The Metalog Distributions

Shape Flexibility

Metalog distributions have virtually unlimited shape flexibility. This is guaranteed by the metalog flexibility theorem, which states that any probability distribution with a continuous quantile function can be approximated arbitrarily closely with a metalog. Like a Taylor series, the metalog quantile function may have any number of terms. Each additional term defines a different probability distribution and adds shape flexibility.

The original paper showed that metalogs parameterized by 105 cumulative distribution function (CDF)-data points from each of 30 traditional source distributions (including normal, lognormal, Weibull, gamma, chi-squared, beta, extreme value, and student-t) converged rapidly to those source distributions as the number of terms increases.  Moreover, the source vs metalog PDFs for all 30 distributions become visually indistinguishable as the number of metalog terms approaches ten. Here we further explore how two-ten term metalogs approximate traditional probability distributions and shape themselves to empirical data sets. 

Below is an example. We used 105 points from CDF of a standard normal distribution to parameterize unbounded metalog distributions. These 105 points correspond to y-probabilities of (0.001, 0.003, 0.006, 0.01, 0.02, ... , 0.99, 0.994, 0.997, 0.999). As the number of terms increases from 2 to 10, the metalog PDF and CDF become visually indistinguishable from the source distribution. 

2-10 term metalog distributions parameterized by 105 points from standard normal CDF
2-10 term metalog approximations of standard normal distribution
To view the normal PDF animation in more detail, click here. For similar animations and illustrations, click on any of the following: extreme value, bimodal beta, GammasWeibulls, lognormals, sum-of-lognormals, fish biology empirical data, hydrology empirical data. In all these cases where the source distribution is known, the metalog PDF becomes virtually indistinguishable from the source-distribution PDF as the number of terms approaches ten.