Mountain Scholar :: Browsing by Author "Meyer, Mary, committee member"

Browsing by Author "Meyer, Mary, committee member"

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Open Access
Analysis of structured data and big data with application to neuroscience
(Colorado State University. Libraries, 2015) Sienkiewicz, Ela, author; Wang, Haonan, advisor; Meyer, Mary, committee member; Breidt, F. Jay, committee member; Hayne, Stephen, committee member
Neuroscience research leads to a remarkable set of statistical challenges, many of them due to the complexity of the brain, its intricate structure and dynamical, non-linear, often non-stationary behavior. The challenge of modeling brain functions is magnified by the quantity and inhomogeneity of data produced by scientific studies. Here we show how to take advantage of advances in distributed and parallel computing to mitigate memory and processor constraints and develop models of neural components and neural dynamics. First we consider the problem of function estimation and selection in time-series functional dynamical models. Our motivating application is on the point-process spiking activities recorded from the brain, which poses major computational challenges for modeling even moderately complex brain functionality. We present a big data approach to the identification of sparse nonlinear dynamical systems using generalized Volterra kernels and their approximation using B-spline basis functions. The performance of the proposed method is demonstrated in experimental studies. We also consider a set of unlabeled tree objects with topological and geometric properties. For each data object, two curve representations are developed to characterize its topological and geometric aspects. We further define the notions of topological and geometric medians as well as quantiles based on both representations. In addition, we take a novel approach to define the Pareto medians and quantiles through a multi-objective optimization problem. In particular, we study two different objective functions which measure the topological variation and geometric variation respectively. Analytical solutions are provided for topological and geometric medians and quantiles, and in general, for Pareto medians and quantiles the genetic algorithm is implemented. The proposed methods are applied to analyze a data set of pyramidal neurons.
Open Access
Heavy tail analysis for functional and internet anomaly data
(Colorado State University. Libraries, 2021) Kim, Mihyun, author; Kokoszka, Piotr, advisor; Cooley, Daniel, committee member; Meyer, Mary, committee member; Pinaud, Olivier, committee member
This dissertation is concerned with the asymptotic theory of statistical tools used in extreme value analysis of functional data and internet anomaly data. More specifically, we study four problems associated with analyzing the tail behavior of functional principal component scores in functional data and interarrival times of internet traffic anomalies, which are available only with a round-off error. The first problem we consider is the estimation of the tail index of scores in functional data. We employ the Hill estimator for the tail index estimation and derive conditions under which the Hill estimator computed from the sample scores is consistent for the tail index of the unobservable population scores. The second problem studies the dependence between extremal values of functional scores using the extremal dependence measure (EDM). After extending the EDM defined for positive bivariate observations to multivariate observations, we study conditions guaranteeing that a suitable estimator of the EDM based on these scores converges to the population EDM and is asymptotically normal. The third and last problems investigate the asymptotic and finite sample behavior of the Hill estimator applied to heavy-tailed data contaminated by errors. For the third one, we show that for time series models often used in practice, whose non–contaminated marginal distributions are regularly varying, the Hill estimator is consistent. For the last one, we formulate conditions on the errors under which the Hill and Harmonic Moment estimators applied to i.i.d. data continue to be asymptotically normal. The results of large and finite sample investigations are applied to internet anomaly data.
Open Access
Model selection and nonparametric estimation for regression models
(Colorado State University. Libraries, 2014) He, Zonglin, author; Opsomer, Jean, advisor; Breidt, F. Jay, committee member; Meyer, Mary, committee member; Elder, John, committee member
In this dissertation, we deal with two different topics in statistics. The first topic in survey sampling deals with variable selection for linear regression model from which we will sample with a possibly informative design. Under the assumption that the finite population is generated by a multivariate linear regression model from which we will sample with a possibly informative design, we particularly study the variable selection criterion named predicted residual sums of squares in the sampling context theoretically. We examine the asymptotic properties of weighted and unweighted predicted residual sums of squares under weighted least squares regression estimation and ordinary least squares regression estimation. One simulation study for the variable selection criteria are provided, with the purpose of showing their ability to select the correct model in the practical situation. For the second topic, we are interested in fitting a nonparametric regression model to data for the situation in which some of the covariates are categorical. In the univariate case where the covariate is a ordinal variable, we extend the local polynomial estimator, which normally requires continuous covariates, to a local polynomial estimator that allows for ordered categorical covariates. We derive the asymptotic conditional bias and variance for the local polynomial estimator with ordinal covariate, under the assumption that the categories correspond to quantiles of an unobserved continuous latent variable. We conduct a simulation study with two patterns of ordinal data to evaluate our estimator. In the multivariate case where the covariates contain a mixture of continuous, ordinal, and nominal variables, we use a Nadaraya-Watson estimator with generalized product kernel. We derive the asymptotic conditional bias and variance for the Nadaraya-Watson estimator with continuous, ordinal, and nominal covariates, under the assumption that the categories of the ordinal covariate correspond to quantiles of an unobserved continuous latent variable. We conduct a multivariate simulation study to evaluate our Nadaraya-Watson estimator with generalized product kernel.
