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diff --git a/doc/fitmle.1.html b/doc/fitmle.1.html new file mode 100644 index 0000000..863099c --- /dev/null +++ b/doc/fitmle.1.html @@ -0,0 +1,215 @@ +<!DOCTYPE html> +<html> +<head> + <meta http-equiv='content-type' value='text/html;charset=utf8'> + <meta name='generator' value='Ronn/v0.7.3 (http://github.com/rtomayko/ronn/tree/0.7.3)'> + <title>fitmle(1) - Fit a set of values with a power-law distribution</title> + <style type='text/css' media='all'> + /* style: man */ + body#manpage {margin:0} + .mp {max-width:100ex;padding:0 9ex 1ex 4ex} + .mp p,.mp pre,.mp ul,.mp ol,.mp dl {margin:0 0 20px 0} + .mp h2 {margin:10px 0 0 0} + .mp > p,.mp > pre,.mp > ul,.mp > ol,.mp > dl {margin-left:8ex} + .mp h3 {margin:0 0 0 4ex} + .mp dt {margin:0;clear:left} + .mp dt.flush {float:left;width:8ex} + .mp dd {margin:0 0 0 9ex} + .mp h1,.mp h2,.mp h3,.mp h4 {clear:left} + .mp pre {margin-bottom:20px} + .mp pre+h2,.mp pre+h3 {margin-top:22px} + .mp h2+pre,.mp h3+pre {margin-top:5px} + .mp img {display:block;margin:auto} + .mp h1.man-title {display:none} + .mp,.mp code,.mp pre,.mp tt,.mp kbd,.mp samp,.mp h3,.mp h4 {font-family:monospace;font-size:14px;line-height:1.42857142857143} + .mp h2 {font-size:16px;line-height:1.25} + .mp h1 {font-size:20px;line-height:2} + .mp {text-align:justify;background:#fff} + .mp,.mp code,.mp pre,.mp pre code,.mp tt,.mp kbd,.mp samp {color:#131211} + .mp h1,.mp h2,.mp h3,.mp h4 {color:#030201} + .mp u {text-decoration:underline} + .mp code,.mp strong,.mp b {font-weight:bold;color:#131211} + .mp em,.mp var {font-style:italic;color:#232221;text-decoration:none} + .mp a,.mp a:link,.mp a:hover,.mp a code,.mp a pre,.mp a tt,.mp a kbd,.mp a samp {color:#0000ff} + .mp b.man-ref {font-weight:normal;color:#434241} + .mp pre {padding:0 4ex} + .mp pre code {font-weight:normal;color:#434241} + .mp h2+pre,h3+pre {padding-left:0} + ol.man-decor,ol.man-decor li {margin:3px 0 10px 0;padding:0;float:left;width:33%;list-style-type:none;text-transform:uppercase;color:#999;letter-spacing:1px} + ol.man-decor {width:100%} + ol.man-decor li.tl {text-align:left} + ol.man-decor li.tc {text-align:center;letter-spacing:4px} + ol.man-decor li.tr {text-align:right;float:right} + </style> + <style type='text/css' media='all'> + /* style: toc */ + .man-navigation {display:block !important;position:fixed;top:0;left:113ex;height:100%;width:100%;padding:48px 0 0 0;border-left:1px solid #dbdbdb;background:#eee} + .man-navigation a,.man-navigation a:hover,.man-navigation a:link,.man-navigation a:visited {display:block;margin:0;padding:5px 2px 5px 30px;color:#999;text-decoration:none} + .man-navigation a:hover {color:#111;text-decoration:underline} + </style> +</head> +<!-- + The following styles are deprecated and will be removed at some point: + div#man, div#man ol.man, div#man ol.head, div#man ol.man. + + The .man-page, .man-decor, .man-head, .man-foot, .man-title, and + .man-navigation should be used instead. +--> +<body id='manpage'> + <div class='mp' id='man'> + + <div class='man-navigation' style='display:none'> + <a href="#NAME">NAME</a> + <a href="#SYNOPSIS">SYNOPSIS</a> + <a href="#DESCRIPTION">DESCRIPTION</a> + <a href="#PARAMETERS">PARAMETERS</a> + <a href="#OUTPUT">OUTPUT</a> + <a href="#EXAMPLES">EXAMPLES</a> + <a href="#SEE-ALSO">SEE ALSO</a> + <a href="#REFERENCES">REFERENCES</a> + <a href="#AUTHORS">AUTHORS</a> + </div> + + <ol class='man-decor man-head man head'> + <li class='tl'>fitmle(1)</li> + <li class='tc'>www.complex-networks.net</li> + <li class='tr'>fitmle(1)</li> + </ol> + + <h2 id="NAME">NAME</h2> +<p class="man-name"> + <code>fitmle</code> - <span class="man-whatis">Fit a set of values with a power-law distribution</span> +</p> + +<h2 id="SYNOPSIS">SYNOPSIS</h2> + +<p><code>fitmle</code> <var>data_in</var> [<var>tol</var> [TEST [<var>num_test</var>]]]</p> + +<h2 id="DESCRIPTION">DESCRIPTION</h2> + +<p><code>fitmle</code> fits the data points contained in the file <var>data_in</var> with a +power-law function P(k) ~ k<sup>-gamma</sup>, using the Maximum-Likelihood +Estimator (MLE). In particular, <code>fitmle</code> finds the exponent <code>gamma</code> +and the minimum of the values provided on input for which the +power-law behaviour holds. The second (optional) argument <var>tol</var> sets +the acceptable statistical error on the estimate of the exponent.