combinatorial insights into distributions of wealth, size, and abundance ken locey
TRANSCRIPT
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Combinatorial insights into distributions of wealth, size,
and abundance
Ken Locey
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Rank-abundance curve (RAC)
Rank in abundance
Abun
danc
e
Frequency distribution
Species abundance distribution (SAD)
Abundance class
freq
uenc
y
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Ranked curve (RC)
Rank in abundance,wealth, or size
Abun
danc
e/w
ealth
/size
Frequency distribution
Distributions of wealth, size, abundance
Abundance, wealth, or size class
freq
uenc
y
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Wheat Production (tons)
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Poverty in Rural America, 2008
Percent in Poverty
54 – 25.1 25 – 20.1 20 – 14.1 14 – 12.1 12 – 10.1 10 – 3.1
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Distributions used to predict variation in wealth, size, & abundance
1. Pareto (80-20 rule)2. Log-normal3. Log-series4. Geometric series5. Dirichlet6. Negative binomial7. Zipf8. Zipf-Mandelbrot
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Rank-abundance curve (RAC)
Rank in abundance
Abun
danc
e
Frequency distribution
Predicting, modeling, & explaining the Species abundance distribution (SAD)
Abundance class
freq
uenc
y
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Rank in abundance
Abun
danc
e104
103
102
101
100
ObservedResourcepartitioningDemographic stochasticity
Predicting, modeling, & explaining the Species abundance distribution (SAD)
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Rank in abundance
Abun
danc
e104
103
102
101
100
N = 1,700S = 17
Predicting, modeling, & explaining the Species abundance distribution (SAD)
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How many forms of the SAD for a given N and S?
Rank in abundance
Abun
danc
e104
103
102
101
100
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Integer Partitioning
Integer partition: A positive integer expressed as the sum of unordered positive integers
e.g. 6 = 3+2+1 = 1+2+3 = 2+1+3
Written in non-increasing (lexical) ordere.g. 3+2+1
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Rank-abundance curves are integer partitions
Rank-abundance curve
N = total abundanceS = species richness
S unlabeled abundancesthat sum to N
Integer partition
N = positive integerS = number of parts
S unordered +integersthat sum to N=
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Combinatorial Explosion
N S Shapes of the SAD
1000 10 > 886 trillion
1000 100 > 302 trillion trillion
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Random integer partitions
Goal: Random partitions for N = 5, S = 3:
54+13+23+1+12+2+12+1+1+11+1+1+1+1
Nijenhuis and Wilf (1978) Combinatorial Algorithms for Computer and Calculators. Academic Press, New York.
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SAD feasible sets aredominated by hollow curves
Fre
quen
cy
log2(abundance)
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The SAD feasible setln
(abu
ndan
ce)
Rank in abundance
N=1000, S=40
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Can we explain variation in abundance based on how N and S constrain
observable variation?
Question
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Dataset communities
Christmas Bird Count 129
North American Breeding Bird Survey 1586
Gentry’s Forest Transect 182
Forest Inventory & Analysis 7359
Mammal Community Database 42
Indoor Fungal Communities 124
Terrestrial metagenomes 92
Aquatic metagenomes 48
TOTAL 9562
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The center of the feasible setln
(abu
ndan
ce)
Rank in abundance
N=1000, S=40
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Obs
erve
d ab
unda
nce
100 101 102
Abundance at the center of the feasible set
102
101
100R2 per site
R2 = 1.0
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Obs
erve
d ab
unda
nce
R2 = 0.93
Breeding Bird Survey (1,583 sites)
100 101 102
R2 per site
Abundance at the center of the feasible set
102
101
100
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Abundance at center of the feasible set
Obs
erve
d ab
unda
nce
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Obs
erve
d ab
unda
nce
Abundance at center of the feasible set
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Public code and data repository
https://github.com/weecology/feasiblesets
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Center of the feasible set
Obs
erve
d ho
me
runs
0.93 0.88
0.91 0.91
0.94 0.93
http://mlb.mlb.com
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Combinatorics is one only way to examine feasible sets
Other (more common) ways:Mathematical optimizationLinear programming
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Dataset total sites analyzable sites
Christmas Bird Count 1992 129 (6.5%)
North American Breeding Bird Survey 2769 1586 (57%)
Gentry’s Forest Transect 222 182 (82%)
Forest Inventory & Analysis 10356 7359 (71%)
Mammal Community Database 103 42 (41%)
Indoor Fungal Communities 128 124 (97%)
Terrestrial metagenomes 128 92 (72%)
Aquatic metagenomes 252 48 (19%)
TOTAL 15950 9562 (60%)
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Efficient algorithms for generating random integer partition with
restricted numbers of parts
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Random integer partitions
Goal: Random partitions for N = 5, S = 3:
54+13+23+1+12+2+12+1+1+11+1+1+1+1
Nijenhuis and Wilf (1978) Combinatorial Algorithms for Computer and Calculators. Academic Press, New York.
