statistical inference for partially observed markov processes via … › pdf ›...

Statistical Inference for Partially Observed Markov

Processes via the R Package pomp

Aaron A King Dao Nguyen Edward L IonidesUniversity of Michigan

October 23 2015

Abstract

Partially observed Markov process (POMP) models also known as hidden Markovmodels or state space models are ubiquitous tools for time series analysis The R packagepomp provides a very flexible framework for Monte Carlo statistical investigations usingnonlinear non-Gaussian POMP models A range of modern statistical methods for POMPmodels have been implemented in this framework including sequential Monte Carlo it-erated filtering particle Markov chain Monte Carlo approximate Bayesian computationmaximum synthetic likelihood estimation nonlinear forecasting and trajectory match-ing In this paper we demonstrate the application of these methodologies using somesimple toy problems We also illustrate the specification of more complex POMP mod-els using a nonlinear epidemiological model with a discrete population seasonality andextra-demographic stochasticity We discuss the specification of user-defined models andthe development of additional methods within the programming environment provided bypompThis document is a version of a manuscript in press at the Journal of Statistical SoftwareIt is provided under the Creative Commons Attribution License

Keywords Markov processes hidden Markov model state space model stochastic dynamicalsystem maximum likelihood plug-and-play time series mechanistic model sequential MonteCarlo R

1 Introduction

A partially observed Markov process (POMP) model consists of incomplete and noisy mea-surements of a latent unobserved Markov process The far-reaching applicability of this classof models has motivated much software development (Commandeur et al 2011) It has beena challenge to provide a software environment that can effectively handle broad classes ofPOMP models and take advantage of the wide range of statistical methodologies that havebeen proposed for such models The pomp software package (King et al 2015b) differs fromprevious approaches by providing a general and abstract representation of a POMP modelTherefore algorithms implemented within pomp are necessarily applicable to arbitrary POMPmodels Moreover models formulated with pomp can be analyzed using multiple method-ologies in search of the most effective method or combination of methods for the problemat hand However since pomp is designed for general POMP models methods that exploitadditional model structure have yet to be implemented In particular when linear Gaussian

arX

iv1

509

0050

3v2

[st

atM

E]

21

Oct

201

5

2 Partially observed Markov processes

approximations are adequate for onersquos purposes or when the latent process takes values in asmall discrete set methods that exploit these additional assumptions to advantage such asthe extended and ensemble Kalman filter methods or exact hidden-Markov-model methodsare available but not yet as part of pomp It is the class of nonlinear non-Gaussian POMPmodels with large state spaces upon which pomp is focused

A POMP model may be characterized by the transition density for the Markov process and themeasurement density1 However some methods require only simulation from the transitiondensity whereas others require evaluation of this density Still other methods may not workwith the model itself but with an approximation such as a linearization Algorithms forwhich the dynamic model is specified only via a simulator are said to be plug-and-play (Bretoet al 2009 He et al 2010) Plug-and-play methods can be employed once one has ldquopluggedrdquo amodel simulator into the inference machinery Since many POMP models of scientific interestare relatively easy to simulate the plug-and-play property facilitates data analysis Evenif one candidate model has tractable transition probabilities a scientist will frequently wishto consider alternative models for which these probabilities are intractable In a plug-and-play methodological environment analysis of variations in the model can often be achieved bychanging a few lines of the model simulator codes The price one pays for the flexibility of plug-and-play methodology is primarily additional computational effort which can be substantialNevertheless plug-and-play methods implemented using pomp have proved capable for stateof the art inference problems (eg King et al 2008 Bhadra et al 2011 Shrestha et al2011 2013 Earn et al 2012 Roy et al 2012 Blackwood et al 2013ab He et al 2013Breto 2014 Blake et al 2014) The recent surge of interest in plug-and-play methodologyfor POMP models includes the development of nonlinear forecasting (Ellner et al 1998)iterated filtering (Ionides et al 2006 2015) ensemble Kalman filtering (Shaman and Karspeck2012) approximate Bayesian computation (ABC) (Sisson et al 2007) particle Markov chainMonte Carlo (PMCMC) (Andrieu et al 2010) probe matching (Kendall et al 1999) andsynthetic likelihood (Wood 2010) Although the pomp package provides a general environmentfor methods with and without the plug-and-play property development of the package to datehas emphasized plug-and-play methods

The pomp package is philosophically neutral as to the merits of Bayesian inference It en-ables a POMP model to be supplemented with prior distributions on parameters and severalBayesian methods are implemented within the package Thus pomp is a convenient environ-ment for those who wish to explore both Bayesian and non-Bayesian data analyses

The remainder of this paper is organized as follows Section 2 defines mathematical notationfor POMP models and relates this to their representation as objects of class lsquopomprsquo in thepomp package Section 3 introduces several of the statistical methods currently implementedin pomp Section 4 constructs and explores a simple POMP model demonstrating the use ofthe available statistical methods Section 5 illustrates the implementation of more complexPOMPs using a model of infectious disease transmission as an example Finally Section 6discusses extensions and applications of pomp

1We use the term ldquodensityrdquo in this article encompass both the continuous and discrete cases Thus in thelatter case ie when state variables andor measured quantities are discrete one could replace ldquoprobabilitydensity functionrdquo with ldquoprobability mass functionrdquo

Aaron A King Dao Nguyen Edward L Ionides 3

2 POMP models and their representation in pomp

Let θ be a p-dimensional real-valued parameter θ isin Rp For each value of θ let X(t θ) t isinT be a Markov process with X(t θ) taking values in Rq The time index set T sub R maybe an interval or a discrete set Let ti isin T i = 1 N be the times at which X(t θ)is observed and t0 isin T be an initial time Assume t0 le t1 lt t2 lt middot middot middot lt tN We writeXi = X(ti θ) and Xij = (Xi Xi+1 Xj) The process X0N is only observed by way ofanother process Y1N = (Y1 YN ) with Yn taking values in Rr The observable randomvariables Y1N are assumed to be conditionally independent given X0N The data ylowast1N =(ylowast1 y

lowastN ) are modeled as a realization of this observation process and are considered

fixed We suppose that X0N and Y1N have a joint density fX0N Y1N(x0n y1n θ) The

POMP structure implies that this joint density is determined by the initial density fX0(x0 θ)together with the conditional transition probability density fXn|Xnminus1

(xn |xnminus1 θ) and themeasurement density fYn|Xn

(yn |xn θ) for 1 le n le N In particular we have

fX0N Y1N(x0N y1N θ) = fX0(x0 θ)

Nprodn=1

fXn|Xnminus1(xn|xnminus1 θ) fYn|Xn

(yn|xn θ) (1)

Note that this formalism allows the transition density fXn|Xnminus1 and measurement density

fYn|Xn to depend explicitly on n

21 Implementation of POMP models

pomp is fully object-oriented in the package a POMP model is represented by an S4 object(Chambers 1998 Genolini 2008) of class lsquopomprsquo Slots in this object encode the components ofthe POMP model and can be filled or changed using the constructor function pomp and variousother convenience functions Methods for the class lsquopomprsquo class use these components to carryout computations on the model Table 1 gives the mathematical notation corresponding tothe elementary methods that can be executed on a class lsquopomprsquo object

The rprocess dprocess rmeasure and dmeasure arguments specify the transition probabil-ities fXn|Xnminus1

(xn |xnminus1 θ) and measurement densities fYn|Xn(yn |xn θ) Not all of these ar-

guments must be supplied for any specific computation In particular plug-and-play method-ology by definition never uses dprocess An empty dprocess slot in a class lsquopomprsquo objectis therefore acceptable unless a non-plug-and-play algorithm is attempted In the packagethe data and corresponding measurement times are considered necessary parts of a classlsquopomprsquo object whilst specific values of the parameters and latent states are not Applying thesimulate function to an object of class lsquopomprsquo returns another object of class lsquopomprsquo withinwhich the data ylowast1N have been replaced by a stochastic realization of Y1N the correspondingrealization of X0N is accessible via the states method and the params slot has been filledwith the value of θ used in the simulation

To illustrate the specification of models in pomp and the use of the packagersquos inference algo-rithms we will use a simple example The Gompertz (1825) model can be constructed via

Rgt library(pomp)

Rgt pompExample(gompertz)


Method Argument to the Mathematical terminologypomp constructor

rprocess rprocess Simulate from fXn|Xnminus1(xn |xnminus1 θ)

dprocess dprocess Evaluate fXn|Xnminus1(xn |xnminus1 θ)

rmeasure rmeasure Simulate from fYn|Xn(yn |xn θ)

dmeasure dmeasure Evaluate fYn|Xn(yn |xn θ)

rprior rprior Simulate from the prior distribution π(θ)dprior dprior Evaluate the prior density π(θ)initstate initializer Simulate from fX0(x0 θ)timezero t0 t0time times t1N

obs data ylowast1N

states mdash x0N

coef params θ

Table 1 Constituent methods for class lsquopomprsquo objects and their translation into mathematicalnotation for POMP models For example the rprocess method is set using the rprocess

argument to the pomp constructor function

which results in the creation of an object of class lsquopomprsquo named gompertz in the workspaceThe structure of this model and its implementation in pomp is described below in Section 4One can view the components of gompertz listed in Table 1 by executing

Rgt obs(gompertz)

Rgt states(gompertz)

Rgt asdataframe(gompertz)

Rgt plot(gompertz)

Rgt timezero(gompertz)

Rgt time(gompertz)

Rgt coef(gompertz)

Rgt initstate(gompertz)

Executing pompExample() lists other examples provided with the package

22 Initial conditions

In some experimental situations fX0(x0 θ) corresponds to a known experimental initializa-tion but in general the initial state of the latent process will have to be inferred If thetransition density for the dynamic model fXn|Xnminus1

(xn |xnminus1 θ) does not depend on timeand possesses a unique stationary distribution it may be natural to set fX0(x0 θ) to be thisstationary distribution Otherwise and more commonly in the authorsrsquo experience no clearscientifically motivated choice of fX0(x0 θ) exists and one can proceed by treating the valueof X0 as a parameter to be estimated In this case fX0(x0 θ) concentrates at a point thelocation of which depends on θ


23 Covariates

Scientifically one may be interested in the role of a vector-valued covariate process Z(t)in explaining the data Modeling and inference conditional on Z(t) can be carried outwithin the general framework for nonhomogeneous POMP models since the arbitrary densitiesfXn|Xnminus1

fX0 and fYn|Xncan depend on the observed process Z(t) For example it may

be the case that fXn|Xnminus1(xn |xnminus1 θ) depends on n only through Z(t) for tnminus1 le t le tn

The covar argument in the pomp constructor allows for time-varying covariates measured attimes specified in the tcovar argument A example using covariates is given in Section 5

3 Methodology for POMP models

Data analysis typically involves identifying regions of parameter space within which a postu-lated model is statistically consistent with the data Additionally one frequently desires toassess the relative merits of alternative models as explanations of the data Once the userhas encoded one or more POMP models as objects of class lsquopomprsquo the package provides avariety of algorithms to assist with these data analysis goals Table 2 provides an overviewof several inference methodologies for POMP models Each method may be categorized asfull-information or feature-based Bayesian or frequentist and plug-and-play or not plug-and-play

Approaches that work with the full likelihood function whether in a Bayesian or frequentistcontext can be called full-information methods Since low-dimensional sufficient statisticsare not generally available for POMP models methods which take advantage of favorablelow-dimensional representations of the data typically lose some statistical efficiency We usethe term ldquofeature-basedrdquo to describe all methods not based on the full likelihood since suchmethods statistically emphasize some features of the data over others

Many Monte Carlo methods of inference can be viewed as algorithms for the exploration ofhigh-dimensional surfaces This view obtains whether the surface in question is the likelihoodsurface or that of some other objective function The premise behind many recent method-ological developments in Monte Carlo methods for POMP models is that generic stochasticnumerical analysis tools such as standard Markov chain Monte Carlo and Robbins-Monrotype methods are effective only on the simplest models For many models of scientific interesttherefore methods that leverage the POMP structure are needed Though pomp has suffi-cient flexibility to encode arbitrary POMP models and methods and therefore also provides aplatform for the development of novel POMP inference methodology pomprsquos development todate has focused on plug-and-play methods However the package developers welcome con-tributions and collaborations to further expand pomprsquos functionality in non-plug-and-playdirections also In the remainder of this section we describe and discuss several inferencemethods all currently implemented in the package

31 The likelihood function and sequential Monte Carlo

The log likelihood for a POMP model is `(θ) = log fY1N(ylowast1N θ) which can be written as a

sum of conditional log likelihoods

`(θ) =

Nsumn=1

`n|1nminus1(θ) (2)


(a) Plug-and-play

Frequentist Bayesian

Full information Iterated filtering (mifSection 32)

PMCMC (pmcmc Section 33)

Feature-based Nonlinear forecasting (nlfSection 36)

ABC (abc Section 35)

synthetic likelihood(probematch Section 34)

(b) Not plug-and-play


Full information EM and Monte Carlo EM MCMCKalman filter

Feature-based Trajectory matching(trajmatch)

Extended Kalman filter

extended Kalman filterYule-Walker equations

Table 2 Inference methods for POMP models For those currently implemented in pompfunction name and a reference for description are provided in parentheses StandardExpectation-Maximization (EM) and Markov chain Monte Carlo (MCMC) algorithms arenot plug-and-play since they require evaluation of fXn|Xnminus1

(xn |xnminus1 θ) The Kalman filterand extended Kalman filter are not plug-and-play since they cannot be implemented basedon a model simulator The Kalman filter provides the likelihood for a linear Gaussian modelThe extended Kalman filter employs a local linear Gaussian approximation which can be usedfor frequentist inference (via maximization of the resulting quasi-likelihood) or approximateBayesian inference (by adding the parameters to the state vector) The Yule-Walker equationsfor ARMA models provide an example of a closed-form method of moments estimator

where`n|1nminus1(θ) = log fYn|Y1nminus1

(ylowastn | ylowast1nminus1 θ) (3)

and we use the convention that ylowast10 is an empty vector The structure of a POMP modelimplies the representation

`n|1nminus1(θ) = log

intfYn|Xn

(ylowastn|xn θ)fXn|Y1nminus1(xn | ylowast1nminus1 θ) dxn (4)

