power.law.fit: Fitting a power-law distribution function to discrete data

Description

power.law.fit fits a power-law distribution to a data set.

Usage

power.law.fit(x, xmin=NULL, start=2, force.continuous=FALSE,
              implementation=c("plfit", "R.mle"), ...)

Arguments

The data to fit, a numeric vector. For implementation R.mle the data must be integer values. For the plfit implementation non-integer values might be present and then a continuous pow

xmin

Numeric scalar, or NULL. The lower bound for fitting the power-law. If NULL, the smallest value in x will be used for the R.mle implementation, and its value will be automati

start

Numeric scalar. The initial value of the exponent for the minimizing function, for the R.mle implementation. Ususally it is safe to leave this untouched.

force.continuous

Logical scalar. Whether to force a continuous distribution for the plfit implementation, even if the sample vector contains integer values only (by chance). If this argument is false, igraph will assume a continuo

implementation

Character scalar. Which implementation to use. See details below.

...

Additional arguments, passed to the maximum likelihood optimizing function, mle, if the R.mle implementation is chosen. It is ignored by the plfit

`Value`

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:
continuousLogical scalar, whether the fitted power-law
  distribution was continuous or discrete.
alphaNumeric scalar, the exponent of the fitted power-law
  distribution.
xminNumeric scalar, the minimum value from which the power-law
    distribution was fitted. In other words, only the values larger than
    xmin were used from the input vector.
logLikNumeric scalar, the log-likelihood of the fitted
    parameters.
KS.statNumeric scalar, the test statistic of a
    Kolmogorov-Smirnov test that compares the fitted distribution with
    the input vector. Smaller scores denote better fit.
KS.pNumeric 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.

`concept`

Power-law

`Details`

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 $x_{min}$. The goal of this function
  is to determine $\alpha$ if $x_{min}$ is given,
  or to determine $x_{min}$ and the corresponding value of
  $\alpha$.
  power.law.fit 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 $x_{min}$
  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
  $x_{min}$; When $x_{min}$ 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.

`References`

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.

`See Also`

mle

`Examples`

Run this code# 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 <- power.law.fit(d+1, 10)
fit2 <- power.law.fit(d+1, 10, implementation="R.mle")

fit1$alpha
coef(fit2)
fit1$logLik
logLik(fit2)
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