A new diversity index

Abstract

We introduce here a new index of diversity based on consideration of reasonable propositions that such an index should have in order to represent diversity. The behaviour of the index is compared with that of the Gini–Simpson diversity index, and is found to predict more realistic values of diversity for small communities, in particular when each species is equally represented and for small communities. The index correctly provides a measure of true diversity that is equal to the species richness across all values of species and organism numbers when all species are equally represented, as well as Hill’s more stringent ‘doubling’ criterion when they are not. In addition, a new graphical interpretation is introduced that permits a straightforward visual comparison of pairs of indices across a wide range within a parameter space based on species and organism numbers.

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Phys. Biol. 18 (2021) 066004 https://doi.org/10.1088/1478-3975/ac264e
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PAPER
Anewdiversityindex
ATAugousti
1,∗,NAtkins
1,ABen-Naim
2,SBignall
3,GHunter
1, M Tunnicliffe1
and A Radosz4
1Faculty of Science, Engineering and Computing, Kingston University London, United Kingdom
2Department of Physical Chemistry,The Hebrew University, Jerusalem 91904, Israel
3The Portland Hospital, London, United Kingdom
4Wroclaw University of Science and Technology, Wroclaw, Poland
∗Author to whom any correspondence should be addressed.
E-mail: augousti@kingston.ac.uk
Keywords: diversity index, Gini– Simpson index, biodiversity, species richness, entropy
Abstract
We introduce here a new index of diversity based on consideration of reasonable propositions that
such an index should have in order to represent diversity. The behaviour of the index is compared
with that of the Gini–Simpson diversity index, and is found to predict more realistic values of
diversity for small communities, in particular when each species is equally represented and for small
communities. The index correctly provides a measure of true diversity that is equal to the species
richness across all values of species and organism numbers when all species are equally represented,
as well as Hill’s more stringent ‘doubling’ criterion when they are not. In addition, a new graphical
interpretation is introduced that permits a straightforward visual comparison of pairs of indices
across a wide range within a parameter space based on species and organism numbers.
1. Introduction
The issue of measuring diversity in an ecosystem has
been a topic of vigorous discussion for many years.
Although many measures for measurement of this
diversity have been introduced over the years, it is fair
to say that most of these have been imported from
other fields. For example, a commonly used index,
the Gini– Simpson index, was originally introduced
by Gini in 1912 [1] as a statistical measure to indi-
cate the variability in distributions of both contin-
uous and discrete variables, the latter including, for
example, four categories of hair colour. Simpson [2]
later presented essentially the same index to repre-
sent ‘...a measure of concentration in terms of pop-
ulation constants’, in order to overcome difficulties
introduced by measuresdefined by Yule [3]andFisher
et al [4] which depended on sample data rather than
population constants.
The diversity has also been related to Shannon’s
measure of information (SMI) based on the idea that
the greater the diversity in the number of species and
the evenness of the distribution of organisms between
them, then the greater is the uncertainty in selection,
the latter being measured by the SMI (see, for instance
Macarthur [5]). The use of the SMI for such an appli-
cation is not without controversy [6] (as indeed is the
application of the SMI in a wide variety of applica-
tions with varying degrees of suitability (for a discus-
sion of this in the context of entropy see, for example,
Ben-Naim [7]). The concept of diversity has also been
extended to relate to diversity that is measured at mul-
tiple locations and then combined [8,9], and appli-
cations of the concept of diversity range across many
areas of biology, information theory, physics, eco-
nomics and even psychology [10], where the term
emodiversity has been used to describe the variety and
relative abundance of individuals’ emotional experi-
ences. At a microscopic level, the idea of diversity has
also been used to determine the impact of a class of
antibiotics known as macrolides used for therapeutic
purposes on the gut microbiota of children [11], for
instance.
In a very clear analysis Jost [12]distinguishes
between the use of uncertainty measures, which he
terms entropies, and diversities, which are intended
to provide a measure of relative abundance, and he
defines the term ‘true diversity’—an ‘effective num-
ber of species’ which would give the same value as the
diversity index determined for a specific distribution
if all of the species present were equally represented
by organisms. His use of the term ‘true diversity’ is
somewhat in opposition to the views of Hoffman and
Hoffman [13]. Despite the plethora of indices that
© 2021 The Author(s). Published by IOP Publishing Ltd

