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Title: Recent Advances in Statistical Ecology using Computationally Intensive Methods


1
Recent Advances in Statistical Ecology using
Computationally Intensive Methods
  • Ruth King

2
Overview
  • Introduction to wildlife populations and
    identification of questions of interest.
  • Motivating example.
  • Issues to be addressed
  • Missing data
  • Model discrimination.
  • Summary.
  • Future research.

3
Wildlife Populations
  • In recent years there has been increasing
    interest in wildlife populations.
  • Often we may be interested in population changes
    over time, e.g. if there is a declining
    population. (Steve Buckland)
  • Alternatively, we may be interested in the
    underlying dynamics of the system, in order to
    obtain a better understanding of the population.
  • We shall concentrate on this latter problem, with
    particular focus on identifying factors that
    affect demographic rates.

4
Data Collection
  • Data are often collected via some form of
    capture-recapture study.
  • Observers go out into the field and record all
    animals that are seen at a series of capture
    events.
  • Animals may be recorded via simply resightings or
    recaptures (of live animals) and recoveries (of
    dead animals).
  • At each capture event all unmarked animals are
    uniquely marked all observed animals are
    recorded and subsequently released back into the
    population.

5
Data
  • Each animal is uniquely identifiable so our data
    consist of the capture histories for each
    individual observed in the study.
  • A typical capture history may look like
  • 0 1 1 0 0 1 2
  • 0/1 corresponds to the individual being
    unobserved/observed at that capture time and
  • 2 denotes an individual is recovered dead.
  • We can then explicitly write down the
    corresponding likelihood as a function of
    survival (?), recapture (p) and recovery (?)
    rates.

6
Likelihood
  • The likelihood is the product over all
    individuals of the probability of their
    corresponding capture history, conditional on
    their initial capture.
  • For example, for an individual with history
  • 0 1 1 0 0 1 2
  • the contribution to the likelihood is
  • ?2 p3 ?3 (1-p4) ?4 (1-p5) ?5 p6 (1-?6) ?7.
  • Then, we can use this likelihood to estimate the
    parameter values (either MLEs or posterior
    distributions).

7
Covariates
  • Covariates are often used to explain temporal
    heterogeneity within the parameters.
  • Typically these are environmental factors, such
    as resource availability, weather conditions or
    human intervention.
  • Alternatively, heterogeneity in a population can
    often be explained via different (individual)
    covariates.
  • For example, sex, condition or breeding status.
  • We shall consider the survival rates to be
    possibly dependent on the different covariates.

8
Case Study Soay sheep
  • We consider mark-recapture-recovery (MRR) data
    relating to Soay sheep on the island of Hirta.
  • The sheep are free from human activity and
    external competition/predation.
  • Thus, this population is ideal for investigating
    the impact of different environmental and/or
    individual factors on the sheep.
  • We consider annual data from 1986-2000, collected
    on female sheep (1079 individuals).

This is joint work with Steve Brooks and Tim
Coulson
9
Covariate Information
  • Individual covariates
  • Coat type (1dark, 2light)
  • Horn type (1polled, 2scurred, 3classical)
  • Birth weight (real normalised)
  • Age (in years)
  • Weight (real - normalised)
  • Number of lambs born to the sheep in the spring
    prior to summer census (0, 1, 2)
  • And in the spring following the census (0, 1, 2).
  • Environmental covariates
  • NAO index population size March rainfall
    Autumn rainfall and March temperature.

10
Survival Rates - ?
  • Let the set of environmental covariate values at
    time t be denoted by xt.
  • The survival rate for animal i, of age a, at time
    t, is given by,
  • logit ?i,a,t ?a ?aT xt ?aT yi ?aT zi,t
    ?a,t
  • Here, yi denotes the set of time-independent
    covariates and zi,t the time varying covariates
  • ?a,t N(0,?a2) denotes a random effect.
  • There arise two issues here missing covariate
    values and model choice.