(cf Eq 1) Although `(θ) typically has no closed form it can frequently be computedby Monte Carlo methods Sequential Monte Carlo (SMC) builds up a representation offXn|Y1nminus1

(xn | ylowast1nminus1 θ) that can be used to obtain an estimate ˆn|1nminus1(θ) of `n|1nminus1(θ) and

hence an approximation ˆ(θ) to `(θ) SMC (a basic version of which is presented as Algo-rithm 1) is also known as the particle filter since it is conventional to describe the MonteCarlo sample XF

nj j in 1 J as a swarm of particles representing fXn|Y1n(xn | ylowast1n θ) The

swarm is propagated forward according to the dynamic model and then assimilated to thenext data point Using an evolutionary analogy the prediction step (line 3) mutates theparticles in the swarm and the filtering step (line 7) corresponds to selection SMC is imple-mented in pomp in the pfilter function The basic particle filter in Algorithm 1 possesses


Algorithm 1 Sequential Monte Carlo (SMC or particle filter) pfilter( P Np = J)using notation from Table 1 where P is a class lsquopomprsquo object with definitions for rprocessdmeasure initstate coef and obs

input Simulator for fXn|Xnminus1(xn |xnminus1 θ) evaluator for fYn|Xn

(yn |xn θ) simulator forfX0(x0 θ) parameter θ data ylowast1N number of particles J

1 Initialize filter particles simulate XF0j sim fX0 ( middot θ) for j in 1 J

2 for n in 1N do3 Simulate for prediction XP

nj sim fXn|Xnminus1

(middot |XF

nminus1j θ)

for j in 1 J

4 Evaluate weights w(n j) = fYn|Xn(ylowastn|XP

nj θ) for j in 1 J

5 Normalize weights w(n j) = w(n j)sumJ

m=1w(nm)6 Apply Algorithm 2 to select indices k1J with P [kj = m] = w(nm)7 Resample set XF

nj = XPnkj

for j in 1 J

8 Compute conditional log likelihood ˆn|1nminus1 = log

(Jminus1

sumJm=1w(nm)

)

9 end

output Log likelihood estimate ˆ(θ) =sumN

n=1ˆn|1nminus1 filter sample XF

n1J for n in 1 N complexity O(J)

the plug-and-play property Many variations and elaborations to SMC have been proposedthese may improve numerical performance in appropriate situations (Cappe et al 2007) buttypically lose the plug-and-play property Arulampalam et al (2002) Doucet and Johansen(2009) and Kantas et al (2015) have written excellent introductory tutorials on the particlefilter and particle methods more generally

Basic SMC methods fail when an observation is extremely unlikely given the model Thisleads to the situation that at most a few particles are consistent with the observation inwhich case the effective sample size (Liu 2001) of the Monte Carlo sample is small and theparticle filter is said to suffer from particle depletion Many elaborations of the basic SMCalgorithm have been proposed to ameliorate this problem However it is often preferable toremedy the situation by seeking a better model The plug-and-play property assists in thisprocess by facilitating investigation of alternative models

In line 6 of Algorithm 1 systematic resampling (Algorithm 2) is used in preference to multi-nomial resampling Algorithm 2 reduces Monte Carlo variability while resampling with theproper marginal probability In particular if all the particle weights are equal then Algo-rithm 2 has the appropriate behavior of leaving the particles unchanged As pointed outby (Douc et al 2005) stratified resampling performs better than multinomial sampling andAlgorithm 2 is in practice comparable in performance to stratified resampling and somewhatfaster


Algorithm 2 Systematic resampling Line 6 of Algorithm 1

input Weights w1J normalized so thatsumJ

j=1 wj = 1

1 Construct cumulative sum cj =sumj

m=1 wm for j in 1 J 2 Draw a uniform initial sampling point U1 sim Uniform(0 Jminus1)3 Construct evenly spaced sampling points Uj = U1 + (j minus 1)Jminus1 for j in 2 J 4 Initialize set p = 15 for j in 1 J do6 while Uj gt cp do7 Step to the next resampling index set p = p+ 18 end9 Assign resampling index set kj = p

10 endoutput Resampling indices k1J complexity O(J)

32 Iterated filtering

Iterated filtering techniques maximize the likelihood obtained by SMC (Ionides et al 20062011 2015) The key idea of iterated filtering is to replace the model we are interested infittingmdashwhich has time-invariant parametersmdashwith a model that is just the same except thatits parameters take a random walk in time Over multiple repetitions of the filtering procedurethe intensity of this random walk approaches zero and the modified model approaches theoriginal one Adding additional variability in this way has four positive effects

A1 It smooths the likelihood surface which facilitates optimization

A2 It combats particle depletion by adding diversity to the population of particles

A3 The additional variability can be exploited to explore the likelihood surface and estimateof the gradient of the (smoothed) likelihood based on the same filtering procedure thatis required to estimate of the value of the likelihood (Ionides et al 2006 2011)

A4 It preserves the plug-and-play property inherited from the particle filter

Iterated filtering is implemented in the mif function By default mif carries out the procedureof Ionides et al (2006) The improved iterated filtering algorithm (IF2) of Ionides et al (2015)has shown promise A limited version of IF2 is available via the method=mif2 option a fullversion of this algorithm will be released soon In all iterated filtering methods by analogywith annealing the random walk intensity can be called a temperature which is decreasedaccording to a prescribed cooling schedule One strives to ensure that the algorithm willfreeze at the maximum of the likelihood as the temperature approaches zero

The perturbations on the parameters in lines 2 and 7 of Algorithm 3 follow a normal distri-bution with each component [θ]i of the parameter vector perturbed independently Neithernormality nor independence are necessary for iterated filtering but rather than varying theperturbation distribution one can transform the parameters to make these simple algorithmicchoices reasonable


Algorithm 3 Iterated filtering mif(P start = θ0 Nmif =M Np = J rwsd = σ1p

iclag =L varfactor =C coolingfactor = a) using notation from Table 1 where P isa class lsquopomprsquo object with defined rprocess dmeasure initstate and obs components

input Starting parameter θ0 simulator for fX0(x0 θ) simulator forfXn|Xnminus1

(xn |xnminus1 θ) evaluator for fYn|Xn(yn |xn θ) data ylowast1N labels

I sub 1 p designating IVPs fixed lag L for estimating IVPs number ofparticles J number of iterations M cooling rate 0 lt a lt 1 perturbation scalesσ1p initial scale multiplier C gt 0

1 for m in 1M do2 Initialize parameters [ΘF

0j ]i sim Normal([θmminus1]i (Ca

mminus1σi)2)

for i in 1 p j in 1 J

3 Initialize states simulate XF0j sim fX0

(middot ΘF

0j

)for j in 1 J

4 Initialize filter mean for parameters θ0 = θmminus15 Define [V1]i = (C2 + 1)a2mminus2σ2

i 6 for n in 1 N do

7 Perturb parameters[ΘPnj

]isim Normal

([ΘFnminus1j

]i (amminus1σi)

2)

for i 6isin I j in 1 J

8 Simulate prediction particles XPnj sim fXn|Xnminus1

(middot |XF

nminus1j ΘPnj

)for j in 1 J


nj ΘPnj) for j in 1 J


u=1w(n u)11 Apply Algorithm 2 to select indices k1J with P [ku = j] = w (n j)12 Resample particles XF

nj = XPnkj

and ΘFnj = ΘP

nkjfor j in 1 J

13 Filter mean[θn]i

=sumJ

j=1 w(n j)[ΘPnj

]i

for i 6isin I

14 Prediction variance [Vn+1]i = (amminus1σi)2 +

sumj w(n j)

([ΘP

nj ]i minus [θn]i)2

for i 6isin I

15 end

16 Update non-IVPs [θm]i = [θmminus1]i + [V1]isumN

n=1[Vn]minus1i

([θn]i minus [θnminus1]i

)for i 6isin I

17 Update IVPs [θm]i = 1J

sumj

[ΘFLj

]i

for i isin I

18 endoutput Monte Carlo maximum likelihood estimate θM complexity O(JM)

Algorithm 3 gives special treatment to a subset of the components of the parameter vectortermed initial value parameters (IVPs) which arise when unknown initial conditions aremodeled as parameters These IVPs will typically be inconsistently estimable as the length ofthe time series increases since for a stochastic process one expects only early time points tocontain information about the initial state Searching the parameter space using time-varyingparameters is inefficient in this situation and so Algorithm 3 perturbs these parameters onlyat time zero

Lines 7ndash12 of Algorithm 3 are exactly an application of SMC (Algorithm 1) to a modifiedPOMP model in which the parameters are added to the state space This approach hasbeen used in a variety of previously proposed POMP methodologies (Kitagawa 1998 Liu andWest 2001 Wan and Van Der Merwe 2000) but iterated filtering is distinguished by havingtheoretical justification for convergence to the maximum likelihood estimate (Ionides et al2011)


33 Particle Markov chain Monte Carlo

Algorithm 4 Particle Markov Chain Monte Carlo pmcmc(P start = θ0

Nmcmc =M Np = J proposal = q) using notation from Table 1 where P is a class lsquopomprsquoobject with defined methods for rprocess dmeasure initstate dprior and obs Thesupplied proposal samples from a symmetric but otherwise arbitrary MCMC proposal dis-tribution q(θP | θ)input Starting parameter θ0 simulator for fX0(x0 | θ) simulator for

fXn|Xnminus1(xn |xnminus1 θ) evaluator for fYn|Xn

(yn |xn θ) simulator for q(θP | θ) dataylowast1N number of particles J number of filtering operations M perturbation scalesσ1p evaluator for prior fΘ(θ)

1 Initialization compute ˆ(θ0) using Algorithm 1 with J particles2 for m in 1M do3 Draw a parameter proposal θPm from the proposal distribution ΘP

m sim q ( middot | θmminus1)

4 Compute ˆ(θPm) using Algorithm 1 with J particles5 Generate U sim Uniform(0 1)

6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt

fΘ(θPm) exp(ˆ(θPm))

fΘ(θmminus1) exp(ˆ(θmminus1))(

θmminus1 ˆ(θmminus1)) otherwise

7 endoutput Samples θ1M representing the posterior distribution fΘ|Y1N

(θ | ylowast1N )complexity O(JM)

Full-information plug-and-play Bayesian inference for POMP models is enabled by particleMarkov chain Monte Carlo (PMCMC) algorithms (Andrieu et al 2010) PMCMC methodscombine likelihood evaluation via SMC with MCMC moves in the parameter space Thesimplest and most widely used PMCMC algorithm termed particle marginal Metropolis-Hastings (PMMH) is based on the observation that the unbiased likelihood estimate providedby SMC can be plugged in to the Metropolis-Hastings update procedure to give an algorithmtargeting the desired posterior distribution for the parameters (Andrieu and Roberts 2009)PMMH is implemented in pmcmc as described in Algorithm 4 In part because it gainsonly a single likelihood evaluation from each particle-filtering operation PMCMC can becomputationally relatively inefficient (Bhadra 2010 Ionides et al 2015) Nevertheless itsinvention introduced the possibility of full-information plug-and-play Bayesian inferences insome situations where they had been unavailable

34 Synthetic likelihood of summary statistics

Some motivations to estimate parameter based on features rather than the full likelihoodinclude

B1 Reducing the data to sensibly selected and informative low-dimensional summary statis-tics may have computational advantages (Wood 2010)

B2 The scientific goal may be to match some chosen characteristics of the data rather thanall aspects of it Acknowledging the limitations of all models this limited aspiration may


Algorithm 5 Synthetic likelihood evaluation probe(P nsim = J probes = S) usingnotation from Table 1 where P is a class lsquopomprsquo object with defined methods for rprocessrmeasure initstate and obs

input Simulator for fXn|Xnminus1(xn |xnminus1 θ) simulator for fX0(x0 θ) simulator for

fYn|Xn(yn |xn θ) parameter θ data ylowast1N number of simulations J vector of

summary statistics or probes S = (S1 Sd)1 Compute observed probes slowasti = Si(ylowast1N ) for i in 1 d

2 Simulate J datasets Y j1N sim fY1N

( middot θ) for j in 1 J

3 Compute simulated probes sij = Si(Y j1N ) for i in 1 d and j in 1 J

4 Compute sample mean microi = Jminus1sumJ

j=1 sij for i in 1 d

5 Compute sample covariance Σik = (J minus 1)minus1sumJ

j=1(sij minus microi)(skj minus microk) for i and k in 1 d

6 Compute the log synthetic likelihood

ˆS(θ) = minus1

2(slowast minus micro)TΣminus1(slowast minus micro)minus 1

2log |Σ| minus d

2log(2π) (5)

output Synthetic likelihood ˆS(θ)complexity O(J)

be all that can reasonably be demanded (Kendall et al 1999 Wood 2001)

B3 In conjunction with full-information methodology consideration of individual features hasdiagnostic value to determine which aspects of the data are driving the full-informationinferences (Reuman et al 2006)

B4 Feature-based methods for dynamic models typically do not require the POMP modelstructure However that benefit is outside the scope of the pomp package

B5 Feature-based methods are typically doubly plug-and-play meaning that they requiresimulation but not evaluation for both the latent process transition density and themeasurement model

When pursuing goal B1 one aims to find summary statistics which are as close as possibleto sufficient statistics for the unknown parameters Goals B2 and B3 deliberately look forfeatures which discard information from the data in this context the features have beencalled probes (Kendall et al 1999) The features are denoted by a collection of functionsS = (S1 Sd) where each Si maps an observed time series to a real number We writeS = (S1 Sd) for the vector-valued random variable with S = S(Y1N ) with fS(s θ)being the corresponding joint density The observed feature vector is slowast where slowasti = Si(ylowast1N )and for any parameter set one can look for parameter values for which typical features forsimulated data match the observed features One can define a likelihood function `S(θ) =fS(slowast θ) Arguing that S should be approximately multivariate normal for suitable choicesof the features Wood (2010) proposed using simulations to construct a multivariate normalapproximation to `S(θ) and called this a synthetic likelihood


