Inference for Dynamic and Latent Variable Models Via Iterated, Perturbed Bayes Maps

Total Page:16

File Type:pdf, Size:1020Kb

Inference for Dynamic and Latent Variable Models Via Iterated, Perturbed Bayes Maps Inference for dynamic and latent variable models via iterated, perturbed Bayes maps Edward L. Ionidesa,1, Dao Nguyena, Yves Atchadéa, Stilian Stoeva, and Aaron A. Kingb,c Departments of aStatistics, bEcology and Evolutionary Biology, and cMathematics, University of Michigan, Ann Arbor, MI 48109 Edited by Peter J. Bickel, University of California, Berkeley, CA, and approved December 9, 2014 (received for review June 6, 2014) Iterated filtering algorithms are stochastic optimization procedures Algorithms with this property have been called plug-and-play (12, for latent variable models that recursively combine parameter 16). Various other plug-and-play methods for POMP models have perturbations with latent variable reconstruction. Previously, been recently proposed (17–20), due largely to the convenience of theoretical support for these algorithms has been based on this property in scientific applications. the use of conditional moments of perturbed parameters to ap- proximate derivatives of the log likelihood function. Here, a theo- An Algorithm and Related Questions retical approach is introduced based on the convergence of A general POMP model consists of an unobserved stochastic an iterated Bayes map. An algorithm supported by this theory process fXðtÞ; t ≥ t0g with observations Y1; ...; YN made at times displays substantial numerical improvement on the computa- dimðXÞ t1; ...; tN . We suppose that XðtÞ takes values in X ⊂ R , Yn tional challenge of inferring parameters of a partially observed takesvaluesinY ⊂ RdimðYÞ, and there is an unknown param- Markov process. dimðΘÞ eter θ taking values in Θ ⊂ R . We adopt notation ym:n = ym; ym+1; ...; yn for integers m ≤ n, so we write the collection of sequential Monte Carlo | particle filter | maximum likelihood | observations as Y1:N .WritingXn = XðtnÞ, the joint density of Markov process X0:N and Y1:N is assumed to exist, and the Markovian prop- STATISTICS erty of X0:N together with the conditional independence of n iterated filtering algorithm was originally proposed for the observation process means that this joint density can be Amaximum likelihood inference on partially observed Markov written as process (POMP) models by Ionides et al. (1). Variations on the original algorithm have been proposed to extend it to f ; ðx : ; y : ; θÞ general latent variable models (2) and to improve numerical X0:N Y1:N 0 N 1 N QN performance (3, 4). In this paper, we study an iterated filtering = f ðx ; θÞ f ðx jx − ; θÞf ðy jx ; θÞ: X0 0 XnjXn − 1 n n 1 YnjXn n n ECOLOGY algorithm that generalizes the data cloning method (5, 6) and n=1 is therefore also related to other Monte Carlo methods for likelihood-based inference (7–9). Data cloning methodology The data consist of a sequence of observations, y1*:N. We write is based on the observation that iterating a Bayes map con- ; θ fY1:N ðy1:N Þ for the marginal density of Y1:N , and the likelihood verges to a point mass at the maximum likelihood estimate. ℓ θ = * ; θ function is defined to be ð Þ fY1:N ðy1:N Þ. We look for a max- Combining such iterations with perturbations of model θ^ ℓ θ parameters improves the numerical stability of data cloning imum likelihood estimate (MLE), i.e., a value maximizing ð Þ. and provides a foundation for stable algorithms in which the Bayes map is numerically approximated by sequential Monte Significance Carlo computations. We investigate convergence of a sequential Monte Carlo im- Many scientific challenges involve the study of stochastic dy- plementation of an iterated filtering algorithm that combines namic systems for which only noisy or incomplete measure- data cloning, in the sense of Lele et al. (5), with the stochastic ments are available. Inference for partially observed Markov parameter perturbations used by the iterated filtering algorithm process models provides a framework for formulating and of (1). Lindström et al. (4) proposed a similar algorithm, termed answering questions about these systems. Except when the fast iterated filtering, but the theoretical support for that algo- system is small, or approximately linear and Gaussian, state- rithm involved unproved conjectures. We present convergence of-the-art statistical methods are required to make efficient use results for our algorithm, which we call IF2. Empirically, it can of available data. Evaluation of the likelihood for a partially dramatically outperform the previous iterated filtering algorithm observed Markov process model can be formulated