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 • Shannon’s 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. Shannon’s 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 > S then Eq,T > 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@bluffton.edu