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

aif

i N

aa

n

i

M

f za

iz

ifaa dzPpzxPaPL

11 1

1

0

)|(),|(),|(

N

a

n

i

aaq

a

zqHpxzPEpF1 1

),|,(),|,,(log),,,(

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(

)exp(

)exp(

w

w

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