Phys. Biol. 18 (2021) 066004 ATAugoustiet al
have been developed [14], new indices continue to be
introduced [15], since most of the existing indices suf-
fer from weaknesses that cause them to be misleading
under particular situations, typically when the num-
ber of species or organisms tends to very large or small
limits. Xu et al [16] provide a helpful review that iden-
tifies the connections and differences in a wide range
of indices and entropies.
The aim of this work was to address the develop-
ment of an index by consideration of first principles,
beginning with a series of ‘desirable’ features that such
an index would possess. Some of these desirable fea-
tures seek to address weaknesses perceived in existing
measures, as well as those based on uncertainty.
Section 2details the features expected in such
an index, and offers a possible functional form that
exhibits these features. Section 3compares the results
calculated using this index with that of the widely-
used Gini–Simpson index, with which it shares some
characteristics and introduces a novel graphical rep-
resentation that permits a visual comparison of any
index in a convenient manner. Section 4presents and
discusses these results, and section 5concludes with
a summary and possible future developments. Refer-
ences are presented in reference section, appendix A
provides details of an improved algorithm for iden-
tification of acceptable distributions given particu-
lar values for the number of organisms and species.
Appendix Bprovides a derivation of the form of the
‘true’ diversity (see section 4for details) for this new
index and appendix Cprovides a proof supporting the
discussion in section 4, namely compliance with Hill’s
doubling criterion (see section 4for details).
2. Considerations for construction of a
new index
The purpose of the index is to provide a measure of
diversity— based on the number of species and organ-
isms that inhabit an environment—that gives rela-
tively intuitive results, especially with regard to partic-
ular limits, such as very large or very small numbers
ofspeciesororganisms,aswellasforparticularlyeven
distributions. The following terms will be used
mnumber of species
ninumber of organisms of species i(i=1,
2, .., m)
Ntotal number of organisms
Γdiversity index
Γifunctional form of Γwith respect to species i
n0the ‘ideal’ average number of organisms
per species if all organisms are equally distributed
(=N/m)
kϕscaling factor for azimuthal angle in graphical
representation
kθscaling factor for elevation angle in graphical
representation
pNm
It is clear that
m

i=1
ni=N.(1)
Let the diversity function have the following prop-
erties.
(a) It should depend on the distribution of organ-
isms, the total number of organisms and the total
number of species
Γ=Γ(ni,m,N).
(b) It should have a finite range—this can permit
comparison with other indices
0Γ1.
(c) The index should be maximised when the organ-
isms are evenly distributed among species, so for
ni=N/m=n0
Γ=Γ
max
n0and ∂Γ
∂ni
=0.
(d) When organisms are evenly distributed among
species, the index should be maximal as the num-
ber of organisms increases
lim
N→∞ Γ=1.
Given these initial constraints, what are the
possible functional forms for Γ?Aformthat
fulfills condition (c) is
Γi=f(ni)e−(ni−n0),(2)
where fdenotes an arbitrary polynomial func-
tion of the argument (ni). The simplest choice
here is to a scaled form in which variable niis
replaced by ni/n0and a linear form is chosen for
f,hence
Γi=ni
n0
e1−ni
n0.(3)
Such a functional form for Γihas a peak value
of 1 at ni=n0and diminishes monotonically
to zero as nitends to zero or infinity. This sug-
gests a form for Γitself, which is simply a nor-
malised sum of these contributions arising from
each species Γi
Γ= 1
m
m