Simulation-based evaluation of a feature matching criterion is implemented by probe (Algo-rithm 5) The feature matching criterion requires a scale and a natural scale to use is theempirical covariance of the simulations Working on this scale as implemented by probe thereis no substantial difference between the probe approaches of Kendall et al (1999) and Wood(2010) Numerical optimization of the synthetic likelihood is implemented by probematchwhich offers a choice of the subplex method (Rowan 1990 King 2008) or any method providedby optim or the nloptr package (Johnson 2014 Ypma 2014)

35 Approximate Bayesian computation (ABC)

Algorithm 6 Approximate Bayesian Computation abc(P start = θ0 Nmcmc =M

probes = S scale = τ1d proposal = q epsilon = ε) using notation from Table 1 where P

is a class lsquopomprsquo object with defined methods for rprocess rmeasure initstate dpriorand obsinput Starting parameter θ0 simulator for fX0(x0 θ) simulator for

fXn|Xnminus1(xn |xnminus1 θ) simulator for fYn|Xn

(yn |xn θ) simulator for q(θP | θ) dataylowast1N number of proposals M vector of probes S = (S1 Sd) perturbationscales σ1p evaluator for prior fΘ(θ) feature scales τ1d tolerance ε

1 Compute observed probes slowasti = Si(ylowast1N ) for i in 1 d2 for m in 1 M do3 Draw a parameter proposal θPm from the proposal distribution ΘP

m sim q ( middot | θmminus1)4 Simulate dataset Y1N sim fY1N

( middot θPm)5 Compute simulated probes si = Si(Y1N ) for i in 1 d6 Generate U sim Uniform(0 1)

7 Set θm =

θPm if

dsumi=1

(si minus slowastiτi

)2

lt ε2 and U ltfΘ(θPm)

fΘ(θmminus1)

θmminus1 otherwise

8 endoutput Samples θ1M representing the posterior distribution fΘ|S1d

(θ | slowast1d)complexity Nominally O(M) but performance will depend on the choice of ε τi and σi

as well as on the choice of probes S

ABC algorithms are Bayesian feature-matching techniques comparable to the frequentistgeneralized method of moments (Marin et al 2012) The vector of summary statistics S thecorresponding random variable S and the value slowast = S(ylowast1N ) are defined as in Section 34The goal of ABC is to approximate the posterior distribution of the unknown parametersgiven S = slowast ABC has typically been motivated by computational considerations as inpoint B1 of Section 34 (Sisson et al 2007 Toni et al 2009 Beaumont 2010) Points B2and B3 also apply (Ratmann et al 2009)

The key theoretical insight behind ABC algorithms is that an unbiased estimate of the like-lihood can be substituted into a Markov chain Monte Carlo algorithm to target the requiredposterior the same result that justifies PMCMC (Andrieu and Roberts 2009) HoweverABC takes a different approach to approximating the likelihood The likelihood of the ob-served features `S(θ) = fS(slowast θ) has an approximately unbiased estimate based on a single


Monte Carlo realization Y1N sim fY1N( middot θ) given by

ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)

where Bd is the volume of the d-dimensional unit ball and τi is a scaling chosen for theith feature The likelihood approximation in Eq 6 differs from the synthetic likelihood inAlgorithm 5 in that only a single simulation is required As ε become small the bias inEq 6 decreases but the Monte Carlo variability increases The ABC implementation abc

(presented in Algorithm 6) is a random walk Metropolis implementation of ABC-MCMC(Algorithm 3 of Marin et al 2012) In the same style as iterated filtering and PMCMC weassume a Gaussian random walk in parameter space the package supports alternative choicesof proposal distribution

36 Nonlinear forecasting

Nonlinear forecasting (NLF) uses simulations to build up an approximation to the one-stepprediction distribution that is then evaluated on the data We saw in Section 31 that SMCevaluates the prediction density for the observation fYn|Y1nminus1

(ylowastn | ylowast1nminus1 θ) by first buildingan approximation to the prediction density of the latent process fXn|Y1nminus1

(xn | ylowast1nminus1 θ)NLF by contrast uses simulations to fit a linear regression model which predicts Yn basedon a collection of L lagged variables Ynminusc1 YnminuscL The prediction errors when thismodel is applied to the data give rise to a quantity called the quasi-likelihood which behavesfor many purposes like a likelihood (Smith 1993) The implementation in nlf maximises thequasi-likelihood computed in Algorithm 7 using the subplex method (Rowan 1990 King2008) or any other optimizer offerered by optim The construction of the quasi-likelihood innlf follows the specific recommendations of Kendall et al (2005) In particular the choiceof radial basis functions fk in line 5 and the specification of mk and s in lines 3 and 4were proposed by Kendall et al (2005) based on trial and error The quasi-likelihood ismathematically most similar to a likelihood when min(c1L) = 1 so that `Q(θ) approximatesthe factorization of the likelihood in Eq 2 With this in mind it is natural to set c1L = 1 LHowever Kendall et al (2005) found that a two-step prediction criterion with min(c1L) = 2led to improved numerical performance It is natural to ask when one might choose to usequasi-likelihood estimation in place of full likelihood estimation implemented by SMC Someconsiderations follow closely related to the considerations for synthetic likelihood and ABC(Sections 34 and 35)

C1 NLF benefits from stationarity since (unlike SMC) it uses all time points in the simulationto build a prediction rule valid at all time points Indeed NLF has not been consideredapplicable for non-stationary models and on account of this nlf is not appropriate if themodel includes time-varying covariates An intermediate scenario between stationarityand full non-stationarity is seasonality where the dynamic model is forced by cyclicalcovariates and this is supported by nlf (cf B1 in Section 34)

C2 Potentially quasi-likelihood could be preferable to full likelihood in some situations Ithas been argued that a two-step prediction criterion may sometimes be more robust thanthe likelihood to model misspecification (Xia and Tong 2011) (cf B2)


Algorithm 7 Simulated quasi log likelihood for NLF Pseudocode for thequasi-likelihood function optimized by nlf( P start = θ0 nasymp = J nconverge =Bnrbf =K lags = c1L) Using notation from Table 1 P is a class lsquopomprsquo object with de-fined methods for rprocess rmeasure initstate and obs


fYn|Xn(yn |xn θ) parameter θ data ylowast1N collection of lags c1L length of

discarded transient B length of simulation J number of radial basis functions K1 Simulate long stationary time series Y1(B+J) sim fY1(B+J)

( middot θ)2 Set Ymin = minY(B+1)(B+J) Ymax = maxY(B+1)(B+J) and R = Ymax minus Ymin

3 Locations for basis functions mk = Ymin +Rtimes [12times (k minus 1)(K minus 1)minus1 minus 01] for k in 1 K4 Scale for basis functions s = 03timesR 5 Define radial basis functions fk(x) = exp(xminusmk)

22s2 for k in 1 K

6 Define prediction function H(ynminusc1 ynminusc2 ynminuscL) =sumL

j=1

sumKk=1 ajkfk(ynminuscj )

7 Compute ajk j isin 1L k isin 1K to minimize

σ2 =1

J

B+Jsumn=B+1

[Yn minusH(Ynminusc1 Ynminusc2 YnminuscL)

]2 (7)

8 Compute the simulated quasi log likelihood

ˆQ(θ) = minusN minus c

2log 2πσ2 minus

Nsumn=1+c

[ylowastn minusH(ylowastnminusc1 y

lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)

output Simulated quasi log likelihood ˆQ(θ)

complexity O(B) +O(J)

C3 Arguably two-step prediction should be viewed as a diagnostic tool that can be used tocomplement full likelihood analysis rather than replace it (Ionides 2011) (cf B3)

C4 NLF does not require that the model be Markovian (cf B4) although the pomp imple-mentation nlf does

C5 NLF is doubly plug-and-play (cf B5)

C6 The regression surface reconstruction carried out by NLF does not scale well with thedimension of the observed data NLF is recommended only for low-dimensional timeseries observations

NLF can be viewed as an estimating equation method and so standard errors can be com-puted by standard sandwich estimator or bootstrap techniques (Kendall et al 2005) Theoptimization in NLF is typically carried out with a fixed seed for the random number gen-erator so the simulated quasi-likelihood is a deterministic function If rprocess dependssmoothly on the random number sequence and on the parameters and the number of calls to


the random number generator does not depend on the parameters then fixing the seed resultsin a smooth objective function However some common components to model simulatorssuch as rnbinom make different numbers of calls to the random number generator dependingon the arguments which introduces nonsmoothness into the objective function

4 Model construction and data analysis Simple examples

41 A first example The Gompertz model

The plug-and-play methods in pomp were designed to facilitate data analysis based on com-plicated models but we will first demonstrate the basics of pomp using simple discrete-timemodels the Gompertz and Ricker models for population growth (Reddingius 1971 Ricker1954) The Ricker model will be introduced in Section 45 and used in Section 46 the re-mainder of Section 4 will use the Gompertz model The Gompertz model postulates that thedensity Xt+∆t of a population of organisms at time t + ∆t depends on the density Xt attime t according to

Xt+∆t = K1minuseminusr ∆tXeminusr ∆t

t εt (9)

In Eq 9 K is the carrying capacity of the population r is a positive parameter and theεt are independent and identically-distributed lognormal random variables with log εt simNormal(0 σ2) Additionally we will assume that the population density is observed witherrors in measurement that are lognormally distributed

log Yt sim Normal(logXt τ

2) (10)

Taking a logarithmic transform of Eq 9 gives

logXt+∆t sim Normal((

1minus eminusr∆t)

logK + eminusr∆t logXt σ2) (11)

On this transformed scale the model is linear and Gaussian and so we can obtain exactvalues of the likelihood function by applying the Kalman filter Plug-and-play methods arenot strictly needed this example therefore allows us to compare the results of generallyapplicable plug-and-play methods with exact results from the Kalman filter Later we willlook at the Ricker model and a continuous-time model for which no such special tricks areavailable

The first step in implementing this model in pomp is to construct an R object of class lsquopomprsquothat encodes the model and the data This involves the specification of functions to do some orall of rprocess rmeasure and dmeasure along with data and (optionally) other informationThe documentation (pomp) spells out the usage of the pomp constructor including detailedspecifications for all its arguments and links to several examples

To begin we will write a function that implements the process model simulator This is afunction that will simulate a single step (trarr t+ ∆t) of the unobserved process (Eq 9)

Rgt gompertzprocsim lt- function(x t params deltat )

+ eps lt- exp(rnorm(n = 1 mean = 0 sd = params[sigma]))

+ S lt- exp(-params[r] deltat)

+ setNames(params[K]^(1 - S) x[X]^S eps X)

+


The translation from the mathematical description (Eq 9) to the simulator is straightforwardWhen this function is called the argument x contains the state at time t The parameters(including K r and σ) are passed in the argument params Notice that x and params arenamed numeric vectors and that the output must likewise be a named numeric vector withnames that match those of x The argument deltat species the time-step size In this casethe time-step will be 1 unit we will see below how this is specified

Next we will implement a simulator for the observation process Eq 10

Rgt gompertzmeassim lt- function(x t params )

+ setNames(rlnorm(n = 1 meanlog = log(x[X]) sd = params[tau]) Y)

+

Again the translation from the measurement model Eq 10 is straightforward When the func-tion gompertzmeassim is called the named numeric vector x will contain the unobservedstates at time t params will contain the parameters as before This return value will be anamed numeric vector containing a single draw from the observation process (Eq 10)

Complementing the measurement model simulator is the corresponding measurement modeldensity which we implement as follows

Rgt gompertzmeasdens lt- function(y x t params log )

+ dlnorm(x = y[Y] meanlog = log(x[X]) sdlog = params[tau]

+ log = log)

+

We will need this later on for inference using pfilter mif and pmcmc When gompertzmeasdens

is called y will contain the observation at time t x and params will be as before and the pa-rameter log will indicate whether the likelihood (log=FALSE) or the log likelihood (log=TRUE)is required

With the above in place we build an object of class lsquopomprsquo via a call to pomp

Rgt gompertz lt- pomp(data = dataframe(time = 1100 Y = NA) times = time

+ rprocess = discretetimesim(stepfun = gompertzprocsim deltat = 1)

+ rmeasure = gompertzmeassim t0 = 0)

The first argument (data) specifies a data frame that holds the data and the times at whichthe data were observed Since this is a toy problem we have as yet no data in a momentwe will generate some simulated data The second argument (times) specifies which of thecolumns of data is the time variable The rprocess argument specifies that the process modelsimulator will be in discrete time with each step of duration deltat taken by the functiongiven in the stepfun argument The rmeasure argument specifies the measurement modelsimulator function t0 fixes t0 for this model here we have chosen this to be one time unitprior to the first observation

It is worth noting that implementing the rprocess rmeasure and dmeasure components as Rfunctions as we have done above leads to needlessly slow computation As we will see belowpomp provides facilities for specifying the model in C which can accelerate computationsmanyfold


Figure 1 Simulated data from the Gompertz model (Eqs 9 and 10) This figure shows theresult of executing plot(gompertz variables = Y)

Before we can simulate from the model we need to specify some parameter values Theparameters must be a named numeric vector containing at least all the parameters referencedby the functions gompertzprocsim and gompertzmeassim The parameter vector needsto determine the initial condition X(t0) as well Let us take our parameter vector to be

Rgt theta lt- c(r = 01 K = 1 sigma = 01 tau = 01 X0 = 1)

The parameters r K σ and τ appear in gompertzprocsim and gompertzmeassim Theinitial condition X0 is also given in theta The fact that the initial condition parameterrsquosname ends in 0 is significant it tells pomp that this is the initial condition of the state variableX This use of the 0 suffix is the default behavior of pomp one can however parameterize theinitial condition distribution arbitrarily using pomprsquos optional initializer argument

We can now simulate the model at these parameters

Rgt gompertz lt- simulate(gompertz params = theta)

Now gompertz is identical to what it was before except that the missing data have beenreplaced by simulated data The parameter vector (theta) at which the simulations wereperformed has also been saved internally to gompertz We can plot the simulated data via

Rgt plot(gompertz variables = Y)

Fig 1 shows the results of this operation

42 Computing likelihood using SMC

As discussed in Section 3 some parameter estimation algorithms in the pomp package aredoubly plug-and-play in that they require only rprocess and rmeasure These include the


nonlinear forecasting algorithm nlf the probe-matching algorithm probematch and ap-proximate Bayesian computation via abc The plug-and-play full-information methods inpomp however require dmeasure ie the ability to evaluate the likelihood of the data giventhe unobserved state The gompertzmeasdens above does this but we must fold it intothe class lsquopomprsquo object in order to use it We can do this with another call to pomp

