This document summarizes Colin Gillespie's presentation on using Bayesian inference methods and moment closure approximations to model stochastic population dynamics, with a case study on cotton aphid data. The presentation covers the cotton aphid data set, deterministic and stochastic models for population growth, simulating the stochastic model using the Gillespie algorithm, estimating model parameters using moment closure approximations, and fitting the model to real cotton aphid count data.
Bayesian inference for stochastic population models with application to aphids
1. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Bayesian inference for stochastic population
models with application to aphids
Colin Gillespie
Joint work with
Andrew Golightly
School of Mathematics & Statistics, Newcastle University
December 2, 2009
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
2. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Talk Outline
Cotton aphid data set
Deterministic & stochastic models
Moment closure
Parameter estimation
Simulation study
Real data
Conclusion
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
3. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Cotton Aphids
Aphid infestation
A cotton aphid infestation of a cotton plant can result in:
leaves that curl and pucker
seedling plants become stunted and may die
a late season infestation can result in stained cotton
cotton aphids have developed resistance to many chemical
treatments and so can be difficult to treat
Basically it costs someone a lot of money
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
4. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Cotton Aphids
Aphid infestation
A cotton aphid infestation of a cotton plant can result in:
leaves that curl and pucker
seedling plants become stunted and may die
a late season infestation can result in stained cotton
cotton aphids have developed resistance to many chemical
treatments and so can be difficult to treat
Basically it costs someone a lot of money
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
5. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Cotton Aphids
The data consists of
five observations at each plot;
the sampling times are t=0, 1.14, 2.29, 3.57 and 4.57
weeks (i.e. every 7 to 8 days);
three blocks, each being in a distinct area;
three irrigation treatments (low, medium and high);
three nitrogen levels (blanket, variable and none);
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
6. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
The Data
2004 Cotton Aphid data set
0 1 2 3 4
Nitrogen (Z) Nitrogen (Z) Nitrogen (Z)
Water (H) Water (L) Water (M)
2500
2000
1500
q
1000 q q
q
500 q q q
q q
0 q q q q q q
Nitrogen (V) Nitrogen (V) Nitrogen (V)
Water (H) Water (L) Water (M)
Aphid Population
2500
2000
1500
q q q 1000
q q q 500
q q q q
q q q q q 0
Nitrogen (B) Nitrogen (B) Nitrogen (B)
Water (H) Water (L) Water (M)
2500
2000
1500 q q
1000 q
500 q q
q q q
q q q q q q q
0
0 1 2 3 4 0 1 2 3 4
Time
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
7. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Cotton Aphid data set
The Data
0 1 2 3 4
Nitrogen (Z) Nitrogen (Z)
Water (H) Water (L)
2500
2000
1500
q
1000 q
q
500 q q
q
0 q q q q q
Nitrogen (V) Nitrogen (V)
Water (H) Water (L)
q q
q q q
q q
q q q q
Nitrogen (B)
Colin Gillespie — Nottingham 2009 Nitrogen (B)
Bayesian inference for stochastic population models
8. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Some Notation
Let
n(t) to be the size of the aphid population at time t
c(t) to be the cumulative aphid population at time t
1 We observe n(t) at discrete time points
2 We don’t observe c(t)
3 c(t) ≥ n(t)
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
9. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
The Model
We assume, based on previous modelling (Matis et al., 2004)
an aphid birth rate of λn(t)
an aphid death rate of µn(t)c(t)
So extinction is certain, as eventually µnc > λn for large t
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
10. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
The Model
Deterministic Representation
Previous modelling efforts have focused on deterministic
models:
dn(t)
= λn(t) − µc(t)n(t)
dt
dc(t)
= λn(t)
dt
Some Problems
Initial and final aphid populations are quite small
No allowance for ‘natural’ random variation
Solution: use a stochastic model
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
11. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
The Model
Deterministic Representation
Previous modelling efforts have focused on deterministic
models:
dn(t)
= λn(t) − µc(t)n(t)
dt
dc(t)
= λn(t)
dt
Some Problems
Initial and final aphid populations are quite small
No allowance for ‘natural’ random variation
Solution: use a stochastic model
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
12. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
The Model
Stochastic Representation
Let pn,c (t) denote the probability:
there are n aphids in the population at time t
a cumulative population size of c at time t
This gives the forward Kolmogorov equation
dpn,c (t)
= λ(n − 1)pn−1,c−1 (t) + µc(n + 1)pn+1,c (t)
dt
− n(λ + µc)pn,c (t)
Even though this equation is fairly simple, it still can’t be
solved exactly.