11
Issue 1 Missing Data
  • The capture histories and time-invariant
    covariate values data are presented in the form
  • Note that there are also many missing values for
    the time-dependent weight covariate.

12
Problem
  • Given the set of covariate values, the
    corresponding survival rate can be obtained, and
    hence the likelihood calculated.
  • However, a complexity arises when there are
    unknown (i.e. missing) covariate values, removing
    the simple and explicit expression for the
    likelihood.

13
Missing Data Classical Approaches
  • Typical classical approaches to missing data
    problems include
  • Ignoring individuals with missing covariate
    values
  • EM algorithm (can be difficult to implement and
    computationally expensive)
  • Imputation of missing values using some
    underlying model (e.g. Gompertz curve).
  • Conditional approach for time-varying covariates
    (Catchpole, Morgan and Tavecchia, 2006 in
    submission)
  • We consider a Bayesian approach, where we assume
    an underlying model for the missing data which
    allows us to account for the corresponding
    uncertainty of the missing values.

14
Bayesian Approach
  • Suppose that we wish to make inference on the
    parameters ?, and the data observed corresponds
    to capture histories, c, and covariate values,
    vobs.
  • Then, Bayes theorem states
  • ?(?c, vobs) Ç L(c, vobs ?) p(?)
  • The posterior distribution is very complex and so
    we use Markov chain Monte Carlo (MCMC) to obtain
    estimates of the posterior statistics of
    interest.
  • However, in our case the likelihood is
    analytically intractable, due to the missing
    covariate values.

15
Auxiliary Variables
  • We treat the missing covariate values (vmis) as
    parameters or auxiliary variables (AVs).
  • We then form the joint posterior distribution
    over the parameters, ?, and AVs, given the
    capture histories c and observed covariate values
    yobs
  • ?(?, vmis c, vobs) Ç L(c ?, vmis , vobs)
  • f(vmis, vobs ?) p(?)
  • We can now sample from the joint posterior
    distribution ?(?, vmis c, vobs).
  • We can integrate out over the missing covariate
    values, vmis, within the MCMC algorithm to obtain
    a sample from ?(? c, vobs).

16
f(vmis, vobs ?) Categorical Data
  • For categorical data (coat type and horn type),
    we assume the following model.
  • Let y1,i denote the horn type of individual i.
    Then, y1,i 2 1,2,3, and we assume that,
  • y1,i Multinomial(1,q),
  • where q q1, q2, q3.
  • Thus, we have additional parameters, q, which can
    be regarded as the underlying probability of each
    horn type.
  • The qs are updated within the MCMC algorithm, as
    well as the y1,is which are unknown.
  • We assume the analogous model for coat type.

17
f(vmis, vobs ?) Continuous Data
  • Let y3,i denote the birth weight of individual i.
  • Then, a possible model is to assume that,
  • y3,i N(?, ?2),
  • where ? and ?2 are to be estimated.
  • For the weight of individual i, aged a at time t,
    denoted by z1,i,t we set,
  • z1,i,t N(z1,i,t-1 ?a ?t, ?w2),
  • where the parameters ?a, ?t and ?w2 are to be
    estimated.
  • In general, the modelling assumptions will depend
    on the system under study.

18
Practical Implications
  • Within each step of the MCMC algorithm, we not
    only need to update the parameter values ?, but
    also impute the missing covariate values and
    random effects (if present).
  • This can be computationally expensive for large
    amounts of missing data.
  • The posterior results may depend on the
    underlying model for the covariates a
    sensitivity analysis can be performed using
    different underlying models.
  • (State-space modelling can also be implemented
    using similar ideas see Steves talk).

19
Issue 2 Model Selection
  • For the sheep data set we can now deal with the
    issue of missing covariate values.
  • But.. how do we decide which covariates to use
    and/or the age structure? often there may be a
    large number of possible covariates and/or age
    structures.
  • Discriminating between different models can often
    be of particular interest, since they represent
    competing biological hypotheses.
  • Model choice can also be important for future
    predictions of the system.