Rgt gompertz lt- pomp(gompertz dmeasure = gompertzmeasdens)

The result of the above is a new class lsquopomprsquo object gompertz in every way identical to theone we had before but with the measurement-model density function dmeasure now specified

To estimate the likelihood of the data we can use the function pfilter an implementation ofAlgorithm 1 We must decide how many concurrent realizations (particles) to use the largerthe number of particles the smaller the Monte Carlo error but the greater the computationalburden Here we run pfilter with 1000 particles to estimate the likelihood at the trueparameters

Rgt pf lt- pfilter(gompertz params = theta Np = 1000)

Rgt logliktruth lt- logLik(pf)

Rgt logliktruth

[1] 3627102

Since the true parameters (ie the parameters that generated the data) are stored within theclass lsquopomprsquo object gompertz and can be extracted by the coef function we could have done

Rgt pf lt- pfilter(gompertz params = coef(gompertz) Np = 1000)

or simply

Rgt pf lt- pfilter(gompertz Np = 1000)

Now let us compute the log likelihood at a different point in parameter space one for whichr K and σ are each 50 higher than their true values

Rgt thetaguess lt- thetatrue lt- coef(gompertz)

Rgt thetaguess[c(r K sigma)] lt- 15 thetatrue[c(r K sigma)]

Rgt pf lt- pfilter(gompertz params = thetaguess Np = 1000)

Rgt loglikguess lt- logLik(pf)

Rgt loglikguess

[1] 2519585

In this case the Kalman filter computes the exact log likelihood at the true parameters tobe 3601 while the particle filter with 1000 particles gives 3627 Since the particle filtergives an unbiased estimate of the likelihood the difference is due to Monte Carlo error inthe particle filter One can reduce this error by using a larger number of particles andorby re-running pfilter multiple times and averaging the resulting estimated likelihoods The


latter approach has the advantage of allowing one to estimate the Monte Carlo error itselfwe will demonstrate this in Section 43

43 Maximum likelihood estimation via iterated filtering

Let us use the iterated filtering approach described in Section 32 to obtain an approximatemaximum likelihood estimate for the data in gompertz Since the parameters of Eqs 9 and 10are constrained to be positive when estimating we transform them to a scale on which theyare unconstrained The following encodes such a transformation and its inverse

Rgt gompertzlogtf lt- function(params ) log(params)

Rgt gompertzexptf lt- function(params ) exp(params)

We add these to the existing class lsquopomprsquo object via

Rgt gompertz lt- pomp(gompertz toEstimationScale = gompertzlogtf

+ fromEstimationScale = gompertzexptf)

The following codes initialize the iterated filtering algorithm at several starting points aroundthetatrue and estimate the parameters r τ and σ

Rgt estpars lt- c(r sigma tau)

Rgt library(foreach)

Rgt mif1 lt- foreach(i = 110 combine = c) dopar

+ thetaguess lt- thetatrue

+ rlnorm(n = length(estpars) meanlog = log(thetaguess[estpars])

+ sdlog = 1) -gt thetaguess[estpars]

+ mif(gompertz Nmif = 100 start = thetaguess transform = TRUE

+ Np = 2000 varfactor = 2 coolingfraction = 07

+ rwsd = c(r = 002 sigma = 002 tau = 002))

+

Rgt pf1 lt- foreach(mf = mif1 combine = c) dopar

+ pf lt- replicate(n = 10 logLik(pfilter(mf Np = 10000)))

+ logmeanexp(pf)

+

Note that we have set transform = TRUE in the call to mif above this causes the parame-ter transformations we have specified to be applied to enforce the positivity of parametersNote also that we have used the foreach package (Revolution Analytics and Weston 2014) toparallelize the computations

Each of the 10 mif runs ends up at a different point estimate (Fig 2) We focus on thatwith the highest estimated likelihood having evaluated the likelihood several times to reducethe Monte Carlo error in the likelihood evaluation The particle filter produces an unbiasedestimate of the likelihood therefore we will average the likelihoods not the log likelihoods


Figure 2 Convergence plots can be used to help diagnose convergence of the iterated filtering(IF) algorithm These and additional diagnostic plots are produced when plot is applied toa class lsquomifrsquo or class lsquomifListrsquo object

Rgt mf1 lt- mif1[[whichmax(pf1)]]

Rgt thetamif lt- coef(mf1)

Rgt loglikmif lt- replicate(n = 10 logLik(pfilter(mf1 Np = 10000)))

Rgt loglikmif lt- logmeanexp(loglikmif se = TRUE)

Rgt thetatrue lt- coef(gompertz)

Rgt logliktrue lt- replicate(n = 10 logLik(pfilter(gompertz Np = 20000)))

Rgt logliktrue lt- logmeanexp(logliktrue se = TRUE)

For the calculation above we have replicated the iterated filtering search made a carefulestimation of the log likelihood ˆ and its standard error using pfilter at each of the resultingpoint estimates and then chosen the parameter corresponding to the highest likelihood as ournumerical approximation to the MLE Taking advantage of the Gompertz modelrsquos tractabilitywe also use the Kalman filter to maximize the exact likelihood ` and evaluate it at theestimated MLE obtained by mif The resulting estimates are shown in Table 3 Usually thelast row and column of Table 3 would not be available even for a simulation study validating


r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788

Table 3 Results of estimating parameters r σ and τ of the Gompertz model (Eqs 9 and 10)by maximum likelihood using iterated filtering (Algorithm 3) compared with the exact MLEand with the true value of the parameter The first three columns show the estimated valuesof the three parameters The next two columns show the log likelihood ˆ estimated by SMC(Algorithm 1) and its standard error respectively The exact log likelihood ` is shown in therightmost column An ideal likelihood-ratio 95 confidence set not usually computationallyavailable includes all parameters having likelihood within qchisq(095df=3)2 = 391 ofthe exact MLE We see that both the mif MLE and the truth are in this set In this examplethe mif MLE is close to the exact MLE so it is reasonable to expect that profile likelihoodconfidence intervals and likelihood ratio tests constructed using the mif MLE have statisticalproperties similar to those based on the exact MLE

the inference methodology for a known POMP model In this case we see that the mif

procedure is successfully maximizing the likelihood up to an error of about 01 log units

44 Full-information Bayesian inference via PMCMC

To carry out Bayesian inference we need to specify a prior distribution on unknown parame-ters The pomp constructor function provides the rprior and dprior arguments which can befilled with functions that simulate from and evaluate the prior density respectively Methodsbased on random-walk Metropolis-Hastings require evaluation of the prior density (dprior)but not simulation (rprior) so we specify dprior for the Gompertz model as follows

Rgt hyperparams lt- list(min = coef(gompertz)10 max = coef(gompertz) 10)

Rgt gompertzdprior lt- function (params log)

+ f lt- sum(dunif(params min = hyperparams$min max = hyperparams$max

+ log = TRUE))

+ if (log) f else exp(f)

+

The PMCMC algorithm described in Section 33 can then be applied to draw a sample fromthe posterior Recall that for each parameter proposal PMCMC pays the full price of aparticle-filtering operation in order to obtain the Metropolis-Hastings acceptance probabilityFor the same price iterated filtering obtains in addition an estimate of the derivative anda probable improvement of the parameters For this reason PMCMC is relatively inefficientat traversing parameter space When Bayesian inference is the goal it is therefore advisableto first locate a neighborhood of the MLE using for example iterated filtering PMCMCcan then be initialized in this neighborhood to sample from the posterior distribution Thefollowing adopts this approach running 5 independent PMCMC chains using a multivariatenormal random-walk proposal (with diagonal variance-covariance matrix see mvndiagrw)


Figure 3 Diagnostic plots for the PMCMC algorithm The trace plots in the left column showthe evolution of 5 independent MCMC chains after a burn-in period of length 20000 Kerneldensity estimates of the marginal posterior distributions are shown at right The effectivesample size of the 5 MCMC chains combined is lowest for the r variable and is 250 the useof 40000 proposal steps in this case is a modest number The density plots at right show theestimated marginal posterior distributions The vertical line corresponds to the true value ofeach parameter

Rgt pmcmc1 lt- foreach(i=15combine=c) dopar

+ pmcmc(pomp(gompertz dprior = gompertzdprior) start = thetamif

+ Nmcmc = 40000 Np = 100 maxfail = Inf

+ proposal=mvndiagrw(c(r = 001 sigma = 001 tau = 001)))

+

Comparison with the analysis of Section 43 reinforces the observation of Bhadra (2010) thatPMCMC can require orders of magnitude more computation than iterated filtering Iteratedfiltering may have to be repeated multiple times while computing profile likelihood plotswhereas one successful run of PMCMC is sufficient to obtain all required posterior inferencesHowever in practice multiple runs from a range of starting points is always good practicesince convergence cannot be reliably assessed on the basis of a single chain To verify theconvergence of the approach or to compare the performance with other approaches we canuse diagnostic plots produced by the plot command (see Fig 3)


45 A second example The Ricker model

In Section 46 we will illustrate probe matching (Section 34) using a stochastic version of theRicker map (Ricker 1954) We switch models to allow direct comparison with Wood (2010)whose synthetic likelihood computations are reproduced below In particular the results ofSection 46 demonstrate frequentist inference using synthetic likelihood and also show thatthe full likelihood is both numerically tractable and reasonably well behaved contrary to theclaim of Wood (2010) We will also take the opportunity to demonstrate features of pompthat allow acceleration of model codes through the use of Rrsquos facilities for compiling anddynamically linking C code

The Ricker model is another discrete-time model for the size of a population The populationsize Nt at time t is postulated to obey

Nt+1 = r Nt exp(minusNt + et) etsimNormal(0 σ2

) (12)

In addition we assume that measurements Yt of Nt are themselves noisy according to

YtsimPoisson(φNt) (13)

where φ is a scaling parameter As before we will need to implement the modelrsquos state-process simulator (rprocess) We have the option of writing these functions in R as we didwith the Gompertz model However we can realize manyfold speed-ups by writing these inC In particular pomp allows us to write snippets of C code that it assembles compiles anddynamically links into a running R session To begin the process we will write snippets forthe rprocess rmeasure and dmeasure components

Rgt rickersim lt-

+ e = rnorm(0 sigma)

+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-

+ lik = dpois(y phi N give_log)

+

Note that in this implementation both N and e are state variables The logical flag give_log

requests the likelihood when FALSE the log likelihood when TRUE Notice that in thesesnippets we never declare the variables pomp will construct the appropriate declarationsautomatically

In a similar fashion we can add transformations of the parameters to enforce constraints

Rgt logtrans lt-

+ Tr = log(r)

+ Tsigma = log(sigma)

+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)

+ Tsigma = exp(sigma)

+ Tphi = exp(phi)

+ TN_0 = exp(N_0)

Note that in the foregoing C snippets the prefix T designates the transformed version of theparameter A full set of rules for using Csnippets including illustrative examples is given inthe package help system (Csnippet)

Now we can construct a class lsquopomprsquo object as before and fill it with simulated data

Rgt pomp(data = dataframe(time = seq(0 50 by = 1) y = NA)

+ rprocess = discretetimesim(stepfun = Csnippet(rickersim)

+ deltat = 1) rmeasure = Csnippet(rickerrmeas)

+ dmeasure = Csnippet(rickerdmeas)

+ toEstimationScale = Csnippet(logtrans)

+ fromEstimationScale = Csnippet(exptrans)

+ paramnames = c(r sigma phi N0 e0)

+ statenames = c(N e) times = time t0 = 0

+ params = c(r = exp(38) sigma = 03 phi = 10

+ N0 = 7 e0 = 0)) -gt ricker

Rgt ricker lt- simulate(rickerseed=73691676L)

46 Feature-based synthetic likelihood maximization

In pomp probes are simply functions that can be applied to an array of real or simulated datato yield a scalar or vector quantity Several functions that create useful probes are includedwith the package including those recommended by Wood (2010) In this illustration we willmake use of these probes probemarginal probeacf and probenlar probemarginal

regresses the data against a sample from a reference distribution the probersquos values are thoseof the regression coefficients probeacf computes the auto-correlation or auto-covariance ofthe data at specified lags probenlar fits a simple nonlinear (polynomial) autoregressivemodel to the data again the coefficients of the fitted model are the probersquos values Weconstruct a list of probes

Rgt plist lt- list(probemarginal(y ref = obs(ricker) transform = sqrt)

+ probeacf(y lags = c(0 1 2 3 4) transform = sqrt)

+ probenlar(y lags = c(1 1 1 2) powers = c(1 2 3 1)

+ transform = sqrt))

Each element of plist is a function of a single argument Each of these functions can beapplied to the data in ricker and to simulated data sets Calling pomprsquos function probe

results in the application of these functions to the data and to each of some large numbernsim of simulated data sets and finally to a comparison of the two [Note that probefunctions may be vector-valued so a single probe taking values in Rk formally corresponds


to a collection of k probe functions in the terminology of Section 34] Here we will applyprobe to the Ricker model at the true parameters and at a wild guess

Rgt pbtruth lt- probe(ricker probes = plist nsim = 1000 seed = 1066L)

Rgt guess lt- c(r = 20 sigma = 1 phi = 20 N0 = 7 e0 = 0)

Rgt pbguess lt- probe(ricker params = guess probes = plist nsim = 1000

+ seed = 1066L)

Results summaries and diagnostic plots showing the model-data agreement and correlationsamong the probes can be obtained by

Rgt summary(pbtruth)

Rgt summary(pbguess)

Rgt plot(pbtruth)

Rgt plot(pbguess)

An example of a diagnostic plot (using a smaller set of probes) is shown in Fig 4 Among thequantities returned by summary is the synthetic likelihood (Algorithm 5) One can attemptto identify parameters that maximize this quantity this procedure is referred to in pompas ldquoprobe matchingrdquo Let us now attempt to fit the Ricker model to the data using probe-matching

Rgt pm lt- probematch(pbguess est = c(r sigma phi) transform = TRUE

+ method = Nelder-Mead maxit = 2000 seed = 1066L reltol = 1e-08)

