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Power-Expected-Posterior Priors for generalized linear models

BAYESIAN ANALYSIS 2018年第3期13卷 721-748页

作者： Fouskakis, Dimitris Ntzoufras, Ioannis Perrakis, Konstantinos Natl Tech Univ Athens Dept Math Athens 15780 Greece Athens Univ Econ & Business Dept Stat Athens 10434 Greece German Ctr Neurodegenerat Dis DZNE D-53127 Bonn Germany

The power-expected-posterior (PEP) prior provides an objective, automatic, consistent and parsimonious model selection procedure. At the same time it resolves the conceptual and computational problems due to the use of imaginary data. Namely, (i) it dispenses with the need to select and average across all possible minimal imaginary samples, and (ii) it diminishes the effect that the imaginary data have upon the posterior distribution. These attributes allow for large sample approximations, when needed, in order to reduce the computational burden under more complex models. In this work we generalize the applicability of the PEP methodology, focusing on the framework of generalized linear models (GLMs), by introducing two new PEP definitions which are in effect applicable to any general model setting. Hyper-prior extensions for the power parameter that regulates the contribution of the imaginary data are introduced. We further study the validity of the predictive matching and of the model selection consistency, providing analytical proofs for the former and empirical evidence supporting the latter. For estimation of posterior model and inclusion probabilities we introduce a tuning-free Gibbs-based variable selection sampler. Several simulation scenarios and one real life example are considered in order to evaluate the performance of the proposed methods compared to other commonly used approaches based on mixtures of g-priors. Results indicate that the GLM-PEP priors are more effective in the identification of sparse and parsimonious model formulations.

关键词： expected-posterior prior g-prior generalized linear models hyper-g priors imaginary data objective Bayesian model selection power-prior

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Learning High-Dimensional generalized linear Autoregressive models

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IEEE TRANSACTIONS ON INFORMATION THEORY 2019年第4期65卷 2401-2422页

作者： Hall, Eric C. Raskutti, Garvesh Willett, Rebecca M. Univ Wisconsin Wisconsin Inst Discovery Madison WI 53706 USA Univ Wisconsin Dept Stat Madison WI 53706 USA Univ Chicago Chicago IL 60637 USA

Vector autoregressive models characterize a variety of time series in which linear combinations of current and past observations can be used to accurately predict future observations. For instance, each element of an observation vector could correspond to a different node in a network, and the parameters of an autoregressive model would correspond to the impact of the network structure on the time series evolution. Often, these models are used successfully in practice to learn the structure of social, epidemiological, financial, or biological neural networks. However, little is known about statistical guarantees on the estimates of such models in non-Gaussian settings. This paper addresses the inference of the autoregressive parameters and associated network structure within a generalized linear model framework that includes Poisson and Bernoulli autoregressive processes. At the heart of this analysis is a sparsity-regularized maximum likelihood estimator. While sparsity-regularization is well-studied in the statistics and machine learning communities, those analysis methods cannot be applied to autoregressive generalized linear models because of the correlations and potential heteroscedasticity inherent in the observations. Sample complexity bounds are derived using a combination of martingale concentration inequalities and modern empirical process techniques for dependent random variables. These bounds, which are supported by several simulation studies, characterize the impact of various network parameters on the estimator performance.

关键词： Autoregressive processes generalized linear models statistical learning structured learning

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Likelihood-based inference for generalized linear mixed models: Inference with the R package glmm

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STAT 2021年第1期10卷

作者： Knudson, Christina Benson, Sydney Geyer, Charles Jones, Galin Univ St Thomas Dept Math OSS 2012115 Summit Ave St Paul MN 55105 USA Univ Minnesota Div Biostat Minneapolis MN 55455 USA Univ Minnesota Sch Stat Minneapolis MN 55455 USA

The R package glmm enables likelihood-based inference for generalized linear mixed models with a canonical link. No other publicly available software accurately conducts likelihood-based inference for generalized linear mixed models with crossed random effects. glmm is able to do so by approximating the likelihood function and two derivatives using importance sampling. The importance sampling distribution is an essential piece of Monte Carlo likelihood approximation, and developing a good one is the main challenge in implementing it. The package glmm uses the data to tailor the importance sampling distribution and is constructed to ensure finite Monte Carlo standard errors. In the context of the generalized linear mixed model, the salamander model with crossed random effects has become a benchmark example. We use this model to illustrate the complexities of the likelihood function and to demonstrate the use of the R package glmm.