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
13. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Stochastic Simulation:
Kendall, 1950 or the ‘Gillespie’ Algorithm
1 Initialise system;
2 Calculate rate = λn + µnc;
3 Time to next event: t ∼ Exp(rate);
4 Choose a birth or death event proportional to the rate;
5 Update n, c & time;
6 If time > maxtime stop, else go to 2.
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
14. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
The Model
Some simulations - Deterministic solution
1000
750
Aphid pop.
500
250
0
0 5 10
Time (days)
Parameters: n(0) = c(0) = 1, λ = 1.7 and µ = 0.001
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
15. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
The Model
Some simulations - Stochastic realisations
1000
750
Aphid pop.
500
250
0
0 5 10
Time (days)
Parameters: n(0) = c(0) = 1, λ = 1.7 and µ = 0.001
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
16. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
The Model
Some simulations - Stochastic realisations
1000
750
Aphid pop.
500
250
0
0 5 10
Time (days)
Parameters: n(0) = c(0) = 1, λ = 1.7 and µ = 0.001
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
17. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
The Model
Some simulations - 90% IQR Range
1000
750
Aphid pop.
500
250
0
0 5 10
Time (days)
Parameters: n(0) = c(0) = 1, λ = 1.7 and µ = 0.001
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
18. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Stochastic Parameter Estimation
Let X(tu ) = (n(tu ), c(tu )) be the vector of observed aphid
counts and unobserved cumulative population size at time
tu ;
To infer λ and µ, we need to estimate
Pr[X(tu )| X(tu−1 ), λ, µ]
i.e. the solution of the forward Kolmogorov equation
We will use moment closure to estimate this distribution
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
19. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Stochastic Parameter Estimation
Let X(tu ) = (n(tu ), c(tu )) be the vector of observed aphid
counts and unobserved cumulative population size at time
tu ;
To infer λ and µ, we need to estimate
Pr[X(tu )| X(tu−1 ), λ, µ]
i.e. the solution of the forward Kolmogorov equation
We will use moment closure to estimate this distribution
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
20. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Moment Closure
The bivariate moment generating function is defined as:
∞
M(θ, φ; t) ≡ enθ ecφ pn,c (t)
n,c=0
The associated cumulant generating function is:
∞
θ n φc
K (θ, φ; t) ≡ log[M(θ, φ; t)] = κnc (t)
n! c!
n,c=0
For the first few moments, cumulants are convenient:
κ10 and κ01 are the marginal means of n(t) and c(t)
{κ20 , κ02 , κ11 } are the marginal variances and covariances,
respectively.
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
21. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Moment Closure
The bivariate moment generating function is defined as:
∞
M(θ, φ; t) ≡ enθ ecφ pn,c (t)
n,c=0
The associated cumulant generating function is:
∞
θ n φc
K (θ, φ; t) ≡ log[M(θ, φ; t)] = κnc (t)
n! c!
n,c=0
For the first few moments, cumulants are convenient:
κ10 and κ01 are the marginal means of n(t) and c(t)
{κ20 , κ02 , κ11 } are the marginal variances and covariances,
respectively.