as a filter- of ref. 1, which we refer to as IF1. Although IF1 and IF2 both in- ing problem. Iterated filtering algorithms carry out repeated volve recursively filtering through the data, the theoretical justifi- Monte Carlo filtering operations to maximize the likelihood. cation and practical implementations of these algorithms are We develop a new theoretical framework for iterated filtering fundamentally different. IF1 approximates the Fisher score and construct a new algorithm that dramatically outperforms previous approaches on a challenging inference problem in function, whereas IF2 implements an iterated Bayes map. IF1 disease ecology. has been used in applications for which no other computa- tionally feasible algorithm for statistically efficient, likelihood- Author contributions: E.L.I., D.N., Y.A., and A.A.K. designed research; E.L.I., D.N., Y.A., S.S., based inference was known (10–15). The extra capabilities of- and A.A.K. performed research; E.L.I., D.N., Y.A., and A.A.K. analyzed data; and E.L.I., fered by IF2 open up further possibilities for drawing infer- D.N., Y.A., S.S., and A.A.K. wrote the paper. ences about nonlinear partially observed stochastic dynamic The authors declare no conflict of interest. models from time series data. This article is a PNAS Direct Submission. Iterated filtering algorithms implemented using basic sequential 1To whom correspondence should be addressed. Email: [email protected]. Monte Carlo techniques have the property that they do not need This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. to evaluate the transition density of the latent Markov process. 1073/pnas.1410597112/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1410597112 PNAS | January 20, 2015 | vol. 112 | no. 3 | 719–724 Downloaded by guest on September 27, 2021 Z ^ Algorithm IF2. Iterated filtering ℓ ðθ0:N Þhðθ0:N jφ ; σÞfðφÞdφ dθ0:N−1 σ θ = Z ; [1] input: T fð N Þ ^ θ ℓ θ : h θ : φ ; σ f φ dφ dθ : Simulator for fX0 ðx0 ; Þ ð 0 N Þ ð 0 N j Þ ð Þ 0 N θ : Simulator for fXnjXn − 1 ðxnjxn−1 ; Þ, n in 1 N Evaluator for f ðy jx ; θÞ, n in 1 : N YnjXn n n with f and Tσ f approximating the initial and final density of the * Data, y1:N parameter swarm. For our theoretical analysis, we consider the Number of iterations, M case when the SD of the parameter perturbations is held fixed at Number of particles, J σm = σ > 0 for m = 1; ...; M. In this case, IF2 is a Monte Carlo Initial parameter swarm, Θ0,j in 1 : J f j g TM f θ σ θ φ σ : approximation to σ ð Þ. We call the fixed version of IF2 Perturbation density, hnð j ; Þ, n in 1 N “ ” σ homogeneous iterated filtering since each iteration imple- Perturbation sequence, 1:M ments the same map. For any fixed σ, one cannot expect a pro- ΘM : output: Final parameter swarm, f j ,j in 1 Jg cedure such as IF2 to converge to a point mass at the MLE. For m in 1 : M However, for fixed but small σ, we show that IF2 does approx- ΘF,m ∼ θ Θm−1 σ : imately maximize the likelihood, with an error that shrinks to 0,j h0ð j j ; mÞ for j in 1 J F,m ∼ ΘF,m : zero in a limit as σ → 0 and M → ∞. An immediate motivation for X0,j fX0 ðx0; 0,j Þ for j in 1 J For n in 1 : N studying the homogeneous case is simplicity; it turns out that ΘP,m ∼ h θ ΘF,m ,σ for j in 1 : J even with this simplifying assumption, the theoretical analysis n,j nð j nÀ−1,j mÞ Á XP,m ∼ f x jXF,m ; ΘP,m for j in 1 : J is not entirely straightforward. Moreover, the homogeneous anal- n,j XnjXn− 1 n n−1,j j ysis gives at least as much insight as an asymptotic analysis into wm = f y* XP,m; ΘP,m for j in 1 : J σ n,j Yn jXn n n,j n,j P the practical properties of IF2, when m decreases down to some P = = m = J m σ > Draw k1:J with ðkj iÞ wn,i u=1wn,u positive level 0 but never completes the asymptotic limit F,m P,m F,m P,m Θ = Θ and X = X for j in 1 : J σm → 0. Iterated filtering algorithms have been primarily devel- n,j n,kj n,j n,kj End For oped in the context of making progress on complex models for Θm = ΘF,m : Set j N,j for j in 1 J which successfully achieving and validating global likelihood End For optimization is challenging. In such situations, it is advisable to run multiple searches and continue each search up to the limits of available computation (25). If no single search can The IF2 algorithm defined above provides a plug-and-play reliably locate the global maximum, a theory assuring conver- θ^ Monte Carlo approach to obtaining . A simplification of IF2 gence to a neighborhood of the maximum is as relevant as a arises when N = 1, in which case, iterated filtering is called theory assuring convergence to the maximum itself in a prac- iterated importance sampling (2) (SI Text, Iterated Importance tically unattainable limit.
