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Jan Hannig

Publications and source records attributed to Jan Hannig.

50 records · Page 3Linked to original sources
arXiv2018

A Bayesian Approach to Multi-State Hidden Markov Models: Application to Dementia Progression

People are living longer than ever before, and with this arises new complications and challenges for humanity. Among the most pressing of these challenges is of understanding the role of aging in the development of dementia. This paper is motivated by the Mayo Clinic Study of Aging data for 4742 subjects since 2004, and how it can be used to draw inference on the role of aging in the development of dementia. We construct a hidden Markov model (HMM) to represent progression of dementia from states associated with the buildup of amyloid plaque in the brain, and the loss of cortical thickness. A hierarchical Bayesian approach is taken to estimate the parameters of the HMM with a truly time-inhomogeneous infinitesimal generator matrix, and response functions of the continuous-valued biomarker measurements are cut-point agnostic. A Bayesian approach with these features could be useful in many disease progression models. Additionally, an approach is illustrated for correcting a common bias in delayed enrollment studies, in which some or all subjects are not observed at baseline. Standard software is incapable of accounting for this critical feature, so code to perform the estimation of the model described below is made available online.

stat.ME↗
arXiv2018

Nonparametric generalized fiducial inference for survival functions under censoring

Fiducial Inference, introduced by Fisher in the 1930s, has a long history, which at times aroused passionate disagreements. However, its application has been largely confined to relatively simple parametric problems. In this paper, we present what might be the first time fiducial inference, as generalized by Hannig et al. (2016), is systematically applied to estimation of a nonparametric survival function under right censoring. We find that the resulting fiducial distribution gives rise to surprisingly good statistical procedures applicable to both one sample and two sample problems. In particular, we use the fiducial distribution of a survival function to construct pointwise and curvewise confidence intervals for the survival function, and propose tests based on the curvewise confidence interval. We establish a functional Bernstein-von Mises theorem, and perform thorough simulation studies in scenarios with different levels of censoring. The proposed fiducial based confidence intervals maintain coverage in situations where asymptotic methods often have substantial coverage problems. Furthermore, the average length of the proposed confidence intervals is often shorter than the length of competing methods that maintain coverage. Finally, the proposed fiducial test is more powerful than various types of log-rank tests and sup log-rank tests in some scenarios. We illustrate the proposed fiducial test comparing chemotherapy against chemotherapy combined with radiotherapy using data from the treatment of locally unresectable gastric cancer.

stat.ME↗
arXiv2018

Angle-Based Joint and Individual Variation Explained

Integrative analysis of disparate data blocks measured on a common set of experimental subjects is a major challenge in modern data analysis. This data structure naturally motivates the simultaneous exploration of the joint and individual variation within each data block resulting in new insights. For instance, there is a strong desire to integrate the multiple genomic data sets in The Cancer Genome Atlas to characterize the common and also the unique aspects of cancer genetics and cell biology for each source. In this paper we introduce Angle-Based Joint and Individual Variation Explained capturing both joint and individual variation within each data block. This is a major improvement over earlier approaches to this challenge in terms of a new conceptual understanding, much better adaption to data heterogeneity and a fast linear algebra computation. Important mathematical contributions are the use of score subspaces as the principal descriptors of variation structure and the use of perturbation theory as the guide for variation segmentation. This leads to an exploratory data analysis method which is insensitive to the heterogeneity among data blocks and does not require separate normalization. An application to cancer data reveals different behaviors of each type of signal in characterizing tumor subtypes. An application to a mortality data set reveals interesting historical lessons. Software and data are available at GitHub .

stat.ML↗
arXiv2018

Non-penalized variable selection in high-dimensional linear model settings via generalized fiducial inference

