An Elementary Diversity Index Developed Using

Taylor Series and Lagrange Multipliers

by

Donald E. Hooley

Bluffton College

Bluffton, OH

www.bluffton.edu/mat/seminar/

An Elementary Diversity Index

• A Diversity Question

• Properties of Diversity Indices

• Shannon’s Diversity Index

• Taylor Series Approximation

• Lagrange Multipliers and Equiprobability

• Results

• Project Ideas

A Diversity Measurement Question

Which classroom is more diverse?

1) Half Caucasian, Half African-American

2) 1 Indian, 1 Mexican, 1 Russian, 1 French, 1 Laotian, 15 Caucasian

Diversity Indices

Berger-Parker Nmax/N

Margalef (S-1)/ln N

McIntosh

U=

N = number of individuals in total population

Nmax = number of individuals in most populous species

S = total number of species

)/()( NNUN 2

in

Diversity Indices

Shannon

Simpson

pi = proportion of species i in population

ni = number of individuals of species i

N = number of individuals in total population

S = total number of species in population

i

S

ii pp ln

1

S

i

ii

NN

nn

1 )1(

)1(/1

Minimal Diversity Property

Classroom Calculator Use

All Some Never

S1) 1 0 0

S2) .20 .40 .40

S3) 0 0 1

Maximum Diversity

Flower Garden

Tulips Roses Mums

G 1) .80 .10 .10

G 2) .30 .40 .30

G 3) .33 .33 .33

Equiprobability

Given population proportions pi, i = 1 to n, maximum diversity occurs when

p1 = p2 = … = pn = 1/n

Maximum Diversity(again)

Categories

A B C D

G 1) .33 .33 .33

G 2) .25 .25 .25 .25

Diversity Index Properties

Property 1 When only one category is represented diversity equals 0.

Property 2 (Equiprobability) Maximum diversity occurs when each category is represented equally.

Property 3 Diversity at equiprobability is greater when the number of categories is greater.

Shannon’s Diversity Index

pi = proportion of species i in population

S = total number of species

)ln(1

i

S

ii ppH

Taylor Series Approximation

f(p) = - p ln p near p = .5

so define

2

2

69.25.

)5.(2

)5(.)5.)(5(.)5(.

pp

pf

pff

)( 2

1i

S

ii ppE

An Elementary Diversity Index

Minimal Diversity Property

If p1 = 1 then E = 0

Note

)( 2

1i

S

ii ppE

*10 E

Equiprobability Property

Maximize

subject to

Lagrange multipliers

1 – 2pi = for all i

thus pi = pj for all

)(),,( 2

11 i

S

iiS ppppE

Sji ,1

1),,(1

1

S

iiS pppg

gE

Increasing Categories Property

so if T > S

then Eq,T > EqS

S

iSq SS

E1

2

,

11

S

SSS

SSS

11

11

1

112

Results Index Class 1 Class 2

(.5,.5) (.05,.05,.05,.05,.05,.75)

.69 .96

.50 .43

)ln(1

i

S

ii ppH

)( 2

1i

S

ii ppE

Results Index Class1 Class 2

(.5,.5) (.05,.05,.05,.05,.05,.75)

.69 .96

.50 .43

.80 1.925

.35 .71

)ln(1

i

S

ii ppH

)( 2

1i

S

ii ppE

)7.3(. 2

1i

S

ii ppQAE

)5.4.1(. 2

1i

S

ii ppSLE

Diversity Projects for Inside

• Shoe type – male/female

• Clothing type

• Watches/glasses –student/fac.

• Writing implement choice

• Hair/Eye color

• Lunch Beverage choice

• Club membership

• Opinions on issues

• Discussion topics

Biological Diversity Projects

• Flower type

• Leaf shape

• Species – lawn/woodlot park/woods

garden/weedlot

• Insect – lawn/log

Sampling – select area

adjust size to pop.

equal organism count

Sociological Diversity Projects

• Automobile color

• Automobile type

• Automobile license origin

- student/faculty lots

- campus/town lots

- mall/grocery

• Topics of discussion

- town meeting/PTA

- concert intermission/club

BibliographyAbramson, Norman. 1963. Information Theory and Coding, New York: McGraw-Hill.

Ash, Robert. 1965. Information Theory, New York: John Wiley and Sons.

Gold, Harvey. 1977. Mathematical Modeling of Biological Systems – An Introductory Guidebook, New York: John Wiley and Sons.

Kolmes, Steven, and Mitchell, Kevin. 1991. Information Theory and Biological Diversity, UMAP Module 705, Arlington, MA: Comap, Inc.

Magurran, Anne E. 1988. Ecological Diversity and Its Measurment, Princeton, NJ: Princeton University Press.

www.bluffton.edu/mat/seminar/

Email: hooleyd@bluffton.edu