Download - Ceratina on Dianthus flower
Ceratina on Dianthus flower
Graphical Models and Pollination
- Ayesha Ali
University of Guelph
With: Tom Woodcock, Liam Callaghan,
Catherine Crea.
TIES 2010 June 23, 1010
Outline Motivation: Pollination Ecology
Qualitative Pollination Webs- Feature Extraction
Quantitative Pollination Webs
- Driving Mechanisms
Hierarchical graphical models
Motivation: Mutualistic relationship
Plants need to be pollinated by birds and insects for reproduction
Offer rewards for being visited, (e.g. pollen, nectar, oil) Halictidae on Queen Anne’s Lace
Motivation: Species decline
Recent years has seen a decline in some insect species (e.g. bees)
Forest fragmentation has led to a decline in some plant species Andrena – native wild bee
Motivation: Species decline Extinction of given plant may adversely
affect survival of given insect, and vice versa (e.g. Mauna Kea silversword )
Need to maintain
species abundance /
diversity in ecosystem
Ans: Pollination webs?
Orthonevra drinking nectar on HopTree
Pollination Webs: bi-partite graph Nodes are plant and insect species
Edges from insects to plants represent plant-insect interaction
Often called “interaction” or “visitation” web
Only small fraction of interactions observed
Similar to food webs, except role of pollinator and pollinated never change
Pollinators (Insects) Pollinated (Plants)
Pollination Webs: bi-partite graph
Pollination ecologist approach
Use adjacency matrix I (N x M)
I AF = 1 if animal A visited flower F
0 otherwise
Given a pollination web, what are the important features that characterize the plant-pollinator interactions?
Pollination Webs
Pollinators (Insects) Pollinated (Plants)
Ecosystem Interventions Can we infer consequence of eco-system
disturbances (eg. removal of a player due to forest fragmentation)?
Which plants or animals are vulnerable to presence of non-natives?
Problem: Quantification of connection strength, and Understanding mechanism behind interactions
Quantified Pollination Webs
Let Xij = frequency of ij-interactions observed
Conditional on the total number of counts,
X ~ Multinomial(p)
Proportions are correlated within insect species
Observed interactions are actually a mixture of pollination visits, and non-pollination visits
Quantified Pollination Webs
We can use graphical models to represent the data generating mechanism
Two main issues: How to incorporate Visit typeDriving force behind interactions?
Use hierarchical graphical model, with probability that an insect-plant pair interact depending on other variables
Hierarchical Pollination Model I Insects visit one of M floral species, with
probability based on the unobserved visit type
Use a variational EM-algorithm to get a generative model of the process, by incorporating the unobserved visit types
Similar idea in AI user rating profile models: Users rate each of M items, based on some
unobserved attitude toward each item
Hierarchical Pollination Model I
XZθ
α
na
For each specie:
X | z,p ~ Multin(pz)
Z | θ ~ Bern(θ) θ ~ Beta()
p
Z is an unobserved random variable that is 1 if pollination visit, 0 otherwise
pafz = Pr(insect a visits plant f | visit type z)
M
Hierarchical Pollination Model I
Free energy maximization (Neal and Hinton) E-step: compute
M-step: maximize free energy wrt variational and model parameters (fixed-point iteration or Newton-Raphson)
A
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i N
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iz
ifaa dzPpzxPaPL
11 1
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N
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zqHpxzPEpF1 1
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Hierarchical Pollination Model II Borrow from econometrics choice models:
Consumers assign a utility to each of M items
Conditional on the total number of counts,
X ~ Multinomial(p)
ifafafaT
ifaU w
M
f fafa
fafa
M
f fafaT
fafaT
fap11
)exp(
)exp(
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w
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Hierarchical Pollination Model II
Xη
δ
na
For each specie a:
X | p ~ Multin(p) exp(ηjg)| δa ~
Gamma(δa-1λfa, δa
-1)
p ~ Dirichlet(δa-1λa)
pM
β
w
p follows a Dirichlet-multinomial regression: Space, time, phenotypic and/or phylogenetic
traits of pollinators or flowers or both
Hierarchical Pollination Model II
Fitting presents no computational issues – Newton-Raphson can converge quickly
Can use existing software to fit model (LIMDEP, Stata, etc.: negative binomial with fixed effects for panel count data)
Vasquez et al. (2009) present a non-stochastic version of this framework
Conclusions Pollination webs can help to understand
insect-floral interactions
Hierarchical models provide a framework for incorporating covariates into the generative model
Provide insights into where conservation efforts should be placed
Future Works
Learn linkage rules: mine bootstrapped samples of data
Overdispersion due to “real” zero-interactions
Modify error distribution for utilities in order to study competition between insects
THANKS!
CANPOLIN Tom Woodcock Elizabeth Elle Peter Kevan
Syrphidae Pt Pelee