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
22. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Moment Closure
On multiplying the forward Kolmogorov equation by enθ ecφ
and summing over {n, c}, we get
∂K ∂K ∂2K ∂K ∂K
= λ(eθ+φ − 1) + µ(e−θ − 1) +
∂t ∂θ ∂θ∂φ ∂θ ∂φ
Differentiating wrt to θ, and setting θ = φ = 0 gives an ODE
for κ10
Differentiating wrt to φ and setting θ = φ = 0 gives an ODE
for κ01
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
23. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Moment Closure
On multiplying the forward Kolmogorov equation by enθ ecφ
and summing over {n, c}, we get
∂K ∂K ∂2K ∂K ∂K
= λ(eθ+φ − 1) + µ(e−θ − 1) +
∂t ∂θ ∂θ∂φ ∂θ ∂φ
Differentiating wrt to θ, and setting θ = φ = 0 gives an ODE
for κ10
Differentiating wrt to φ and setting θ = φ = 0 gives an ODE
for κ01
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
24. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Moment Equations for the Means
dκ10
= λκ10 − µ(κ10 κ01 + κ11 )
dt
dκ01
= λκ10
dt
The equation for the κ10 depends on the
κ11 = Cov(n(t), c(t))
remember that κ10 = E[n(t)]
Setting κ11 =0 gives the deterministic model
We can think of the deterministic version as a ‘first order’
approximation
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
25. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Moment Equations for the Means
dκ10
= λκ10 − µ(κ10 κ01 + κ11 )
dt
dκ01
= λκ10
dt
The equation for the κ10 depends on the
κ11 = Cov(n(t), c(t))
remember that κ10 = E[n(t)]
Setting κ11 =0 gives the deterministic model
We can think of the deterministic version as a ‘first order’
approximation
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
26. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Second Order Moment Equations
dκ20
= µ(κ11 − 2κ10 κ11 − 2κ21 + κ01 (κ10 − 2κ20 ))
dt
+ λ(κ10 + 2κ20 )
dκ11
= λ(κ10 + κ20 + κ11 ) − µ(κ10 κ02 + κ01 κ11 + κ12 )
dt
dκ02
= λ(κ10 + 2κ11 ) .
dt
In turn, the covariance ODE contains higher order terms
In general the i th equation depends on the (i + 1)th equation
To circumvent this dependency problem, we need to close
the equations
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
27. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Second Order Moment Equations
dκ20
= µ(κ11 − 2κ10 κ11 − 2κ21 + κ01 (κ10 − 2κ20 ))
dt
+ λ(κ10 + 2κ20 )
dκ11
= λ(κ10 + κ20 + κ11 ) − µ(κ10 κ02 + κ01 κ11 + κ12 )
dt
dκ02
= λ(κ10 + 2κ11 ) .
dt
In turn, the covariance ODE contains higher order terms
In general the i th equation depends on the (i + 1)th equation
To circumvent this dependency problem, we need to close
the equations
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
28. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Closing the Moment Equations
The easiest option is to assume an underlying Normal
distribution, i.e. κi = 0 for i > 2
But we could also use the Poisson distribution
κi = κi−1
or the Lognormal
3
3 E[X 2 ]
E[X ] =
E[X ]
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
29. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Comments on the Moment Closure Approximation
For this model:
the means and variances are estimated with an error rate
less than 2.5%
Solving five ODEs is much faster than multiple simulations
In general,
the approximation works well when the stochastic mean
and deterministic solutions are similar
the approximation usually breaks in an obvious manner, i.e.
negative variances
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
30. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Comments on the Moment Closure Approximation
For this model:
the means and variances are estimated with an error rate
less than 2.5%
Solving five ODEs is much faster than multiple simulations
In general,
the approximation works well when the stochastic mean
and deterministic solutions are similar
the approximation usually breaks in an obvious manner, i.e.
