</var> of the weighted
model <em>y = X c</em> for the observations <var>y</var> with weights <var>w</var>
and the matrix of predictor variables <var>X</var>, using the preallocated
workspace provided in <var>work</var>.  The <em>p</em>-by-<em>p</em> covariance matrix of the model
parameters <var>cov</var> is computed as <em>(X^T W X)^{-1}</em>. The weighted
sum of squares of the residuals from the best-fit, <em>\chi^2</em>, is
returned in <var>chisq</var>. If the coefficient of determination is
desired, it can be computed from the expression <em>R^2 = 1 - \chi^2
/ WTSS</em>, where the weighted total sum of squares (WTSS) of the
observations <var>y</var> may be computed from <code>gsl_stats_wtss</code>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005fwlinear_005ftsvd"></a>Function: <em>int</em> <strong>gsl_multifit_wlinear_tsvd</strong> <em>(const gsl_matrix * <var>X</var>, const gsl_vector * <var>w</var>, const gsl_vector * <var>y</var>, const double <var>tol</var>, gsl_vector * <var>c</var>, gsl_matrix * <var>cov</var>, double * <var>chisq</var>, size_t * <var>rank</var>, gsl_multifit_linear_workspace * <var>work</var>)</em></dt>
<dd><p>This function computes the best-fit parameters <var>c</var> of the weighted
model <em>y = X c</em> for the observations <var>y</var> with weights <var>w</var>
and the matrix of predictor variables <var>X</var>, using a truncated SVD expansion.
Singular values which satisfy <em>s_i \le tol \times s_0</em>
are discarded from the fit, where <em>s_0</em> is the largest singular value.
The <em>p</em>-by-<em>p</em> covariance matrix of the model
parameters <var>cov</var> is computed as <em>(X^T W X)^{-1}</em>. The weighted
sum of squares of the residuals from the best-fit, <em>\chi^2</em>, is
returned in <var>chisq</var>. The effective rank of the system (number of
singular values used in the solution) is returned in <var>rank</var>.
If the coefficient of determination is
desired, it can be computed from the expression <em>R^2 = 1 - \chi^2
/ WTSS</em>, where the weighted total sum of squares (WTSS) of the
observations <var>y</var> may be computed from <code>gsl_stats_wtss</code>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005flinear_005fest"></a>Function: <em>int</em> <strong>gsl_multifit_linear_est</strong> <em>(const gsl_vector * <var>x</var>, const gsl_vector * <var>c</var>, const gsl_matrix * <var>cov</var>, double * <var>y</var>, double * <var>y_err</var>)</em></dt>
<dd><p>This function uses the best-fit multilinear regression coefficients
<var>c</var> and their covariance matrix
<var>cov</var> to compute the fitted function value
<var>y</var> and its standard deviation <var>y_err</var> for the model <em>y = x.c</em> 
at the point <var>x</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005flinear_005fresiduals"></a>Function: <em>int</em> <strong>gsl_multifit_linear_residuals</strong> <em>(const gsl_matrix * <var>X</var>, const gsl_vector * <var>y</var>, const gsl_vector * <var>c</var>, gsl_vector * <var>r</var>)</em></dt>
<dd><p>This function computes the vector of residuals <em>r = y - X c</em> for
the observations <var>y</var>, coefficients <var>c</var> and matrix of predictor
variables <var>X</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmultifit_005flinear_005frank"></a>Function: <em>size_t</em> <strong>gsl_multifit_linear_rank</strong> <em>(const double <var>tol</var>, const gsl_multifit_linear_workspace * <var>work</var>)</em></dt>
<dd><p>This function returns the rank of the matrix <em>X</em> which must first have its
singular value decomposition computed. The rank is computed by counting the number
of singular values <em>\sigma_j</em> which satisfy <em>\sigma_j &gt; tol \times \sigma_0</em>,
where <em>\sigma_0</em> is the largest singular value.
</p></dd></dl>

<hr>
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Next: <a href="Regularized-regression.html#Regularized-regression" accesskey="n" rel="next">Regularized regression</a>, Previous: <a href="Linear-regression.html#Linear-regression" accesskey="p" rel="previous">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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