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
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
|
<html lang="en">
<head>
<title>Weighted Samples - 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 4.11">
<link title="Top" rel="start" href="index.html#Top">
<link rel="up" href="Statistics.html" title="Statistics">
<link rel="prev" href="Correlation.html" title="Correlation">
<link rel="next" href="Maximum-and-Minimum-values.html" title="Maximum and Minimum values">
<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, 2008, 2009, 2010 The GSL Team.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 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 the freedom to copy and modify this
GNU Manual.''-->
<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="Weighted-Samples"></a>
Next: <a rel="next" accesskey="n" href="Maximum-and-Minimum-values.html">Maximum and Minimum values</a>,
Previous: <a rel="previous" accesskey="p" href="Correlation.html">Correlation</a>,
Up: <a rel="up" accesskey="u" href="Statistics.html">Statistics</a>
<hr>
</div>
<h3 class="section">21.7 Weighted Samples</h3>
<p>The functions described in this section allow the computation of
statistics for weighted samples. The functions accept an array of
samples, x_i, with associated weights, w_i. Each sample
x_i is considered as having been drawn from a Gaussian
distribution with variance \sigma_i^2. The sample weight
w_i is defined as the reciprocal of this variance, w_i =
1/\sigma_i^2. Setting a weight to zero corresponds to removing a
sample from a dataset.
<div class="defun">
— Function: double <b>gsl_stats_wmean</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fwmean-1944"></a></var><br>
<blockquote><p>This function returns the weighted mean of the dataset <var>data</var> with
stride <var>stride</var> and length <var>n</var>, using the set of weights <var>w</var>
with stride <var>wstride</var> and length <var>n</var>. The weighted mean is defined as,
<pre class="example"> \Hat\mu = (\sum w_i x_i) / (\sum w_i)
</pre>
</blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wvariance</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fwvariance-1945"></a></var><br>
<blockquote><p>This function returns the estimated variance of the dataset <var>data</var>
with stride <var>stride</var> and length <var>n</var>, using the set of weights
<var>w</var> with stride <var>wstride</var> and length <var>n</var>. The estimated
variance of a weighted dataset is calculated as,
<pre class="example"> \Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2)))
\sum w_i (x_i - \Hat\mu)^2
</pre>
<p class="noindent">Note that this expression reduces to an unweighted variance with the
familiar 1/(N-1) factor when there are N equal non-zero
weights.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wvariance_m</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n, double wmean</var>)<var><a name="index-gsl_005fstats_005fwvariance_005fm-1946"></a></var><br>
<blockquote><p>This function returns the estimated variance of the weighted dataset
<var>data</var> using the given weighted mean <var>wmean</var>.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wsd</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fwsd-1947"></a></var><br>
<blockquote><p>The standard deviation is defined as the square root of the variance.
This function returns the square root of the corresponding variance
function <code>gsl_stats_wvariance</code> above.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wsd_m</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n, double wmean</var>)<var><a name="index-gsl_005fstats_005fwsd_005fm-1948"></a></var><br>
<blockquote><p>This function returns the square root of the corresponding variance
function <code>gsl_stats_wvariance_m</code> above.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wvariance_with_fixed_mean</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n, const double mean</var>)<var><a name="index-gsl_005fstats_005fwvariance_005fwith_005ffixed_005fmean-1949"></a></var><br>
<blockquote><p>This function computes an unbiased estimate of the variance of the weighted
dataset <var>data</var> when the population mean <var>mean</var> of the underlying
distribution is known <em>a priori</em>. In this case the estimator for
the variance replaces the sample mean \Hat\mu by the known
population mean \mu,
<pre class="example"> \Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) / (\sum w_i)
</pre>
</blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wsd_with_fixed_mean</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n, const double mean</var>)<var><a name="index-gsl_005fstats_005fwsd_005fwith_005ffixed_005fmean-1950"></a></var><br>
<blockquote><p>The standard deviation is defined as the square root of the variance.
This function returns the square root of the corresponding variance
function above.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wtss</b> (<var>const double w</var>[]<var>, const size_t wstride, const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fwtss-1951"></a></var><br>
— Function: double <b>gsl_stats_wtss_m</b> (<var>const double w</var>[]<var>, const size_t wstride, const double data</var>[]<var>, size_t stride, size_t n, double wmean</var>)<var><a name="index-gsl_005fstats_005fwtss_005fm-1952"></a></var><br>
<blockquote><p>These functions return the weighted total sum of squares (TSS) of
<var>data</var> about the weighted mean. For <code>gsl_stats_wtss_m</code> the
user-supplied value of <var>wmean</var> is used, and for <code>gsl_stats_wtss</code>
it is computed using <code>gsl_stats_wmean</code>.
<pre class="example"> TSS = \sum w_i (x_i - wmean)^2
</pre>
</blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wabsdev</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fwabsdev-1953"></a></var><br>
<blockquote><p>This function computes the weighted absolute deviation from the weighted
mean of <var>data</var>. The absolute deviation from the mean is defined as,
<pre class="example"> absdev = (\sum w_i |x_i - \Hat\mu|) / (\sum w_i)
</pre>
</blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wabsdev_m</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n, double wmean</var>)<var><a name="index-gsl_005fstats_005fwabsdev_005fm-1954"></a></var><br>
<blockquote><p>This function computes the absolute deviation of the weighted dataset
<var>data</var> about the given weighted mean <var>wmean</var>.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wskew</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fwskew-1955"></a></var><br>
<blockquote><p>This function computes the weighted skewness of the dataset <var>data</var>.
<pre class="example"> skew = (\sum w_i ((x_i - \Hat x)/\Hat \sigma)^3) / (\sum w_i)
</pre>
</blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wskew_m_sd</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n, double wmean, double wsd</var>)<var><a name="index-gsl_005fstats_005fwskew_005fm_005fsd-1956"></a></var><br>
<blockquote><p>This function computes the weighted skewness of the dataset <var>data</var>
using the given values of the weighted mean and weighted standard
deviation, <var>wmean</var> and <var>wsd</var>.
</p></blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wkurtosis</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n</var>)<var><a name="index-gsl_005fstats_005fwkurtosis-1957"></a></var><br>
<blockquote><p>This function computes the weighted kurtosis of the dataset <var>data</var>.
<pre class="example"> kurtosis = ((\sum w_i ((x_i - \Hat x)/\Hat \sigma)^4) / (\sum w_i)) - 3
</pre>
</blockquote></div>
<div class="defun">
— Function: double <b>gsl_stats_wkurtosis_m_sd</b> (<var>const double w</var>[]<var>, size_t wstride, const double data</var>[]<var>, size_t stride, size_t n, double wmean, double wsd</var>)<var><a name="index-gsl_005fstats_005fwkurtosis_005fm_005fsd-1958"></a></var><br>
<blockquote><p>This function computes the weighted kurtosis of the dataset <var>data</var>
using the given values of the weighted mean and weighted standard
deviation, <var>wmean</var> and <var>wsd</var>.
</p></blockquote></div>
<hr>The GNU Scientific Library - a free numerical library licensed under the GNU GPL<br>Back to the <a href="/software/gsl/">GNU Scientific Library Homepage</a></body></html>
|
