arXiv subjects

Jonathan R. Bradley

Publications and source records attributed to Jonathan R. Bradley.

30 records · Page 2Linked to original sources

arXiv2019

What is the best predictor that you can compute in five minutes using a given Bayesian hierarchical model?

The goal of this paper is to provide a way for statisticians to answer the question posed in the title of this article using any Bayesian hierarchical model of their choosing and without imposing additional restrictive model assumptions. We are motivated by the fact that the rise of ``big data'' has created difficulties for statisticians to directly apply their methods to big datasets. We introduce a ``data subset model'' to the popular ``data model, process model, and parameter model'' framework used to summarize Bayesian hierarchical models. The hyperparameters of the data subset model are specified constructively in that they are chosen such that the implied size of the subset satisfies pre-defined computational constraints. Thus, these hyperparameters effectively calibrates the statistical model to the computer itself to obtain predictions/estimations in a pre-specified amount of time. Several properties of the data subset model are provided including: propriety, partial sufficiency, and semi-parametric properties. Furthermore, we show that subsets of normally distributed data are asymptotically partially sufficient under reasonable constraints. Results from a simulated dataset will be presented across different computers, to show the effect of the computer on the statistical analysis. Additionally, we provide a joint spatial analysis of two different environmental datasets.

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arXiv2019

Latent Multivariate Log-Gamma Models for High-Dimensional Multi-Type Responses with Application to Daily Fine Particulate Matter and Mortality Counts

Tracking and estimating Daily Fine Particulate Matter (PM2.5) is very important as it has been shown that PM2.5 is directly related to mortality related to lungs, cardiovascular system, and stroke. That is, high values of PM2.5 constitute a public health problem in the US, and it is important that we precisely estimate PM2.5 to aid in public policy decisions. Thus, we propose a Bayesian hierarchical model for high-dimensional "multi-type" responses. By "multi-type" responses we mean a collection of correlated responses that have different distributional assumptions (e.g., continuous skewed observations, and count-valued observations). The Centers for Disease Control and Prevention (CDC) database provides counts of mortalities related to PM2.5 and daily averaged PM2.5 which are both treated as responses in our analysis. Our model capitalizes on the shared conjugate structure between the Weibull (to model PM2.5), Poisson (to model diseases mortalities), and multivariate log-gamma distributions, and we use dimension reduction to aid with computation. Our model can also be used to improve the precision of estimates and estimate values at undisclosed/missing counties. We provide a simulation study to illustrate the performance of the model, and give an in-depth analysis of the CDC dataset.

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arXiv2019

Bayesian Hierarchical Models with Conjugate Full-Conditional Distributions for Dependent Data from the Natural Exponential Family

We introduce a Bayesian approach for analyzing (possibly) high-dimensional dependent data that are distributed according to a member from the natural exponential family of distributions. This problem requires extensive methodological advancements, as jointly modeling high-dimensional dependent data leads to the so-called "big n problem." The computational complexity of the "big n problem" is further exacerbated when allowing for non-Gaussian data models, as is the case here. Thus, we develop new computationally efficient distribution theory for this setting. In particular, we introduce the "conjugate multivariate distribution," which is motivated by the univariate distribution introduced in Diaconis and Ylvisaker (1979). Furthermore, we provide substantial theoretical and methodological development including: results regarding conditional distributions, an asymptotic relationship with the multivariate normal distribution, conjugate prior distributions, and full-conditional distributions for a Gibbs sampler. To demonstrate the wide-applicability of the proposed methodology, we provide two simulation studies and three applications based on an epidemiology dataset, a federal statistics dataset, and an environmental dataset, respectively.

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arXiv2018

Spatio-Temporal Models for Big Multinomial Data using the Conditional Multivariate Logit-Beta Distribution

We introduce a Bayesian approach for analyzing high-dimensional multinomial data that are referenced over space and time. In particular, the proportions associated with multinomial data are assumed to have a logit link to a latent spatio-temporal mixed effects model. This strategy allows for covariances that are nonstationarity in both space and time, asymmetric, and parsimonious. We also introduce the use of the conditional multivariate logit-beta distribution into the dependent multinomial data setting, which leads to conjugate full-conditional distributions for use in a collapsed Gibbs sampler. We refer to this model as the multinomial spatio-temporal mixed effects model (MN-STM). Additionally, we provide methodological developments including: the derivation of the associated full-conditional distributions, a relationship with a latent Gaussian process model, and the stability of the non-stationary vector autoregressive model. We illustrate the MN-STM through simulations and through a demonstration with public-use Quarterly Workforce Indicators (QWI) data from the Longitudinal Employer Household Dynamics (LEHD) program of the U.S. Census Bureau.

