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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

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
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

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

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 indices5

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

Minimal Diversity Property

Classroom Calculator Use

All Some Never

S1) 1 0 0

S2) .20 .40 .40

S3) 0 0 1

maximum diversity

Maximum Diversity

Flower Garden

Tulips Roses Mums

G 1) .80 .10 .10

G 2) .30 .40 .30

G 3) .33 .33 .33

equiprobability

Equiprobability

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

p1 = p2 = … = pn = 1/n

maximum diversity again

Maximum Diversity(again)

Categories

A B C D

G 1) .33 .33 .33

G 2) .25 .25 .25 .25

diversity index properties

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

Shannon’s Diversity Index

pi = proportion of species i in population

S = total number of species

taylor series approximation

Taylor Series Approximation

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

so define

an elementary diversity index13

An Elementary Diversity Index

Minimal Diversity Property

If p1 = 1 then E = 0

Note

equiprobability property

Equiprobability Property

Maximize

subject to

Lagrange multipliers

1 – 2pi = for all i

thus pi = pj for all

increasing categories property

Increasing Categories Property

so if T > S

then Eq,T > EqS

results

Results

Index Class 1 Class 2

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

.69 .96

.50 .43

results17

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
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
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
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

bibliography

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