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fitmle(1) -- Fit a set of values with a power-law distribution
======
## SYNOPSIS
`fitmle` <data_in> [<tol> [TEST [<num\_test>]]]
## DESCRIPTION
`fitmle` fits the data points contained in the file <data_in> with a
power-law function P(k) ~ k^(-gamma), using the Maximum-Likelihood
Estimator (MLE). In particular, `fitmle` finds the exponent `gamma`
and the minimum of the values provided on input for which the
power-law behaviour holds. The second (optional) argument <tol> sets
the acceptable statistical error on the estimate of the exponent.
If `TEST` is provided, the program associates a p-value to the
goodness of the fit, based on the Kolmogorov-Smirnov statistics
computed on <num_test> sampled distributions from the theoretical
power-law function. If <num_test> is not provided, the test is based
on 100 sampled distributions.
## PARAMETERS
* <data_in>:
Set of values to fit. If is equal to `-` (dash), read the set from
STDIN.
* <tol>:
The acceptable statistical error on the estimation of the
exponent. If omitted, it is set to 0.1.
* TEST:
If the third parameter is `TEST`, the program computes an estimate
of the p-value associated to the best-fit, based on <num_test>
synthetic samples of the same size of the input set.
* <num_test>:
Number of synthetic samples to use for the estimation of the
p-value of the best fit.
## OUTPUT
If `fitmle` is given less than three parameters (i.e., if `TEST` is
not specified), the output is a line in the format:
gamma k_min ks
where `gamma` is the estimate for the exponent, `k_min` is the
smallest of the input values for which the power-law behaviour holds,
and `ks` is the value of the Kolmogorov-Smirnov statistics of the
best-fit.
If `TEST` is specified, the output line contains also the estimate of
the p-value of the best fit:
gamma k_min ks p-value
where `p-value` is based on <num_test> samples (or just 100, if
<num_test> is not specified) of the same size of the input, obtained
from the theoretical power-law function computed as a best fit.
## EXAMPLES
Let us assume that the file `AS-20010316.net_degs` contains the degree
sequence of the data set `AS-20010316.net` (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:
$ fitmle AS-20010316.net_degs
Using discrete fit
2.06165 6 0.031626 0.17
$
where `2.06165` is the estimated value of the exponent `gamma`, `6` is
the minimum degree value for which the power-law behaviour holds, and
`0.031626` 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
`AS-20010316.net_degs` are integers. The latter information is printed
to STDERR.
It is possible to compute the p-value of the estimate by running:
$ fitmle AS-20010316.net_degs 0.1 TEST
Using discrete fit
2.06165 6 0.031626 0.17
$
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 <num_test> was not provided. If we allow a
slightly larger value of the statistical error on the estimate of the
exponent `gamma`, we obtain different values of `gamma` and `k_min`,
and a much higher p-value:
$ fitmle AS-20010316.net_degs 0.15 TEST 1000
Using discrete fit
2.0585 19 0.0253754 0.924
$
Notice that in this case, the p-value of the estimate is equal to
0.924, and is based on 1000 synthetic samples.
## SEE ALSO
deg_seq(1), power_law(1)
## REFERENCES
* A\. Clauset, C. R. Shalizi, and M. E. J. Newman. "Power-law
distributions in empirical data". SIAM Rev. 51, (2007), 661-703.
* V\. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles,
Methods and Applications", Chapter 5, Cambridge University Press
(2017)
## AUTHORS
(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 `<v.nicosia@qmul.ac.uk>`.
|
