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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 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
Increasing Categories Property 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 adjust size to pop. equal organism count
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 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