Statistical Inference for Partially Observed Markov Processes via the R Package pomp

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, iterated filtering, particle Markov chain Monte Carlo, approximate Bayesian computation, maximum synthetic likelihood estimation, nonlinear forecasting, and trajectory matching. In this paper, we demonstrate the application of these methodologies using some simple toy problems. We also illustrate the specification of more complex POMP models, 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.Comment: In press at the Journal of Statistical Software. A version of this paper is provided at the pomp package website: http://kingaa.github.io/pom

Comparison of Filtering Methods for the Modeling and Retrospective Forecasting of Influenza Epidemics

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. Results suggest that when using the SIRS model the ensemble filters and the basic PF are more capable of faithfully recreating historical influenza incidence time series, while the MIF and pMCMC do not perform as well for multimodal outbreaks. For forecast of the week with the highest influenza activity, the accuracies of the six model-filter frameworks are comparable; the three particle filters perform slightly better predicting peaks 1–5 weeks in the future; the ensemble filters are more accurate predicting peaks in the past

Particle filtering in compartmental projection models

Simulation models are important tools for real-time forecasting of pandemics. Models help health decision makers examine interventions and secure strong guidance when anticipating outbreak evolution. However, models usually diverge from the real observations. Stochastics involved in pandemic systems, such as changes in human contact patterns play a substantial role in disease transmissions and are not usually captured in traditional dynamic models. In addition, models of emerging diseases face the challenge of limited epidemiological knowledge about the natural history of disease. Even when the information about natural history is available -- for example for endemic seasonal diseases -- transmission models are often simplified and are involved with omissions. Availability of data streams can provide a view of early days of a pandemic, but fail to predict how the pandemic will evolve. Recent developments of computational statistics algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo, provide the possibility of creating models based on historical data as well as re-grounding models based on ongoing data observations. The objective of this thesis is to combine particle filtering -- a Sequential Monte Carlo algorithm -- with system dynamics models of pandemics. We developed particle filtering models that can recurrently be re-grounded as new observations become available. To this end, we also examined the effectiveness of this arrangement which is subject to specifics of the configuration (e.g., frequency of data sampling). While clinically-diagnosed cases are valuable incoming data stream during an outbreak, new generation of geo-spatially specific data sources, such as search volumes can work as a complementary data resource to clinical data. As another contribution, we used particle filtering in a model which can be re-grounded based on both clinical and search volume data. Our results indicate that the particle filtering in combination with compartmental models provides accurate projection systems for the estimation of model states and also model parameters (particularly compared to traditional calibration methodologies and in the context of emerging communicable diseases). The results also suggest that more frequent sampling from clinical data improves predictive accuracy outstandingly. The results also present that assumptions to make regarding the parameters associated with the particle filtering itself and changes in contact rate were robust across adequacy of empirical data since the beginning of the outbreak and inter-observation interval. The results also support the use of data from Google search API along with clinical data

Particle Gibbs with Ancestor Sampling

Particle Markov chain Monte Carlo (PMCMC) is a systematic way of combining the two main tools used for Monte Carlo statistical inference: sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC). We present a novel PMCMC algorithm that we refer to as particle Gibbs with ancestor sampling (PGAS). PGAS provides the data analyst with an off-the-shelf class of Markov kernels that can be used to simulate the typically high-dimensional and highly autocorrelated state trajectory in a state-space model. The ancestor sampling procedure enables fast mixing of the PGAS kernel even when using seemingly few particles in the underlying SMC sampler. This is important as it can significantly reduce the computational burden that is typically associated with using SMC. PGAS is conceptually similar to the existing PG with backward simulation (PGBS) procedure. Instead of using separate forward and backward sweeps as in PGBS, however, we achieve the same effect in a single forward sweep. This makes PGAS well suited for addressing inference problems not only in state-space models, but also in models with more complex dependencies, such as non-Markovian, Bayesian nonparametric, and general probabilistic graphical models

