Bayesian inference for stochastic population models with application to aphids

Data & Model Moment Closure Model Fitting Simulation Study Cotton Aphids Conclusion

Bayesian inference for stochastic populationmodels with application to aphids

Colin Gillespie

Joint work with

Andrew GolightlySchool of Mathematics & Statistics, Newcastle University

December 2, 2009

Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models


Talk OutlineCotton aphid data setDeterministic & stochastic modelsMoment closureParameter estimation

Simulation studyReal data

Conclusion



Cotton Aphids

Aphid infestationA cotton aphid infestation of a cotton plant can result in:

leaves that curl and puckerseedling plants become stunted and may diea late season infestation can result in stained cottoncotton aphids have developed resistance to many chemicaltreatments and so can be difficult to treatBasically it costs someone a lot of money



Cotton Aphids

Aphid infestationA cotton aphid infestation of a cotton plant can result in:

leaves that curl and puckerseedling plants become stunted and may diea late season infestation can result in stained cottoncotton aphids have developed resistance to many chemicaltreatments and so can be difficult to treatBasically it costs someone a lot of money



Cotton Aphids

The data consists offive observations at each plot;the sampling times are t=0, 1.14, 2.29, 3.57 and 4.57weeks (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);



The Data

2004 Cotton Aphid data set

Time

Aphi

d Po

pula

tion

0500

1000150020002500

0 1 2 3 4

● ●

●

●

●

Water (H)

Nitrogen (B)

● ●●

●

●

Water (L)

Nitrogen (B)

0 1 2 3 4

● ●●

●

●

Water (M)

Nitrogen (B)

● ●●

●

●

Water (H)

Nitrogen (V)

● ●●

●

●

Water (L)

Nitrogen (V)

05001000150020002500

● ●●

●

●

Water (M)

Nitrogen (V)0

5001000150020002500

● ●●

●

●

Water (H)

Nitrogen (Z)

0 1 2 3 4

● ●●

●

●

Water (L)

Nitrogen (Z)

● ●●

●

●

Water (M)

Nitrogen (Z)



The Data2004 Cotton Aphid data set

Time

Aphi

d Po

pula

tion

0500

1000150020002500

0 1 2 3 4

● ●

●

●

●

Water (H)

Nitrogen (B)

● ●●

●

●

Water (L)

Nitrogen (B)

0 1 2 3 4

● ●●

●

●

Water (M)

Nitrogen (B)

● ●●

●

●

Water (H)

Nitrogen (V)

● ●●

●

●

Water (L)

Nitrogen (V)

05001000150020002500

● ●●

●

●

Water (M)

Nitrogen (V)0

5001000150020002500

● ●●

●

●

Water (H)

Nitrogen (Z)

0 1 2 3 4

● ●●

●

●

Water (L)

Nitrogen (Z)

● ●●

●

●

Water (M)

Nitrogen (Z)



Some Notation

Letn(t) to be the size of the aphid population at time tc(t) to be the cumulative aphid population at time t

1 We observe n(t) at discrete time points2 We don’t observe c(t)3 c(t) ≥ n(t)



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



The Model

Deterministic RepresentationPrevious modelling efforts have focused on deterministicmodels:

dn(t)dt

= λn(t)− µc(t)n(t)

dc(t)dt

= λn(t)

Some ProblemsInitial and final aphid populations are quite smallNo allowance for ‘natural’ random variationSolution: use a stochastic model



The Model

Deterministic RepresentationPrevious modelling efforts have focused on deterministicmodels:

dn(t)dt

= λn(t)− µc(t)n(t)

dc(t)dt

= λn(t)

Some ProblemsInitial and final aphid populations are quite smallNo allowance for ‘natural’ random variationSolution: use a stochastic model



The Model

Stochastic Representation

Let pn,c(t) denote the probability:there are n aphids in the population at time ta cumulative population size of c at time tThis gives the forward Kolmogorov equation

dpn,c(t)dt

= λ(n − 1)pn−1,c−1(t) + µc(n + 1)pn+1,c(t)

− n(λ+ µc)pn,c(t)

Even though this equation is fairly simple, it still can’t besolved exactly.