20
Possible Models
  • We only want to consider biologically plausible
    models.
  • We have uncertainty on the age structure of the
    survival rates, and consider models of the form
  • ?1a ?a1b ?j
  • k is unknown a priori and also the covariate
    dependence for each age group.
  • We consider similar age-type models for p and ?
    with possible arbitrary time dependence.
  • E.g. ?1(N,BW), ?27(W,L),?8(N,P)/p(t)/?1,?2(t)
  • The number of possible models is immense!!

21
Bayesian Approach
  • We treat the model itself to be an unknown
    parameter to be estimated.
  • Then, applying Bayes Theorem we obtain the
    posterior distribution over both parameter and
    model space
  • ?(?m, m data) Ç L(data ?m, m) p(?m) p(m).
  • Here ?m denotes the parameters in model m.

22
Reversible Jump MCMC
  • The reversible jump (RJ)MCMC algorithm allows us
    to construct a Markov chain with stationary
    distribution equal to the posterior distribution.
  • It is simply an extension of the
    Metropolis-Hastings algorithm that allows moves
    between different dimensions.
  • This algorithm is needed because the number of
    parameters, ?m, in model m, may differ between
    models.
  • Note that this algorithm needs only one Markov
    chain, regardless of the number of models.

23
Posterior Model Probabilities
  • Posterior model probabilities are able to
    quantitatively compare different models.
  • The posterior probability of model m is defined
    as,
  • ?(m data) s ?(?m,m data) d?m.
  • These are simply estimated within the RJMCMC
    algorithm as the proportion of the time the chain
    is in the given model.
  • We are also able to obtain model-averaged
    estimates of parameters, which takes into account
    both parameter and model uncertainty.

24
General Comments
  • The RJMCMC algorithm is the most widely used
    algorithm to explore and summarise a posterior
    distribution defined jointly over parameter and
    model space.
  • The posterior model probabilities can be
    sensitive to the priors specified on the
    parameters (p(?)).
  • The acceptance probabilities for reversible jump
    moves are typically lower than MH updates.
  • Longer simulations are generally needed to
    explore the posterior distribution.
  • Only a single Markov chain is necessary,
    irrespective of the number of possible models!!

25
Problem 1
  • Constructing efficient RJ moves can be difficult.
  • This is particularly the case when updating the
    number of age groups for the survival rates.
  • This step involves
  • Proposing a new age structure (typically
    add/remove a change-point)
  • Proposing a covariate dependence for the new age
    structure
  • Proposing new parameter values for this new
    model.
  • It can be very difficult to construct the Markov
    chain with (reasonably) high acceptance rates.

26
Example Reversibility Problem
  • One obvious (and tried!) method for adding a
    change-point would be as follows.
  • Suppose that we propose to split age group ac.
  • We propose new age groups ab and b1c.
  • Consider a small perturbation for each (non-zero)
    regression parameter, e.g.,
  • ?ab ?ac ? and ?b1c ?ac - ?.
  • where ? N(0,??2).
  • However, to satisfy the reversibility
    constraints, a change-point can only be removed
    when the covariate dependence structure is the
    same for two consecutive age groups

27
Example High Posterior Mass
  • An alternative proposal would be to set
  • ?ab ?ac (i.e. keep all parameters same for
    ab).
  • Propose a model (in terms of covariates) for
    ?b1c
  • Choose each possible model with equal probability
    (reverse move always possible)
  • Choose model that differs by at most one
    individual and one environmental covariate.
  • The problem now lies in proposing sensible
    parameter values for the new model.
  • One approach is to use posterior estimates of the
    parameters from a saturated model (full
    covariate dependence for some age structure) as
    the mean of the proposal distribtion.

