Statistical inference for random processes in epidemiology and ecology
29th April 2010
10:00 - 12:00 Ecology
Paul Blackwell
Large-scale spatial modelling and prediction of the
presence/absence of Montastraea coral
Matthew Spencer
Parameter estimation for cellular automata models
Kamila Zychaluk
Semi-parametric models for evolution of coral reef
2:00 - 4:00 Epidemiology
Philip O'Neill
Inference for epidemics with three levels of mixing
Damian Clancy
Bayesian estimation of the basic reproduction number in
stochastic epidemic models
Discussion
University of Liverpool, Mathematical Sciences Building, Room G16
Each talk will last approximately 30 minutes. All welcome.
Supported by the University of Liverpool Research Centre in Mathematics and Modelling
Contact Information
If you have any questions please write to Kamila Zychaluk e-mail: zychaluk@liverpool.ac.uk
or Damian Clancy dclancy@liverpool.ac.uk
Abstracts
Large-scale spatial modelling and prediction of the
presence/absence of Montastraea coral
Paul Blackwell
Often marine biologists may be interested in mapping coral, or at least
estimating whether it is present or absent at a fine scale, over a
larger region than is feasible by direct observation. One approach is to
try to model this presence/absence statistically, based on covariates
that can be observed or derived relatively cheaply. In doing so, there
is strong spatial autocorrelation that is not accounted for by the
covariates, even though they themselves are strongly autocorrelated.
I will look at some modelling based on a set of sites where detailed
presence/absence data for Montastraea coral ARE available, along with
observations of depth and various covariates such as exposure to wave
damage which can be constructed from information that is relatively easy
to obtain. I will consider some possible approaches to implementing
spatial models in this setting, bearing in mind the large sample sizes,
the strong autocorrelation and the desire to make predictions given
partial information on coral presence.
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Parameter estimation for cellular automata models
Matthew Spencer
Cellular automata are a popular way of incorporating space into ecological models.
Space is represented by a set of cells, each of which has one of a finite set of states. Typically, each cell is a unit of
space, and the state of the cell is the organism occupying that cell. Time may be treated either as discrete or continuous.
In discrete-time cellular automata, the state of every cell is updated simultaneously at each time step according to a set
of deterministic or probabilistic rules, which generally depend on the states of the cell and its immediate neighbours.
In continuous-time cellular automata, the state of one cell at a time is updated, typically using a set of probabilistic
rules to generate the distribution of waiting times until the next state change in each cell.
Probabilistic cellular automata are often used as fairly abstract models of ecological systems.
However, there have been only a few attempts to estimate their parameters from real data on the pattern
of states observed at a set of time points. A probabilistic cellular automata model corresponds to a Markov
chain with a very large number of states. For discrete-time cellular automata, parameter estimation is straightforward,
although the number of parameters may be large. For continuous-time cellular automata, the problem is much more difficult.
Although simulation is straightforward, the number of states is usually too large for simulation-based approaches such
as importance sampling and Markov Chain Monte Carlo to perform well. The most straightforward method may be to discard
the full spatial information and attempt to find parameters that reproduce the frequency distribution of state pairs in
adjacent cells. This has two additional benefits: we can obtain the frequencies of state pairs using a pair approximation
instead of simulation; and we can apply our method to cases where we cannot relocate points in space at each sampling time.
We describe the potential application of our method to photographic surveys of coral reefs in Thailand.
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Semi-parametric models for
coral reef dynamics
Kamila Zychaluk
There are many mathematical models for the dynamics of coral reefs.
Typically, these models assume the functional relationships that are
responsible for changes in the reef community but there is often little
evidence on which to choose the functional relationships. Furthermore, the
parameters of such models are difficult to estimate. Instead, we propose a
statistical model based on many data but relatively few assumptions. We use
a large database of repeated observations of the composition of coral
communities to make predictions about the dynamics of reef composition. It
has been suggested that there may be a regional dynamic equilibrium in reef
composition. We use our model to estimate this equilibrium.
We have observations of the proportion of space occupied by three components
(hard corals, macroalgae, and others). These observations were made in
consecutive years for periods between 2 and 11 years, at 69 Caribbean and 55
Great Barrier Reef sites. We assume that the state of the reef after one
year follows a Dirichlet distribution with parameters dependent on the
current state of the reef. These parameters are estimated using a local
linear estimator with cross-validation bandwidth. These estimates are then
used in a transition equation to obtain the stationary distribution of reef
composition. The stationary distributions for the Caribbean and Great
Barrier reef appear very different, in accordance with biological knowledge.
These stationary distributions correspond to the dynamic equilibria for the
two regions, if conditions remain as they are now. In addition to making
predictions, our semi-parametric models provide a summary of the major
features of reef dynamics, which more mechanistic models should be able to
reproduce
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Inference for epidemics with three levels of
mixing
Philip O'Neill
When seeking to contain on-going epidemics, it is often of practical importance to understand the transmission potential given by different types of social mixing (e.g. via household members, via the community at large, via workplaces etc). In practice, intermediate-level
mixing (e.g. schools, workplaces) is the least well-understood, and yet is important for mitigation strategies (e.g. school closures). We explore the feasibility of estimating transmission parameters given different kinds
of data using a model with three levels of mixing. Our findings are illustrated by reference to data from a measles outbreak.
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Bayesian estimation of the basic reproduction number in
stochastic epidemic models
Damian Clancy
In Bayesian inference for epidemic models, it has become commonplace to treat unobserved data (such as times of infection) as extra parameters to be estimated, typically using MCMC. Instead of this, we derive bounds (in the sense of stochastic ordering) on the posterior distribution of the basic reproduction number R0, allowing the unobserved infection times to vary across their whole feasible region. Using linear programming, we can find such bounds quickly and easily, providing a diagnostic check on MCMC results. We apply the method to some epidemic data from the literature.
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