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.



The Model

Some simulations - Deterministic solution

Time (days)

Aph

id p

op.

0

250

500

750

1000

0 5 10

Parameters: n(0) = c(0) = 1, λ = 1.7 and µ = 0.001



The Model

Some simulations - Stochastic realisations

Time (days)

Aph

id p

op.

0

250

500

750

1000

0 5 10




The Model

Some simulations - Stochastic realisations

Time (days)

Aph

id p

op.

0

250

500

750

1000

0 5 10




The Model

Some simulations - 90% IQR Range

Time (days)

Aph

id p

op.

0

250

500

750

1000

0 5 10




Stochastic Parameter Estimation

Let X(tu) = (n(tu), c(tu))′ be the vector of observed aphidcounts and unobserved cumulative population size at timetu;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 equationWe will use moment closure to estimate this distribution



Stochastic Parameter Estimation

Let X(tu) = (n(tu), c(tu))′ be the vector of observed aphidcounts and unobserved cumulative population size at timetu;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 equationWe will use moment closure to estimate this distribution



Moment Closure

The bivariate moment generating function is defined as:

M(θ, φ; t) ≡∞∑

n,c=0

enθecφpn,c(t)

The associated cumulant generating function is:

K (θ, φ; t) ≡ log[M(θ, φ; t)] =∞∑

n,c=0

θn

n!

φc

c!κnc(t)

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.



Moment Closure

The bivariate moment generating function is defined as:

M(θ, φ; t) ≡∞∑

n,c=0

enθecφpn,c(t)

The associated cumulant generating function is:

K (θ, φ; t) ≡ log[M(θ, φ; t)] =∞∑

n,c=0

θn

n!

φc

c!κnc(t)

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.



Moment Closure

On multiplying the forward Kolmogorov equation by enθecφ

and summing over {n, c}, we get

∂K∂t

= λ(eθ+φ − 1)∂K∂θ

+ µ(e−θ − 1)

(∂2K∂θ∂φ

+∂K∂θ

∂K∂φ

)Differentiating wrt to θ, and setting θ = φ = 0 gives an ODEfor κ10

Differentiating wrt to φ and setting θ = φ = 0 gives an ODEfor κ01



Moment Closure

On multiplying the forward Kolmogorov equation by enθecφ

and summing over {n, c}, we get

∂K∂t

= λ(eθ+φ − 1)∂K∂θ

+ µ(e−θ − 1)

(∂2K∂θ∂φ

+∂K∂θ

∂K∂φ

)Differentiating wrt to θ, and setting θ = φ = 0 gives an ODEfor κ10

Differentiating wrt to φ and setting θ = φ = 0 gives an ODEfor κ01



Moment Equations for the Means

dκ10

dt= λκ10 − µ(κ10κ01 + κ11)

dκ01

dt= λκ10

The equation for the κ10 depends on theκ11 = Cov(n(t), c(t))

remember that κ10 = E[n(t)]

Setting κ11=0 gives the deterministic modelWe can think of the deterministic version as a ‘first order’approximation



Moment Equations for the Means

dκ10

dt= λκ10 − µ(κ10κ01 + κ11)

dκ01

dt= λκ10

The equation for the κ10 depends on theκ11 = Cov(n(t), c(t))

remember that κ10 = E[n(t)]

Setting κ11=0 gives the deterministic modelWe can think of the deterministic version as a ‘first order’approximation



Second Order Moment Equations

dκ20

dt= µ(κ11 − 2κ10κ11 − 2κ21 + κ01(κ10 − 2κ20))

+ λ(κ10 + 2κ20)

dκ11

dt= λ(κ10 + κ20 + κ11)− µ(κ10κ02 + κ01κ11 + κ12)

dκ02

dt= λ(κ10 + 2κ11) .