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arXiv2017

Multi-rubric Models for Ordinal Spatial Data with Application to Online Ratings from Yelp

Interest in online rating data has increased in recent years in which ordinal ratings of products or local businesses are provided by users of a website, such as Yelp or Amazon. One source of heterogeneity in ratings is that users apply different standards when supplying their ratings; even if two users benefit from a product the same amount, they may translate their benefit into ratings in different ways. In this article we propose an ordinal data model, which we refer to as a multi-rubric model, which treats the criteria used to convert a latent utility into a rating as user-specific random effects, with the distribution of these random effects being modeled nonparametrically. We demonstrate that this approach is capable of accounting for this type of variability in addition to usual sources of heterogeneity due to item quality, user biases, interactions between items and users, and the spatial structure of the users and items. We apply the model developed here to publicly available data from the website Yelp and demonstrate that it produces interpretable clusterings of users according to their rating behavior, in addition to providing better predictions of ratings and better summaries of overall item quality.

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arXiv2016

Multivariate spatio-temporal models for high-dimensional areal data with application to Longitudinal Employer-Household Dynamics

Many data sources report related variables of interest that are also referenced over geographic regions and time; however, there are relatively few general statistical methods that one can readily use that incorporate these multivariate spatio-temporal dependencies. Additionally, many multivariate spatio-temporal areal data sets are extremely high dimensional, which leads to practical issues when formulating statistical models. For example, we analyze Quarterly Workforce Indicators (QWI) published by the US Census Bureau's Longitudinal Employer-Household Dynamics (LEHD) program. QWIs are available by different variables, regions, and time points, resulting in millions of tabulations. Despite their already expansive coverage, by adopting a fully Bayesian framework, the scope of the QWIs can be extended to provide estimates of missing values along with associated measures of uncertainty. Motivated by the LEHD, and other applications in federal statistics, we introduce the multivariate spatio-temporal mixed effects model (MSTM), which can be used to efficiently model high-dimensional multivariate spatio-temporal areal data sets. The proposed MSTM extends the notion of Moran's I basis functions to the multivariate spatio-temporal setting. This extension leads to several methodological contributions, including extremely effective dimension reduction, a dynamic linear model for multivariate spatio-temporal areal processes, and the reduction of a high-dimensional parameter space using a novel parameter model.

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arXiv2015

Computationally Efficient Distribution Theory for Bayesian Inference of High-Dimensional Dependent Count-Valued Data

We introduce a Bayesian approach for multivariate spatio-temporal prediction for high-dimensional count-valued data. Our primary interest is when there are possibly millions of data points referenced over different variables, geographic regions, and times. This problem requires extensive methodological advancements, as jointly modeling correlated data of this size leads to the so-called "big n problem." The computational complexity of prediction in this setting is further exacerbated by acknowledging that count-valued data are naturally non-Gaussian. Thus, we develop a new computationally efficient distribution theory for this setting. In particular, we introduce a multivariate log-gamma distribution and provide substantial theoretical development including: results regarding conditional distributions, marginal distributions, an asymptotic relationship with the multivariate normal distribution, and full-conditional distributions for a Gibbs sampler. To incorporate dependence between variables, regions, and time points, a multivariate spatio-temporal mixed effects model (MSTM) is used. The results in this manuscript are extremely general, and can be used for data that exhibit fewer sources of dependency than what we consider (e.g., multivariate, spatial-only, or spatio-temporal-only data). Hence, the implications of our modeling framework may have a large impact on the general problem of jointly modeling correlated count-valued data. We show the effectiveness of our approach through a simulation study. Additionally, we demonstrate our proposed methodology with an important application analyzing data obtained from the Longitudinal Employer-Household Dynamics (LEHD) program, which is administered by the U.S. Census Bureau.

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arXiv2015

Regionalization of Multiscale Spatial Processes using a Criterion for Spatial Aggregation Error

The modifiable areal unit problem and the ecological fallacy are known problems that occur when modeling multiscale spatial processes. We investigate how these forms of spatial aggregation error can guide a regionalization over a spatial domain of interest. By "regionalization" we mean a specification of geographies that define the spatial support for areal data. This topic has been studied vigorously by geographers, but has been given less attention by spatial statisticians. Thus, we propose a criterion for spatial aggregation error (CAGE), which we minimize to obtain an optimal regionalization. To define CAGE we draw a connection between spatial aggregation error and a new multiscale representation of the Karhunen-Loeve (K-L) expansion. This relationship between CAGE and the multiscale K-L expansion leads to illuminating theoretical developments including: connections between spatial aggregation error, squared prediction error, spatial variance, and a novel extension of Obled-Creutin eigenfunctions. The effectiveness of our approach is demonstrated through an analysis of two datasets, one using the American Community Survey and one related to environmental ocean winds.