Open Access
Non-asymptotic properties of spectral decomposition of large gram-type matrices with applications to high-dimensional inference
(Colorado State University. Libraries, 2020) Zhang, Lyuou, author; Zhou, Wen, advisor; Wang, Haonan, advisor; Breidt, Jay, committee member; Meyer, Mary, committee member; Yang, Liuqing, committee member
Jointly modeling a large and possibly divergent number of temporally evolving subjects arises ubiquitously in statistics, econometrics, finance, biology, and environmental sciences. To circumvent the challenges due to the high dimesionality as well as the temporal and/or contemporaneous dependence, the factor model and its variants have been widely employed. In general, they model the large scale temporally dependent data using some low dimensional structures that capture variations shared across dimensions. In this dissertation, we investigate the non-asymptotic properties of spectral decomposition of high-dimensional Gram-type matrices based on factor models. Specifically, we derive the exponential tail bound for the first and second moments of the deviation between the empirical and population eigenvectors to the right Gram matrix as well as the Berry-Esseen type bound to characterize the Gaussian approximation of these deviations. We also obtain the non-asymptotic tail bound of the ratio between eigenvalues of the left Gram matrix, namely the sample covariance matrix, and their population counterparts regardless of the size of the data matrix. The documented non-asymptotic properties are further demonstrated in a suite of applications, including the non-asymptotic characterization of the estimated number of latent factors in factor models and related machine learning problems, the estimation and forecasting of high-dimensional time series, the spectral properties of large sample covariance matrix such as perturbation bounds and inference on the spectral projectors, and low-rank matrix denoising from temporally dependent data. Next, we consider the estimation and inference of a flexible subject-specific heteroskedasticity model for large scale panel data, which employs latent semiparametric factor structure to simultaneously account for the heteroskedasticity across subjects and contemporaneous and/or serial correlations. Specifically, the subject-specific heteroskedasticity is modeled by the product of unobserved factor process and subject-specific covariate effect. Serving as the loading, the covariate effect is further modeled via additive models. We propose a two-step procedure for estimation. Theoretical validity of this procedure is documented. By scrupulously examining the non-asymptotic rates for recovering the latent factor process and its loading, we show the consistency and asymptotic efficiency of our regression coefficient estimator in addition to the asymptotic normality. This leads to a more efficient confidence set for the regression coefficient. Using a comprehensive simulation study, we demonstrate the finite sample performance of our procedure, and numerical results corroborate the theoretical findings. Finally, we consider the factor model-assisted variable clustering for temporally dependent data. The population level clusters are characterized by the latent factors of the model. We combine the approximate factor model with population level clusters to give an integrative group factor model as a background model for variable clustering. In this model, variables are loaded on latent factors and the factors are the same for variables from a common cluster and are different for variables from different groups. The commonality among clusters is modeled by common factors and the clustering structure is modeled by unique factors of each cluster. We quantify the difficulty of clustering data generated from integrative group factor model in terms of a permutation-invariant clustering error. We develop an algorithm to recover clustering assignments and study its minimax-optimality. The analysis of integrative group factor model and our proposed algorithm partitions a two-dimensional phase space into three regions showing the impact of parameters on the possibility of clustering in integrative group factor model and the statistical guarantee of our proposed algorithm. We also obtain the non-asymptotic characterization of the estimated number of latent factors. The model can be extended to the case of diverging number of clusters with similar results.