In turn, the covariance ODE contains higher order termsIn general the i th equation depends on the (i + 1)th equationTo circumvent this dependency problem, we need to closethe equations



Second Order Moment Equations

dκ20

dt= µ(κ11 − 2κ10κ11 − 2κ21 + κ01(κ10 − 2κ20))

+ λ(κ10 + 2κ20)

dκ11

dt= λ(κ10 + κ20 + κ11)− µ(κ10κ02 + κ01κ11 + κ12)

dκ02

dt= λ(κ10 + 2κ11) .

In turn, the covariance ODE contains higher order termsIn general the i th equation depends on the (i + 1)th equationTo circumvent this dependency problem, we need to closethe equations



Closing the Moment Equations

The easiest option is to assume an underlying Normaldistribution, i.e. κi = 0 for i > 2But we could also use the Poisson distribution

κi = κi−1

or the Lognormal

E[X 3] =

(E[X 2]

E[X ]

)3



Comments on the Moment Closure Approximation

For this model:the means and variances are estimated with an error rateless than 2.5%Solving five ODEs is much faster than multiple simulations

In general,the approximation works well when the stochastic meanand deterministic solutions are similarthe approximation usually breaks in an obvious manner, i.e.negative variances



Comments on the Moment Closure Approximation

For this model:the means and variances are estimated with an error rateless than 2.5%Solving five ODEs is much faster than multiple simulations

In general,the approximation works well when the stochastic meanand deterministic solutions are similarthe approximation usually breaks in an obvious manner, i.e.negative variances



Parameter Inference

Giventhe parameters: {λ, µ}the initial states: X(tu−1) = (n(tu−1), c(tu−1));

We haveX(tu) |X(tu−1), λ, µ ∼ N(ψu−1,Σu−1)

where ψu−1 and Σu−1 are calculated using the moment closureapproximation



Parameter Inference

Summarising our beliefs about {λ, µ} and the unobservedcumulative 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

p (λ, µ,c |n) ∝ p(λ, µ) p (c(t0))4∏

u=1

p (x(tu) |x(tu−1), λ, µ)

For the results shown, we used a simple random walk MHstep to explore the parameter and state spacesWe did investigate more sophisticated schemes, but themixing properties were similar



Parameter Inference

Summarising our beliefs about {λ, µ} and the unobservedcumulative 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

p (λ, µ,c |n) ∝ p(λ, µ) p (c(t0))4∏

u=1

p (x(tu) |x(tu−1), λ, µ)

For the results shown, we used a simple random walk MHstep to explore the parameter and state spacesWe did investigate more sophisticated schemes, but themixing properties were similar



Simulation Study

Three treatments & two blocksBaseline birth and death rates: {λ = 1.75, µ = 0.00095}Treatment 2 increases µ by 0.0004Treatment 3 increases λ by 0.35The block effect reduces µ by 0.0003

Treatment 1 Treatment 2 Treatment 3Block 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}



Simulation Study





Simulation Study





Simulation Study





Simulated Data

Time

Aph

id P

opul

atio

n

0

500

1000

0 1 2 3 4

●

●

●

●

●

Treament 1

0 1 2 3 4

●●

●

●

●

Treatment 2

0 1 2 3 4

●

●

●

●

●

Treatment 3



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 = 0So 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+ α∗i + β∗k



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 = 0So 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+ α∗i + β∗k



MCMC Scheme

Using the MCMC scheme described previously, wegenerated 2M iterates and thinned by 1KThis took a few hours and convergence was fairly quickWe used independent proper uniform priors for theparametersFor the initial unobserved cumulative population, we had

c(t0) = n(t0) + ε

where ε has a Gamma distribution with shape 1 and scale10.This set up mirrors the scheme that we used for the realdata set



