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authorKatolaZ <katolaz@freaknet.org>2017-09-27 15:06:31 +0100
committerKatolaZ <katolaz@freaknet.org>2017-09-27 15:06:31 +0100
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+ <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>&lt;v.nicosia@qmul.ac.uk&gt;</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>