Open Access
Parametric and semiparametric model estimation and selection in geostatistics
(Colorado State University. Libraries, 2012) Chu, Tingjin, author; Wang, Haonan, advisor; Zhu, Jun, advisor; Meyer, Mary, committee member; Luo, J. Rockey, committee member
This dissertation is focused on geostatistical models, which are useful in many scientific disciplines, such as climatology, ecology and environmental monitoring. In the first part, we consider variable selection in spatial linear models with Gaussian process errors. Penalized maximum likelihood estimation (PMLE) that enables simultaneous variable selection and parameter estimation is developed and for ease of computation, PMLE is approximated by one-step sparse estimation (OSE). To further improve computational efficiency particularly with large sample sizes, we propose penalized maximum covariance-tapered likelihood estimation (PMLET) and its one-step sparse estimation (OSET). General forms of penalty functions with an emphasis on smoothly clipped absolute deviation are used for penalized maximum likelihood. Theoretical properties of PMLE and OSE, as well as their approximations PMLET and OSET using covariance tapering are derived, including consistency, sparsity, asymptotic normality, and the oracle properties. For covariance tapering, a by-product of our theoretical results is consistency and asymptotic normality of maximum covariance-tapered likelihood estimates. Finite-sample properties of the proposed methods are demonstrated in a simulation study and for illustration, the methods are applied to analyze two real data sets. In the second part, we develop a new semiparametric approach to geostatistical modeling and inference. In particular, we consider a geostatistical model with additive components, where the covariance function of the spatial random error is not pre-specified and thus flexible. A novel, local Karhunen-Loève expansion is developed and a likelihood-based method devised for estimating the model parameters. In addition, statistical inference, including spatial interpolation and variable selection, is considered. Our proposed computational algorithm utilizes Newton-Raphson on a Stiefel manifold and is computationally efficient. A simulation study demonstrates sound finite-sample properties and a real data example is given to illustrate our method. While the numerical results are comparable to maximum likelihood estimation under the true model, our method is shown to be more robust against model misspecification and is computationally far more efficient for larger sample sizes. Finally, the theoretical properties of the estimates are explored and in particular, a consistency result is established.
Open Access
Semiparametric regression in the presence of complex variance structures arising from small angle x-ray scattering data
(Colorado State University. Libraries, 2014) Bugbee, Bruce D., author; Breidt, F. Jay, advisor; Estep, Don, advisor; Meyer, Mary, committee member; Hoeting, Jennifer, committee member; Luger, Karolin, committee member
An ongoing problem in structural biology is how best to infer structural information for complex, biological macromolecules from indirect observational data. Molecular shape dictates functionality but is not always directly observable. There exists a wide class of experimental methods whose data can be used for indirectly inferring molecular shape features with varying degrees of resolution. Of these methods, small angle X-ray scattering (SAXS) is desirable due to low requirements on the sample of interest. However, SAXS data suffers numerous statistical problems that require the development of novel methodologies. A primary concern is the impact of radially reducing two-dimensional sensor data to a series of smooth mean and variance curves. Additionally, pronounced heteroskedasticity is often observed near sensor boundaries. The work presented here focuses on developing general model frameworks and implementation methods appropriate for SAXS data. Semiparametric regression refers to models that combine known parametric structures with flexible nonparametric components. Three semiparametric regression model frameworks that are well-suited for handling smooth data are presented. The first model introduced is the standard semiparametric regression model, described as a mixed model with low rank penalized splines as random effects. The second model extends the first to the case of heteroskedastic errors, which violate standard model assumptions. The latent variance function in the model is estimated through an additional semiparametric regression, allowing for appropriate uncertainty estimation at the mean level. The final model considers a data structure unique to SAXS experiments. This model incorporates both radial mean and radial variance data in hopes to better infer three-dimensional shape properties and understand experimental effects by including all available data. Each of the three model frameworks is structured hierarchically. Bayesian inference is appealing in this context, as it provides efficient and generalized modeling frameworks in a unified way. The main statistical contributions of this thesis are from the specific methods developed to address the computational challenges of Bayesian inference for these models. The contributions include new Markov Chain Monte Carlo (MCMC) procedures for numerical approximation of posterior distributions and novel variational approximations that are extremely fast and accurate. For the heteroskedastic semiparametric case, known form posterior conditionals are available for all model parameters save for the regression coefficients controlling the latent model variance function. A novel implementation of a multivariate delayed rejection adaptive Metropolis (DRAM) procedure is used to sample from this posterior conditional distribution. The joint model for radial mean and radial variance data is shown to be of comparable structure to the heteroskedastic case and the new DRAM methodology is extended to handle this case. Simulation studies of all three methods are provided, showing that these models provide accurate fits of observed data and latent variance functions. The demands of scientific data processing in the context of SAXS, where large data sets are rapidly attained, lead to consideration of fast approximations as alternatives to MCMC. {Variational approximations} or {Variational Bayes} describes a class of approximation methods where the posterior distribution of the parameters is