Fitting a power-law distribution function to discrete data
fit_power_law fits a power-law distribution to a data set.
fit_power_law(
x,
xmin = NULL,
start = 2,
force.continuous = FALSE,
implementation = c("plfit", "R.mle"),
...
)x |
The data to fit, a numeric vector. For implementation
‘ |
xmin |
Numeric scalar, or |
start |
Numeric scalar. The initial value of the exponent for the
minimizing function, for the ‘ |
force.continuous |
Logical scalar. Whether to force a continuous
distribution for the ‘ |
implementation |
Character scalar. Which implementation to use. See details below. |
... |
Additional arguments, passed to the maximum likelihood
optimizing function, |
This function fits a power-law distribution to a vector containing samples from a distribution (that is assumed to follow a power-law of course). In a power-law distribution, it is generally assumed that P(X=x) is proportional to x^-alpha, where x is a positive number and alpha is greater than 1. In many real-world cases, the power-law behaviour kicks in only above a threshold value xmin. The goal of this function is to determine alpha if xmin is given, or to determine xmin and the corresponding value of alpha.
fit_power_law provides two maximum likelihood implementations. If
the implementation argument is ‘R.mle’, then the BFGS
optimization (see mle) algorithm is applied. The additional
arguments are passed to the mle function, so it is possible to change the
optimization method and/or its parameters. This implementation can
not to fit the xmin argument, so use the
‘plfit’ implementation if you want to do that.
The ‘plfit’ implementation also uses the maximum likelihood
principle to determine alpha for a given xmin;
When xmin is not given in advance, the algorithm will attempt
to find itsoptimal value for which the p-value of a Kolmogorov-Smirnov
test between the fitted distribution and the original sample is the largest.
The function uses the method of Clauset, Shalizi and Newman to calculate the
parameters of the fitted distribution. See references below for the details.
Depends on the implementation argument. If it is
‘R.mle’, then an object with class ‘mle’. It can
be used to calculate confidence intervals and log-likelihood. See
mle-class for details.
If implementation is ‘plfit’, then the result is a
named list with entries:
continuous |
Logical scalar, whether the fitted power-law distribution was continuous or discrete. |
alpha |
Numeric scalar, the exponent of the fitted power-law distribution. |
xmin |
Numeric scalar, the minimum value from which the
power-law distribution was fitted. In other words, only the values larger
than |
logLik |
Numeric scalar, the log-likelihood of the fitted parameters. |
KS.stat |
Numeric scalar, the test statistic of a Kolmogorov-Smirnov test that compares the fitted distribution with the input vector. Smaller scores denote better fit. |
KS.p |
Numeric scalar, the p-value of the Kolmogorov-Smirnov test. Small p-values (less than 0.05) indicate that the test rejected the hypothesis that the original data could have been drawn from the fitted power-law distribution. |
Tamas Nepusz ntamas@gmail.com and Gabor Csardi csardi.gabor@gmail.com
Power laws, Pareto distributions and Zipf's law, M. E. J. Newman, Contemporary Physics, 46, 323-351, 2005.
Aaron Clauset, Cosma R .Shalizi and Mark E.J. Newman: Power-law distributions in empirical data. SIAM Review 51(4):661-703, 2009.
# This should approximately yield the correct exponent 3 g <- barabasi.game(1000) # increase this number to have a better estimate d <- degree(g, mode="in") fit1 <- fit_power_law(d+1, 10) fit2 <- fit_power_law(d+1, 10, implementation="R.mle") fit1$alpha stats4::coef(fit2) fit1$logLik stats4::logLik(fit2)
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