SlideShare ist ein Scribd-Unternehmen logo
1 von 53
Downloaden Sie, um offline zu lesen
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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

Weitere ähnliche Inhalte

Andere mochten auch

Stochastic Calculus Main Results
Stochastic Calculus Main ResultsStochastic Calculus Main Results
Stochastic Calculus Main ResultsSSA KPI
 
Stochastic Analysis of Resource Plays: Maximizing Portfolio Value and Mitiga...
Stochastic Analysis of Resource Plays:  Maximizing Portfolio Value and Mitiga...Stochastic Analysis of Resource Plays:  Maximizing Portfolio Value and Mitiga...
Stochastic Analysis of Resource Plays: Maximizing Portfolio Value and Mitiga...Portfolio Decisions
 
Research internship on optimal stochastic theory with financial application u...
Research internship on optimal stochastic theory with financial application u...Research internship on optimal stochastic theory with financial application u...
Research internship on optimal stochastic theory with financial application u...Asma Ben Slimene
 
Distributed Stochastic Gradient MCMC
Distributed Stochastic Gradient MCMCDistributed Stochastic Gradient MCMC
Distributed Stochastic Gradient MCMCKaede Hayashi
 
Matlab tme series benni
Matlab tme series benniMatlab tme series benni
Matlab tme series bennidvbtunisia
 
Stochastic Alternating Direction Method of Multipliers
Stochastic Alternating Direction Method of MultipliersStochastic Alternating Direction Method of Multipliers
Stochastic Alternating Direction Method of MultipliersTaiji Suzuki
 
Stochastic Modeling for Valuation and Risk Management
Stochastic Modeling for Valuation and Risk ManagementStochastic Modeling for Valuation and Risk Management
Stochastic Modeling for Valuation and Risk ManagementRoderick Powell
 
Matlab practical and lab session
Matlab practical and lab sessionMatlab practical and lab session
Matlab practical and lab sessionDr. Krishna Mohbey
 
Stochastic Approximation and Simulated Annealing
Stochastic Approximation and Simulated AnnealingStochastic Approximation and Simulated Annealing
Stochastic Approximation and Simulated AnnealingSSA KPI
 
The Stochastic Simulation Algorithm
The Stochastic Simulation AlgorithmThe Stochastic Simulation Algorithm
The Stochastic Simulation AlgorithmStephen Gilmore
 
Hierarchical Stochastic Neighbor Embedding
Hierarchical Stochastic Neighbor EmbeddingHierarchical Stochastic Neighbor Embedding
Hierarchical Stochastic Neighbor EmbeddingNicola Pezzotti
 
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...Volatility
 
Practical Aspects of Stochastic Modeling.pptx
Practical Aspects of Stochastic Modeling.pptxPractical Aspects of Stochastic Modeling.pptx
Practical Aspects of Stochastic Modeling.pptxRon Harasym
 
Stochastic Optimization: Solvers and Tools
Stochastic Optimization: Solvers and ToolsStochastic Optimization: Solvers and Tools
Stochastic Optimization: Solvers and ToolsSSA KPI
 
Matlab Overviiew
Matlab OverviiewMatlab Overviiew
Matlab OverviiewNazim Naeem
 
Matlab Functions
Matlab FunctionsMatlab Functions
Matlab FunctionsUmer Azeem
 
Introduction to Image Processing with MATLAB
Introduction to Image Processing with MATLABIntroduction to Image Processing with MATLAB
Introduction to Image Processing with MATLABSriram Emarose
 
short course on Subsurface stochastic modelling and geostatistics
short course on Subsurface stochastic modelling and geostatisticsshort course on Subsurface stochastic modelling and geostatistics
short course on Subsurface stochastic modelling and geostatisticsAmro Elfeki
 

Andere mochten auch (20)

MATLAB INTRODUCTION
MATLAB INTRODUCTIONMATLAB INTRODUCTION
MATLAB INTRODUCTION
 
Stochastic Calculus Main Results
Stochastic Calculus Main ResultsStochastic Calculus Main Results
Stochastic Calculus Main Results
 
Stochastic Analysis of Resource Plays: Maximizing Portfolio Value and Mitiga...
Stochastic Analysis of Resource Plays:  Maximizing Portfolio Value and Mitiga...Stochastic Analysis of Resource Plays:  Maximizing Portfolio Value and Mitiga...
Stochastic Analysis of Resource Plays: Maximizing Portfolio Value and Mitiga...
 
Research internship on optimal stochastic theory with financial application u...
Research internship on optimal stochastic theory with financial application u...Research internship on optimal stochastic theory with financial application u...
Research internship on optimal stochastic theory with financial application u...
 
Distributed Stochastic Gradient MCMC
Distributed Stochastic Gradient MCMCDistributed Stochastic Gradient MCMC
Distributed Stochastic Gradient MCMC
 
Matlab tme series benni
Matlab tme series benniMatlab tme series benni
Matlab tme series benni
 
Stochastic Alternating Direction Method of Multipliers
Stochastic Alternating Direction Method of MultipliersStochastic Alternating Direction Method of Multipliers
Stochastic Alternating Direction Method of Multipliers
 
Stochastic Modeling for Valuation and Risk Management
Stochastic Modeling for Valuation and Risk ManagementStochastic Modeling for Valuation and Risk Management
Stochastic Modeling for Valuation and Risk Management
 
Matlab practical and lab session
Matlab practical and lab sessionMatlab practical and lab session
Matlab practical and lab session
 
Stochastic Approximation and Simulated Annealing
Stochastic Approximation and Simulated AnnealingStochastic Approximation and Simulated Annealing
Stochastic Approximation and Simulated Annealing
 
The Stochastic Simulation Algorithm
The Stochastic Simulation AlgorithmThe Stochastic Simulation Algorithm
The Stochastic Simulation Algorithm
 
Hierarchical Stochastic Neighbor Embedding
Hierarchical Stochastic Neighbor EmbeddingHierarchical Stochastic Neighbor Embedding
Hierarchical Stochastic Neighbor Embedding
 
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...
Efficient Numerical PDE Methods to Solve Calibration and Pricing Problems in ...
 