approximated by minimizing the Kullback-Leibler divergence between the true posterior and a class of distributions under mild structural constraints. Variational approximations have been shown to be good approximations of true posteriors in many cases. A novel variational approximation for the general heteroskedastic semiparametric regression model is derived here. Simulation studies are provided demonstrating fit and coverage properties comparable to the DRAM results at a fraction of the computational cost. A variational approximation for the joint model of radial mean and variance data is also provided but is shown to suffer from poor performance due to high correlation across a subset of regression parameters. The heteroskedastic semiparametric regression framework has some strong structural relationships with a distinct, important problem: spatially adaptive smoothing. A noisy function with different amounts of smoothness over its domain may be systematically under-smoothed or over-smoothed if the smoothing is not spatially adaptive. A novel variational approximation is derived for the problem of spatially adaptive penalized spline regression, and shown to have excellent performance. This approximation method is shown to be able to fit highly oscillatory data while not requiring the traditional tuning and computational resources of standard MCMC implementations. Potential scientific contribution of the statistical methodology developed here are illuminated with SAXS data examples. Analysis of SAXS data typically has two primary concerns: description of experimental effects and estimation of physical shape parameters. Formal statistical procedures for testing the effect of sample concentration and exposure time are presented as alternatives to current methods, in which data sets are evaluated subjectively and often combined in ad hoc ways. Additionally, estimation procedures for the scattering intensity at zero angle, known to be proportional to molecular weight, and the radius of gyration are described along with appropriate measures of uncertainty. Finally, a brief example of the joint radial mean and variance method is provided. Guidelines for extending the models presented here to more complex SAXS problems are also given.
Open Access
Statistical modeling and inferences on directed networks
(Colorado State University. Libraries, 2024) Du, Wenqin, author; Zhou, Wen, advisor; Breidt, F. Jay, committee member; Meyer, Mary, committee member; Pezeshki, Ali, committee member
Network data has received great attention for elucidating comprehensive insights into nodes interactions and underlying network dynamics. This dissertation contributes new modeling tools and inference procedures to the field of network analysis, incorporating the dependence structure inherently introduced by the network data. Our first direction centers on modeling directed edges with count measurements, an area that has received limited attention in the literature. Most existing methods either assume the count edges are derived from continuous random variables or model the edge dependence by parametric distributions. In this dissertation, we develop a latent multiplicative Poisson model for directed network with count edges. Our approach directly models the edge dependence of count data by the pairwise dependence of latent errors, which are assumed to be weakly exchangeable. This assumption not only covers a variety of common network effects, but also leads to a concise representation of the error covariance. In addition, identification and inference of the mean structure, as well as the regression coefficients, depend on the errors only through their covariance, which provides substantial flexibility for our model. We propose a pseudo-likelihood based estimator for the regression coefficients that enjoys consistency and asymptotic normality. We evaluate our method by extensive numerical studies that corroborate the theory and apply our model to a food sharing network data to reveal interesting network effects that are further verified in literature. In the second project, we study the inference procedure of network dependence structures. While much research has targeted network-covariate associations and community detection, the inference of important network effects such as the reciprocity and sender-receiver effects has been largely overlooked. Testing network effects for network data or weighted directed networks is challenging due to the intricate potential edge dependence. Most existing methods are model-based, carrying strong assumptions with restricted applicability. In contrast, we present a novel, fully nonparametric framework that requires only minimal regularity assumptions. While inspired by recent developments in U-statistic literature, our work significantly broadens their scopes. Specifically, we identified and carefully addressed the indeterminate degeneracy inherent in network effect estimators - a challenge that aforementioned tools do not handle. We established Berry-Esseen type bound for the accuracy of type-I error rate control, as well as novel analysis show the minimax optimality of our test's power. Simulations highlight the superiority of our method in computation speed, accuracy, and numerical robustness relative to benchmarks. To showcase the practicality of our methods, we apply them to two real-world relationship networks, one in faculty hiring networks and the other in international trade networks. Finally, this dissertation introduces modeling strategies and corresponding methods for discerning the core-periphery (CP) structure in weighted directed networks. We adopt the signal-plus-noise model, categorizing uniform relational patterns as non-informative, by which we define the sender and receiver peripheries. Furthermore, instead of confining the core component to a specific structure, we consider it complementary to either the sender or receiver peripheries. Based on our definitions of the sender and receiver peripheries, we propose spectral algorithms to identify the CP structure in weighted directed networks. Our algorithm stands out with statistical guarantees, ensuring the identification of sender and receiver peripheries with overwhelmingly probability. Additionally, our methods scale effectively for expansive directed networks. We evaluate the proposed methods in extensive simulation studies and applied it to a faculty hiring network data, revealing captivating insights into the informative and non-informative sender/receiver behaviors.