i=1
ni
n0
e1−ni
n0=
m

i=1
ni
NeN−mni
N.(4)
Such an index is determined primarily by the
distribution of organisms between species rather
than the total number of species themselves. This
dependence on the number of species can be
incorporated by the inclusion of two further con-
straints, thus
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Phys. Biol. 18 (2021) 066004 ATAugoustiet al
(e) If a single species is present only, then the
index should indicate there is no diversity, hence
lim
m→1Γ=0.
(f) And, in the case of equally distributed organisms
(see equation (B.3))
lim
m→∞ Γ=1.
Both of these constraints can be accommo-
dated by including an additional factor of m−1
m
giving the final form of the index as
Γ= m−1
m
m

i=1
ni
NeN−mni
N.(5)
3. Representation of the index and
comparison with the Gini–Simpson
index
One of the challenges in this area is to find an appro-
priate representation which displays compactly values
of Γfor a wide variety of values of N,m,andni.Such
a representation would also permit an easier compar-
ison with other diversity indices, once they have been
normalised to unity if necessary.
Even for relatively low values of both Nand
m, the possible range and enumeration of distribu-
tions becomes a challenging combinatorial problem
to represent adequately. For example, even for eight
species with 15 organisms to be distributed between
them, this represents 3432 distinct combinations.
This paper introduces, to the best of our knowledge,
a novel graphical transformation which permits the
compression and visualisation of the vast range of
combinations within the positive octant of a three-
dimensional spherical coordinate space.
The starting point is a unit line, along which will
be represented values of the index for a systematic list-
ing of distributions defined by niand m. The system-
atic listing is described further below. The line will be
oriented in the positive octant using a spherical coor-
dinate system, such that the azimuthal angle ϕand the
elevation θare defined by
ϕ=kϕtan−1(m−1) (6)
θ=kθtan−1(N−1).(7)
The values of kϕand kθare chosen to provide a
broader fill of the available space, otherwise if these
are omitted then incrementing mor Nfrom starting
values of 1 will result in the next lines being oriented
at 45◦to the x-axis and the x–yplane respectively.
The specific value of the index for each distribution
is colour coded according to a defined colour scale,
with specific colours representing values between 0
and 1, occasionally referred to as a ‘heat map’. So
overall each possible configuration is a set of discrete
coloured points along a notional line whose orien-
tation depends on the total number of species and
organisms. In this way, an unbounded range of val-
ues for these quantities can be mapped and visualised
relatively easily.
The representation of each distribution along each
line is achieved systematically using the following
scheme. Firstly, the unit line is divided by the total
number of distributions which are possible given spe-
cific numbers of species and organisms. One way
to perform this systematic listing is by using mod-
ulo arithmetic and performing a checksum on the
sum of digits. This is most easily illustrated by way
of an example using a low number of species and
organisms.
Consider an environment with three species and a
total of five organisms. Since three species are defined,
there must exist at least one organism representing
each of these species. This leaves two further organ-
isms to be distributed among the three species. The
difference between the number of organisms and
species can be represented by p,hencep=2 here. If
one represents each distribution as an m-digit num-
ber to modulo p+1, then the possible distributions
of remaining organisms are enumerated by counting
as
000
001
002
010
011
012
020
021
022
100
101
102
110
111
120
120
122
200
Where the possible distributions whose checksum is
equal to p=2 are shown in bold. Hence there are
six ways in total to distribute five organisms between
three species. In this instance the unit line would be
subdivided into steps of 1/6, and the value of the diver-
sity index plotted at each of these locations along the
line for the distributions shown below
113
122
131
212
221
311
This describes the initial strategy that was adopted
to enumerate and display the possible distributions.
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Phys. Biol. 18 (2021) 066004 ATAugoustiet al
However, such an approach based on incrementing
a count uniformly and then checking for the cor-
rect digit sum is highly inefficient as the values of m
and pincrease. For example, for ten species and 20
organisms, the count would need to be in excess of
2.3 ×1010 in order to identify the 92 378 accept-
able distributions. Appendix Aprovides details of
an improved counting strategy which, by using non-
uniform increments, reduces the count to the exact
number of acceptable distributions; in the example
cited above this represents an improvement of over
255 000-fold.
4. Results and discussion
The algorithms described above were coded in Mat-
lab, and figure 1displays an example of a comparison
between the present diversity index (PDI), and a well-
known conventional index used in environmental sci-
ence—the Gini–Simpson index (GSDI)—which is
given by
Gini −Simpson index =1−
m

