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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 ... A Diversity Measurement Question. Which ... Gold, Harvey. 1977. ... – PowerPoint PPT presentation

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Title: An Elementary Diversity Index Developed Using Taylor Series and Lagrange Multipliers

1
An Elementary Diversity Index Developed Using
Taylor Series and Lagrange Multipliers
• by
• Donald E. Hooley
• Bluffton College
• Bluffton, OH
• www.bluffton.edu/mat/seminar/

2
An Elementary Diversity Index
• A Diversity Question
• Properties of Diversity Indices
• Shannons Diversity Index
• Taylor Series Approximation
• Lagrange Multipliers and Equiprobability
• Results
• Project Ideas

3
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

4
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

5
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

6
Minimal Diversity Property
• Classroom Calculator Use
• All Some Never
• S1) 1 0 0
• S2) .20 .40 .40
• S3) 0 0 1

7
Maximum Diversity
• Flower Garden
• Tulips Roses Mums
• G 1) .80 .10 .10
• G 2) .30 .40 .30
• G 3) .33 .33 .33

8
Equiprobability
• Given population proportions pi, i 1 to n,
maximum diversity occurs when
• p1 p2 pn 1/n

9
Maximum Diversity(again)
• Categories
• A B C D
• G 1) .33 .33 .33
• G 2) .25 .25 .25 .25

10
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.

11
Shannons Diversity Index
• pi proportion of species i in population
• S total number of species

12
Taylor Series Approximation
• f(p) - p ln p near p .5
• so define

13
An Elementary Diversity Index
• Minimal Diversity Property
• If p1 1 then E 0
• Note

14
Equiprobability Property
• Maximize
• subject to
• Lagrange multipliers
• 1 2pi for all i
• thus pi pj for all

15
Increasing Categories Property
• so if T gt S
• then Eq,T gt EqS

16
Results
• Index Class 1 Class 2
• (.5,.5) (.05,.05,.05,.05,.05,
.75)
• .69 .96
• .50 .43

17
Results
• Index Class1 Class 2
• (.5,.5)
(.05,.05,.05,.05,.05,.75)
• .69 .96
• .50 .43
• .80 1.925
• .35 .71

18
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

19
Biological Diversity Projects
• Flower type
• Leaf shape
• Species lawn/woodlot
• park/woods
• garden/weedlot
• Insect lawn/log
• Sampling select area
• equal organism count

20
Sociological Diversity Projects
• Automobile color
• Automobile type
• - student/faculty lots
• - campus/town lots
• - mall/grocery
• Topics of discussion
• - town meeting/PTA
• - concert intermission/club

21
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_at_bluffton.edu