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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
  • adjust size to pop.
  • equal organism count

20
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

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
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