Recommended publications
  • Comparison of Filtering Methods for the Modeling and Retrospective Forecasting of Influenza Epidemics
    Comparison of Filtering Methods for the Modeling and Retrospective Forecasting of Influenza Epidemics Wan Yang1*, Alicia Karspeck2, Jeffrey Shaman1 1 Department of Environmental Health Sciences, Mailman School of Public Health, Columbia University, New York, New York, United States of America, 2 Climate and Global Dynamics Division, National Center for Atmospheric Research, Boulder, Colorado, United States of America Abstract A variety of filtering methods enable the recursive estimation of system state variables and inference of model parameters. These methods have found application in a range of disciplines and settings, including engineering design and forecasting, and, over the last two decades, have been applied to infectious disease epidemiology. For any system of interest, the ideal filter depends on the nonlinearity and complexity of the model to which it is applied, the quality and abundance of observations being entrained, and the ultimate application (e.g. forecast, parameter estimation, etc.). Here, we compare the performance of six state-of-the-art filter methods when used to model and forecast influenza activity. Three particle filters— a basic particle filter (PF) with resampling and regularization, maximum likelihood estimation via iterated filtering (MIF), and particle Markov chain Monte Carlo (pMCMC)—and three ensemble filters—the ensemble Kalman filter (EnKF), the ensemble adjustment Kalman filter (EAKF), and the rank histogram filter (RHF)—were used in conjunction with a humidity-forced susceptible-infectious-recovered-susceptible (SIRS) model and weekly estimates of influenza incidence. The modeling frameworks, first validated with synthetic influenza epidemic data, were then applied to fit and retrospectively forecast the historical incidence time series of seven influenza epidemics during 2003–2012, for 115 cities in the United States.
    [Show full text]
  • Application of Iterated Filtering to Stochastic Volatility Models Based on Non-Gaussian Ornstein-Uhlenbeck Process
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Szczepocki, Piotr Article Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process Statistics in Transition New Series Provided in Cooperation with: Polish Statistical Association Suggested Citation: Szczepocki, Piotr (2020) : Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process, Statistics in Transition New Series, ISSN 2450-0291, Exeley, New York, Vol. 21, Iss. 2, pp. 173-187, http://dx.doi.org/10.21307/stattrans-2020-019 This Version is available at: http://hdl.handle.net/10419/236769 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ www.econstor.eu STATISTICS IN TRANSITION new series, June 2020 Vol.