This code runs optimrsquos Nelder-Mead optimizer from the starting parameters guess in anattempt to maximize the synthetic likelihood based on the probes in plist Both the startingparameters and the list of probes are stored internally in pbguess which is why we need notspecify them explicitly here While probematch provides substantial flexibility in choice ofoptimization algorithm for situations requiring greater flexibility pomp provides the functionprobematchobjfun which constructs an objective function suitable for use with arbitraryoptimization routines

By way of putting the synthetic likelihood in context let us compare the results of estimatingthe Ricker model parameters using probe-matching and using iterated filtering (IF) which isbased on likelihood The following code runs 600 IF iterations starting at guess

Rgt mf lt- mif(ricker start = guess Nmif = 100 Np = 1000 transform = TRUE

+ coolingfraction = 095^50 varfactor = 2 iclag = 3

+ rwsd=c(r = 01 sigma = 01 phi = 01) maxfail = 50)

Rgt mf lt- continue(mf Nmif = 500 maxfail = 20)

Table 4 compares parameters Monte Carlo likelihoods (ˆ) and synthetic likelihoods (ˆS basedon the probes in plist) at each of (a) the guess (b) the truth (c) the MLE from mif and(d) the maximum synthetic likelihood estimate (MSLE) from probematch These resultsdemonstrate that it is possible and indeed not difficult to maximize the likelihood for thismodel contrary to the claim of Wood (2010) Since synthetic likelihood discards some of the


Figure 4 Results of plot on a class lsquoprobedpomprsquo-class object Above the diagonal thepairwise scatterplots show the values of the probes on each of 1000 data sets The red linesshow the values of each of the probes on the data The panels along the diagonal show thedistributions of the probes on the simulated data together with their values on the data anda two-sided p value The numbers below the diagonal are the Pearson correlations betweenthe corresponding pairs of probes


r σ φ ˆ se(ˆ) ˆS se(ˆS)

guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002

Table 4 Parameter estimation by means of maximum synthetic likelihood (Algorithm 5) vsby means of maximum likelihood via iterated filtering (Algorithm 3) The row labeled ldquoguessrdquocontains the point at which both algorithms were initialized That labeledldquotruthrdquocontains thetrue parameter value ie that at which the data were generated The rows labeled ldquoMLErdquoand ldquoMSLErdquo show the estimates obtained using iterated filtering and maximum syntheticlikelihood respectively Parameters r σ and τ were estimated all others were held at theirtrue values The columns labeled ˆ and ˆS are the Monte Carlo estimates of the log likelihoodand the log synthetic likelihood respectively their Monte Carlo standard errors are alsoshown While likelihood maximization results in an estimate for which both ˆ and ˆS exceedtheir values at the truth the value of ˆat the MSLE is smaller than at the truth an indicationof the relative statistical inefficiency of maximum synthetic likelihood

information in the data it is not surprising that Table 4 also shows the statistical inefficiencyof maximum synthetic likelihood relative to that of likelihood

47 Bayesian feature matching via ABC

Whereas synthetic likelihood carries out many simulations for each likelihood estimation ABC(as described in Section 35) uses only one Each iteration of ABC is therefore much quickeressentially corresponding to the cost of SMC with a single particle or synthetic likelihoodwith a single simulation A consequence of this is that ABC cannot determine a good relativescaling of the features within each likelihood evaluation and this must be supplied in advanceOne can imagine an adaptive version of ABC which modifies the scaling during the course ofthe algorithm but here we do a preliminary calculation to accomplish this We return to theGompertz model to faciliate comparison between ABC and PMCMC

Rgt plist lt- list(probemean(var = Y transform = sqrt)

+ probeacf(Y lags = c(0 5 10 20))

+ probemarginal(Y ref = obs(gompertz)))

+ psim lt- probe(gompertz probes = plist nsim = 500)

+ scaledat lt- apply(psim$simvals 2 sd)

Rgt abc1 lt- foreach(i = 15 combine = c) dopar

+ abc(pomp(gompertz dprior = gompertzdprior) Nabc = 4e6

+ probes = plist epsilon = 2 scale = scaledat


+

The effective sample size of the ABC chains is lowest for the r parameter (as was the casefor PMCMC) and is 450 as compared to 250 for pmcmc in Section 44 The total compu-tational effort allocated to abc here matches that for pmcmc since pmcmc used 100 particles


Figure 5 Marginal posterior distributions using full information via pmcmc (solid line) andpartial information via abc (dashed line) Kernel density estimates are shown for the posteriormarginal densities of log10(r) (left panel) log10(σ) (middle panel) and log10(τ) (right panel)The vertical lines indicate the true values of each parameter

for each likelihood evaluation but is awarded 100 times fewer Metropolis-Hastings steps Inthis example we conclude that abc mixes somewhat more rapidly (as measured by totalcomputational effort) than pmcmc Fig 5 investigates the statistical efficiency of abc on thisexample We see that abc gives rise to somewhat broader posterior distributions than thefull-information posteriors from pmcmc As in all numerical studies of this kind one cannotreadily generalize from one particular example even for this specific model and dataset theconclusions might be sensitive to the algorithmic settings However one should be aware ofthe possibility of losing substantial amounts of information even when the features are basedon reasoned scientific argument (Shrestha et al 2011 Ionides 2011) Despite this loss of sta-tistical efficiency points B2ndashB5 of Section 34 identify situations in which ABC may be theonly practical method available for Bayesian inference

48 Parameter estimation by simulated quasi-likelihood

Within the pomp environment it is fairly easy to try a quick comparison to see how nlf (Sec-tion 36) compares with mif (Section 32) on the Gompertz model Carrying out a simulationstudy with a correctly specified POMP model is appropriate for assessing computational andstatistical efficiency but does not contribute to the debate on the role of two-step predictioncriteria to fit misspecified models (Xia and Tong 2011 Ionides 2011) The nlf implementationwe will use to compare to the mif call from Section 43 is

Rgt nlf1 lt- nlf(gompertz nasymp = 1000 nconverge = 1000 lags = c(2 3)

+ start = c(r = 1 K = 2 sigma = 05 tau = 05 X0 = 1)

+ est = c(r sigma tau) transform = TRUE)


Figure 6 Comparison of mif and nlf for 10 simulated datasets using two criteria In bothplots the maximum likelihood estimate (MLE) θ obtained using iterated filtering is com-pared with the maximum simulated quasi-likelihood (MSQL) estimate θ obtained usingnonlinear forecasting (A) Improvement in estimated log likelihood ˆ at point estimate overthat at the true parameter value θ (B) Improvement in simulated log quasi-likelihood ˆ

Qat point estimate over that at the true parameter value θ In both panels the diagonal lineis the 1-1 line

where the first argument is the class lsquopomprsquo object start is a vector containing modelparameters at which nlfrsquos search will begin est contains the names of parameters nlf willestimate and lags specifies which past values are to be used in the autoregressive model Thetransform = TRUE setting causes the optimization to be performed on the transformed scaleas in Section 43 In the call above lags = c(2 3) specifies that the autoregressive modelpredicts each observation yt using ytminus2 and ytminus3 as recommended by Kendall et al (2005)The quasi-likelihood is optimized numerically so the reliability of the optimization should beassessed by doing multiple fits with different starting parameter values the results of a smallexperiment (not shown) indicate that on these simulated data repeated optimization is notneeded nlf defaults to optimization by the subplex method (Rowan 1990 King 2008) thoughall optimization methods provided by optim are available as well nasymp sets the length ofthe simulation on which the quasi-likelihood is based larger values will give less variableparameter estimates but will slow down the fitting process The computational demand ofnlf is dominated by the time required to generate the model simulations so efficient codingof rprocess is worthwhile

Fig 6 compares the true parameter θ with the maximum likelihood estimate (MLE) θfrom mif and the maximized simulated quasi-likelihood (MSQL) θ from nlf Fig 6A plotsˆ(θ) minus ˆ(θ) against ˆ(θ) minus ˆ(θ) showing that the MSQL estimate can fall many units of loglikelihood short of the MLE Fig 6B plots ˆ

Q(θ) minus ˆQ(θ) against ˆ

Q(θ) minus ˆQ(θ) showing

that likelihood-based inference is almost as good as nlf at optimizing the simulated quasi-likelihood criterion which nlf targets Fig 6 suggests that the MSQL may be inefficient since


S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro

Figure 7 Diagram of the SIR epidemic model The host population is divided into threeclasses according to infection status S susceptible hosts I infected (and infectious) hostsR recovered and immune hosts Births result in new susceptibles and all individuals have acommon death rate micro Since the birth rate equals the death rate the expected populationsize P = S+I+R remains constant The SrarrI rate λ called the force of infection dependson the number of infectious individuals according to λ(t) = β IN The IrarrR or recoveryrate is γ The case reports C result from a process by which new infections are recorded withprobability ρ Since diagnosed cases are treated with bed-rest and hence removed infectionsare counted upon transition to R

it can give estimates with poor behavior according to the statistically efficient criterion oflikelihood Another possibility is that this particular implementation of nlf was unfortunateEach mif optimization took 269 sec to run compared to 37 sec for nlf and it is possiblethat extra computer time or other algorithmic adjustments could substantially improve eitheror both estimators It is hard to ensure a fair comparison between methods and in practicethere is little substitute for some experimentation with different methods and algorithmicsettings on a problem of interest If the motivation for using NLF is preference for 2-stepprediction in place of the likelihood a comparison with SMC-based likelihood evaluation andmaximization is useful to inform the user of the consequences of that preference

5 A more complex example Epidemics in continuous time

51 A stochastic seasonal SIR model

A mainstay of theoretical epidemiology the SIR model describes the progress of a contagiousimmunizing infection through a population of hosts (Kermack and McKendrick 1927 An-derson and May 1991 Keeling and Rohani 2008) The hosts are divided into three classesaccording to their status vis-a-vis the infection (Fig 7) The susceptible class (S) containsthose that have not yet been infected and are thereby still susceptible to it the infected class(I) comprises those who are currently infected and by assumption infectious the removedclass (R) includes those who are recovered or quarantined as a result of the infection Individ-uals in R are assumed to be immune against reinfection We let S(t) I(t) and R(t) representthe numbers of individuals within the respective classes at time t

It is natural to formulate this model as a continuous-time Markov process In this process thenumbers of individuals within each class change through time in whole-number increments as


discrete births deaths and passages between compartments occur Let NAB be the stochasticcounting process whose value at time t is the number of individuals that have passed fromcompartment A to compartment B during the interval [t0 t) where t0 is an arbitrary startingpoint not later than the first observation We use the notation NmiddotS to refer to births and NAmiddotto refer to deaths from compartment A Let us assume that the per capita birth and deathrates and the rate of transition γ from I to R are constants The S to I transition rate theso-called force of infection λ(t) however should be an increasing function of I(t) For manyinfections it is reasonable to assume that the λ(t) is jointly proportional to the fraction ofthe population infected and the rate at which an individual comes into contact with othersHere we will make these assumptions writing λ(t) = β I(t)P where β is the transmissionrate and P = S + I +R is the population size We will go further and assume that birth anddeath rates are equal and independent of infection status we will let micro denote the commonrate A consequence is that the expected population size remains constant

The continuous-time Markov process is fully specified by the infinitesimal increment proba-bilities Specifically writing ∆N(t) = N(t+ h)minusN(t) we have

P[∆NmiddotS(t)=1 |S(t) I(t) R(t)

]= microP (t)h+ o(h)

P [∆NSI(t)=1 |S(t) I(t) R(t)] = λ(t)S(t)h+ o(h)

P [∆NIR(t)=1 |S(t) I(t) R(t)] = γ I(t)h+ o(h)

P[∆NSmiddot(t)=1 |S(t) I(t) R(t)

]= microS(t)h+ o(h)

P[∆NImiddot(t)=1 |S(t) I(t) R(t)

]= micro I(t)h+ o(h)

P[∆NRmiddot(t)=1 |S(t) I(t) R(t)

]= microR(t)h+ o(h)

(14)

together with statement that all events of the form

∆NAB(t)gt 1 and ∆NAB(t)=1∆NCD(t)=1

for A B C D with (AB) 6= (CD) have probability o(h) The counting processes arecoupled to the state variables (Breto and Ionides 2011) via the following identities

∆S(t) = ∆NmiddotS(t)minus∆NSI(t)minus∆NSmiddot(t)∆I(t) = ∆NSI(t)minus∆NIR(t)minus∆NImiddot(t)∆R(t) = ∆NIR(t)minus∆NRmiddot(t)

(15)

Taking expectations of Eqs 14 and 15 dividing through by h and taking a limit as h darr 0one obtains a system of nonlinear ordinary differential equations which is known as the de-terministic skeleton of the model (Coulson et al 2004) Specifically the SIR deterministicskeleton is

dS

dt= micro (P minus S)minus β I

PS

dI

dt= β

I

PS minus γ I minus micro I

dR

dt= γ I minus microR

(16)

It is typically impossible to monitor S I and R directly It sometimes happens howeverthat public health authorities keep records of cases ie individual infections The number of


cases C(t1 t2) recorded within a given reporting interval [t1 t2) might perhaps be modeledby a negative binomial process

C(t1 t2) sim NegBin(ρ∆NSI(t1 t2) θ) (17)

where ∆NSI(t1 t2) is the true incidence (the accumulated number of new infections that haveoccured over the [t1 t2) interval) ρ is the reporting rate (the probability that an infection isobserved and recorded) θ is the negative binomialldquosizerdquoparameter and the notation is meantto indicate that E [C(t1 t2) |∆NSI(t1 t2) = H] = ρH and Var [C(t1 t2) |∆NSI(t1 t2) = H] =ρH + ρ2H2θ The fact that the observed data are linked to an accumulation as opposed toan instantaneous value introduces a slight complication which we discuss below

52 Implementing the SIR model in pomp

As before we will need to write functions to implement some or all of the SIR modelrsquosrprocess rmeasure and dmeasure components As in Section 45 we will write thesecomponents using pomprsquos Csnippets Recall that these are snippets of C code that pompautomatically assembles compiles and dynamically loads into the running R session

To start with we will write snippets that specify the measurement model (rmeasure anddmeasure)

Rgt rmeas lt-

+ cases = rnbinom_mu(theta rho H)

+

Rgt dmeas lt-

+ lik = dnbinom_mu(cases theta rho H give_log)