关键词： computationally intensive methods generalized linear models likelihood regression statistical inference statistical modelling

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A coefficient of determination (R²) for generalized linear mixed models

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BIOMETRICAL JOURNAL 2019年第4期61卷 860-872页

作者： Piepho, Hans-Peter Univ Hohenheim Inst Crop Sci Biostat Unit D-70593 Stuttgart Germany

Extensions of linear models are very commonly used in the analysis of biological data. Whereas goodness of fit measures such as the coefficient of determination (R-2) or the adjusted R-2 are well established for linear models, it is not obvious how such measures should be defined for generalized linear and mixed models. There are by now several proposals but no consensus has yet emerged as to the best unified approach in these settings. In particular, it is an open question how to best account for heteroscedasticity and for covariance among observations present in residual error or induced by random effects. This paper proposes a new approach that addresses this issue and is universally applicable for arbitrary variance-covariance structures including spatial models and repeated measures. It is exemplified using three biological examples.

关键词： generalized linear mixed models generalized linear models goodness-of-fit linear mixed models semivariogram total variance

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generalized linear models, with Applications in Fisheries Research

Generalized Linear Models, with Applications in Fisheries Re...

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作者： Bonelwa Sidumo Rhodes University

学位级别：硕士

Gambusia affinis (G. affinis) is an invasive fish species found in the Sundays River Valley of the Eastern Cape, South Africa, The relative abundance and population dynamics of G. affinis were quantified in five interconnected impoundments within the Sundays River Valley, This study utilised a G. affinis data set to demonstrate various, classical ANOVA models. generalized linear models were used to standardize catch per unit effort (CPUE) estimates and to determine environmental variables which influenced the CPUE, Based on the generalized linear model results dam age, mean temperature, Oreochromis mossambicus abundance and Glossogobius callidus abundance had a significant effect on the G. affinis CPUE. The Albany Angling Association collected data during fishing tag and release events. These data were utilized to demonstrate repeated measures designs. Mixed-effects models provided a powerful and flexible tool for analyzing clustered data such as repeated measures data and nested data, lienee it has become tremendously popular as a framework for the analysis of bio-behavioral experiments. The results show that the mixed-effects methods proposed in this study are more efficient than those based on generalized linear models. These data were better modeled with mixed-effects models due to their flexibility in handling missing data.

关键词： Gambusia affinis catch per unit effort A NOVA generalized linear models repeated measures designs mixed-effects models generalized mixed effects models

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Replica analysis of overfitting in generalized linear regression models

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JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 2020年第36期53卷

作者： Coolen, A. C. C. Sheikh, M. Mozeika, A. Aguirre-Lopez, F. Antenucci, F. Radboud Univ Nijmegen Dept Biophys NL-6525 AJ Nijmegen Netherlands Saddle Point Sci Ltd 35A South St London W1K 2XF England London Inst Math Sci 35A South St London W1K 2XF England Kings Coll London Dept Math London WC2R 2LS England

Nearly all statistical inference methods were developed for the regime where the number N of data samples is much larger than the data dimension p. Inference protocols such as maximum likelihood (ML) or maximum a posteriori probability (MAP) are unreliable if p = O(N), due to overfitting. This limitation has for many disciplines with increasingly high-dimensional data become a serious bottleneck. We recently showed that in Cox regression for time-to-event data the overfitting errors are not just noise but take mostly the form of a bias, and how with the replica method from statistical physics one can model and predict this bias and the noise statistics. Here we extend our approach to arbitrary generalized linear regression models (GLM), with possibly correlated covariates. We analyse overfitting in ML/MAP inference without having to specify data types or regression models, relying only on the GLM form, and derive generic order parameter equations for the case of L2 priors. Second, we derive the probabilistic relationship between true and inferred regression coefficients in GLMs, and show that, for the relevant hyperparameter scaling and correlated covariates, the L2 regularization causes a predictable direction change of the coefficient vector. Our results, illustrated by application to linear, logistic, and Cox regression, enable one to correct ML and MAP inferences in GLMs systematically for overfitting bias, and thus extend their applicability into the hitherto forbidden regime p=O(N).