    [Show full text]
  • Iterated Filtering 3
    The Annals of Statistics 2011, Vol. 39, No. 3, 1776–1802 DOI: 10.1214/11-AOS886 c Institute of Mathematical Statistics, 2011 ITERATED FILTERING1 By Edward L. Ionides, Anindya Bhadra, Yves Atchade´ and Aaron King University of Michigan and Fogarty International Center, National Institutes of Health, University of Michigan, University of Michigan, and University of Michigan and Fogarty International Center, National Institutes of Health Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engi- neering applications. Iterated filtering algorithms maximize the likeli- hood function for partially observed Markov process models by solving a recursive sequence of filtering problems. We present new theoretical results pertaining to the convergence of iterated filtering algorithms implemented via sequential Monte Carlo filters. This theory comple- ments the growing body of empirical evidence that iterated filtering algorithms provide an effective inference strategy for scientific mod- els of nonlinear dynamic systems. The first step in our theory involves studying a new recursive approach for maximizing the likelihood func- tion of a latent variable model, when this likelihood is evaluated via importance sampling. This leads to the consideration of an iterated importance sampling algorithm which serves as a simple special case of iterated filtering, and may have applicability in its own right. 1. Introduction. Partially observed Markov process (POMP) models are of widespread importance throughout science and engineering. As such, they have been studied under various names including state space models [Durbin Received September 2010; revised March 2011. 1Supported by NSF Grants DMS-08-05533 and EF-04-30120, the Graham Environ- arXiv:0902.0347v2 [math.ST] 22 Nov 2012 mental Sustainability Institute, the RAPIDD program of the Science & Technology Direc- torate, Department of Homeland Security, and the Fogarty International Center, National Institutes of Health.
    [Show full text]
  • Comparison of Filtering Methods for the Modeling and Retrospective Forecasting of Influenza Epidemics
    Comparison of Filtering Methods for the Modeling and Retrospective Forecasting of Influenza Epidemics Wan Yang1*, Alicia Karspeck2, Jeffrey Shaman1 1 Department of Environmental Health Sciences, Mailman School of Public Health, Columbia University, New York, New York, United States of America, 2 Climate and Global Dynamics Division, National Center for Atmospheric Research, Boulder, Colorado, United States of America Abstract A variety of filtering methods enable the recursive estimation of system state variables and inference of model parameters. These methods have found application in a range of disciplines and settings, including engineering design and forecasting, and, over the last two decades, have been applied to infectious disease epidemiology. For any system of interest, the ideal filter depends on the nonlinearity and complexity of the model to which it is applied, the quality and abundance of observations being entrained, and the ultimate application (e.g. forecast, parameter estimation, etc.). Here, we compare the performance of six state-of-the-art filter methods when used to model and forecast influenza activity. Three particle filters— a basic particle filter (PF) with resampling and regularization, maximum likelihood estimation via iterated filtering (MIF), and particle Markov chain Monte Carlo (pMCMC)—and three ensemble filters—the ensemble Kalman filter (EnKF), the ensemble adjustment Kalman filter (EAKF), and the rank histogram filter (RHF)—were used in conjunction with a humidity-forced susceptible-infectious-recovered-susceptible (SIRS) model and weekly estimates of influenza incidence. The modeling frameworks, first validated with synthetic influenza epidemic data, were then applied to fit and retrospectively forecast the historical incidence time series of seven influenza epidemics during 2003–2012, for 115 cities in the United States.
    [Show full text]
  • Iterated Filtering Methods for Markov Process Epidemic Models Arxiv
    Iterated filtering methods for Markov process epidemic models Theresa Stocks1 1Department of Mathematics, Stockholm University, Sweden This manuscript is a preprint of a Chapter to appear in the Handbook of Infectious Disease Data Analysis, Held, L., Hens, N., O’Neill, P.D. and Wallinga, J. (Eds.). Chapman & Hall/CRC, 2018. Please use the book for possible citations. Chapter Abstract Dynamic epidemic models have proven valuable for public health decision makers as they provide useful insights into the understanding and prevention of infectious diseases. However, inference for these types of models can be difficult because the disease spread is typically only partially observed e.g. in form of reported incidences in given time periods. This chapter discusses how to perform likelihood- based inference for partially observed Markov epidemic models when it is relatively easy to generate samples from the Markov transmission model while the likelihood function is intractable. The first part of the chapter reviews the theoretical back- ground of inference for partially observed Markov processes (POMP) via iterated filtering. In the second part of the chapter the performance of the method and as- sociated practical difficulties are illustrated on two examples. In the first example a simulated outbreak data set consisting of the number of newly reported cases aggregated by week is fitted to a POMP where the underlying disease transmis- sion model is assumed to be a simple Markovian SIR model. The second example illustrates possible model extensions such as seasonal forcing and over-dispersion in both, the transmission and observation model, which can be used, e.g., when analysing routinely collected rotavirus surveillance data.
