Statistical inference for partially observed branching processes with immigration

Copyright © 2017 Applied Probability Trust. In the paper we consider the following modification of a discrete-time branching process with stationary immigration. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals may change their offspring distributions. In the subcritical case we investigate the possibility of using the known estimators for the offspring mean and for the mean of the stationary-limiting distribution of the process when the observation of the population sizes is restricted. We prove that, if both the population and the number of immigrants are partially observed, the estimators are still strongly consistent. We also prove that the \u27skipped\u27 version of the estimator for the offspring mean is asymptotically normal and the estimator of the stationary distribution\u27s mean is asymptotically normal under additional assumptions

Inference for Partially Observed Multitype Branching Processes and Ecological Applications

Multitype branching processes with immigration in one type are used to model the dynamics of stage-structured plant populations. Parametric inference is first carried out when count data of all types are observed. Statistical identifiability is proved together with derivation of consistent and asymptotically Gaussian estimators for all the parameters ruling the population dynamics model. However, for many ecological data, some stages (i.e. types) cannot be observed in practice. We study which mechanisms can still be estimated given the model and the data available in this context. Parametric inference is investigated in the case of Poisson distributions. We prove that identifiability holds for only a subset of the parameter set depend- ing on the number of generations observed, together with consistent and asymptotic properties of estimators. Finally, simulations are performed to study the behaviour of the estimators when the model is no longer Poisson. Quite good results are obtained for a large class of models with distributions having mean and variance within the same order of magnitude, leading to some stability results with respect to the Poisson assumption.Comment: 31 pages, 1 figur

Likelihood-Based Inference for Discretely Observed Birth-Death-Shift Processes, with Applications to Evolution of Mobile Genetic Elements

Continuous-time birth-death-shift (BDS) processes are frequently used in stochastic modeling, with many applications in ecology and epidemiology. In particular, such processes can model evolutionary dynamics of transposable elements - important genetic markers in molecular epidemiology. Estimation of the effects of individual covariates on the birth, death, and shift rates of the process can be accomplished by analyzing patient data, but inferring these rates in a discretely and unevenly observed setting presents computational challenges. We propose a mutli-type branching process approximation to BDS processes and develop a corresponding expectation maximization (EM) algorithm, where we use spectral techniques to reduce calculation of expected sufficient statistics to low dimensional integration. These techniques yield an efficient and robust optimization routine for inferring the rates of the BDS process, and apply more broadly to multi-type branching processes where rates can depend on many covariates. After rigorously testing our methodology in simulation studies, we apply our method to study intrapatient time evolution of IS6110 transposable element, a frequently used element during estimation of epidemiological clusters of Mycobacterium tuberculosis infections.Comment: 31 pages, 7 figures, 1 tabl

Relative frequencies in multitype branching processes

This paper considers the relative frequencies of distinct types of individuals in multitype branching processes. We prove that the frequencies are asymptotically multivariate normal when the initial number of ancestors is large and the time of observation is fixed. The result is valid for any branching process with a finite number of types; the only assumption required is that of independent individual evolutions. The problem under consideration is motivated by applications in the area of cell biology. Specifically, the reported limiting results are of advantage in cell kinetics studies where the relative frequencies but not the absolute cell counts are accessible to measurement. Relevant statistical applications are discussed in the context of asymptotic maximum likelihood inference for multitype branching processes.Comment: Published in at http://dx.doi.org/10.1214/08-AAP539 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Efficient Transition Probability Computation for Continuous-Time Branching Processes via Compressed Sensing

Branching processes are a class of continuous-time Markov chains (CTMCs) with ubiquitous applications. A general difficulty in statistical inference under partially observed CTMC models arises in computing transition probabilities when the discrete state space is large or uncountable. Classical methods such as matrix exponentiation are infeasible for large or countably infinite state spaces, and sampling-based alternatives are computationally intensive, requiring a large integration step to impute over all possible hidden events. Recent work has successfully applied generating function techniques to computing transition probabilities for linear multitype branching processes. While these techniques often require significantly fewer computations than matrix exponentiation, they also become prohibitive in applications with large populations. We propose a compressed sensing framework that significantly accelerates the generating function method, decreasing computational cost up to a logarithmic factor by only assuming the probability mass of transitions is sparse. We demonstrate accurate and efficient transition probability computations in branching process models for hematopoiesis and transposable element evolution.Comment: 18 pages, 4 figures, 2 table

Probabilistic Inference with Generating Functions for Population Dynamics of Unmarked Individuals

Modeling the interactions of different population dynamics (e.g. reproduction, migration) within a population is a challenging problem that underlies numerous ecological research questions. Powerful, interpretable models for population dynamics are key to developing intervention tactics, allocating limited conservation resources, and predicting the impact of uncertain environmental forces on a population. Fortunately, probabilistic graphical models provide a robust mechanistic framework for these kinds of problems. However, in the relatively common case where individuals in the population are unmarked (i.e. indistinguishable from one another), models of the population dynamics naturally contain a deceptively challenging statistical feature: discrete latent variables with unbounded/countably infinite support. Unfortunately, existing inference algorithms for discrete distributions are applicable only for finite distributions and while approximate inference algorithms exist for countably infinite discrete distributions, they are generally unreliable and inefficient. In this work, we develop the first known general-purpose polynomial-time exact inference algorithms for this class of models using a novel representation based on probability generating functions. These methods are flexibe, easy to use, and significantly faster than existing approximate solutions. We also introduce a novel approximation scheme based on this technique that allows it to gracefully scale to populations well beyond the computational limits of any previously known exact or approximate general-purpose inference algorithm for population dynamics. Finally, we conduct an ecological case study on historical data demonstrating the downstream impact of these advances to a large scale population monitoring setting

Robust Estimation in Multitype Branching Processes Based on their Asymptotic Properties

2000 Mathematics Subject Classification: 60J80.In this work we propose two procedures for robust estimation of the individual distributions of multitype discrete-time Galton-Watson branching processes with an increasing number of ancestors, using the relative frequencies of the process and their asymptotic distributions. The study is based on simulations and numerical results.The research was partially supported by appropriated state funds for research allocated to Sofia University (contract 112/2010), Bulgaria

Bootstrap for Critical Branching Process with Non-Stationary Immigration

2000 Mathematics Subject Classification: Primary 60J80, Secondary 62F12, 60G99.In the critical branching process with a stationary immigration the standard parametric bootstrap for an estimator of the offspring mean is invalid. We consider the process with non-stationary immigration, whose mean and variance α(n) and β(n) are finite for each n ≥ 1 and are regularly varying sequences with nonnegative exponents α and β, respectively. It turns out that if α(n) → ∞ and β(n) = o(nα2(n)) as n → ∞, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least squares estimator. We state a theorem which justifies the validity of the bootstrap. By Monte-Carlo and bootstrap simulations for the process we confirm the theoretical findings. The simulation study highlights the validity and utility of the bootstrap in this model as it mimics the Monte-Carlo pivots even when generation size is small