negative variances
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
31. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Parameter Inference
Given
the parameters: {λ, µ}
the initial states: X(tu−1 ) = (n(tu−1 ), c(tu−1 ));
We have
X(tu ) | X(tu−1 ), λ, µ ∼ N(ψu−1 , Σu−1 )
where ψu−1 and Σu−1 are calculated using the moment closure
approximation
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
32. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Parameter Inference
Summarising our beliefs about {λ, µ} and the unobserved
cumulative population c(t0 ) via priors p(λ, µ) and p(c(t0 ))
The joint posterior for parameters and unobserved states
(for a single data set) is
4
p (λ, µ, c | n) ∝ p(λ, µ) p (c(t0 )) p (x(tu ) | x(tu−1 ), λ, µ)
u=1
For the results shown, we used a simple random walk MH
step to explore the parameter and state spaces
We did investigate more sophisticated schemes, but the
mixing properties were similar
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
33. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Parameter Inference
Summarising our beliefs about {λ, µ} and the unobserved
cumulative population c(t0 ) via priors p(λ, µ) and p(c(t0 ))
The joint posterior for parameters and unobserved states
(for a single data set) is
4
p (λ, µ, c | n) ∝ p(λ, µ) p (c(t0 )) p (x(tu ) | x(tu−1 ), λ, µ)
u=1
For the results shown, we used a simple random walk MH
step to explore the parameter and state spaces
We did investigate more sophisticated schemes, but the
mixing properties were similar
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
34. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Simulation Study
Three treatments & two blocks
Baseline birth and death rates: {λ = 1.75, µ = 0.00095}
Treatment 2 increases µ by 0.0004
Treatment 3 increases λ by 0.35
The block effect reduces µ by 0.0003
Treatment 1 Treatment 2 Treatment 3
Block 1 {1.75, 0.00095} {1.75, 0.00135} {2.1, 0.00095}
Block 2 {1.75, 0.00065} {1.75, 0.00105} {2.1, 0.00065}
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
35. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Simulation Study
Three treatments & two blocks
Baseline birth and death rates: {λ = 1.75, µ = 0.00095}
Treatment 2 increases µ by 0.0004
Treatment 3 increases λ by 0.35
The block effect reduces µ by 0.0003
Treatment 1 Treatment 2 Treatment 3
Block 1 {1.75, 0.00095} {1.75, 0.00135} {2.1, 0.00095}
Block 2 {1.75, 0.00065} {1.75, 0.00105} {2.1, 0.00065}
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
36. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Simulation Study
Three treatments & two blocks
Baseline birth and death rates: {λ = 1.75, µ = 0.00095}
Treatment 2 increases µ by 0.0004
Treatment 3 increases λ by 0.35
The block effect reduces µ by 0.0003
Treatment 1 Treatment 2 Treatment 3
Block 1 {1.75, 0.00095} {1.75, 0.00135} {2.1, 0.00095}
Block 2 {1.75, 0.00065} {1.75, 0.00105} {2.1, 0.00065}
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
37. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Simulation Study
Three treatments & two blocks
Baseline birth and death rates: {λ = 1.75, µ = 0.00095}
Treatment 2 increases µ by 0.0004
Treatment 3 increases λ by 0.35
The block effect reduces µ by 0.0003
Treatment 1 Treatment 2 Treatment 3
Block 1 {1.75, 0.00095} {1.75, 0.00135} {2.1, 0.00095}
Block 2 {1.75, 0.00065} {1.75, 0.00105} {2.1, 0.00065}
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
38. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Simulated Data
0 1 2 3 4
Treament 1 Treatment 2 Treatment 3
q
Aphid Population
1000
q
q q
500
q
q q
q
q
q q
q q q
0 q
0 1 2 3 4 0 1 2 3 4
Time
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
39. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Parameter Structure
Let i, k represent the block and treatments level, i ∈ {1, 2}
and k ∈ {1, 2, 3}
For each dataset, we assume birth rates of the form:
λik = λ + αi + βk
where α1 = β1 = 0
So for block 1, treatment 1 we have:
λ11 = λ
and for block 2, treatment 1 we have:
λ21 = λ + α2
A similar structure is used for the death rate:
µik = µ + αi∗ + βk
∗
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
40. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Parameter Structure
Let i, k represent the block and treatments level, i ∈ {1, 2}
and k ∈ {1, 2, 3}
For each dataset, we assume birth rates of the form:
λik = λ + αi + βk
where α1 = β1 = 0
So for block 1, treatment 1 we have:
λ11 = λ
and for block 2, treatment 1 we have:
λ21 = λ + α2
A similar structure is used for the death rate:
µik = µ + αi∗ + βk
∗
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
41. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
MCMC Scheme
Using the MCMC scheme described previously, we
generated 2M iterates and thinned by 1K
This took a few hours and convergence was fairly quick
We used independent proper uniform priors for the
parameters
For the initial unobserved cumulative population, we had
c(t0 ) = n(t0 ) +
where has a Gamma distribution with shape 1 and scale
10.
This set up mirrors the scheme that we used for the real
data set
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
42. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Marginal posterior distributions for λ and µ
20000
6
15000
Density
Density
4
10000
2
5000
0
X 0
X
1.6 1.7 1.8 1.9 2.0 0.00090 0.00095 0.00100
Birth Rate Death Rate
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
43. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
MCMC Scheme
Marginal posterior distributions for λ
−0.2 0.0 0.2 0.4
Block 2 Treatment 2 Treatment 3
6
Density
4
2
0 X X X
−0.2 0.0 0.2 0.4 −0.2 0.0 0.2 0.4
Birth Rate
We obtained similar densities for the death rates.