    [Show full text]
  • Application of Iterated Filtering to Stochastic Volatility Models Based on Non-Gaussian Ornstein-Uhlenbeck Process
    STATISTICS IN TRANSITION new series, June 2020 Vol. 21, No. 2, pp. 173–187, DOI 10.21307/stattrans-2020-019 Submitted – 04.03.2019; accepted for publishing – 31.01.2020 Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process Piotr Szczepocki 1 ABSTRACT Barndorff-Nielsen and Shephard (2001) proposed a class of stochastic volatility models in which the volatility follows the Ornstein–Uhlenbeck process driven by a positive Levy process without the Gaussian component. The parameter estimation of these models is challenging because the likelihood function is not available in a closed-form expression. A large number of estimation techniques have been proposed, mainly based on Bayesian inference. The main aim of the paper is to present an application of iterated filtering for parameter estimation of such models. Iterated filtering is a method for maximum likelihood inference based on a series of filtering operations, which provide a sequence of parameter estimates that converges to the maximum likelihood estimate. An application to S&P500 index data shows the model perform well and diagnostic plots for iterated filtering ensure convergence iterated filtering to maximum likelihood estimates. Empirical application is accompanied by a simulation study that confirms the validity of the approach in the case of Barndorff-Nielsen and Shephard’s stochastic volatility models. Key words: Ornstein–Uhlenbeck process, stochastic volatility, iterated filtering. 1. Introduction Barndorff-Nielsen and Shephard (2001) proposed a continuous-time stochastic volatility model (BN-S model), in which the logarithm of the asset price y t is assumed to be the solution of the following stochastic differential equation: dy t P EV t dt V t dB t (1) where B t tt is the Brownian motion, P R is the drift parameter, E R is the risk premium.
    [Show full text]
  • Inference for Nonlinear Dynamical Systems
    2022-45 Workshop on Theoretical Ecology and Global Change 2 - 18 March 2009 Inference for nonlinear dynamical systems PASCUAL SPADA Maria Mercedes University of Michigan Department of Ecology and Evolutionary Biology 2019 Natural Science Bldg., 830 North University Ave. Ann Arbor MI 48109-1048 U.S.A. Inference for nonlinear dynamical systems E. L. Ionides†‡, C. Breto´ †, and A. A. King§ †Department of Statistics, University of Michigan, 1085 South University Avenue, Ann Arbor, MI 48109-1107; and §Department of Ecology and Evolutionary Biology, University of Michigan, 830 North University Avenue, Ann Arbor, MI 48109-1048 Edited by Lawrence D. Brown, University of Pennsylvania, Philadelphia, PA, and approved September 21, 2006 (received for review April 19, 2006) Nonlinear stochastic dynamical systems are widely used to model (16, 17). Important climatic fluctuations, such as the El Nin˜o systems across the sciences and engineering. Such models are natural Southern Oscillation (ENSO), affect temperature and salinity, to formulate and can be analyzed mathematically and numerically. and operate on a time scale comparable to that associated with However, difficulties associated with inference from time-series data loss of immunity (18, 19). Therefore, it is critical to disentangle about unknown parameters in these models have been a constraint the intrinsic dynamics associated with cholera transmission on their application. We present a new method that makes maximum through the two main pathways and with loss of immunity, from likelihood estimation feasible for partially-observed nonlinear sto- the extrinsic forcing associated with climatic fluctuations (20). chastic dynamical systems (also known as state-space models) where We consider a model for cholera dynamics that is a continuous- this was not previously the case.