</p> + +<p>If <code>TEST</code> is provided, the program associates a p-value to the +goodness of the fit, based on the Kolmogorov-Smirnov statistics +computed on <var>num_test</var> sampled distributions from the theoretical +power-law function. If <var>num_test</var> is not provided, the test is based +on 100 sampled distributions.</p> + +<h2 id="PARAMETERS">PARAMETERS</h2> + +<dl> +<dt class="flush"><var>data_in</var></dt><dd><p> Set of values to fit. If is equal to <code>-</code> (dash), read the set from + STDIN.</p></dd> +<dt class="flush"><var>tol</var></dt><dd><p> The acceptable statistical error on the estimation of the + exponent. If omitted, it is set to 0.1.</p></dd> +<dt class="flush">TEST</dt><dd><p> If the third parameter is <code>TEST</code>, the program computes an estimate + of the p-value associated to the best-fit, based on <var>num_test</var> + synthetic samples of the same size of the input set.</p></dd> +<dt><var>num_test</var></dt><dd><p> Number of synthetic samples to use for the estimation of the + p-value of the best fit.</p></dd> +</dl> + + +<h2 id="OUTPUT">OUTPUT</h2> + +<p>If <code>fitmle</code> is given less than three parameters (i.e., if <code>TEST</code> is +not specified), the output is a line in the format:</p> + +<pre><code> gamma k_min ks +</code></pre> + +<p>where <code>gamma</code> is the estimate for the exponent, <code>k_min</code> is the +smallest of the input values for which the power-law behaviour holds, +and <code>ks</code> is the value of the Kolmogorov-Smirnov statistics of the +best-fit.</p> + +<p>If <code>TEST</code> is specified, the output line contains also the estimate of +the p-value of the best fit:</p> + +<pre><code> gamma k_min ks p-value +</code></pre> + +<p>where <code>p-value</code> is based on <var>num_test</var> samples (or just 100, if +<var>num_test</var> is not specified) of the same size of the input, obtained +from the theoretical power-law function computed as a best fit.</p> + +<h2 id="EXAMPLES">EXAMPLES</h2> + +<p>Let us assume that the file <code>AS-20010316.net_degs</code> contains the degree +sequence of the data set <code>AS-20010316.net</code> (the graph of the Internet +at the AS level in March 2001). The exponent of the best-fit power-law +distribution can be obtained by using:</p> + +<pre><code> $ fitmle AS-20010316.net_degs + Using discrete fit + 2.06165 6 0.031626 0.17 + $ +</code></pre> + +<p>where <code>2.06165</code> is the estimated value of the exponent <code>gamma</code>, <code>6</code> is +the minimum degree value for which the power-law behaviour holds, and +<code>0.031626</code> is the value of the Kolmogorov-Smirnov statistics of the +best-fit. The program is also telling us that it decided to use the +discrete fitting procedure, since all the values in +<code>AS-20010316.net_degs</code> are integers. The latter information is printed +to STDERR.</p> + +<p>It is possible to compute the p-value of the estimate by running:</p> + +<pre><code> $ fitmle AS-20010316.net_degs 0.1 TEST + Using discrete fit + 2.06165 6 0.031626 0.17 + $ +</code></pre> + +<p>which provides a p-value equal to 0.17, meaning that 17% of the +synthetic samples showed a value of the KS statistics larger than that +of the best-fit. The estimation of the p-value here is based on 100 +synthetic samples, since <var>num_test</var> was not provided. If we allow a +slightly larger value of the statistical error on the estimate of the +exponent <code>gamma</code>, we obtain different values of <code>gamma</code> and <code>k_min</code>, +and a much higher p-value:</p> + +<pre><code> $ fitmle AS-20010316.net_degs 0.15 TEST 1000 + Using discrete fit + 2.0585 19 0.0253754 0.924 + $ +</code></pre> + +<p>Notice that in this case, the p-value of the estimate is equal to +0.924, and is based on 1000 synthetic samples.</p> + +<h2 id="SEE-ALSO">SEE ALSO</h2> + +<p><a class="man-ref" href="deg_seq.1.html">deg_seq<span class="s">(1)</span></a>, <a class="man-ref" href="power_law.1.html">power_law<span class="s">(1)</span></a></p> + +<h2 id="REFERENCES">REFERENCES</h2> + +<ul> +<li><p>A. Clauset, C. R. Shalizi, and M. E. J. Newman. "Power-law +distributions in empirical data". SIAM Rev. 51, (2007), 661-703.</p></li> +<li><p>V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles, +Methods and Applications", Chapter 5, Cambridge University Press +(2017)</p></li> +</ul> + + +<h2 id="AUTHORS">AUTHORS</h2> + +<p>(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 <code><v.nicosia@qmul.ac.uk></code>.</p> + + + <ol class='man-decor man-foot man foot'> + <li class='tl'>www.complex-networks.net</li> + <li class='tc'>September 2017</li> + <li class='tr'>fitmle(1)</li> + </ol> + + </div> +</body> +</html> |