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
44. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Application to the Cotton Aphid Data Set
Recall that the data consists of
five observations on twenty randomly chosen leaves in
each plot;
three blocks, each being in a distinct area;
three irrigation treatments (low, medium and high);
three nitrogen levels (blanket, variable and none);
the sampling times are t=0, 1.14, 2.29, 3.57 and 4.57
weeks (i.e. every 7 to 8 days).
Following in the same vein as the simulated data, we are
estimating 38 parameters (including interaction terms) and the
latent cumulative aphid population.
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
45. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Cotton Aphid Data
Marginal posterior distributions for λ and µ
6
15000
Density
Density
4
10000
2 5000
0 0
1.6 1.7 1.8 1.9 2.0 0.00090 0.00095 0.00100
Birth Rate Death Rate
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
46. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Does the Model Fit the Data?
We simulate predictive distributions from the MCMC
output, i.e. we randomly sample parameter values (λ, µ)
and the unobserved state c and simulate forward
We simulate forward using the Gillespie simulator
not the moment closure approximation
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
47. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Does the Model Fit the data?
Predictive distributions for 6 of the 27 Aphid data sets
D 123 D 121 D131
2500
2000
1500
X q
q
q X 1000
q
q
X q
q
q q
Aphid Population
q q
q
q q
q
q q 500
X
q q
q
X
q q q
q
q X
q
q
q
q X X
q
q
q
X
q X q
q
q
X X 0
q
D 112 D 122 D 113
q
q
X
2500
q
q
2000
1500 q
X
q
q q
q
1000
q
q q
q
X q
q X
q
q
q
q
q
q qq
500 X q
X q
q
X q
q
q q
q
X
q
q
q
X X q
X X
q
0
q
1.14 2.29 3.57 4.57 1.14 2.29 3.57 4.57 1.14 2.29 3.57 4.57
Time
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
48. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Summarising the Results
Consider the additional number of aphids per treatment
combination
Set c(0) = n(0) = 1 and tmax = 6
We now calculate the number of aphids we would see for
each parameter combination in addition to the baseline
For example, the effect due to medium water:
∗
λ211 = λ + αWater (M) and µ211 = µ + αWater (M)
So
i i
Additional aphids = cWater (M) − cbaseline
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
49. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Aphids over Baseline
Main Effects
0 2000 6000 10000
Nitrogen (V) Water (H) Water (M)
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
Density
Block 3 Block 2 Nitrogen (Z)
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
0 2000 6000 10000 0 2000 6000 10000
Aphids
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
50. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Aphids over Baseline
Interactions
0 2000 6000 10000 0 2000 6000 10000
W(H) N(Z) W(M) N(Z) W(H) N(V) W(M) N(V)
0.003
0.002
0.001
0.000
B3 W(H) B2 W(H) B3 W(M) B2 W(M)
0.003
Density
0.002
0.001
0.000
B3 N(Z) B2 N(Z) B3 N(V) B2 N(V)
0.003
0.002
0.001
0.000
0 2000 6000 10000 0 2000 6000 10000
Aphids
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
51. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Conclusions
The 95% credible intervals for the baseline birth and death
rates are (1.64, 1.86) and (0.000904, 0.000987).
Main effects have little effect by themselves
However block 2 appears to have a very strong interaction
with nitrogen
Moment closure parameter inference is a very useful
technique for estimating parameters in stochastic
population models
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
52. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Future Work
Other data sets suggest that there is aphid immigration in
the early stages
Model selection for stochastic models
Incorporate measurement error
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models
53. Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion
Acknowledgements
Andrew Golightly Richard Boys
Peter Milner
Darren Wilkinson Jim Matis (Texas A & M)
References
Gillespie, C. S., Golightly, A. Bayesian inference for generalized
stochastic population growth models with application to aphids,
Journal of the Royal Statistical Society, Series C, 2010.
Gillespie, C.S. Moment closure approximations for mass-action
models. IET Systems Biology 2009.
Milner, P., Gillespie, C. S., Wilkinson, D. J. Parameter estimation
via moment closure stochastic models, in preparation.
Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models