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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?
1) Half Caucasian, Half African-American
2) 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
)/()( NNUN 2
in
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
i
S
ii pp ln
1
S
i
ii
NN
nn
1 )1(
)1(/1
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
)ln(1
i
S
ii ppH
Taylor Series Approximation
f(p) = - p ln p near p = .5
so define
2
2
69.25.
)5.(2
)5(.)5.)(5(.)5(.
pp
pf
pff
)( 2
1i
S
ii ppE
An Elementary Diversity Index
Minimal Diversity Property
If p1 = 1 then E = 0
Note
)( 2
1i
S
ii ppE
*10 E
Equiprobability Property
Maximize
subject to
Lagrange multipliers
1 – 2pi = for all i
thus pi = pj for all
)(),,( 2
11 i
S
iiS ppppE
Sji ,1
1),,(1
1
S
iiS pppg
gE
Increasing Categories Property
so if T > S
then Eq,T > EqS
S
iSq SS
E1
2
,
11
S
SSS
SSS
11
11
1
112
Results Index Class 1 Class 2
(.5,.5) (.05,.05,.05,.05,.05,.75)
.69 .96
.50 .43
)ln(1
i
S
ii ppH
)( 2
1i
S
ii ppE
Results Index Class1 Class 2
(.5,.5) (.05,.05,.05,.05,.05,.75)
.69 .96
.50 .43
.80 1.925
.35 .71
)ln(1
i
S
ii ppH
)( 2
1i
S
ii ppE
)7.3(. 2
1i
S
ii ppQAE
)5.4.1(. 2
1i
S
ii ppSLE
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
BibliographyAbramson, 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: [email protected]