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Graphical Models and
Pollination
- Ayesha Ali
University of Guelph
With: Tom Woodcock,
Liam Callaghan,
Catherine Crea.
TIES 2010
June 23, 1010
Ceratina on Dianthus flower
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
Pollination Webs: bi-partite graph
Pollinators (Insects)
Pollinated (Plants)
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 type
 Driving 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
α
p
θ
Z
For each specie:
X
M
na
 X | z,p ~ Multin(pz)
 Z | θ ~ Bern(θ)
 θ ~ Beta()
 Z is an unobserved random variable that is
1 if pollination visit, 0 otherwise
 pafz = Pr(insect a visits plant f | visit type z)
Hierarchical Pollination Model I
i

M
1


i
i
L     P( a | a,  a )   P( x f | z , pz ) P( z | a) 
f 1  z 0

 A i 1 

n
f




 ia
N
 d
a
a 1
 Free energy maximization (Neal and Hinton)
 E-step: compute
N
na


F ( ,  ,  , p)   Eq log P( , z, x |  , p)  H q( , z |  a ,  a )
a 1 i 1
 M-step: maximize free energy wrt variational and
model parameters (fixed-point iteration or NewtonRaphson)

Hierarchical Pollination Model II
 Borrow from econometrics choice models:
 Consumers assign a utility to each of M items
Uifa   w fa  fa   ifa
T
 Conditional on the total number of counts,
X ~ Multinomial(p)
exp(  w fa   fa )
T
p fa 

M
f 1
exp(  w fa   fa )
T

 fa exp( fa )

M
f 1
 fa exp( fa )
Hierarchical Pollination Model II
δ
β
η
For each specie a:
p
 X | p ~ Multin(p)
 exp(ηjg)| δa ~
X
M
w
Gamma(δa-1λfa, δa-1)
na  p ~ Dirichlet(δ -1λ )
a
a
 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 nonstochastic 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” zerointeractions
 Modify error distribution for utilities in order
to study competition between insects
THANKS!




CANPOLIN
Tom Woodcock
Elizabeth Elle
Peter Kevan
Syrphidae Pt Pelee