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- 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

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