Distribution Fitting

This R code uses the R poweRlaw package to determine (estimate) which distribution fits best to a given data-set of a graph. The R poweRlaw package is an implementation of maximum likelihood estimators that supports power-law, log-normal, Poisson, and exponential distributions.

Steps

1. Estimate xmin: As most distributions only apply for values greater than some minimum xmin, first the lower threshold xmin (i.e. the estimated lower cutoff) is estimated using the estimate_xmin function, which implements a Kolmogorov-Smirnov (KS) approach. The KS statistic is a two sample test which is used to test whether two samples come from the same distribution. The KS statistic is the maximum distance between the data and the fitted model Cumulative Distribution Functions (CDFs).
2. Uncertainty of xmin: In order to quantify the uncertainty of the estimated thresholds xmin the poweRlaw bootstrap function is used.
3. Estimate parameters: After updating the xmin values with the respective estimates, the parameters of the distributions are estimated using the estimate_pars function. This function estimates the parameters by using their maximum likelihood estimator.
4. Standard deviations of parameters: The standard deviations for the parameters are derived using again the bootstrap and the sd functions.
5. Goodness of fit: A goodness-of-fit test is performed, which generates a p-value that quantifies the plausibility of the hypothesis that the graph follows a certain distribution. Therefore, the poweRlaw bootstrap_p function was used. This function is based on the measurement of the distance between the distribution of the data-set and the hypothesized model, which is then compared with the distance measurements for comparable synthetic data-sets. Large p-values (i.e. p-values close to 1) mean that the difference between the observed data and the respective model can be attributed to statistical fluctuations alone and imply hence, that the model is a plausible fit. A small p-value, on the other hand, means that the model is not a plausible fit for the data.
6. Likelihood-ratio test: In order to draw more precise conclusion, a likelihood-ratio test using the poweRlaw compare_distributions function is used. This function implements the likelihood-ratio test as introduced by Vuong in 1989. In particular, this function compares two given models with the null hypothesis being that both classes of distributions are equally far from the true distribution. If the null hypothesis is true, the test statistic R (i.e. the logarithm of the ratio of the two likelihoods of the competing distributions) is equal to 0. If, however, one model is closer to the true distribution, the test statistic takes positive or negative values thereby indicating which class of distribution is the better fit. However, if the test statistic R takes a value close to zero, then statistical fluctuations could change the sign of the ratio, which means hence that the results of the test cannot be trusted. In order to overcome this issue, the method proposed by Vuong gives a p-value that indicates the statistical significance of R. A small p-value (p < 0:1) indicates that the sign is a reliable indicator of which model is the better fit.

References

[1] A. Clauset, C. R. Shalizi, and M. E. J. Newman. Power-Law Distributions in Empirical Data. Society for Industrial and Applied Mathematics, 51(4):661–703, 2009.

[2] C. Gillespie. Package “poweRlaw”. Analysis of Heavy Tailed Distributions. CRAN, 0.70.0 edition, 2016.

[3] Q. H. Vuong. Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses. Econometrica, 57(2):307–333, 1989.