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Register a new userMCEM and SAEM Algorithms for Geostatistical Models under Preferential Sampling
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Douglas Silva
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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The problem of preferential sampling in geostatistics arises when the choise of location to be sampled is made with information about the phenomena in the study. The geostatistical model under preferential sampling deals with this problem, but parameter estimation is challenging because the likelihood function has no closed form. We developed an MCEM and an SAEM algorithm for finding the maximum likelihood estimators of parameters of the model and compared our methodology with the existing ones: Monte Carlo likelihood approximation and Laplace approximation. Simulated studies were realized to assess the quality of the proposed methods and showed good parameter estimation and prediction in preferential sampling. Finally, we illustrate our findings on the well known moss data from Galicia.
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Nowadays, the confidentiality of data and information is of great importance for many companies and organizations. For this reason, they may prefer not to release exact data, but instead to grant researchers access to approximate data. For example, rather than providing the exact income of their clients, they may only provide researchers with grouped data, that is, the number of clients falling in each of a set of non-overlapping income intervals. The challenge is to estimate the mean and variance structure of the hidden ungrouped data based on the observed grouped data. To tackle this problem, this work considers the exact observed data likelihood and applies the Expectation-Maximization (EM) and Monte-Carlo EM (MCEM) algorithms for cases where the hidden data follow a univariate, bivariate, or multivariate normal distribution. The results are then compared with the case of ignoring the grouping and applying regular maximum likelihood. The well-known Galton data and simulated datasets are used to evaluate the properties of the proposed EM and MCEM algorithms.
Under measurement constraints, responses are expensive to measure and initially unavailable on most of records in the dataset, but the covariates are available for the entire dataset. Our goal is to sample a relatively small portion of the dataset where the expensive responses will be measured and the resultant sampling estimator is statistically efficient. Measurement constraints require the sampling probabilities can only depend on a very small set of the responses. A sampling procedure that uses responses at most only on a small pilot sample will be called response-free. We propose a response-free sampling procedure mbox{(OSUMC)} for generalized linear models (GLMs). Using the A-optimality criterion, i.e., the trace of the asymptotic variance, the resultant estimator is statistically efficient within a class of sampling estimators. We establish the unconditional asymptotic distribution of a general class of response-free sampling estimators. This result is novel compared with the existing conditional results obtained by conditioning on both covariates and responses. Under our unconditional framework, the subsamples are no longer independent and new martingale techniques are developed for our asymptotic theory. We further derive the A-optimal response-free sampling distribution. Since this distribution depends on population level quantities, we propose the Optimal Sampling Under Measurement Constraints (OSUMC) algorithm to approximate the theoretical optimal sampling. Finally, we conduct an intensive empirical study to demonstrate the advantages of OSUMC algorithm over existing methods in both statistical and computational perspectives.
This paper presents a general model framework for detecting the preferential sampling of environmental monitors recording an environmental process across space and/or time. This is achieved by considering the joint distribution of an environmental process with a site--selection process that considers where and when sites are placed to measure the process. The environmental process may be spatial, temporal or spatio--temporal in nature. By sharing random effects between the two processes, the joint model is able to establish whether site placement was stochastically dependent of the environmental process under study. The embedding into a spatio--temporal framework also allows for the modelling of the dynamic site---selection process itself. Real--world factors affecting both the size and location of the network can be easily modelled and quantified. Depending upon the choice of population of locations to consider for selection across space and time under the site--selection process, different insights about the precise nature of preferential sampling can be obtained. The general framework developed in the paper is designed to be easily and quickly fit using the R-INLA package. We apply this framework to a case study involving particulate air pollution over the UK where a major reduction in the size of a monitoring network through time occurred. It is demonstrated that a significant response--biased reduction in the air quality monitoring network occurred. We also show that the network was consistently unrepresentative of the levels of particulate matter seen across much of GB throughout the operating life of the network. Finally we show that this may have led to a severe over-reporting of the population--average exposure levels experienced across GB. This could have great impacts on estimates of the health effects of black smoke levels.
Phylodynamics seeks to estimate effective population size fluctuations from molecular sequences of individuals sampled from a population of interest. One way to accomplish this task formulates an observed sequence data likelihood exploiting a coalescent model for the sampled individuals genealogy and then integrating over all possible genealogies via Monte Carlo or, less efficiently, by conditioning on one genealogy estimated from the sequence data. However, when analyzing sequences sampled serially through time, current methods implicitly assume either that sampling times are fixed deterministically by the data collection protocol or that their distribution does not depend on the size of the population. Through simulation, we first show that, when sampling times do probabilistically depend on effective population size, estimation methods may be systematically biased. To correct for this deficiency, we propose a new model that explicitly accounts for preferential sampling by modeling the sampling times as an inhomogeneous Poisson process dependent on effective population size. We demonstrate that in the presence of preferential sampling our new model not only reduces bias, but also improves estimation precision. Finally, we compare the performance of the currently used phylodynamic methods with our proposed model through clinically-relevant, seasonal human influenza examples.
Modern data sets in various domains often include units that were sampled non-randomly from the population and have a latent correlation structure. Here we investigate a common form of this setting, where every unit is associated with a latent variable, all latent variables are correlated, and the probability of sampling a unit depends on its response. Such settings often arise in case-control studies, where the sampled units are correlated due to spatial proximity, family relations, or other sources of relatedness. Maximum likelihood estimation in such settings is challenging from both a computational and statistical perspective, necessitating approximations that take the sampling scheme into account. We propose a family of approximate likelihood approaches which combine composite likelihood and expectation propagation. We demonstrate the efficacy of our solutions via extensive simulations. We utilize them to investigate the genetic architecture of several complex disorders collected in case-control genetic association studies, where hundreds of thousands of genetic variants are measured for every individual, and the underlying disease liabilities of individuals are correlated due to genetic similarity. Our work is the first to provide a tractable likelihood-based solution for case-control data with complex dependency structures.
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