1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
|
<html lang="en">
<head>
<title>Computing the covariance matrix of best fit parameters - GNU Scientific Library -- Reference Manual</title>
<meta http-equiv="Content-Type" content="text/html">
<meta name="description" content="GNU Scientific Library -- Reference Manual">
<meta name="generator" content="makeinfo 4.8">
<link title="Top" rel="start" href="index.html#Top">
<link rel="up" href="Nonlinear-Least_002dSquares-Fitting.html" title="Nonlinear Least-Squares Fitting">
<link rel="prev" href="Minimization-Algorithms-without-Derivatives.html" title="Minimization Algorithms without Derivatives">
<link rel="next" href="Example-programs-for-Nonlinear-Least_002dSquares-Fitting.html" title="Example programs for Nonlinear Least-Squares Fitting">
<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
<!--
Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 The GSL Team.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2 or
any later version published by the Free Software Foundation; with the
Invariant Sections being ``GNU General Public License'' and ``Free Software
Needs Free Documentation'', the Front-Cover text being ``A GNU Manual'',
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled ``GNU Free Documentation
License''.
(a) The Back-Cover Text is: ``You have freedom to copy and modify this
GNU Manual, like GNU software.''-->
<meta http-equiv="Content-Style-Type" content="text/css">
<style type="text/css"><!--
pre.display { font-family:inherit }
pre.format { font-family:inherit }
pre.smalldisplay { font-family:inherit; font-size:smaller }
pre.smallformat { font-family:inherit; font-size:smaller }
pre.smallexample { font-size:smaller }
pre.smalllisp { font-size:smaller }
span.sc { font-variant:small-caps }
span.roman { font-family:serif; font-weight:normal; }
span.sansserif { font-family:sans-serif; font-weight:normal; }
--></style>
</head>
<body>
<div class="node">
<p>
<a name="Computing-the-covariance-matrix-of-best-fit-parameters"></a>
Next: <a rel="next" accesskey="n" href="Example-programs-for-Nonlinear-Least_002dSquares-Fitting.html">Example programs for Nonlinear Least-Squares Fitting</a>,
Previous: <a rel="previous" accesskey="p" href="Minimization-Algorithms-without-Derivatives.html">Minimization Algorithms without Derivatives</a>,
Up: <a rel="up" accesskey="u" href="Nonlinear-Least_002dSquares-Fitting.html">Nonlinear Least-Squares Fitting</a>
<hr>
</div>
<h3 class="section">37.8 Computing the covariance matrix of best fit parameters</h3>
<p><a name="index-best_002dfit-parameters_002c-covariance-2419"></a><a name="index-least-squares_002c-covariance-of-best_002dfit-parameters-2420"></a><a name="index-covariance-matrix_002c-nonlinear-fits-2421"></a>
<div class="defun">
— Function: int <b>gsl_multifit_covar</b> (<var>const gsl_matrix * J, double epsrel, gsl_matrix * covar</var>)<var><a name="index-gsl_005fmultifit_005fcovar-2422"></a></var><br>
<blockquote><p>This function uses the Jacobian matrix <var>J</var> to compute the covariance
matrix of the best-fit parameters, <var>covar</var>. The parameter
<var>epsrel</var> is used to remove linear-dependent columns when <var>J</var> is
rank deficient.
<p>The covariance matrix is given by,
<pre class="example"> covar = (J^T J)^{-1}
</pre>
<p class="noindent">and is computed by QR decomposition of J with column-pivoting. Any
columns of R which satisfy
<pre class="example"> |R_{kk}| <= epsrel |R_{11}|
</pre>
<p class="noindent">are considered linearly-dependent and are excluded from the covariance
matrix (the corresponding rows and columns of the covariance matrix are
set to zero).
<p>If the minimisation uses the weighted least-squares function
f_i = (Y(x, t_i) - y_i) / \sigma_i then the covariance
matrix above gives the statistical error on the best-fit parameters
resulting from the gaussian errors \sigma_i on
the underlying data y_i. This can be verified from the relation
\delta f = J \delta c and the fact that the fluctuations in f
from the data y_i are normalised by \sigma_i and
so satisfy <!-- {$\langle \delta f \delta f^T \rangle = I$} -->
<\delta f \delta f^T> = I.
<p>For an unweighted least-squares function f_i = (Y(x, t_i) -
y_i) the covariance matrix above should be multiplied by the variance
of the residuals about the best-fit \sigma^2 = \sum (y_i - Y(x,t_i))^2 / (n-p)
to give the variance-covariance
matrix \sigma^2 C. This estimates the statistical error on the
best-fit parameters from the scatter of the underlying data.
<p>For more information about covariance matrices see <a href="Fitting-Overview.html">Fitting Overview</a>.
</p></blockquote></div>
<!-- ============================================================ -->
</body></html>
|
