An Elementary Diversity Index Developed Using Taylor Series and Lagrange Multipliers by Donald E....
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Transcript of An Elementary Diversity Index Developed Using Taylor Series and Lagrange Multipliers by Donald E....
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]