Prediction of Infectious Disease outbreaks based on limited information

The last two decades have seen several large-scale epidemics of international impact, including human, animal and plant epidemics. Policy makers face health challenges that require epidemic predictions based on limited information. There is therefore a pressing need to construct models that allow us to frame all available information to predict an emerging outbreak and to control it in a timely manner. The aim of this thesis is to develop an early-warning modelling approach that can predict emerging disease outbreaks. Based on Bayesian techniques ideally suited to combine information from different sources into a single modelling and estimation framework, I developed a suite of approaches to epidemiological data that can deal with data from different sources and of varying quality. The SEIR model, particle filter algorithm and a number of influenza-related datasets were utilised to examine various models and methodologies to predict influenza outbreaks. The data included a combination of consultations and diagnosed influenza-like illness (ILI) cases for five influenza seasons. I showed that for the pandemic season, different proxies lead to similar behaviour of the effective reproduction number. For influenza datasets, there exists a strong relationship between consultations and diagnosed datasets, especially when considering time-dependent models. Individual parameters for different influenza seasons provided similar values, thereby offering an opportunity to utilise such information in future outbreaks. Moreover, my findings showed that when the temperature drops below 14°C, this triggers the first substantial rise in the number of ILI cases, highlighting that temperature data is an important signal to trigger the start of the influenza epidemic. Further probing was carried out among Maltese citizens and estimates on the under-reporting rate of the seasonal influenza were established. Based on these findings, a new epidemiological model and framework were developed, providing accurate real-time forecasts with a clear early warning signal to the influenza outbreak. This research utilised a combination of novel data sources to predict influenza outbreaks. Such information is beneficial for health authorities to plan health strategies and control epidemics

Efficient real-time monitoring of an emerging influenza pandemic: How feasible?

A prompt public health response to a new epidemic relies on the ability to monitor and predict its evolution in real time as data accumulate. The 2009 A/H1N1 outbreak in the UK revealed pandemic data as noisy, contaminated, potentially biased and originating from multiple sources. This seriously challenges the capacity for real-time monitoring. Here, we assess the feasibility of real-time inference based on such data by constructing an analytic tool combining an age-stratified SEIR transmission model with various observation models describing the data generation mechanisms. As batches of data become available, a sequential Monte Carlo (SMC) algorithm is developed to synthesise multiple imperfect data streams, iterate epidemic inferences and assess model adequacy amidst a rapidly evolving epidemic environment, substantially reducing computation time in comparison to standard MCMC, to ensure timely delivery of real-time epidemic assessments. In application to simulated data designed to mimic the 2009 A/H1N1 epidemic, SMC is shown to have additional benefits in terms of assessing predictive performance and coping with parameter nonidentifiability

Incorporating Particle Filtering and System Dynamic Modelling in Infection Transmission of Measles and Pertussis

Childhood viral and bacterial infections remain an important public problem, and research into their dynamics has broader scientific implications for understanding both dynamical systems and associated methodologies at the population level. Measles and pertussis are two important childhood infectious diseases. Measles is a highly transmissible disease and is one of the leading causes of death among young children under 5 globally. Pertussis (whooping cough) is another common childhood infectious disease, which is most harmful for babies and young children and can be deadly. While the use of ongoing surveillance data and - recently - dynamic models offer insight on measles (or pertussis) dynamics, both suffer notable shortcomings when applied to measles (or pertussis) outbreak prediction. In this thesis, I apply the Sequential Monte Carlo approach of particle filtering, incorporating reported measles and pertussis incidence for Saskatchewan during the pre-vaccination era, using an adaptation of a previously contributed measles and pertussis compartmental models. To secure further insight, I also perform particle filtering on age structured adaptations of the models. For some models, I further consider two different methods of configuring the contact matrix. The results indicate that, when used with a suitable dynamic model, particle filtering can offer high predictive capacity for measles and pertussis dynamics and outbreak occurrence in a low vaccination context. Based on the most competitive model as evaluated by predictive accuracy, I have performed prediction and outbreak classification analysis. The prediction results demonstrated that the most competitive models could predict the measles and pertussis outbreak patterns and classify whether there will be an outbreak or not in the next month (Area under the ROC Curve of measles is 0.89, while pertussis is 0.91). I conclude that anticipating the outbreak dynamics of measles and pertussis in low vaccination regions by applying particle filtering with simple measles and pertussis transmission models, and incorporating time series of reported case counts, is a valuable technique to assist public health authorities in estimating risk and magnitude of measles and pertussis outbreaks. Such approach offers particularly strong value proposition for other pathogens with little-known dynamics, important latent drivers, and in the context of the growing number of high-velocity electronic data sources. Strong additional benefits are also likely to be realized from extending the application of this technique to highly vaccinated populations