    [Show full text]
  • Statistical Inference for Partially Observed Markov Processes Via the R Package Pomp
    Statistical Inference for Partially Observed Markov Processes via the R Package pomp Aaron A. King, Dao Nguyen, Edward L. Ionides University of Michigan October 23, 2015 Abstract Partially observed Markov process (POMP) models, also known as hidden Markov models or state space models, are ubiquitous tools for time series analysis. The R package pomp provides a very flexible framework for Monte Carlo statistical investigations using nonlinear, non-Gaussian POMP models. A range of modern statistical methods for POMP models have been implemented in this framework including sequential Monte Carlo, it- erated filtering, particle Markov chain Monte Carlo, approximate Bayesian computation, maximum synthetic likelihood estimation, nonlinear forecasting, and trajectory match- ing. In this paper, we demonstrate the application of these methodologies using some simple toy problems. We also illustrate the specification of more complex POMP mod- els, using a nonlinear epidemiological model with a discrete population, seasonality, and extra-demographic stochasticity. We discuss the specification of user-defined models and the development of additional methods within the programming environment provided by pomp. *This document is a version of a manuscript in press at the Journal of Statistical Software. It is provided under the Creative Commons Attribution License. Keywords: Markov processes, hidden Markov model, state space model, stochastic dynamical system, maximum likelihood, plug-and-play, time series, mechanistic model, sequential Monte Carlo, 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 class arXiv:1509.00503v2 [stat.ME] 21 Oct 2015 of models has motivated much software development (Commandeur et al.
    [Show full text]
  • Iterated Filtering Algorithms
    Iterated Filtering Algorithms Yves Atchad´e Department of Statistics, University of Michigan Joint work with Edward Ionides, Dao Nguyen, and Aaron King University of Michigan. May 15, 2015 Introduction I Discrete state space model: X1 ∼ νθ; Xk jFk−1 ∼ pθ(·|Xk−1); and Yk jfFk−1; Xk g ∼ pθ(·|Xk ); k ≥ 1: I The Markov chain fXk ; k ≥ 1g is latent (unobserved). We observe fYk ; k ≥ 1g. p I The goal is to estimate the parameter θ 2 R from the observation fyk ; 1 ≤ k ≤ ng of fYk ; 1 ≤ k ≤ ng. I I E. L. Ionides, A. Bhadra, Y. F. Atchad´eand A. King (2011). Iterated Filtering. Annals of Statistics Vol. 39, 1776-1802. I E. L. Ionides, D. Nguyen, Y. F. Atchad´e,S. Stove, and A. King (2015). Inference for dynamic and latent variable models via iterated, perturbed Bayes maps. Proceedings of the National Academy of Science 112 (3) 719-724. Introduction I Given data y1:n, the likelihood function is def `n(θ) = log pθ(y1:n); where pθ(y1:n) is the density of Y1:n: Z pθ(y1:n) = pθ(x0:n)pθ(y1:njx1:n)dx1:n: I Our goal is to compute the mle ^ θn = Argmaxθ2Θ `n(θ): I Standard but challenging computational problem. I We are particularly interested in the case where the transition densities fθ(xk jxk−1) are intractable. Introduction Example. I dXt = µθ(Xt )dt + Cθ(Xt )dWt ; R t Yt = 0 hθ(Xs )ds + t : I We observe fYt1 ;:::; Ytn g at some non-random time t1 < : : : < tn.
    [Show full text]
  • Statistical Inference for Partially Observed Markov Processes Via the R Package Pomp
    Statistical Inference for Partially Observed Markov Processes via the R Package pomp Aaron A. King Dao Nguyen Edward L. Ionides University of Michigan University of Michigan University of Michigan Abstract Partially observed Markov process (POMP) models, also known as hidden Markov models or state space models, are ubiquitous tools for time series analysis. The R package pomp provides a very flexible framework for Monte Carlo statistical investigations using nonlinear, non-Gaussian POMP models. A range of modern statistical methods for POMP models have been implemented in this framework including sequential Monte Carlo, it- erated filtering, particle Markov chain Monte Carlo, approximate Bayesian computation, maximum synthetic likelihood estimation, nonlinear forecasting, and trajectory match- ing. In this paper, we demonstrate the application of these methodologies using some simple toy problems. We also illustrate the specification of more complex POMP mod- els, using a nonlinear epidemiological model with a discrete population, seasonality, and extra-demographic stochasticity. We discuss the specification of user-defined models and the development of additional methods within the programming environment provided by pomp. *This document is an updated version of Journal of Statistical Software 69(12): 1–43. It is provided under the Creative Commons Attribution License. Keywords: Markov processes, hidden Markov model, state space model, stochastic dynamical system, maximum likelihood, plug-and-play, time series, mechanistic model, sequential Monte Carlo, 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 class of models has motivated much software development (Commandeur et al.