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arXiv2015

Spatio-Temporal Change of Support with Application to American Community Survey Multi-Year Period Estimates

We present hierarchical Bayesian methodology to perform spatio-temporal change of support (COS) for survey data with Gaussian sampling errors. This methodology is motivated by the American Community Survey (ACS), which is an ongoing survey administered by the U.S. Census Bureau that provides timely information on several key demographic variables. The ACS has published 1-year, 3-year, and 5-year period-estimates, and margins of errors, for demographic and socio-economic variables recorded over predefined geographies. The spatio-temporal COS methodology considered here provides data users with a way to estimate ACS variables on customized geographies and time periods, while accounting for sampling errors. Additionally, 3-year ACS period estimates are to be discontinued, and this methodology can provide predictions of ACS variables for 3-year periods given the available period estimates. The methodology is based on a spatio-temporal mixed effects model with a low-dimensional spatio-temporal basis function representation, which provides multi-resolution estimates through basis function aggregation in space and time. This methodology includes a novel parameterization that uses a target dynamical process and recently proposed parsimonious Moran's I propagator structures. Our approach is demonstrated through two applications using public-use ACS estimates, and is shown to produce good predictions on a holdout set of 3-year period estimates.

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arXiv2014

Bayesian Spatial Change of Support for Count-Valued Survey Data

We introduce Bayesian spatial change of support methodology for count-valued survey data with known survey variances. Our proposed methodology is motivated by the American Community Survey (ACS), an ongoing survey administered by the U.S. Census Bureau that provides timely information on several key demographic variables. Specifically, the ACS produces 1-year, 3-year, and 5-year "period-estimates," and corresponding margins of errors, for published demographic and socio-economic variables recorded over predefined geographies within the United States. Despite the availability of these predefined geographies it is often of interest to data users to specify customized user-defined spatial supports. In particular, it is useful to estimate demographic variables defined on "new" spatial supports in "real-time." This problem is known as spatial change of support (COS), which is typically performed under the assumption that the data follows a Gaussian distribution. However, count-valued survey data is naturally non-Gaussian and, hence, we consider modeling these data using a Poisson distribution. Additionally, survey-data are often accompanied by estimates of error, which we incorporate into our analysis. We interpret Poisson count-valued data in small areas as an aggregation of events from a spatial point process. This approach provides us with the flexibility necessary to allow ACS users to consider a variety of spatial supports in "real-time." We demonstrate the effectiveness of our approach through a simulated example as well as through an analysis using public-use ACS data.

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arXiv2014

A Comparison of Spatial Predictors when Datasets Could be Very Large

In this article, we review and compare a number of methods of spatial prediction. To demonstrate the breadth of available choices, we consider both traditional and more-recently-introduced spatial predictors. Specifically, in our exposition we review: traditional stationary kriging, smoothing splines, negative-exponential distance-weighting, Fixed Rank Kriging, modified predictive processes, a stochastic partial differential equation approach, and lattice kriging. This comparison is meant to provide a service to practitioners wishing to decide between spatial predictors. Hence, we provide technical material for the unfamiliar, which includes the definition and motivation for each (deterministic and stochastic) spatial predictor. We use a benchmark dataset of $\mathrm{CO}_{2}$ data from NASA's AIRS instrument to address computational efficiencies that include CPU time and memory usage. Furthermore, the predictive performance of each spatial predictor is assessed empirically using a hold-out subset of the AIRS data.

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arXiv2014

Mixed Effects Modeling for Areal Data that Exhibit Multivariate-Spatio-Temporal Dependencies

There are many data sources available that report related variables of interest that are also referenced over geographic regions and time; however, there are relatively few general statistical methods that one can readily use that incorporate these multivariate-spatio-temporal dependencies. As such, we introduce the multivariate-spatio-temporal mixed effects model (MSTM) to analyze areal data with multivariate-spatio-temporal dependencies. The proposed MSTM extends the notion of Moran's I basis functions to the multivariate-spatio-temporal setting. This extension leads to several methodological contributions including extremely effective dimension reduction, a dynamic linear model for multivariate-spatio-temporal areal processes, and the reduction of a high-dimensional parameter space using a novel parameter model. Several examples are used to demonstrate that the MSTM provides an extremely viable solution to many important problems found in different and distinct corners of the spatio-temporal statistics literature including: modeling nonseparable and nonstationary covariances, combing data from multiple repeated surveys, and analyzing massive multivariate-spatio-temporal datasets.

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