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<title>fitmle(1) - Fit a set of values with a power-law distribution</title>
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<a href="#NAME">NAME</a>
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<a href="#SYNOPSIS">SYNOPSIS</a>
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<a href="#DESCRIPTION">DESCRIPTION</a>
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<a href="#PARAMETERS">PARAMETERS</a>
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<a href="#OUTPUT">OUTPUT</a>
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<a href="#EXAMPLES">EXAMPLES</a>
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<a href="#SEE-ALSO">SEE ALSO</a>
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<a href="#REFERENCES">REFERENCES</a>
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<ol class='man-decor man-head man head'>
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<li class='tl'>fitmle(1)</li>
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<li class='tc'>www.complex-networks.net</li>
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<li class='tr'>fitmle(1)</li>
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</ol>
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<h2 id="NAME">NAME</h2>
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<p class="man-name">
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<code>fitmle</code> - <span class="man-whatis">Fit a set of values with a power-law distribution</span>
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</p>
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<h2 id="SYNOPSIS">SYNOPSIS</h2>
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<p><code>fitmle</code> <var>data_in</var> [<var>tol</var> [TEST [<var>num_test</var>]]]</p>
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<h2 id="DESCRIPTION">DESCRIPTION</h2>
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<p><code>fitmle</code> fits the data points contained in the file <var>data_in</var> with a
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power-law function P(k) ~ k<sup>-gamma</sup>, using the Maximum-Likelihood
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Estimator (MLE). In particular, <code>fitmle</code> finds the exponent <code>gamma</code>
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and the minimum of the values provided on input for which the
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power-law behaviour holds. The second (optional) argument <var>tol</var> sets
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the acceptable statistical error on the estimate of the exponent.</p>
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<p>If <code>TEST</code> is provided, the program associates a p-value to the
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goodness of the fit, based on the Kolmogorov-Smirnov statistics
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computed on <var>num_test</var> sampled distributions from the theoretical
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power-law function. If <var>num_test</var> is not provided, the test is based
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on 100 sampled distributions.</p>
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<h2 id="PARAMETERS">PARAMETERS</h2>
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<dl>
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<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
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STDIN.</p></dd>
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<dt class="flush"><var>tol</var></dt><dd><p> The acceptable statistical error on the estimation of the
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exponent. If omitted, it is set to 0.1.</p></dd>
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<dt class="flush">TEST</dt><dd><p> If the third parameter is <code>TEST</code>, the program computes an estimate
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of the p-value associated to the best-fit, based on <var>num_test</var>
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synthetic samples of the same size of the input set.</p></dd>
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<dt><var>num_test</var></dt><dd><p> Number of synthetic samples to use for the estimation of the
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p-value of the best fit.</p></dd>
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</dl>
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<h2 id="OUTPUT">OUTPUT</h2>
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<p>If <code>fitmle</code> is given less than three parameters (i.e., if <code>TEST</code> is
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not specified), the output is a line in the format:</p>
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<pre><code> gamma k_min ks
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</code></pre>
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<p>where <code>gamma</code> is the estimate for the exponent, <code>k_min</code> is the
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smallest of the input values for which the power-law behaviour holds,
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and <code>ks</code> is the value of the Kolmogorov-Smirnov statistics of the
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best-fit.</p>
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<p>If <code>TEST</code> is specified, the output line contains also the estimate of
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the p-value of the best fit:</p>
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<pre><code> gamma k_min ks p-value
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</code></pre>
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<p>where <code>p-value</code> is based on <var>num_test</var> samples (or just 100, if
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<var>num_test</var> is not specified) of the same size of the input, obtained
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from the theoretical power-law function computed as a best fit.</p>
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<h2 id="EXAMPLES">EXAMPLES</h2>
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<p>Let us assume that the file <code>AS-20010316.net_degs</code> contains the degree
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sequence of the data set <code>AS-20010316.net</code> (the graph of the Internet
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at the AS level in March 2001). The exponent of the best-fit power-law
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distribution can be obtained by using:</p>
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<pre><code> $ fitmle AS-20010316.net_degs
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Using discrete fit
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2.06165 6 0.031626 0.17
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$
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</code></pre>
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<p>where <code>2.06165</code> is the estimated value of the exponent <code>gamma</code>, <code>6</code> is
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the minimum degree value for which the power-law behaviour holds, and
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<code>0.031626</code> is the value of the Kolmogorov-Smirnov statistics of the
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best-fit. The program is also telling us that it decided to use the
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discrete fitting procedure, since all the values in
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<code>AS-20010316.net_degs</code> are integers. The latter information is printed
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to STDERR.</p>
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<p>It is possible to compute the p-value of the estimate by running:</p>
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<pre><code> $ fitmle AS-20010316.net_degs 0.1 TEST
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Using discrete fit
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2.06165 6 0.031626 0.17
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$
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</code></pre>
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<p>which provides a p-value equal to 0.17, meaning that 17% of the
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synthetic samples showed a value of the KS statistics larger than that
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of the best-fit. The estimation of the p-value here is based on 100
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synthetic samples, since <var>num_test</var> was not provided. If we allow a
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slightly larger value of the statistical error on the estimate of the
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exponent <code>gamma</code>, we obtain different values of <code>gamma</code> and <code>k_min</code>,
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and a much higher p-value:</p>
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<pre><code> $ fitmle AS-20010316.net_degs 0.15 TEST 1000
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Using discrete fit
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2.0585 19 0.0253754 0.924
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$
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</code></pre>
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<p>Notice that in this case, the p-value of the estimate is equal to
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0.924, and is based on 1000 synthetic samples.</p>
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<h2 id="SEE-ALSO">SEE ALSO</h2>
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<p><span class="man-ref">deg_seq<span class="s">(1)</span></span>, <span class="man-ref">power_law<span class="s">(1)</span></span></p>
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<h2 id="REFERENCES">REFERENCES</h2>
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<ul>
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<li><p>A. Clauset, C. R. Shalizi, and M. E. J. Newman. "Power-law
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distributions in empirical data". SIAM Rev. 51, (2007), 661-703.</p></li>
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<li><p>V. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles,
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Methods and Applications", Chapter 5, Cambridge University Press
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(2017)</p></li>
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</ul>
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<h2 id="AUTHORS">AUTHORS</h2>
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<p>(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 <code><v.nicosia@qmul.ac.uk></code>.</p>
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<ol class='man-decor man-foot man foot'>
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<li class='tl'>www.complex-networks.net</li>
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<li class='tc'>September 2017</li>
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<li class='tr'>fitmle(1)</li>
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</ol>
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