Brillouin’s Diversity Index

Basic Concepts

For a population (or a non-random sample), we can use Brillouin’s index of diversity, instead of Shannon’s index. Brillouin’s index is defined as

where n_i is the number of observations from the sample in the i^th of k (non-empty) categories and n is the sample size

The maximum value of H is

where c = INT(n/k) and d = MOD(n, k).

Finally, Brillouin’s index of relative diversity is J = H/H_max.

Example

Example 1: Find Brillouin’s index of diversity and index of relative diversity for a population distributed among five categories as shown in range B4:F4 of Figure 1.

Figure 1 – Population Index of Diversity

Figure 1 shows these two indices. Here, cell G4 contains the formula =SUM(B4:F4), cell B5 contains the formula =LOG10(FACT(B4)), cell G5 contains =LOG10(FACT(G4)), cell B7 contains the formula =(G5–SUM(B5:F5))/G4, cell B8 contains =COUNTA(B3:F3), cell B9 contains =INT(G4/B8), cell B10 contains =MOD(G4,B8) and cell B11 contains the formula

=(LOG10(FACT(G4))–(B8-B10)*LOG10(FACT(B9))–B10*LOG10(FACT(B9+1)))/G4

and cell B12 contains the formula B7/B11.

Calculation refinements

Note that the Excel function =FACT(n) only works for non-negative integer values for n ≤ 170. For larger values, we need to take advantage of the fact that n! = Γ(n+1) where Γ(x) is the gamma function, as described in Gamma Function. In particular, we can calculate ln(n!), the natural log of n!, in Excel by the formula =GAMMALN(n+1). The log base 10 of n! can be calculated by =GAMMALN(n+1)/LN(n+1).

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Krebs, C. J. (2014) Species diversity measures
https://www.zoology.ubc.ca/~krebs/downloads/krebs_chapter_13_2017.pdf