Standard penalized methods of variable selection and parameter estimation rely on the magnitude of coefficient estimates to decide which variables to include in the final model. However, coefficient estimates are unreliable when the design matrix is collinear. To overcome this challenge an entirely new perspective on variable selection is presented within a generalized fiducial inference framework. This new procedure is able to effectively account for linear dependencies among subsets of covariates in a high-dimensional setting where $p$ can grow almost exponentially in $n$, as well as in the classical setting where $p \le n$. It is shown that the procedure very naturally assigns small probabilities to subsets of covariates which include redundancies by way of explicit $L_{0}$ minimization. Furthermore, with a typical sparsity assumption, it is shown that the proposed method is consistent in the sense that the probability of the true sparse subset of covariates converges in probability to 1 as $n \to \infty$, or as $n \to \infty$ and $p \to \infty$. Very reasonable conditions are needed, and little restriction is placed on the class of possible subsets of covariates to achieve this consistency result.

stat.ME↗
arXiv2017

Covariance Estimation via Fiducial Inference

As a classical problem, covariance estimation has drawn much attention from the statistical community for decades. Much work has been done under the frequentist and the Bayesian frameworks. Aiming to quantify the uncertainty of the estimators without having to choose a prior, we have developed a fiducial approach to the estimation of covariance matrix. Built upon the Fiducial Berstein-von Mises Theorem (Sonderegger and Hannig 2014), we show that the fiducial distribution of the covariate matrix is consistent under our framework. Consequently, the samples generated from this fiducial distribution are good estimators to the true covariance matrix, which enable us to define a meaningful confidence region for the covariance matrix. Lastly, we also show that the fiducial approach can be a powerful tool for identifying clique structures in covariance matrices.

stat.ME↗
arXiv2016

Higher order asymptotics of Generalized Fiducial Distribution

Generalized Fiducial Inference (GFI) is motivated by R.A. Fisher's approach of obtaining posterior-like distributions when there is no prior information available for the unknown parameter. Without the use of Bayes' theorem GFI proposes a distribution on the parameter space using a technique called increasing precision asymptotics \cite{hannig2013generalized}. In this article we analyzed the regularity conditions under which the Generalized Fiducial Distribution (GFD) will be first and second order exact in a frequentist sense. We used a modification of an ingenious technique named "Shrinkage method" \cite{bickel1990decomposition}, which has been extensively used in the probability matching prior contexts, to find the higher order expansion of the frequentist coverage of Fiducial quantile. We identified when the higher order terms of one-sided coverage of Fiducial quantile will vanish and derived a workable recipe for obtaining such GFDs. These ideas are demonstrated on several examples.

math.ST↗
arXiv2016

Non-iterative Joint and Individual Variation Explained

Integrative analysis of disparate data blocks measured on a common set of experimental subjects is one major challenge in modern data analysis. This data structure naturally motivates the simultaneous exploration of the joint and individual variation within each data block resulting in new insights. For instance, there is a strong desire to integrate the multiple genomic data sets in The Cancer Genome Atlas (TCGA) to characterize the common and also the unique aspects of cancer genetics and cell biology for each source. In this paper we introduce Non-iterative Joint and Individual Variation Explained (Non-iterative JIVE), capturing both joint and individual variation within each data block. This is a major improvement over earlier approaches to this challenge in terms of a new conceptual understanding, much better adaption to data heterogeneity and a fast linear algebra computation. Important mathematical contributions are the use of score subspaces as the principal descriptors of variation structure and the use of perturbation theory as the guide for variation segmentation. This leads to a method which is robust against the heterogeneity among data blocks without a need for normalization. An application to TCGA data reveals different behaviors of each type of signal in characterizing tumor subtypes. An application to a mortality data set reveals interesting historical lessons.

stat.ME↗
arXiv2016

A Note on Automatic Data Transformation

Modern data analysis frequently involves variables with highly non-Gaussian marginal distributions. However, commonly used analysis methods are most effective with roughly Gaussian data. This paper introduces an automatic transformation that improves the closeness of distributions to normality. For each variable, a new family of parametrizations of the shifted logarithm transformation is proposed, which is unique in treating the data as real-valued, and in allowing transformation for both left and right skewness within the single family. This also allows an automatic selection of the parameter value (which is crucial for high dimensional data with many variables to transform) by minimizing the Anderson-Darling test statistic of the transformed data. An application to image features extracted from melanoma microscopy slides demonstrate the utility of the proposed transformation in addressing data with excessive skewness, heteroscedasticity and influential observations.