+

Here we are using cases to refer to the data (number of reported cases) and H to refer tothe true incidence over the reporting interval The negative binomial simulator rnbinom_mu

and density function dnbinom_mu are provided by R The logical flag give_log requeststhe likelihood when FALSE the log likelihood when TRUE Notice that in these snippets wenever declare the variables pomp will ensure that the state variable (H) observable (cases)parameters (theta rho) and likelihood (lik) are defined in the contexts within which thesesnippets are executed

For the rprocess portion we could simulate from the continuous-time Markov process exactly(Gillespie 1977) the pomp function gillespiesim implements this algorithm However forpractical purposes the exact algorithm is often prohibitively slow If we are willing to livewith an approximate simulation scheme we can use the so-called ldquotau-leaprdquo algorithm oneversion of which is implemented in pomp via the eulersim plug-in This algorithm holds thetransition rates λ micro γ constant over a small interval of time ∆t and simulates the numbersof births deaths and transitions that occur over that interval It then updates the statevariables S I R accordingly increments the time variable by ∆t recomputes the transitionrates and repeats Naturally as ∆t rarr 0 this approximation to the true continuous-timeprocess becomes better and better The critical feature from the inference point of viewhowever is that no relationship need be assumed between the Euler simulation interval ∆tand the reporting interval which itself need not even be the same from one observation tothe next


Under the above assumptions the number of individuals leaving any of the classes by allavailable routes over a particular time interval is a multinomial process For example if∆NSI and ∆NS are the numbers of S individuals acquiring infection and dying respectivelyover the Euler simulation interval [t t+ ∆t) then

(∆NSI∆NS S minus∆NSI minus∆NS) sim Multinom (S(t) pSrarrI pSrarr 1minus pSrarrI minus pSrarr) (18)

where

pSrarrI =λ(t)

λ(t) + micro

(1minus eminus(λ(t)+micro) ∆t

)pSrarr =

micro

λ(t) + micro


)

(19)

By way of shorthand we say that the random variable (∆NSI∆NS) in Eq 18 has an Euler-multinomial distribution Such distributions arise with sufficient frequency in compartmen-tal models that pomp provides convenience functions for them Specifically the functionsreulermultinom and deulermultinom respectively draw random deviates from and evaluatethe probability mass function of such distributions As the help pages relate reulermultinomand deulermultinom parameterize the Euler-multinomial distributions by the size (S(t) inEq 18) rates (λ(t) and micro) and time interval ∆t Obviously the Euler-multinomial distribu-tions generalize to an arbitrary number of exit routes

The help page (eulersim) informs us that to use eulersim we need to specify a functionthat advances the states from t to t + ∆t Again we will write this in C to realize fasterrun-times

Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R

+ rate[0] = mu P birth

+ rate[1] = beta I P transmission

+ rate[2] = mu death from S

+ rate[3] = gamma recovery

+ rate[4] = mu death from I

+ rate[5] = mu death from R

+ dN[0] = rpois(rate[0] dt)

+ reulermultinom(2 S amprate[1] dt ampdN[1])

+ reulermultinom(2 I amprate[3] dt ampdN[3])

+ reulermultinom(1 R amprate[5] dt ampdN[5])

+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+

As before pomp will ensure that the undeclared state variables and parameters are definedin the context within which the snippet is executed Note however that in the above we


do declare certain local variables In particular the rate and dN arrays hold the rates andnumbers of transition events respectively Note too that we make use of pomprsquos C interfaceto reulermultinom documented in the package help pages (reulermultinom) The packagehelp system (Csnippet) includes instructions for and examples of the use of Csnippets

Two significant wrinkles remains to be explained First notice that in sirstep the variable Hsimply accumulates the numbers of new infections H is a counting process that is nondecreas-ing with time In fact the incidence within an interval [t1 t2) is ∆NSI(t1 t2) = H(t2)minus H(t1)This leads to a technical difficulty with the measurement process however in that the dataare assumed to be records of new infections occurring within the latest reporting intervalwhile the process model tracks the accumulated number of new infections since time t0 Wecan get around this difficulty by re-setting H to zero immediately after each observation Wecause pomp to do this via the pomp functionrsquos zeronames argument as we will see in a mo-ment The section on ldquoaccumulator variablesrdquo in the pomp help page (pomp) discusses this inmore detail

The second wrinkle has to do with the initial conditions ie the states S(t0) I(t0) R(t0)By default pomp will allow us to specify these initial states arbitrarily For the model tobe consistent they should be positive integers that sum to the population size N We canenforce this constraint by customizing the parameterization of our initial conditions We dothis in by furnishing a custom initializer in the call to pomp Let us construct it now andfill it with simulated data

Rgt pomp(data = dataframe(cases = NA time = seq(0 10 by=152))

+ times = time t0 = -152 dmeasure = Csnippet(dmeas)

+ rmeasure = Csnippet(rmeas) rprocess = eulersim(

+ stepfun = Csnippet(sirstep) deltat = 15220)

+ statenames = c(S I R H)

+ paramnames = c(gamma mu theta beta popsize

+ rho S0 I0 R0) zeronames=c(H)

+ initializer=function(params t0 )

+ fracs lt- params[c(S0 I0 R0)]

+ setNames(c(round(params[popsize]fracssum(fracs))0)

+ c(SIRH))

+ params = c(popsize = 500000 beta = 400 gamma = 26

+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1

Rgt simulate(sir1 seed = 1914679908L) -gt sir1

Notice that we are assuming here that the data are collected weekly and use an Euler step-sizeof 120 wk Here we have assumed an infectious period of 2 wk (1γ = 126 yr) and a basicreproductive number R0 of β(γ + micro) asymp 15 We have assumed a host population size of500000 and 10 reporting efficiency Fig 8 shows one realization of this process


Figure 8 Result of plot(sir1) The class lsquopomprsquo object sir1 contains the SIR model withsimulated data


53 Complications Seasonality imported infections extra-demographicstochasticity

To illustrate the flexibility afforded by pomprsquos plug-and-play methods let us add a bit ofreal-world complexity to the simple SIR model We will modify the model to take four factsinto account

1 For many infections the contact rate is seasonal β = β(t) varies in more or less periodicfashion with time

2 The host population may not be truly closed imported infections arise when infectedindividuals visit the host population and transmit

3 The host population need not be constant in size If we have data for example on thenumbers of births occurring in the population we can incorporate this directly into themodel

4 Stochastic fluctuation in the rates λ micro and γ can give rise to extrademographic stochas-ticity ie random process variability beyond the purely demographic stochasticity wehave included so far

To incorporate seasonality we would like to assume a flexible functional form for β(t) Herewe will use a three-coefficient Fourier series

log β(t) = b0 + b1 cos 2πt+ b2 sin 2πt (20)

There are a variety of ways to account for imported infections Here we will simply assumethat there is some constant number ι of infected hosts visiting the population Putting thistogether with the seasonal contact rate results in a force of infection λ(t) = β(t) (I(t) + ι) N

To incorporate birth-rate information let us suppose we have data on the number of birthsoccurring each month in this population and that these data are in the form of a data framebirthdat with columns time and births We can incorporate the varying birth rate intoour model by passing it as a covariate to the simulation code When we pass birthdat asthe covar argument to pomp we cause a look-up table to be created and made available tothe simulator The package employs linear interpolation to provide a value of each variablein the covariate table at any requisite time from the userrsquos perspective a variable births

will simply be available for use by the model codes

Finally we can allow for extrademographic stochasticity by allowing the force of infection tobe itself a random variable We will accomplish this by assuming a random phase in β

λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)

where the phase Φ satisfies the stochastic differential equation

dΦ = dt+ σ dWt (22)

where dW (t) is a white noise specifically an increment of standard Brownian motion Thismodel assumption attempts to capture variability in the timing of seasonal changes in trans-mission rates As σ varies it can represent anything from a very mild modulation of thetiming of the seasonal progression to much more intense variation

Let us modify the process-model simulator to incorporate these complexities


Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW

+ Beta = exp(b1 + b2 cos(M_2PI Phi) + b3 sin(M_2PI Phi))

+ rate[0] = births birth

+ rate[1] = Beta (I + iota) P infection









+ dW = rnorm(dt sigma sqrt(dt))

+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]

+ noise += (dW - dt) sigma

+

Rgt pomp(sir1 rprocess = eulersim(

+ stepfun = Csnippet(seassirstep) deltat = 15220)

+ dmeasure = Csnippet(dmeas) rmeasure = Csnippet(rmeas)

+ covar = birthdat tcovar = time zeronames = c(H noise)

+ statenames = c(S I R H P Phi noise)

+ paramnames = c(gamma mu popsize rhothetasigma

+ S0 I0 R0 b1 b2 b3 iota)

+ initializer = function(params t0 )


+ setNames(c(round(params[popsize]c(fracssum(fracs)1))000)

+ c(SIRPHPhinoise))

+ params = c(popsize = 500000 iota = 5 b1 = 6 b2 = 02

+ b3 = -01 gamma = 26 mu = 150 rho = 01 theta = 100

+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2


Fig 9 shows the simulated data and latent states The sir2 object we have constructed herecontains all the key elements of models used within the pomp to investigate cholera (Kinget al 2008) measles (He et al 2010) malaria (Bhadra et al 2011) pertussis (Blackwoodet al 2013a Lavine et al 2013) pneumococcal pneumonia (Shrestha et al 2013) rabies(Blackwood et al 2013b) and Ebola virus disease (King et al 2015a)


Figure 9 One realization of the SIR model with seasonal contact rate imported infectionsand extrademographic stochasticity in the force of infection


6 Conclusion

The pomp package is designed to be both a tool for data analysis based on POMP modelsand a sound platform for the development of inference algorithms The model specificationlanguage provided by pomp is very general Implementing a POMP model in pomp makes awide range of inference algorithms available Moreover the separation of model from inferencealgorithm facilitates objective comparison of alternative models and methods The examplesdemonstrated in this paper are relatively simple but the package has been instrumental in anumber of scientific studies (eg King et al 2008 Bhadra et al 2011 Shrestha et al 2011Earn et al 2012 Roy et al 2012 Shrestha et al 2013 Blackwood et al 2013ab Lavine et al2013 He et al 2013 Breto 2014 King et al 2015a)

As a development platform pomp is particularly convenient for implementing algorithms withthe plug-and-play property since models will typically be defined by their rprocess simulatortogether with rmeasure and often dmeasure but can accommodate inference methods basedon other model components such as dprocess and skeleton (the deterministic skeleton of thelatent process) As an open-source project the package readily supports expansion and theauthors invite community participation in the pomp project in the form of additional infer-ence algorithms improvements and extensions of existing algorithms additional modeldataexamples documentation contributions and improvements bug reports and feature requests

Complex models and large datasets can challenge computational resources With this in mindkey components of the pomp are written in C and pomp provides facilities for users to writemodels either in R or for the acceleration that typically proves necessary in applicationsin C Multi-processor computing also becomes necessary for ambitious projects The twomost common computationally intensive tasks are assessment of Monte Carlo variability andinvestigation of the role of starting values and other algorithmic settings on optimizationroutines These analyses require only embarrassingly parallel computations and need nospecial discussion here

The package contains more examples (via pompExample) which can be used as templates forimplementation of new models the R and C code underlying these examples is provided withthe package In addition pomp provides a number of interactive demos (via demo) Furtherdocumentation and an introductory tutorial are provided with the package and on the pompwebsite httpkingaagithubiopomp

Acknowledgments Initial development of pomp was carried out as part of the Inferencefor Mechanistic Models working group supported from 2007 to 2010 by the National Centerfor Ecological Analysis and Synthesis a center funded by the US National Science Foun-dation (Grant DEB-0553768) the University of California Santa Barbara and the State ofCalifornia Participants were C Breto S P Ellner M J Ferrari G J Gibson G HookerE L Ionides V Isham B E Kendall K Koelle A A King M L Lavine K B NewmanD C Reuman P Rohani and H J Wearing As of this writing the pomp development teamis A A King E L Ionides and D Nguyen Financial support was provided by grants DMS-1308919 DMS-0805533 EF-0429588 from the US National Science Foundation and by theResearch and Policy for Infectious Disease Dynamics program of the Science and TechnologyDirectorate US Department of Homeland Security and the Fogarty International CenterUS National Institutes of Health


References

Anderson RM May RM (1991) Infectious Diseases of Humans Oxford University PressOxford

Andrieu C Doucet A Holenstein R (2010) ldquoParticle Markov Chain Monte Carlo MethodsrdquoJournal of the Royal Statistical Society B 72(3) 269ndash342 doi101111j1467-9868

200900736x

Andrieu C Roberts GO (2009) ldquoThe Pseudo-Marginal Approach for Efficient ComputationrdquoThe Annals of Statistics 37(2) 697ndash725 doi10121407-AOS574

Arulampalam MS Maskell S Gordon N Clapp T (2002) ldquoA Tutorial on Particle Filtersfor Online Nonlinear Non-Gaussian Bayesian Trackingrdquo IEEE Transactions on SignalProcessing 50 174 ndash 188

Beaumont MA (2010) ldquoApproximate Bayesian Computation in Evolution and EcologyrdquoAnnual Review of Ecology Evolution and Systematics 41 379ndash406 doi101146

annurev-ecolsys-102209-144621

Bhadra A (2010) ldquoDiscussion of lsquoParticle Markov Chain Monte Carlo Methodsrsquo by C AndrieuA Doucet and R Holensteinrdquo Journal of the Royal Statistical Society B 72 314ndash315 doi101111j1467-9868200900736x

Bhadra A Ionides EL Laneri K Pascual M Bouma M Dhiman R (2011) ldquoMalaria inNorthwest India Data Analysis via Partially Observed Stochastic Differential EquationModels Driven by Levy Noiserdquo Journal of the American Statistical Association 106(494)440ndash451 doi101198jasa2011ap10323

Blackwood JC Cummings DAT Broutin H Iamsirithaworn S Rohani P (2013a) ldquoDeci-phering the Impacts of Vaccination and Immunity on Pertussis Epidemiology in ThailandrdquoProceedings of the National Academy of Sciences of the USA 110(23) 9595ndash9600 doi

