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
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Additional documentation for the optim package for Octave.
Copyright (C) Olaf Till <i7tiol@t-online.de>
You can redistribute this documentation and/or modify it under the terms
of the GNU General Public License as published by the Free Software
Foundation; either version 3 of the License, or (at your option) any
later version.
This documentation is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.
You should have received a copy of the GNU General Public License along
with this documentation; if not, see <http://www.gnu.org/licenses/>. -->
<!-- Created by GNU Texinfo 6.5, http://www.gnu.org/software/texinfo/ -->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>LinearRegression (optim_doc)</title>
<meta name="description" content="LinearRegression (optim_doc)">
<meta name="keywords" content="LinearRegression (optim_doc)">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<link href="index.html#Top" rel="start" title="Top">
<link href="Function-index.html#Function-index" rel="index" title="Function index">
<link href="Residual-optimization.html#Residual-optimization" rel="up" title="Residual optimization">
<link href="wsolve.html#wsolve" rel="next" title="wsolve">
<link href="polyconf.html#polyconf" rel="prev" title="polyconf">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.indentedblock {margin-right: 0em}
blockquote.smallindentedblock {margin-right: 0em; font-size: smaller}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smalllisp {margin-left: 3.2em}
kbd {font-style: oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nolinebreak {white-space: nowrap}
span.roman {font-family: initial; font-weight: normal}
span.sansserif {font-family: sans-serif; font-weight: normal}
ul.no-bullet {list-style: none}
-->
</style>
</head>
<body lang="en">
<a name="LinearRegression"></a>
<div class="header">
<p>
Next: <a href="wsolve.html#wsolve" accesskey="n" rel="next">wsolve</a>, Previous: <a href="polyconf.html#polyconf" accesskey="p" rel="prev">polyconf</a>, Up: <a href="Residual-optimization.html#Residual-optimization" accesskey="u" rel="up">Residual optimization</a> [<a href="Function-index.html#Function-index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Function-LinearRegression"></a>
<h3 class="section">2.13 Function LinearRegression</h3>
<a name="index-LinearRegression-3"></a>
<a name="XREFLinearRegression"></a><dl>
<dt><a name="index-LinearRegression"></a>Function File: <em></em> <strong>LinearRegression</strong> <em>(<var>F</var>, <var>y</var>)</em></dt>
<dt><a name="index-LinearRegression-1"></a>Function File: <em></em> <strong>LinearRegression</strong> <em>(<var>F</var>, <var>y</var>, <var>w</var>)</em></dt>
<dt><a name="index-LinearRegression-2"></a>Function File: <em>[<var>p</var>, <var>e_var</var>, <var>r</var>, <var>p_var</var>, <var>fit_var</var>] =</em> <strong>LinearRegression</strong> <em>(…)</em></dt>
<dd>
<p>general linear regression
</p>
<p>determine the parameters p_j (j=1,2,...,m) such that the function
f(x) = sum_(j=1,...,m) p_j*f_j(x) is the best fit to the given values
y_i by f(x_i) for i=1,...,n, i.e. minimize
sum_(i=1,...,n)(y_i-sum_(j=1,...,m) p_j*f_j(x_i))^2 with respect to p_j
</p>
<p>parameters:
</p><ul>
<li> <var>F</var> is an n*m matrix with the values of the basis functions at
the support points. In column j give the values of f_j at the points
x_i (i=1,2,...,n)
</li><li> <var>y</var> is a column vector of length n with the given values
</li><li> <var>w</var> is a column vector of length n with the weights of
the data points. 1/w_i is expected to be proportional to the
estimated uncertainty in the y values. Then the weighted expression
sum_(i=1,...,n)(w_i^2*(y_i-f(x_i))^2) is minimized.
</li></ul>
<p>return values:
</p><ul>
<li> <var>p</var> is the vector of length m with the estimated values
of the parameters
</li><li> <var>e_var</var> is the vector of estimated variances of the
provided y values. If weights are provided, then the product
e_var_i * w^2_i is assumed to be constant.
</li><li> <var>r</var> is the weighted norm of the residual
</li><li> <var>p_var</var> is the vector of estimated variances of the parameters p_j
</li><li> <var>fit_var</var> is the vector of the estimated variances of the
fitted function values f(x_i)
</li></ul>
<p>To estimate the variance of the difference between future y values
and fitted y values use the sum of <var>e_var</var> and <var>fit_var</var>
</p>
<p>Caution:
do NOT request <var>fit_var</var> for large data sets, as a n by n matrix is
generated
</p>
</dd></dl>
<p>See also <a href="https://www.gnu.org/software/octave/doc/interpreter/XREFols.html#XREFols">(octave)ols</a>, <a href="https://www.gnu.org/software/octave/doc/interpreter/XREFgls.html#XREFgls">(octave)gls</a>,
<a href="https://www.gnu.org/software/octave/doc/interpreter/XREFregress.html#XREFregress">(octave)regress</a>, <a href="leasqr.html#XREFleasqr">leasqr</a>,
<a href="nonlin_005fcurvefit.html#XREFnonlin_005fcurvefit">nonlin_curvefit</a>,
<a href="https://www.gnu.org/software/octave/doc/interpreter/XREFpolyfit.html#XREFpolyfit">(octave)polyfit</a>, <a href="wpolyfit.html#XREFwpolyfit">wpolyfit</a>,
<a href="pronyfit.html#XREFpronyfit">pronyfit</a>.
</p>
<hr>
<div class="header">
<p>
Next: <a href="wsolve.html#wsolve" accesskey="n" rel="next">wsolve</a>, Previous: <a href="polyconf.html#polyconf" accesskey="p" rel="prev">polyconf</a>, Up: <a href="Residual-optimization.html#Residual-optimization" accesskey="u" rel="up">Residual optimization</a> [<a href="Function-index.html#Function-index" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