Marginal posterior distributions for λ and µ

Birth Rate

Den

sity

0

2

4

6

1.6 1.7 1.8 1.9 2.0

X

Death Rate

Den

sity

0

5000

10000

15000

20000

0.00090 0.00095 0.00100

X



MCMC Scheme

Marginal posterior distributions for λ

Birth Rate

Den

sity

0

2

4

6

−0.2 0.0 0.2 0.4

X

Block 2

−0.2 0.0 0.2 0.4

X

Treatment 2

−0.2 0.0 0.2 0.4

X

Treatment 3

We obtained similar densities for the death rates.Colin Gillespie — Nottingham 2009 Bayesian inference for stochastic population models


Application to the Cotton Aphid Data Set

Recall that the data consists offive observations on twenty randomly chosen leaves ineach 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 4.57weeks (i.e. every 7 to 8 days).

Following in the same vein as the simulated data, we areestimating 38 parameters (including interaction terms) and thelatent cumulative aphid population.



Cotton Aphid Data

Marginal posterior distributions for λ and µ

Birth Rate

Den

sity

0

2

4

6

1.6 1.7 1.8 1.9 2.0

Death Rate

Den

sity

0

5000

10000

15000

0.00090 0.00095 0.00100



Does the Model Fit the Data?

We simulate predictive distributions from the MCMCoutput, i.e. we randomly sample parameter values (λ, µ)and the unobserved state c and simulate forwardWe simulate forward using the Gillespie simulator

not the moment closure approximation



Does the Model Fit the data?

Predictive distributions for 6 of the 27 Aphid data sets

Time

Aph

id P

opul

atio

n

0

500

1000

1500

2000

2500

1.14 2.29 3.57 4.57

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Summarising the Results

Consider the additional number of aphids per treatmentcombinationSet c(0) = n(0) = 1 and tmax = 6We now calculate the number of aphids we would see foreach parameter combination in addition to the baselineFor example, the effect due to medium water:

λ211 = λ+ αWater (M) and µ211 = µ+ α∗Water (M)

SoAdditional aphids = c i

Water (M) − c ibaseline



Aphids over Baseline

Main Effects

Aphids

Dens

ity

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0 2000 6000 10000

Block 3 Block 2

0 2000 6000 10000

Nitrogen (Z)

Nitrogen (V)

0 2000 6000 10000

Water (H)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

Water (M)



Aphids over Baseline

Interactions

Aphids

Dens

ity

0.000

0.001

0.002

0.003

0 2000 6000 10000

B3 N(Z) B2 N(Z)

0 2000 6000 10000

B3 N(V) B2 N(V)

B3 W(H) B2 W(H) B3 W(M)

0.000

0.001

0.002

0.003

B2 W(M)

0.000

0.001

0.002

0.003

W(H) N(Z)

0 2000 6000 10000

W(M) N(Z) W(H) N(V)

0 2000 6000 10000

W(M) N(V)



Conclusions

The 95% credible intervals for the baseline birth and deathrates are (1.64,1.86) and (0.000904,0.000987).Main effects have little effect by themselvesHowever block 2 appears to have a very strong interactionwith nitrogenMoment closure parameter inference is a very usefultechnique for estimating parameters in stochasticpopulation models



Future Work

Other data sets suggest that there is aphid immigration inthe early stagesModel selection for stochastic modelsIncorporate measurement error



Acknowledgements

Andrew GolightlyPeter MilnerDarren Wilkinson

Richard Boys

Jim Matis (Texas A & M)

References

Gillespie, C. S., Golightly, A. Bayesian inference for generalizedstochastic population growth models with application to aphids,Journal of the Royal Statistical Society, Series C, 2010.

Gillespie, C.S. Moment closure approximations for mass-actionmodels. IET Systems Biology 2009.

Milner, P., Gillespie, C. S., Wilkinson, D. J. Parameter estimationvia moment closure stochastic models, in preparation.


Documents

Bayesian inference for stochastic population models with application to aphids