i=1
ni(ni−1)
N(N−1) .(8)
Theimagesshowarangeofvaluesofmbetween
3 and 8, and organism numbers ranging between m
+1 and 15. The coloured dots representing values in
lines for larger numbers of possible distributions are
reduced in size in order to avoid overlap and for ease
of viewing. Such an image is best navigated using a
3D interactive format, as is available in Matlab, for
example
Even in a 2D representation as an image a clear
comparison between values calculated according to
the two indices can be made, as well as the variation
of the value within a single index as the organisms
are distributed among them. For ease of viewing, the
size of the coloured dot representing each value has
been scaled, so that unit lines containing more distri-
butions are represented by smaller dots. The values of
kϕand kθselected for use in figure 1are 1/6and1/8
respectively.
The figure primarily serves to illustrate the fact
that the GSDI generally provides larger values for the
diversity index than the PDI, particularly so at lower
values for the number of species and organisms. An
advantage of the PDI is that it gives lower values than
the GSDI in distributions close to or equal to uniform
representation in small communities.
It follows from the definition above that the limit-
ing value for the new DI in the case of equally repre-
sented species is simply m−1
m, contrasting with a value
of N(m−1)
m(N−1) for the GSDI. In the latter case, when all
species have a single organism representing them the
value of the GSDI is 1, which clearly does not present
a realistic picture of diversity, being both indepen-
dent of the number of species present and suggesting
‘total’ diversity. The value of the new DI in this case
depends on the number of species (as required by the
conditions provided for its construction) but is inde-
pendent of the number of organisms present; it may
be argued that this gives a more realistic value that
represents the distribution of the organisms between
the species irrespective of how many (equal) repre-
sentatives there are for each species. For example, it
might reasonably be considered that three species rep-
resented by ten organisms each in fact represents no
greater diversity than three species represented by a
single member each, and this is the sense of diversity
captured in the new DI. In this sense, it is closer to a
measure of ‘species richness’.
Jost [12] introduces the term ‘true
diversity’—referred to with different names by
different authors, for instance the ‘effective number
of species’ by Macarthur [5]andthe‘numbers
equivalent’ by Patil and Tailee [17]ineconomics.
Thus for a given value of the index calculated for
a particular distribution, this corresponds to the
number of species present if all organisms are equally
represented. In this case, the ‘true diversity’ can easily
be shown to be (see appendix B)
1
1−Γ(9)
and this corresponds, in the limit of large Nand ni,
to the same value for the Gini–Simpson index. It is
interesting to note this expression, while true for all
values of species number niand number of organisms
Nfor the new diversity index, is only true in the ‘large
community’ limit for the GS index, and this might
be considered a point in its favour. Indeed, most of
the indices introduced to date may be represented by
expressions of the form
m

i=1ni
Nq
, (10)
where qis an exponent that varies from one index to
another. In consequence, different indices that reduce
to an expression of this kind with the same value of
qare equivalent, and measures of the true diversity
would be the same for such indices, and given by [12]
qD=m