A unified approach to mortality modelling using state-space framework: characterisation, identification, estimation and forecasting

This paper explores and develops alternative statistical representations and estimation approaches for dynamic mortality models. The framework we adopt is to reinterpret popular mortality models such as the Lee-Carter class of models in a general state-space modelling methodology, which allows modelling, estimation and forecasting of mortality under a unified framework. Furthermore, we propose an alternative class of model identification constraints which is more suited to statistical inference in filtering and parameter estimation settings based on maximization of the marginalized likelihood or in Bayesian inference. We then develop a novel class of Bayesian state-space models which incorporate apriori beliefs about the mortality model characteristics as well as for more flexible and appropriate assumptions relating to heteroscedasticity that present in observed mortality data. We show that multiple period and cohort effect can be cast under a state-space structure. To study long term mortality dynamics, we introduce stochastic volatility to the period effect. The estimation of the resulting stochastic volatility model of mortality is performed using a recent class of Monte Carlo procedure specifically designed for state and parameter estimation in Bayesian state-space models, known as the class of particle Markov chain Monte Carlo methods. We illustrate the framework we have developed using Danish male mortality data, and show that incorporating heteroscedasticity and stochastic volatility markedly improves model fit despite an increase of model complexity. Forecasting properties of the enhanced models are examined with long term and short term calibration periods on the reconstruction of life tables.Comment: 46 page

Architectures and GPU-Based Parallelization for Online Bayesian Computational Statistics and Dynamic Modeling

Recent work demonstrates that coupling Bayesian computational statistics methods with dynamic models can facilitate the analysis of complex systems associated with diverse time series, including those involving social and behavioural dynamics. Particle Markov Chain Monte Carlo (PMCMC) methods constitute a particularly powerful class of Bayesian methods combining aspects of batch Markov Chain Monte Carlo (MCMC) and the sequential Monte Carlo method of Particle Filtering (PF). PMCMC can flexibly combine theory-capturing dynamic models with diverse empirical data. Online machine learning is a subcategory of machine learning algorithms characterized by sequential, incremental execution as new data arrives, which can give updated results and predictions with growing sequences of available incoming data. While many machine learning and statistical methods are adapted to online algorithms, PMCMC is one example of the many methods whose compatibility with and adaption to online learning remains unclear. In this thesis, I proposed a data-streaming solution supporting PF and PMCMC methods with dynamic epidemiological models and demonstrated several successful applications. By constructing an automated, easy-to-use streaming system, analytic applications and simulation models gain access to arriving real-time data to shorten the time gap between data and resulting model-supported insight. The well-defined architecture design emerging from the thesis would substantially expand traditional simulation models' potential by allowing such models to be offered as continually updated services. Contingent on sufficiently fast execution time, simulation models within this framework can consume the incoming empirical data in real-time and generate informative predictions on an ongoing basis as new data points arrive. In a second line of work, I investigated the platform's flexibility and capability by extending this system to support the use of a powerful class of PMCMC algorithms with dynamic models while ameliorating such algorithms' traditionally stiff performance limitations. Specifically, this work designed and implemented a GPU-enabled parallel version of a PMCMC method with dynamic simulation models. The resulting codebase readily has enabled researchers to adapt their models to the state-of-art statistical inference methods, and ensure that the computation-heavy PMCMC method can perform significant sampling between the successive arrival of each new data point. Investigating this method's impact with several realistic PMCMC application examples showed that GPU-based acceleration allows for up to 160x speedup compared to a corresponding CPU-based version not exploiting parallelism. The GPU accelerated PMCMC and the streaming processing system can complement each other, jointly providing researchers with a powerful toolset to greatly accelerate learning and securing additional insight from the high-velocity data increasingly prevalent within social and behavioural spheres. The design philosophy applied supported a platform with broad generalizability and potential for ready future extensions. The thesis discusses common barriers and difficulties in designing and implementing such systems and offers solutions to solve or mitigate them