skbio.diversity.alpha.renyi(counts, order=2, base=None)[source]#

Calculate Renyi entropy.

Renyi entropy (\(^qH\)) is a generalization of Shannon index, with an exponent (order) \(q\) instead of 1. It is defined as:

\[^qH = \frac{1}{1-q}\log_b{(\sum_{i=1}^S p_i^q)}\]

where \(S\) is the number of taxa and \(p_i\) is the proportion of the sample represented by taxon \(i\).

counts1-D array_like, int

Vector of counts.

orderint or float, optional

Order (\(q\)). Ranges between 0 and infinity. Default is 2.

baseint or float, optional

Logarithm base to use in the calculation. Default is e.


Renyi entropy.


Renyi entropy was originally defined in [1]. It is a generalization of multiple entropy notions, as determined by the order (\(q\)). Special cases of Renyi entropy include:

  • \(q=0\): Max-entropy (\(\log{S}\)).

  • \(q \to 1\): Shannon entropy (index).

  • \(q=2\): Collision entropy, a.k.a, Renyi’s quadratic entropy, or “Renyi entropy”. Equivalent to the logarithm of inverse Simpson index.

  • \(q \to \infty\): Min-entropy (\(-\log{\max{p}}\)).

Renyi entropy is equivalent to the logarithm of Hill number.



Rényi, A. (1961, January). On measures of entropy and information. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics (Vol. 4, pp. 547-562). University of California Press.