101073pnas1220908110

Blackwood JC Streicker DG Altizer S Rohani P (2013b) ldquoResolving the Roles of ImmunityPathogenesis and Immigration for Rabies Persistence in Vampire Batsrdquo Proceedings of theNational Academy of Sciences of the USA doi101073pnas1308817110

Blake IM Martin R Goel A Khetsuriani N Everts J Wolff C Wassilak S Aylward RBGrassly NC (2014) ldquoThe Role of Older Children and Adults in Wild Poliovirus Trans-missionrdquo Proceedings of the National Academy of Sciences of the USA 111(29) 10604ndash10609 doi101073pnas1323688111

Breto C (2014) ldquoOn Idiosyncratic Stochasticity of Financial Leverage Effectsrdquo Statistics ampProbability Letters 91 20ndash26 ISSN 0167-7152 doi101016jspl201404003

Breto C He D Ionides EL King AA (2009) ldquoTime Series Analysis via Mechanistic ModelsrdquoThe Annals of Applied Statistics 3 319ndash348 doi10121408-AOAS201


Breto C Ionides EL (2011) ldquoCompound Markov Counting Processes and their Applica-tions to Modeling Infinitesimally Over-dispersed Systemsrdquo Stochastic Processes and theirApplications 121(11) 2571ndash2591 doi101016jspa201107005

Cappe O Godsill S Moulines E (2007) ldquoAn Overview of Existing Methods and RecentAdvances in Sequential Monte Carlordquo Proceedings of the IEEE 95(5) 899ndash924 doi

101109JPROC2007893250

Chambers JM (1998) Programming with Data Springer-Verlag New York ISBN 0-387-98503-4 URL httpcmbell-labscomcmmsdepartmentssiaSbook

Commandeur JJF Koopman SJ Ooms M (2011) ldquoStatistical Software for State Space Meth-odsrdquo Journal of Statistical Software 41(1) 1ndash18 URL httpwwwjstatsoftorgv41

i01

Coulson T Rohani P Pascual M (2004) ldquoSkeletons Noise and Population Growth The Endof an Old Debaterdquo Trends in Ecology and Evolution 19 359ndash364

Douc R Cappe O Moulines E (2005) ldquoComparison of Resampling Schemes for Particle Fil-teringrdquo In Proceedings of the 4th International Symposium on Image and Signal Processingand Analysis 2005 pp 64ndash69 IEEE doi101109ISPA2005195385

Doucet A Johansen A (2009) ldquoA Tutorial on Particle Filtering and Smoothing Fifteen YearsLaterrdquo In D Crisan B Rozovsky (eds) Oxford Handbook of Nonlinear Filtering OxfordUniversity Press

Earn DJD He D Loeb MB Fonseca K Lee BE Dushoff J (2012) ldquoEffects of School Closureon Incidence of Pandemic Influenza in Alberta Canadardquo The Annals of Internal Medicine156(3) 173ndash181 doi1073260003-4819-156-3-201202070-00005

Ellner SP Bailey BA Bobashev GV Gallant AR Grenfell BT Nychka DW (1998) ldquoNoiseand Nonlinearity in Measles Epidemics Combining Mechanistic and Statistical Approachesto Population Modelingrdquo American Naturalist 151(5) 425ndash440 doi101086286130

Genolini C (2008) ldquoA (Not So) Short Introduction to S4rdquo Technical report The RProject for Statistical Computing URL httpcranr-projectorgdoccontrib

Genolini-S4tutorialV0-5enpdf

Gillespie DT (1977) ldquoExact Stochastic Simulation of Coupled Chemical Reactionsrdquo Journalof Physical Chemistry 81(25) 2340ndash2361 doi101021j100540a008

Gompertz B (1825) ldquoOn the Nature of the Function Expressive of the Law of Human Mor-tality and on a New Mode of Determining the Value of Life Contingenciesrdquo PhilosophicalTransactions of the Royal Society of London 115 513ndash583 doi101098rstl1825

0026

He D Dushoff J Day T Ma J Earn DJD (2013) ldquoInferring the Causes of the Three Wavesof the 1918 Influenza Pandemic in England and Walesrdquo Proceedings of the Royal Societyof London B 280(1766) 20131345 doi101098rspb20131345


He D Ionides EL King AA (2010) ldquoPlug-and-play Inference for Disease Dynamics Measlesin Large and Small Towns as a Case Studyrdquo Journal of the Royal Society Interface 7(43)271ndash283 doi101098rsif20090151

Ionides EL (2011) ldquoDiscussion on ldquoFeature Matching in Time Series Modelingrdquo by Y Xiaand H Tongrdquo Statistical Science 26 49ndash52 doi10121411-STS345C

Ionides EL Bhadra A Atchade Y King AA (2011) ldquoIterated Filteringrdquo The Annals ofStatistics 39(3) 1776ndash1802 doi10121411-AOS886

Ionides EL Breto C King AA (2006) ldquoInference for Nonlinear Dynamical Systemsrdquo Pro-ceedings of the National Academy of Sciences of the USA 103(49) 18438ndash18443 doi

101073pnas0603181103

Ionides EL Nguyen D Atchade Y Stoev S King AA (2015) ldquoInference for Dynamic andLatent Variable Models via Iterated Perturbed Bayes Mapsrdquo Proceedings of the NationalAcademy of Sciences of the USA doi101073pnas1410597112

Johnson SG (2014) The NLopt Nonlinear-Optimization Package Version 242 URL http

ab-initiomitedunlopt

Kantas N Doucet A Singh SS Maciejowski JM (2015) ldquoOn Particle Methods for ParameterEstimation in State-Space Modelsrdquo ArXiv14128695 URL httparxivorgabs1412

8695

Keeling M Rohani P (2008) Modeling infectious diseases in humans and animals PrincetonUniversity Press Princeton ISBN 0691116172

Kendall BE Briggs CJ Murdoch WW Turchin P Ellner SP McCauley E Nisbet RM WoodSN (1999) ldquoWhy do Populations Cycle A Synthesis of Statistical and Mechanistic Mod-eling Approachesrdquo Ecology 80(6) 1789ndash1805 doi1018900012-9658(1999)080[1789WDPCAS]20CO2

Kendall BE Ellner SP McCauley E Wood SN Briggs CJ Murdoch WW Turchin P (2005)ldquoPopulation Cycles in the Pine Looper Moth Dynamical Tests of Mechanistic HypothesesrdquoEcological Monographs 75(2) 259ndash276 URL httpwwwjstororgstable4539097

Kermack WO McKendrick AG (1927) ldquoA Contribution to the Mathematical Theory ofEpidemicsrdquo Proceedings of the Royal Society of London Series A 115 700ndash721 URLhttpwwwjstororgstable94815

King AA (2008) subplex Subplex Optimization Algorithm R package version 11-4 URLhttpsubplexr-forger-projectorg

King AA Domenech de Celles M Magpantay FMG Rohani P (2015a) ldquoAvoidable Errorsin the Modelling of Outbreaks of Emerging Pathogens with Special Reference to EbolardquoProceedings of the Royal Society of London Series B 282(1806) 20150347 doi101098rspb20150347

King AA Ionides EL Breto CM Ellner SP Ferrari MJ Kendall BE Lavine M NguyenD Reuman DC Wearing H Wood SN (2015b) pomp Statistical Inference for Partially


Observed Markov Processes R package version 065-1 URL httpkingaagithubio

pomp

King AA Ionides EL Pascual M Bouma MJ (2008) ldquoInapparent Infections and CholeraDynamicsrdquo Nature 454(7206) 877ndash880 doi101038nature07084

Kitagawa G (1998) ldquoA Self-Organising State Space Modelrdquo Journal of the American Statis-tical Association 93 1203ndash1215 doi10108001621459199810473780

Lavine JS King AA Andreasen V Bjoslashrnstad ON (2013) ldquoImmune Boosting ExplainsRegime-Shifts in Prevaccine-Era Pertussis Dynamicsrdquo PLoS ONE 8(8) e72086 doi

101371journalpone0072086

Liu J West M (2001) ldquoCombining Parameter and State Estimation in Simulation-BasedFilteringrdquo In A Doucet N de Freitas NJ Gordon (eds) Sequential Monte Carlo Methodsin Practice pp 197ndash224 Springer-Verlag New York

Liu JS (2001) Monte Carlo Strategies in Scientific Computing Springer-Verlag New York

Marin JM Pudlo P Robert CP Ryder RJ (2012) ldquoApproximate Bayesian Com-putational Methodsrdquo Statistics and Computing 22(6) 1167ndash1180 doi101007

s11222-011-9288-2

Ratmann O Andrieu C Wiuf C Richardson S (2009) ldquoModel Criticism Based on Likelihood-free Inference with an Application to Protein Network Evolutionrdquo Proceedings of theNational Academy of Sciences of the USA 106(26) 10576ndash10581 doi101073pnas

0807882106

Reddingius J (1971) Gambling for Existence A Discussion of some Theoretical Problems inAnimal Population Ecology B J Brill Leiden

Reuman DC Desharnais RA Costantino RF Ahmad OS Cohen JE (2006) ldquoPower SpectraReveal the Influence Of Stochasticity on Nonlinear Population Dynamicsrdquo Proceedings ofthe National Academy of Sciences of the USA 103(49) 18860ndash18865 doi101073pnas0608571103

Revolution Analytics Weston S (2014) foreach Foreach Looping Construct for R R packageversion 142 URL httpCRANR-projectorgpackage=foreach

Ricker WE (1954) ldquoStock and Recruitmentrdquo Journal of the Fisheries Research Board ofCanada 11 559ndash623 doi101139f54-039

Rowan T (1990) Functional Stability Analysis of Numerical Algorithms PhD thesis De-partment of Computer Sciences University of Texas at Austin

Roy M Bouma MJ Ionides EL Dhiman RC Pascual M (2012) ldquoThe Potential Eliminationof Plasmodium Vivax Malaria by Relapse Treatment Insights from a Transmission Modeland Surveillance Data from NW Indiardquo PLoS Neglected Tropical Diseases 7(1) e1979doi101371journalpntd0001979

Shaman J Karspeck A (2012) ldquoForecasting Seasonal Outbreaks of Influenzardquo Proceedings ofthe National Academy of Sciences of the USA 109(50) 20425ndash20430 doi101073pnas1208772109


Shrestha S Foxman B Weinberger DM Steiner C Viboud C Rohani P (2013) ldquoIdentifyingthe Interaction between Influenza and Pneumococcal Pneumonia Using Incidence DatardquoScience Translational Medicine 5(191) 191ra84 doi101126scitranslmed3005982

Shrestha S King AA Rohani P (2011) ldquoStatistical Inference for Multi-Pathogen SystemsrdquoPLoS Computational Biology 7 e1002135 doi101371journalpcbi1002135

Sisson SA Fan Y Tanaka MM (2007) ldquoSequential Monte Carlo without LikelihoodsrdquoProceedings of the National Academy of Sciences of the USA 104(6) 1760ndash1765 doi

101073pnas0607208104

Smith AA (1993) ldquoEstimating Nonlinear Time-series Models using Simulated Vector Au-toregressionrdquo Journal of Applied Econometrics 8(S1) S63ndashS84 doi101002jae

3950080506

Toni T Welch D Strelkowa N Ipsen A Stumpf MP (2009) ldquoApproximate Bayesian Compu-tation Scheme for Parameter Inference and Model Selection in Dynamical Systemsrdquo Journalof the Royal Society Interface 6(31) 187ndash202 doi101098rsif20080172

Wan E Van Der Merwe R (2000) ldquoThe Unscented Kalman Filter for Nonlinear EstimationrdquoIn Adaptive Systems for Signal Processing Communications and Control pp 153ndash158doi101109ASSPCC2000882463

Wood SN (2001) ldquoPartially Specified Ecological Modelsrdquo Ecological Monographs 71 1ndash25doi1018900012-9615(2001)071[0001PSEM]20CO2

Wood SN (2010) ldquoStatistical Inference for Noisy Nonlinear Ecological Dynamic SystemsrdquoNature 466(7310) 1102ndash1104 doi101038nature09319

Xia Y Tong H (2011) ldquoFeature Matching in Time Series Modellingrdquo Statistical Science26(1) 21ndash46 doi10121410-STS345

Ypma J (2014) nloptr R Interface to NLopt R package version 100 URL httpcran

r-projectorgwebpackagesnloptr

Affiliation

Aaron A KingDepartments of Ecology amp Evolutionary Biology and MathematicsCenter for the Study of Complex SystemsUniversity of Michigan48109 Michigan USAE-mail kingaaunicheduURL httpkinglabeeblsaumichedu


Dao NguyenDepartment of StatisticsUniversity of Michigan48109 Michigan USAE-mail nguyenxdunichedu

Edward IonidesDepartment of StatisticsUniversity of Michigan48109 Michigan USAE-mail ionidesunicheduURL httpwwwstatlsaumichedu~ionides

Introduction

POMP models and their representation in pomp

Implementation of POMP models

Initial conditions

Covariates

Methodology for POMP models

The likelihood function and sequential Monte Carlo

Iterated filtering

Particle Markov chain Monte Carlo

Synthetic likelihood of summary statistics

Approximate Bayesian computation (ABC)

Nonlinear forecasting

Model construction and data analysis Simple examples

A first example The Gompertz model

Computing likelihood using SMC

Maximum likelihood estimation via iterated filtering

Full-information Bayesian inference via PMCMC

A second example The Ricker model

Feature-based synthetic likelihood maximization

Bayesian feature matching via ABC

Parameter estimation by simulated quasi-likelihood

A more complex example Epidemics in continuous time

A stochastic seasonal SIR model

Implementing the SIR model in pomp

Complications Seasonality imported infections extra-demographic stochasticity

Conclusion


approximations are adequate for onersquos purposes or when the latent process takes values in asmall discrete set methods that exploit these additional assumptions to advantage such asthe extended and ensemble Kalman filter methods or exact hidden-Markov-model methodsare available but not yet as part of pomp It is the class of nonlinear non-Gaussian POMPmodels with large state spaces upon which pomp is focused