stat.ME↗
arXiv2015

Source detection algorithms for dynamic contaminants based on the analysis of a hydrodynamic limit

In this work we propose and numerically analyze an algorithm for detection of a contaminant source using a dynamic sensor network. The algorithm is motivated using a global probabilistic optimization problem and is based on the analysis of the hydrodynamic limit of a discrete time evolution equation on the lattice under a suitable scaling of time and space. Numerical results illustrating the effectiveness of the algorithm are presented.

math.PR↗
arXiv2014

The importance sampling technique for understanding rare events in Erdős-Rényi random graphs

In dense Erdős-Rényi random graphs, we are interested in the events where large numbers of a given subgraph occur. The mean behavior of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is natural to ask, can one develop efficient numerical schemes to estimate the probability of an Erdős-Rényi graph containing an excessively large number of a fixed given subgraph? Using the large deviation principle we study an importance sampling scheme as a method to numerically compute the small probabilities of large triangle counts occurring within Erdős-Rényi graphs. We show that the exponential tilt suggested directly by the large deviation principle does not always yield an optimal scheme. The exponential tilt used in the importance sampling scheme comes from a generalized class of exponential random graphs. Asymptotic optimality, a measure of the efficiency of the importance sampling scheme, is achieved by a special choice of the parameters in the exponential random graph that makes it indistinguishable from an Erdős-Rényi graph conditioned to have many triangles in the large network limit. We show how this choice can be made for the conditioned Erdős-Rényi graphs both in the replica symmetric phase as well as in parts of the replica breaking phase to yield asymptotically optimal numerical schemes to estimate this rare event probability.

math.PR↗
arXiv2013

Generalized Fiducial Inference for Ultrahigh Dimensional Regression

In recent years the ultrahigh dimensional linear regression problem has attracted enormous attentions from the research community. Under the sparsity assumption most of the published work is devoted to the selection and estimation of the significant predictor variables. This paper studies a different but fundamentally important aspect of this problem: uncertainty quantification for parameter estimates and model choices. To be more specific, this paper proposes methods for deriving a probability density function on the set of all possible models, and also for constructing confidence intervals for the corresponding parameters. These proposed methods are developed using the generalized fiducial methodology, which is a variant of Fisher's controversial fiducial idea. Theoretical properties of the proposed methods are studied, and in particular it is shown that statistical inference based on the proposed methods will have exact asymptotic frequentist property. In terms of empirical performances, the proposed methods are tested by simulation experiments and an application to a real data set. Lastly this work can also be seen as an interesting and successful application of Fisher's fiducial idea to an important and contemporary problem. To the best of the authors' knowledge, this is the first time that the fiducial idea is being applied to a so-called "large p small n" problem.

stat.ME↗
arXiv2012

Generalized fiducial inference for normal linear mixed models

While linear mixed modeling methods are foundational concepts introduced in any statistical education, adequate general methods for interval estimation involving models with more than a few variance components are lacking, especially in the unbalanced setting. Generalized fiducial inference provides a possible framework that accommodates this absence of methodology. Under the fabric of generalized fiducial inference along with sequential Monte Carlo methods, we present an approach for interval estimation for both balanced and unbalanced Gaussian linear mixed models. We compare the proposed method to classical and Bayesian results in the literature in a simulation study of two-fold nested models and two-factor crossed designs with an interaction term. The proposed method is found to be competitive or better when evaluated based on frequentist criteria of empirical coverage and average length of confidence intervals for small sample sizes. A MATLAB implementation of the proposed algorithm is available from the authors.

stat.ME↗
arXiv2011

Continuum Limits of Markov Chains with Application to Network Modeling

In this paper we investigate the continuum limits of a class of Markov chains. The investigation of such limits is motivated by the desire to model very large networks. We show that under some conditions, a sequence of Markov chains converges in some sense to the solution of a partial differential equation. Based on such convergence we approximate Markov chains modeling networks with a large number of components by partial differential equations. While traditional Monte Carlo simulation for very large networks is practically infeasible, partial differential equations can be solved with reasonable computational overhead using well-established mathematical tools.

cs.NI↗