Open Access
Statistical models for COVID-19 infection fatality rates and diagnostic test data
(Colorado State University. Libraries, 2023) Pugh, Sierra, author; Wilson, Ander, advisor; Fosdick, Bailey K., advisor; Keller, Kayleigh, committee member; Meyer, Mary, committee member; Gutilla, Molly, committee member
The COVID-19 pandemic has had devastating impacts worldwide. Early in the pandemic, little was known about the emerging disease. To inform policy, it was essential to develop data science tools to inform public health policy and interventions. We developed methods to fill three gaps in the literature. A first key task for scientists at the start of the pandemic was to develop diagnostic tests to classify an individual's disease status as positive or negative and to estimate community prevalence. Researchers rapidly developed diagnostic tests, yet there was a lack of guidance on how to select a cutoff to classify positive and negative test results for COVID-19 antibody tests developed with limited numbers of controls with known disease status. We propose selecting a cutoff using extreme value theory and compared this method to existing methods through a data analysis and simulation study. Second, there lacked a cohesive method for estimating the infection fatality rate (IFR) of COVID-19 that fully accounted for uncertainty in the fatality data, seroprevalence study data, and antibody test characteristics. We developed a Bayesian model to jointly model these data to fully account for the many sources of uncertainty. A third challenge is providing information that can be used to compare seroprevalence and IFR across locations to best allocate resources and target public health interventions. It is particularly important to account for differences in age-distributions when comparing across locations as age is a well-established risk factor for COVID-19 mortality. There is a lack of methods for estimating the seroprevalence and IFR as continuous functions of age, while adequately accounting for uncertainty. We present a Bayesian hierarchical model that jointly estimates seroprevalence and IFR as continuous functions of age, sharing information across locations to improve identifiability. We use this model to estimate seroprevalence and IFR in 26 developing country locations.
Open Access
Testing and adjusting for informative sampling in survey data
(Colorado State University. Libraries, 2014) Herndon, Wade Wilson, author; Breidt, F. Jay, advisor; Opsomer, Jean, advisor; Cooley, Daniel, committee member; Meyer, Mary, committee member; Doherty, Paul, committee member
Fitting models to survey data can be problematic due to the potentially complex sampling mechanism through which the observed data are selected. Survey weights have traditionally been used to adjust for unequal inclusion probabilities under the design-based paradigm of inference, however, this limits the ability of analysts to make inference of a more general kind, such as to characteristics of a superpopulation. The problems induced by the presence of a complex sampling design can be generally contained under the heading of informative sampling. To say that the sampling is informative is to say that the distribution of the data in the sample is different from the distribution of the data in the population. Two major topics relating to analyzing survey data with (potentially) informative sampling are addressed: testing for informativeness, and model building in the presence of informative sampling. First addressed is the problem of running formal tests for informative sampling in survey data. The major contribution contained here is to detail a new test for informative sampling. The test is shown to be widely applicable and straight-forward to implement in practice, and also useful compared to existing tests. The test is illustrated through a variety of empirical studies as well. These applications include a censored regression problem, linear regression, logistic regression, and fitting a gamma mixture model. Results from the analogous bootstrap test are also presented; these results agree with the analytic versions of the test. Alternative tests for informative sampling do in fact exist, however, the existing methods each have significant drawbacks and limitations which may be resolved in some situation with this new methodology, and overall the literature is quite sparse in this area. In a simulation study, the test is shown to have many desirable properties and maintains high power compared to alternative tests. Also included is discussion about the limiting distribution of the test statistic under a sequence of local alternative hypotheses, and some extensions that are useful in connecting the work contained here with some of the previous work in the area. These extensions also help motivate the semiparametric methods considered in the chapter that follows. The next topic explored is semiparametric methods for including design information in a regression model while staying within a model-based inferential framework. The ideas explored here attempt to exploit relationships between design variables (such as the sample inclusion probabilities) and model covariates. In order to account for the complex sampling design and (potential) bias in estimating model parameters, design variables are included as covariates and considered to be functions of the model covariates that can then be estimated in a design-based paradigm using nonparametric methods. The nonparametric method explored here is kernel smoothing with degree zero. In principle, other (and more complex) kinds of estimators could be used to estimate the functions of the design variables conditional on the model covariates, but the framework presented here provides asymptotic results for only the more simple case of kernel smoothing. The method is illustrated via empirical applications and also through a simulation study in which confidence band coverage rates from the semiparametric method are compared to those obtained through regular linear regression. The semiparametric estimator soundly outperforms the regression estimator.

Browsing by Author "Meyer, Mary, committee member"

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