A POMP model may be characterized by the transition density for the Markov process and themeasurement density1 However some methods require only simulation from the transitiondensity whereas others require evaluation of this density Still other methods may not workwith the model itself but with an approximation such as a linearization Algorithms forwhich the dynamic model is specified only via a simulator are said to be plug-and-play (Bretoet al 2009 He et al 2010) Plug-and-play methods can be employed once one has ldquopluggedrdquo amodel simulator into the inference machinery Since many POMP models of scientific interestare relatively easy to simulate the plug-and-play property facilitates data analysis Evenif one candidate model has tractable transition probabilities a scientist will frequently wishto consider alternative models for which these probabilities are intractable In a plug-and-play methodological environment analysis of variations in the model can often be achieved bychanging a few lines of the model simulator codes The price one pays for the flexibility of plug-and-play methodology is primarily additional computational effort which can be substantialNevertheless plug-and-play methods implemented using pomp have proved capable for stateof the art inference problems (eg King et al 2008 Bhadra et al 2011 Shrestha et al2011 2013 Earn et al 2012 Roy et al 2012 Blackwood et al 2013ab He et al 2013Breto 2014 Blake et al 2014) The recent surge of interest in plug-and-play methodologyfor POMP models includes the development of nonlinear forecasting (Ellner et al 1998)iterated filtering (Ionides et al 2006 2015) ensemble Kalman filtering (Shaman and Karspeck2012) approximate Bayesian computation (ABC) (Sisson et al 2007) particle Markov chainMonte Carlo (PMCMC) (Andrieu et al 2010) probe matching (Kendall et al 1999) andsynthetic likelihood (Wood 2010) Although the pomp package provides a general environmentfor methods with and without the plug-and-play property development of the package to datehas emphasized plug-and-play methods

The pomp package is philosophically neutral as to the merits of Bayesian inference It en-ables a POMP model to be supplemented with prior distributions on parameters and severalBayesian methods are implemented within the package Thus pomp is a convenient environ-ment for those who wish to explore both Bayesian and non-Bayesian data analyses

The remainder of this paper is organized as follows Section 2 defines mathematical notationfor POMP models and relates this to their representation as objects of class lsquopomprsquo in thepomp package Section 3 introduces several of the statistical methods currently implementedin pomp Section 4 constructs and explores a simple POMP model demonstrating the use ofthe available statistical methods Section 5 illustrates the implementation of more complexPOMPs using a model of infectious disease transmission as an example Finally Section 6discusses extensions and applications of pomp

1We use the term ldquodensityrdquo in this article encompass both the continuous and discrete cases Thus in thelatter case ie when state variables andor measured quantities are discrete one could replace ldquoprobabilitydensity functionrdquo with ldquoprobability mass functionrdquo










Nprodn=1


(yn|xn θ) (1)









Rgt library(pomp)









obs data ylowast1N

states mdash x0N

coef params θ




Rgt obs(gompertz)



Rgt plot(gompertz)


Rgt time(gompertz)

Rgt coef(gompertz)







23 Covariates












`(θ) =

Nsumn=1

`n|1nminus1(θ) (2)


(a) Plug-and-play



















intfYn|Xn













nj sim fXn|Xnminus1

(middot |XF

nminus1j θ)

for j in 1 J


nj θ) for j in 1 J



nj = XPnkj

for j in 1 J


(Jminus1

sumJm=1w(nm)

)

9 end










j=1 wj = 1




















mminus1σi)2)



(middot ΘF

0j

)for j in 1 J


i 6 for n in 1 N do


]isim Normal

([ΘFnminus1j

]i (amminus1σi)

2)



(middot |XF

nminus1j ΘPnj

)for j in 1 J





nj = XPnkj

and ΘFnj = ΘP

nkjfor j in 1 J


=sumJ

j=1 w(n j)[ΘPnj

]i

for i 6isin I


sumj w(n j)

([ΘP


for i 6isin I

15 end


n=1[Vn]minus1i


)for i 6isin I


sumj

[ΘFLj

]i

for i isin I













6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt
























ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion










Nprodn=1


(yn|xn θ) (1)









Rgt library(pomp)









obs data ylowast1N

states mdash x0N

coef params θ




Rgt obs(gompertz)



Rgt plot(gompertz)


Rgt time(gompertz)

Rgt coef(gompertz)







23 Covariates












`(θ) =

Nsumn=1

`n|1nminus1(θ) (2)


(a) Plug-and-play



















intfYn|Xn













nj sim fXn|Xnminus1

(middot |XF

nminus1j θ)

for j in 1 J


nj θ) for j in 1 J



nj = XPnkj

for j in 1 J


(Jminus1

sumJm=1w(nm)

)

9 end










j=1 wj = 1




















mminus1σi)2)



(middot ΘF

0j

)for j in 1 J


i 6 for n in 1 N do


]isim Normal

([ΘFnminus1j

]i (amminus1σi)

2)



(middot |XF

nminus1j ΘPnj

)for j in 1 J





nj = XPnkj

and ΘFnj = ΘP

nkjfor j in 1 J


=sumJ

j=1 w(n j)[ΘPnj

]i

for i 6isin I


sumj w(n j)

([ΘP


for i 6isin I

15 end


n=1[Vn]minus1i


)for i 6isin I


sumj

[ΘFLj

]i

for i isin I













6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt
























ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion








obs data ylowast1N

states mdash x0N

coef params θ




Rgt obs(gompertz)



Rgt plot(gompertz)


Rgt time(gompertz)

Rgt coef(gompertz)







23 Covariates












`(θ) =

Nsumn=1

`n|1nminus1(θ) (2)


(a) Plug-and-play



















intfYn|Xn













nj sim fXn|Xnminus1

(middot |XF

nminus1j θ)

for j in 1 J


nj θ) for j in 1 J



nj = XPnkj

for j in 1 J


(Jminus1

sumJm=1w(nm)

)

9 end










j=1 wj = 1




















mminus1σi)2)



(middot ΘF

0j

)for j in 1 J


i 6 for n in 1 N do


]isim Normal

([ΘFnminus1j

]i (amminus1σi)

2)



(middot |XF

nminus1j ΘPnj

)for j in 1 J





nj = XPnkj

and ΘFnj = ΘP

nkjfor j in 1 J


=sumJ

j=1 w(n j)[ΘPnj

]i

for i 6isin I


sumj w(n j)

([ΘP


for i 6isin I

15 end


n=1[Vn]minus1i


)for i 6isin I


sumj

[ΘFLj

]i

for i isin I













6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt
























ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion


23 Covariates












`(θ) =

Nsumn=1

`n|1nminus1(θ) (2)


(a) Plug-and-play



















intfYn|Xn













nj sim fXn|Xnminus1

(middot |XF

nminus1j θ)

for j in 1 J


nj θ) for j in 1 J



nj = XPnkj

for j in 1 J


(Jminus1

sumJm=1w(nm)

)

9 end










j=1 wj = 1




















mminus1σi)2)



(middot ΘF

0j

)for j in 1 J


i 6 for n in 1 N do


]isim Normal

([ΘFnminus1j

]i (amminus1σi)

2)



(middot |XF

nminus1j ΘPnj

)for j in 1 J





nj = XPnkj

and ΘFnj = ΘP

nkjfor j in 1 J


=sumJ

j=1 w(n j)[ΘPnj

]i

for i 6isin I


sumj w(n j)

([ΘP


for i 6isin I

15 end


n=1[Vn]minus1i


)for i 6isin I


sumj

[ΘFLj

]i

for i isin I













6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt
























ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion


(a) Plug-and-play



















intfYn|Xn













nj sim fXn|Xnminus1

(middot |XF

nminus1j θ)

for j in 1 J


nj θ) for j in 1 J



nj = XPnkj

for j in 1 J


(Jminus1

sumJm=1w(nm)

)

9 end










j=1 wj = 1




















mminus1σi)2)



(middot ΘF

0j

)for j in 1 J


i 6 for n in 1 N do


]isim Normal

([ΘFnminus1j

]i (amminus1σi)

2)



(middot |XF

nminus1j ΘPnj

)for j in 1 J





nj = XPnkj

and ΘFnj = ΘP

nkjfor j in 1 J


=sumJ

j=1 w(n j)[ΘPnj

]i

for i 6isin I


sumj w(n j)

([ΘP


for i 6isin I

15 end


n=1[Vn]minus1i


)for i 6isin I


sumj

[ΘFLj

]i

for i isin I













6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt
























ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion







nj sim fXn|Xnminus1

(middot |XF

nminus1j θ)

for j in 1 J


nj θ) for j in 1 J



nj = XPnkj

for j in 1 J


(Jminus1

sumJm=1w(nm)

)

9 end










j=1 wj = 1




















mminus1σi)2)



(middot ΘF

0j

)for j in 1 J


i 6 for n in 1 N do


]isim Normal

([ΘFnminus1j

]i (amminus1σi)

2)



(middot |XF

nminus1j ΘPnj

)for j in 1 J





nj = XPnkj

and ΘFnj = ΘP

nkjfor j in 1 J


=sumJ

j=1 w(n j)[ΘPnj

]i

for i 6isin I


sumj w(n j)

([ΘP


for i 6isin I

15 end


n=1[Vn]minus1i


)for i 6isin I


sumj

[ΘFLj

]i

for i isin I













6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt
























ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion




j=1 wj = 1




















mminus1σi)2)



(middot ΘF

0j

)for j in 1 J


i 6 for n in 1 N do


]isim Normal

([ΘFnminus1j

]i (amminus1σi)

2)



(middot |XF

nminus1j ΘPnj

)for j in 1 J





nj = XPnkj

and ΘFnj = ΘP

nkjfor j in 1 J


=sumJ

j=1 w(n j)[ΘPnj

]i

for i 6isin I


sumj w(n j)

([ΘP


for i 6isin I

15 end


n=1[Vn]minus1i


)for i 6isin I


sumj

[ΘFLj

]i

for i isin I













6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt
























ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion









mminus1σi)2)



(middot ΘF

0j

)for j in 1 J


i 6 for n in 1 N do


]isim Normal

([ΘFnminus1j

]i (amminus1σi)

2)



(middot |XF

nminus1j ΘPnj

)for j in 1 J





nj = XPnkj

and ΘFnj = ΘP

nkjfor j in 1 J


=sumJ

j=1 w(n j)[ΘPnj

]i

for i 6isin I


sumj w(n j)

([ΘP


for i 6isin I

15 end


n=1[Vn]minus1i


)for i 6isin I


sumj

[ΘFLj

]i

for i isin I













6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt
























ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion










6 Set(θm ˆ(θm)

)=

(θPm

ˆ(θPm)) if U lt
























ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion














ˆS(θ) = minus1


2log |Σ| minus d

2log(2π) (5)


















7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion












7 Set θm =

θPm if

dsumi=1


)2


fΘ(θmminus1)

θmminus1 otherwise








ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion



ˆABCS (θ) =

εminusdBminus1

d

dprodi=1

τi if

dsumi=1


)2

lt ε2

0 otherwise

(6)
















22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion








22s2 for k in 1 K


j=1



σ2 =1

J

B+Jsumn=B+1


]2 (7)



2log 2πσ2 minus

Nsumn=1+c


lowastnminusc2 y

lowastnminuscL)

]22σ2

(8)

where c = max(c1L)














t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion







t εt (9)



2) (10)



1minus eminusr∆t)









+






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion






+





+ log = log)

+




























Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion




















Rgt logliktruth

[1] 3627102



or simply







Rgt loglikguess

[1] 2519585




















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion



















+ rwsd = c(r = 002 sigma = 002 tau = 002))

+



+ logmeanexp(pf)

+














r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion












r σ τ ˆ se `

truth 01000 01000 01000 3599 003 3601mif MLE 00127 00655 01200 3768 004 3762

exact MLE 00322 00694 01170 3787 002 3788









+ log = TRUE))


+








+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion







+







) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion






) (12)




Rgt rickersim lt-


+ N = r N exp(-N + e)

+

Rgt rickerrmeas lt-

+ y = rpois(phi N)

+

Rgt rickerdmeas lt-


+




Rgt logtrans lt-

+ Tr = log(r)


+ Tphi = log(phi)

+ TN_0 = log(N_0)


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion


Rgt exptrans lt-

+ Tr = exp(r)


+ Tphi = exp(phi)

+ TN_0 = exp(N_0)












+ N0 = 7 e0 = 0)) -gt ricker
















+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion






+ seed = 1066L)




Rgt plot(pbtruth)

Rgt plot(pbguess)















guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion





guess 200 1000 200 -2308 008 -56 016truth 447 0300 100 -1390 003 177 003MLE 450 0186 102 -1372 004 180 004

MSLE 421 0337 113 -1457 003 194 002














+



















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion


















S I R-micro -

λ(t)

6ρC

micro

micro

-γ

micro















]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion









]= microS(t)h+ o(h)




]= microR(t)h+ o(h)

(14)





(15)


dS


PS

dI

dt= β

I


dR


(16)









Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion








Rgt rmeas lt-


+

Rgt dmeas lt-


+







where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion




where

pSrarrI =λ(t)

λ(t) + micro


)pSrarr =

micro

λ(t) + micro


)

(19)



Rgt sirstep lt-

+ double rate[6]

+ double dN[6]

+ double P

+ P = S + I + R











+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ H += dN[1]

+
















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion















+ c(SIRH))


+ mu = 150 rho = 01 theta = 100 S0 = 26400

+ I0 = 0002 R0 = 1)) -gt sir1


















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion
















λ(t) =

(β(Φ(t))

I(t) + ι

N

)(21)






Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion


Rgt seassirstep lt-

+ double rate[6]

+ double dN[6]

+ double Beta

+ double dW













+ S += dN[0] - dN[1] - dN[2]

+ I += dN[1] - dN[3] - dN[4]

+ R += dN[3] - dN[5]

+ P = S + I + R

+ Phi += dW

+ H += dN[1]


+











+ c(SIRPHPhinoise))



+ sigma = 03 S0 = 0055 I0 = 0002 R0 = 094)) -gt sir2






6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion




6 Conclusion







References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion


References



200900736x








101073pnas1220908110








101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion




101109JPROC2007893250



i01










0026







101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion






101073pnas0603181103





8695










pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion



pomp








s11222-011-9288-2


0807882106












101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion





101073pnas0607208104


3950080506








Affiliation





Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion




Introduction



Initial conditions

Covariates



Iterated filtering


















Conclusion

statistical inference for partially observed markov processes via … › pdf ›...

Documents