    [Show full text]
  • Iterated Filtering and Smoothing with Application to Infectious Disease Models
    Iterated Filtering and Smoothing with Application to Infectious Disease Models by Dao X. Nguyen A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Statistics) in The University of Michigan 2016 Doctoral Committee: Professor Edward L. Ionides, Chair Associate Professor Yves F. Atchade Associate Professor Aaron A. King Associate Professor Stilian A. Stoev c Dao X. Nguyen 2016 All Rights Reserved For my dearest people ii ACKNOWLEDGEMENTS First and foremost, I would like to thank my supervisor, Professor Edward Ionides, for his patience, guidance, assistance and encouragement throughout my graduate study. His scientific insight and perspectives have made me understand simulation- based inference, time series analysis, partially observed stochastic dynamic systems, especially computational methods in statistics. Without him, I would not have been able to learn the scientific skills required to be a scientist. Learning with him has been one of the most enriching and fruitful experiences of my life. My sincere thanks goes to the committee members. Professors Yves Atchade, Stilian Stoev and Aaron King have helped me a lot in thinking critically about my work, and most importantly to write good quality of scientific output. I would also like to thank Professors Vijay Nair and Long Nguyen for giving me the opportunity to join the Statistics Department. Thanks are also due to various professors, administrative staff and graduate students, whom I have shared with them, the most productive working environments. Finally, my deepest thanks, go to my parents, my wife, my son and my daughter, for their unconditional support and encouragement throughout many years.
    [Show full text]
  • Pomp’ July 31, 2016 Type Package Title Statistical Inference for Partially Observed Markov Processes Version 1.7 Date 2016-07-26
    Package ‘pomp’ July 31, 2016 Type Package Title Statistical Inference for Partially Observed Markov Processes Version 1.7 Date 2016-07-26 URL http://kingaa.github.io/pomp Description Tools for working with partially observed Markov processes (POMPs, AKA stochas- tic dynamical systems, state-space models). 'pomp' provides facilities for implement- ing POMP models, simulating them, and fitting them to time series data by a variety of frequen- tist and Bayesian methods. It is also a platform for the implementation of new inference methods. Depends R(>= 3.1.2), methods Imports stats, graphics, digest, mvtnorm, deSolve, coda, subplex, nloptr Suggests magrittr, plyr, reshape2, ggplot2, knitr SystemRequirements For Windows users, Rtools (see http://cran.r-project.org/bin/windows/Rtools/). License GPL (>= 2) LazyData true Contact kingaa at umich dot edu BugReports http://github.com/kingaa/pomp/issues Collate aaa.R authors.R bake.R generics.R eulermultinom.R csnippet.R pomp_fun.R plugins.R builder.R parmat.R logmeanexp.R slice_design.R profile_design.R sobol.R bsplines.R sannbox.R pomp_class.R load.R pomp.R pomp_methods.R rmeasure_pomp.R rprocess_pomp.R initstate_pomp.R dmeasure_pomp.R dprocess_pomp.R skeleton_pomp.R dprior_pomp.R rprior_pomp.R simulate_pomp.R trajectory_pomp.R plot_pomp.R pfilter.R pfilter_methods.R minim.R traj_match.R bsmc.R bsmc2.R kalman.R kalman_methods.R mif.R mif_methods.R mif2.R mif2_methods.R proposals.R pmcmc.R pmcmc_methods.R nlf_funcs.R nlf_guts.R nlf_objfun.R nlf.R probe.R probe_match.R basic_probes.R spect.R spect_match.R abc.R abc_methods.R covmat.R example.R 1 2 R topics documented: NeedsCompilation yes Author Aaron A.
    [Show full text]