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Combinatorial Explosion
N S SAD shapes
1000 10 > 886 trillion
1000 1,...,1000 > 2.4x1031
Probability of generating a random partition of 1000 having 10 parts: < 10-17
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Task: Generate random partitions of N=9 having S=4 parts
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4+3+2
Task: Generate random partitions of N=9 having S=4 parts
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4+3+2
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4+3+2
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4+3+2
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3+3+2+14+3+2
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4+3+2
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3+2=5
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4+3+2=9
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3+3+2+14+3+2=9
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1. Generate a random partition of N - S with S or less as the largest
2. Append S to the front
3. Conjugate the partition
4. Let cool & serve with garnish
A recipe for random partitions of N with S parts
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54+13+23+1+12+2+12+1+1+11+1+1+1+1
Generate a random partition of N-S with S or less as the largest part
Divide & Conquer
Multiplicity
Top down
Bottom up
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Un(bias)
Skewness of partitions in a random sample
Den
sity
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Speed
Number of parts (S)
Sag
e/al
gorit
hm
N = 50 N = 100
N = 150 N = 200
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Old Apples: probability of generating a partition for N = 1000 & S = 10: < 10-17
New Oranges: Seconds to generate a partition for N = 1000 & S = 10: 0.07
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Integer partitions
S positive integers that sum to N without respect to order
What if a distribution has zeros?• subplots with 0 individuals• people with 0 income • publications with 0 citations
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Abundance class
freq
uenc
y
0 1 2 3 4 5
Intraspecific spatial abundance distribution (SSAD)N = abundance of a species
S = number of subplots
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Intraspecific spatial abundance distribution (SSAD)
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Public code repository
https://github.com/klocey/partitions
PeerJ Preprint
https://peerj.com/preprints/78/
Locey KJ, McGlinn DJ. (2013) Efficient algorithms for sampling feasible sets of macroecological patterns. PeerJ PrePrints 1:e78v1
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Future Directions in Combinatorial Feasible Sets
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Future Directions: metrics of Evenness, diversity, & inequality
freq
uenc
y
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Future Directions: metrics of Evenness, diversity, & inequality
freq
uenc
y
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Future Directions: metrics of Evenness, diversity, & inequality
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Per
cent
ile in
feas
ible
set
Gini’s coefficient of inequality
Future Directions: metrics of Evenness, diversity, & inequality
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integer composition: all ordered ways that S positive integers can sum to N
Future Directions: New combinatorial feasible sets
6 = 3+2+1 = 1+2+3 = 3+1+2
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Future Directions: New combinatorial feasible sets
Rank
log
abun
danc
e
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Future Directions: New combinatorial feasible sets
Rank
log
abun
danc
e
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Future Directions: New combinatorial feasible sets
Rank
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Pragmatic: explanations & predictions using few inputs
Mathematical: combinatorics can be used to characterize and understand observable variation in nature
System specific: patterns attributed to specific processes are constrained by general variables. What drives the values of the variables?
Policy, management, & philosophy:Would you want to know if the most costly, likely, preferred outcome was 95% similar to 95% of all others? Why?
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http://figshare.com/articles/Combinatorial_insight_into_distributions_of_wealth_size_and_abundance/866822