28
Problem 2
  • Consider the missing covariates vmis.
  • Then, if the covariate is present, we have,
  • ?(vmis vobs, ?, data) / L(data ?, vobs,
    vmis)
  • f(vmis, vobs ?).
  • However, if the covariate is not present in the
    model, we have,
  • ?(vmis vobs, ?, data) / f(vmis, vobs ?).
  • Thus, adding (or removing) the covariate values
    may be difficult, due to (potentially) fairly
    different posterior conditional distributions.
  • One way to avoid this is to simultaneously update
    the missing covariate values in the model move.

29
Soay Sheep Analysis
  • We now use these techniques for analysing the
    (complex) Soay sheep data.
  • Discussion with experts identified several points
    of particular interest
  • the age-dependence of the parameters
  • identification of the covariates influencing the
    survival rates
  • whether random effects are present.
  • We focus on these issues on the results presented.

30
Prior Specification
  • We place vague priors on the parameters present
    in each model.
  • Priors also need to be specified on the models.
  • Placing an equal prior on each model places a
    high prior mass on models with a large number of
    age groups, since the number of models increases
    with the number of age groups.
  • Thus, we specify an equal prior probability on
    each marginal age structure and a flat prior over
    the covariate dependence given the age structure
    or on time-dependence.

31
Results Survival Rates
  • The marginal models for the age groups with
    largest posterior support are
  • Note that with probability 1, lambs have a
    distinct survival rate.
  • Often the models with most posterior support are
    close neighbours of each other.

Age-structure Posterior probability
?1 ?27 ?8 0.943
?1 ?27 ?89 ?10 0.052
32
Covariate Dependence
BF 3
BF 20
BF Bayes factor
33
Influence of Covariates
Weight
Population size
Age 1
Age 10
34
Marginalisations
  • Presenting only marginal results can hide some of
    the more intricate details.
  • This is most clearly seen from another MRR data
    analysis relating to the UK lapwing population.
  • There are two covariates time and fdays a
    measure of the harshness of the winter.
  • Without going into too many details, we had MRR
    data and survey data (estimates of total
    population size).
  • An integrated analysis was performed, jointly
    analysing both data sets.

35
Marginal Models
  • The marginal models with most posterior support
    for the demographic parameters are
  • (a) ?1 1st year survival (b) ?a adult
    survival
  • Model Post. prob. Model Post. prob.
  • ?1(fdays) 0.636 ?a(fdays,t) 0.522
  • ?1 0.331 ?a(fdays) 0.407
  • (c) ? productivity (d) l recovery rate
  • Model Post. prob. Model Post. prob.
  • r 0.497 l(t) 0.730
  • r(t) 0.393 l(fdays,t) 0.270

This is joint work with Steve Brooks, Chiara
Mazzetta and Steve Freeman
36
Warning!
  • The previous (marginal) posterior estimates hides
    some of the intricate details.
  • The marginal models of the adult survival rate
    and productivity rate are highly correlated.
  • In particular, the joint posterior probabilities
    for these parameters are
  • Model Post. prob.
  • ?a(fdays,t), ? 0.45
  • ?a(fdays), ?(t) 0.36
  • Thus, there is strong evidence that either adult
    survival rate OR productivity rate is time
    dependent.

37
Summary
  • Bayesian techniques are widely used, and allow
    complex data analyses.
  • Covariates can be used to explain both temporal
    and individual heterogeneity.
  • However, missing values can add another layer of
    complexity and the need to make additional
    assumptions.
  • The RJMCMC algorithm can explore the possible
    models and discriminate between competing
    biological hypotheses.
  • The analysis of the Soay sheep has stimulated new
    discussion with biologists, in terms of the
    factors that affect their survival rates.

38
Further Research
  • The development of efficient and generic RJMCMC
    algorithms.
  • Assessing the posterior sensitivity on the model
    specification for the covariates.
  • Investigation of the missing at random assumption
    for the covariates in both classical and Bayesian
    frameworks.
  • Prediction in the presence of time-varying
    covariate information.
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