An Elementary Diversity Index Developed Using Taylor Series and Lagrange Multipliers

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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

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## An Elementary Diversity Index Developed Using Taylor Series and Lagrange Multipliers

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### 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?

Half Caucasian, Half African-American

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

### 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

### 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

### Taylor Series Approximation

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

so define

### An Elementary Diversity Index

Minimal Diversity Property

If p1 = 1 then E = 0

Note

### Equiprobability Property

Maximize

subject to

Lagrange multipliers

1 – 2pi = for all i

thus pi = pj for all

so if T > S

then Eq,T > EqS

### Results

Index Class 1 Class 2

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

.69 .96

.50 .43

### Results

Index Class1 Class 2

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

.69 .96

.50 .43

.80 1.925

.35 .71

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

equal organism count

Sociological Diversity Projects
• Automobile color
• Automobile type

- student/faculty lots

- campus/town lots

- mall/grocery

• Topics of discussion

- town meeting/PTA

- concert intermission/club

### Bibliography

Abramson, 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