Practical Aspects of Stochastic Modeling.pptx
Practical Aspects of Stochastic Modeling.pptxPractical Aspects of Stochastic Modeling.pptx
Practical Aspects of Stochastic Modeling.pptx
 
Stochastic Optimization: Solvers and Tools
Stochastic Optimization: Solvers and ToolsStochastic Optimization: Solvers and Tools
Stochastic Optimization: Solvers and Tools
 
Matlab Overviiew
Matlab OverviiewMatlab Overviiew
Matlab Overviiew
 
Matlab Functions
Matlab FunctionsMatlab Functions
Matlab Functions
 
Introduction to Image Processing with MATLAB
Introduction to Image Processing with MATLABIntroduction to Image Processing with MATLAB
Introduction to Image Processing with MATLAB
 
short course on Subsurface stochastic modelling and geostatistics
short course on Subsurface stochastic modelling and geostatisticsshort course on Subsurface stochastic modelling and geostatistics
short course on Subsurface stochastic modelling and geostatistics
 
Matlab basic and image
Matlab basic and imageMatlab basic and image
Matlab basic and image
 

Mehr von Colin Gillespie

Bayesian Experimental Design for Stochastic Kinetic Models
Bayesian Experimental Design for Stochastic Kinetic ModelsBayesian Experimental Design for Stochastic Kinetic Models
Bayesian Experimental Design for Stochastic Kinetic ModelsColin Gillespie
 
The tau-leap method for simulating stochastic kinetic models
The tau-leap method for simulating stochastic kinetic modelsThe tau-leap method for simulating stochastic kinetic models
The tau-leap method for simulating stochastic kinetic modelsColin Gillespie
 
Poster for Information, probability and inference in systems biology (IPISB 2...
Poster for Information, probability and inference in systems biology (IPISB 2...Poster for Information, probability and inference in systems biology (IPISB 2...
Poster for Information, probability and inference in systems biology (IPISB 2...Colin Gillespie
 
Reference classes: a case study with the poweRlaw package
Reference classes: a case study with the poweRlaw packageReference classes: a case study with the poweRlaw package
Reference classes: a case study with the poweRlaw packageColin Gillespie
 
Introduction to power laws
Introduction to power lawsIntroduction to power laws
Introduction to power lawsColin Gillespie
 
Moment Closure Based Parameter Inference of Stochastic Kinetic Models
Moment Closure Based Parameter Inference of Stochastic Kinetic ModelsMoment Closure Based Parameter Inference of Stochastic Kinetic Models
Moment Closure Based Parameter Inference of Stochastic Kinetic ModelsColin Gillespie
 
An introduction to moment closure techniques
An introduction to moment closure techniquesAn introduction to moment closure techniques
An introduction to moment closure techniquesColin Gillespie
 
Speeding up the Gillespie algorithm
Speeding up the Gillespie algorithmSpeeding up the Gillespie algorithm
Speeding up the Gillespie algorithmColin Gillespie
 
Moment closure inference for stochastic kinetic models
Moment closure inference for stochastic kinetic modelsMoment closure inference for stochastic kinetic models
Moment closure inference for stochastic kinetic modelsColin Gillespie
 

Mehr von Colin Gillespie (10)

Bayesian Experimental Design for Stochastic Kinetic Models
Bayesian Experimental Design for Stochastic Kinetic ModelsBayesian Experimental Design for Stochastic Kinetic Models
Bayesian Experimental Design for Stochastic Kinetic Models
 
The tau-leap method for simulating stochastic kinetic models
The tau-leap method for simulating stochastic kinetic modelsThe tau-leap method for simulating stochastic kinetic models
The tau-leap method for simulating stochastic kinetic models
 
Poster for Information, probability and inference in systems biology (IPISB 2...
Poster for Information, probability and inference in systems biology (IPISB 2...Poster for Information, probability and inference in systems biology (IPISB 2...
Poster for Information, probability and inference in systems biology (IPISB 2...
 
Reference classes: a case study with the poweRlaw package
Reference classes: a case study with the poweRlaw packageReference classes: a case study with the poweRlaw package
Reference classes: a case study with the poweRlaw package
 
Introduction to power laws
Introduction to power lawsIntroduction to power laws
Introduction to power laws
 
Moment Closure Based Parameter Inference of Stochastic Kinetic Models
Moment Closure Based Parameter Inference of Stochastic Kinetic ModelsMoment Closure Based Parameter Inference of Stochastic Kinetic Models
Moment Closure Based Parameter Inference of Stochastic Kinetic Models
 
An introduction to moment closure techniques
An introduction to moment closure techniquesAn introduction to moment closure techniques
An introduction to moment closure techniques
 
Speeding up the Gillespie algorithm
Speeding up the Gillespie algorithmSpeeding up the Gillespie algorithm
Speeding up the Gillespie algorithm
 
Moment closure inference for stochastic kinetic models
Moment closure inference for stochastic kinetic modelsMoment closure inference for stochastic kinetic models
Moment closure inference for stochastic kinetic models
 
WCSB 2012
WCSB 2012 WCSB 2012
WCSB 2012
 

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