关键词： generalized linear models overfitting regression replica method

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Satellite-Based Rainfall Retrieval: From generalized linear models to Artificial Neural Networks

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REMOTE SENSING 2018年第6期10卷 939-939页

作者： Beusch, Lea Foresti, Loris Gabella, Marco Hamann, Ulrich MeteoSwiss Via Monti 146 CH-6605 Locarno Switzerland ETHZ Inst Atmospher & Climate Sci CH-8092 Zurich Switzerland

In this study, we develop and compare satellite rainfall retrievals based on generalized linear models and artificial neural networks. Both approaches are used in classification mode in a first step to identify the precipitating areas (precipitation detection) and in regression mode in a second step to estimate the rainfall intensity at the ground (rain rate). The input predictors are geostationary satellite infrared (IR) brightness temperatures and Satellite Application Facility (SAF) nowcasting products which consist of cloud properties, such as cloud top height and cloud type. Additionally, a set of auxiliary location-describing input variables is employed. The output predictand is the ground-based instantaneous rain rate provided by the European-scale radar composite OPERA, that was additionally quality-controlled. We compare our results to a precipitation product which uses a single infrared (IR) channel for the rainfall retrieval. Specifically, we choose the operational PR-OBS-3 hydrology SAF product as a representative example for this type of approach. With generalized linear models, we show that we are able to substantially improve in terms of hits by considering more IR channels and cloud property predictors. Furthermore, we demonstrate the added value of using artificial neural networks to further improve prediction skill by additionally reducing false alarms. In the rain rate estimation, the indirect relationship between surface rain rates and the cloud properties measurable with geostationary satellites limit the skill of all models, which leads to smooth predictions close to the mean rainfall intensity. Probability matching is explored as a tool to recover higher order statistics to obtain a more realistic rain rate distribution.

关键词： MSG SEVIRI geostationary satellite OPERA radar composite rainfall precipitation detection rain rate estimation generalized linear models artificial neural networks

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Quasi-likelihood Bridge estimators for high-dimensional generalized linear models

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COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2017年第10期46卷 8190-8204页

作者： Cui, Xiaohua Chen, Xia Yan, Li Shaanxi Normal Univ Sch Math & Informat Sci Xian 710062 Shaanxi Peoples R China

In this article, we consider the variable selection and estimation for high-dimensional generalized linear models when the number of parameters diverges with the sample size. We propose a penalized quasi-likelihood function with the bridge penalty. The consistency and the Oracle property of the quasi-likeiihood bridge estimators are obtained. Some simulations and a real data analysis are given to illustrate the performance of the proposed method.

关键词： Bridge estimators generalized linear models High-dimensional data Quasi-likelihood Variable selection

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Robust variable selection for generalized linear models with a diverging number of parameters

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2017年第6期46卷 2967-2981页

作者： Guo, Chaohui Yang, Hu Lv, Jing Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China

In this article, we focus on the problem of robust variable selection for high-dimensional generalized linear models. The proposed procedure is based on smooth-threshold estimating equations and a bounded exponential score function with a tuning parameter . The outstanding merit of this new procedure is that it is robust and efficient by selecting automatically the tuning parameter based on the observed data, and its performance is superior to some recently developed methods, in particular, when many outliers are included. Furthermore, under some regularity conditions, we have shown that the resulting estimator is root n/p(n)-consistent and enjoys the oracle property, when the dimension p(n) of the predictors satisfies the condition p(n)(2)/n -> 0, where n is the sample size. Finally, Monte Carlo simulation studies and a real data example are carried out to examine the finite-sample performance of the proposed method.

关键词： Diverging parameters generalized linear models Oracle properties Robust estimating equations Variable selection

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Adaptive Lasso for generalized linear models with a diverging number of parameters

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2017年第23期46卷 11826-11842页

作者： Cui, Yan Chen, Xia Yan, Li Shaanxi Normal Univ Sch Math & Informat Sci Xian 710119 Shaanxi Peoples R China

In this paper, we study the asymptotic properties of the adaptive Lasso estimators in high-dimensional generalized linear models. The consistency of the adaptive Lasso estimator is obtained. We show that, if a reasonable initial estimator is available, under appropriate conditions, the adaptive Lasso correctly selects covariates with non zero coefficients with probability converging to one, and that the estimators of non zero coefficients have the same asymptotic distribution they would have if the zero coefficients were known in advance. Thus, the adaptive Lasso has an Oracle property. The results are examined by some simulations and a real example.

关键词： Adaptive lasso generalized linear models high-dimensional data oracle property variable selection

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