i=1ni
Nq1
1−q
.(11)
These are a generalization of Hill’s numbers [18],
and the exponent qmay be termed the order of the
diversity. Thus different functional forms for a diver-
sity index may be connected in this way, with the true
diversity being dependent only on the order of the
diversity index.
The DI proposed here is not represented in this
way, and therefore is not immediately equivalent to
indices thus far introduced. It therefore also does not
fall under the category of a generalised weighted GS
index as proposed by Guiasu and Guiasu [15,19], and
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Phys. Biol. 18 (2021) 066004 ATAugoustiet al
Figure 1. Comparison of the new diversity index (left) versus the Gini–Simpson index (right). The colour indicates that, for low
values of the number of organisms and species (larger circles, m=6, p=1– 3) the PDI gives values in the 0.53– 0.63, while the
corresponding values for the GSDI are in the range 0.6– 0.83.
hence their analysis of such generalised functions may
not apply here.
Jost [12] introduces a desirable ‘scaling’ test for
an index, proposing that the true diversity calculated
for sixteen equally represented species is represented
by an index which predicts a value of true diversity
twice as large as that for 8 equally represented species.
In other words, the true diversity should be propor-
tional to the number of species present. In common
with indices defined by expression (10)above,the
new DI passes this test. A stiffer requirement is Hill’s
‘doubling property’ [18], which states that for any
particular distribution, halving the number of rep-
resentatives of each species while doubling the num-
ber of species (the example that Jost [12]providesis
to split each species into two equal groups of males
and females and then treat each of these as a separate
species) should double the measure of true diversity.
A proof provided in appendix Bshows that the new DI
also passes this more stringent test, although it could
be argued that strict doubling in this way is not so
meaningful, providing nothing more than a measure
akin to species richness and failing to properly take
into account the relative representation of common
and rare species.
5. Conclusion
A new diversity index has been introduced which is
based on an intuitive set of properties. Values for
the new index have been calculated and compared
with the popular Gini–Simpson index, and it has
been shown that the new index compares favourably
with it in providing more realistic values, in partic-
ular in predicting the ‘true diversity’ at low counts
of organisms and species. In this instance, these low
counts—defining a ‘small community’—are when
the number of species mis 10 or less and the number
of organisms Nis 20 or less. At high counts of both
of these parameters (the ‘large community’ limit) the
two indices tend to coincide for equally represented
species.
The index passes the desirable test of predict true
diversity that is equivalent to species richness in the
case of equally represented species, and passes also a
more stringent test when the species are not equally
represented. As noted above, it could be argued that
this behaviour is similar to a behaviour equivalent
to species richness, which may not be so meaningful
when rare species are only minimally represented.
We introduce here also a novel way of represent-
ing the space of DI values in a convenient form which
permits closer comparison between different formu-
lations of DI in a visual format.
Future work will seek to establish a more effi-
cient counting algorithm for representing the possi-
ble distributions, a comparison with a broader range
of diversity indices, and application to a broader
range of practical problems in order to characterise
the behaviour of the index more fully, including also
potential use in language applications [20]. We will
also seek to extend the work to explore the behaviour
of indices where the linear term in equation (3)is
replaced by a higher order polynomial, as well as to
5

Phys. Biol. 18 (2021) 066004 ATAugoustiet al
extend the application to include measures of phy-
logenetic distance. A further aim of this work is to
explore the behaviour of indices of this kind when
sampling distributions whose frequencies appear to
be predicted on a combinational basis, and which
appear to follow a universal distribution [21].
Data availability statement
All data that support the findings of this study are
included within the article (and any supplementary
files).
Appendix A. Improved algorithm to
identify acceptable distributions for
specific values of pand m
It is probably easiest to describe the algorithm in the
first instance by way of an example. Consider the fol-
lowing example illustrating the distribution of eight
organisms among four species. As described above,
we need only consider here the remaining organisms
noting that four organisms must already respectively
represent each species, hence p=4.
The enumerated distributions follow the pattern
below. Note that these values correspond to digit val-
ues in a modulo-5 counting base. In general, the val-
ues are incremented by a decimal value of 4, and the
decimal increase between the previous row and the
current one is shown only for those values where the
increment differs from 4.
0004
0013
0022
0031
0040
0103 8
0112
0121
0130
0202 12
0211
0220
0301 16
0310
0400 20
1003 28
1012
1021
1030
1102 12
1111
1120
1201 16
1210
1300 20
2002 52
2011
2020
2101 16
2110
2200 20
3001 76
3010
3100 20
4000 120
Recall that these numbers are written in modulo-
(p+1)—in this case modulo-5—and therefore the
digits, counting from the least significant to the
most significant, represent respectively units, p+1,
(p+1)2,(p+1)3etc. Note that each non-uniform
jump of 4 occurs when the units digits is zero, as well
as when this is zero along with higher place value
digits also being zero. Let jrepresent the digit num-
ber, beginning with j=1 as the units digit, and label
the highest non-zero digit as j=qwhen the non-
uniform increment takes place. It may be observed
that the increment in these cases corresponds to
(p+1)q−Zq(p+1)q−1+Zq−1whereZqrepre-
sents the value in the qth digit prior to increment-
ing. This forms the basis of the algorithm coded in
Matlab, whereby the sequence of zero-occupied digits
is checked prior to incrementing, and it enumerates
precisely the exact number of acceptable distributions
without any redundant counts.
Using the example above, it should be clear that,
in general, if one unit increments are used (without
the zero-checking procedure noted above), the count
will reach a maximum of p(p+1)m−1.Usingastars-
and-bars method (used as early as 1915 by Ehrenfest
and Onnes [22]), it is easy to show that the number
of distributions that contain the correct digit sum (p)
is given by (m+p−1)!
p!(m−1)!.Inthecasegiveninthetextof
m=10 and p=10 (thus N=20) then the total count
corresponds to 2.357 947 691 ×1010 and the number
of acceptable distributions is 92 378.
Appendix B. Derivation of the form of
the true diversity for the PDI
We follow here an algorithm based on the generalised
one provided by Jost [12]inordertocalculatethe
‘true diversity’ for any diversity index based on a sum
of powers of ni, the number of organisms of species i,
simplified in this instance due to the specified form
of the PDI. By formulating an expression for the
diversity index Γof mequally represented species
(hence ni=n0=N/m)and then solving for m,the
required formula for the true diversity (termed Dby
Jost and equivalent to min this case) is obtained.
Thus
Γ= m−1
m
m

i=1
ni
NeN−mni
N(B.1)
and using the equal representation condition
ni=N/m=n0
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Phys. Biol. 18 (2021) 066004 ATAugoustiet al
Γ=m−1
m
m

i=1
1
me(1−m
m).(B.2)
The exponent is zero, and the summation total is
1. Hence
Γ= m−1
m(B.3)
which is easily rearranged to give the required formula
for D, the true diversity.
m=1
1−Γ=D.(B.4)
Appendix C. Proof of compliance with
Hill’s doubling property
Beginning from the definition of the DI
Γ= m−1
m
m

i=1
ni
NeN−mni
N(C.1)
one may label the sum in the formula as S,hence
Γ= (m−1)S
m.(C.2)
The true diversity as defined by Jost is given by
D=1
1−Γ(C.3)
which in the case of equally distributed species (i.e.
when Γ= m−1
m,andS=1) easily reduces to a value of
m.
If ‘Hill doubling’ is conducted in the manner
described by Jost (splitting each species into two equal
groups of males and females and then treating each
of these as a separate species) then this may be repre-
sented in the following way
m→2m
ni→ni
2
ni+m=ni
and Nis unchanged.
Then
Γdouble =2m−1
2m
2m

i=1
ni
2NeN−2mni
/2
N
Γdouble =2m−1
2m
2m

i=1
ni
2NeN−mni
N
Γdouble =2m−1
2mm

i=1
ni
2NeN−mni
N
+
2m

i=m+1
ni
2NeN−mni
N(C.4)
and using the property that
ni+m=ni
this may be rewritten
Γdouble =2m−1
2mm

i=1
ni
2NeN−mni
N
+
m

i=1
ni
2NeN−mni
N(C.5)
Γdouble =2m−1
2mS
2+S
2
hence
Γdouble =(2m−1)S
2m.(C.6)
Since
Ddouble =1
1−Γdouble
(C.7)
then simple substitution gives
Ddouble =2m
2m−(2m−1)S.(C.8)
Since Dis defined to be the diversity equivalent
to a distribution of equally represented species, then
S=1andDbecomes simply 2mi.e. double the orig-
inal value and hence the Hill doubling criterion is
fulfilled.
ORCID iDs
ATAugousti https://orcid.org/0000-0003-3000-
9332
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