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While the overarching pattern of biannual avian migration is well understood, there are significant questions pertaining to this phenomenon that invite further study. Necessary to any analysis of these questions is an understanding of how a given species' spatial distribution evolves in time. While studies of animal movement are often conducted using telemetry data, the collection of such data can be time- and resource-intensive, frequently resulting in small sample sizes. Ecological surveys of animal populations are also indicative of species distribution trends, but may be constrained to a limited spatial domain. Within this article we utilize crowd-sourced observations from the eBird database to model the abundance of migratory bird species in space and time. While crowd-sourced observations are individually less reliable than those produced by experts, the sheer size and spatial coverage of the eBird database make it attractive for use in this setting. We introduce a hidden Markov model for observed bird counts utilizing a novel transition structure developed using principles from circuit theory. After illustrating model properties we fit it to observations of Baltimore orioles and yellow-rumped warblers within the eastern United States and discuss insight it provides into the migratory patterns for these species., Comment: 33 pages, 12 figures

Multi-group covariance estimation for matrix-variate data with small within group sample sizes is a key part of many data analysis tasks in modern applications. To obtain accurate group-specific covariance estimates, shrinkage estimation methods which shrink an unstructured, group-specific covariance either across groups towards a pooled covariance or within each group towards a Kronecker structure have been developed. However, in many applications, it is unclear which approach will result in more accurate covariance estimates. In this article, we present a hierarchical prior distribution which flexibly allows for both types of shrinkage. The prior linearly combines shrinkage across groups towards a shared pooled covariance and shrinkage within groups towards a group-specific Kronecker covariance. We illustrate the utility of the proposed prior in speech recognition and an analysis of chemical exposure data., Comment: 28 pages, 7 figures, 5 tables

Spatial models for areal data are often constructed such that all pairs of adjacent regions are assumed to have near-identical spatial autocorrelation. In practice, data can exhibit dependence structures more complicated than can be represented under this assumption. In this article we develop a new model for spatially correlated data observed on graphs, which can flexibly represented many types of spatial dependence patterns while retaining aspects of the original graph geometry. Our method implies an embedding of the graph into Euclidean space wherein covariance can be modeled using traditional covariance functions, such as those from the Mat\'{e}rn family. We parameterize our model using a class of graph metrics compatible with such covariance functions, and which characterize distance in terms of network flow, a property useful for understanding proximity in many ecological settings. By estimating the parameters underlying these metrics, we recover the "intrinsic distances" between graph nodes, which assist in the interpretation of the estimated covariance and allow us to better understand the relationship between the observed process and spatial domain. We compare our model to existing methods for spatially dependent graph data, primarily conditional autoregressive models and their variants, and illustrate advantages of our method over traditional approaches. We fit our model to bird abundance data for several species in North Carolina, and show how it provides insight into the interactions between species-specific spatial distributions and geography., Comment: 36 pages, 7 figures

Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data., Comment: 24 pages, 9 figures

Many analyses of multivariate data focus on evaluating the dependence between two sets of variables, rather than the dependence among individual variables within each set. Canonical correlation analysis (CCA) is a classical data analysis technique that estimates parameters describing the dependence between such sets. However, inference procedures based on traditional CCA rely on the assumption that all variables are jointly normally distributed. We present a semiparametric approach to CCA in which the multivariate margins of each variable set may be arbitrary, but the dependence between variable sets is described by a parametric model that provides low-dimensional summaries of dependence. While maximum likelihood estimation in the proposed model is intractable, we propose two estimation strategies: one using a pseudolikelihood for the model and one using a Markov chain Monte Carlo (MCMC) algorithm that provides Bayesian estimates and confidence regions for the between-set dependence parameters. The MCMC algorithm is derived from a multirank likelihood function, which uses only part of the information in the observed data in exchange for being free of assumptions about the multivariate margins. We apply the proposed Bayesian inference procedure to Brazilian climate data and monthly stock returns from the materials and communications market sectors.

Directional relational event data, such as email data, often contain unicast messages (i.e., messages of one sender towards one receiver) and multicast messages (i.e., messages of one sender towards multiple receivers). The Enron email data that is the focus in this paper consists of 31% multicast messages. Multicast messages contain important information about the roles of actors in the network, which is needed for better understanding social interaction dynamics. In this paper a multiplicative latent factor model is proposed to analyze such relational data. For a given message, all potential receiver actors are placed on a suitability scale, and the actors are included in the receiver set whose suitability score exceeds a threshold value. Unobserved heterogeneity in the social interaction behavior is captured using a multiplicative latent factor structure with latent variables for actors (which differ for actors as senders and receivers) and latent variables for individual messages. A Bayesian computational algorithm, which relies on Gibbs sampling, is proposed for model fitting. Model assessment is done using posterior predictive checks. Based on our analyses of the Enron email data, a mc-amen model with a 2 dimensional latent variable can accurately capture the empirical distribution of the cardinality of the receiver set and the composition of the receiver sets for commonly observed messages. Moreover the results show that actors have a comparable (but not identical) role as a sender and as a receiver in the network., Comment: 56 pages, 41 figures

Quantification of stylistic differences between musical artists is of academic interest to the music community, and is also useful for other applications such as music information retrieval and recommendation systems. Information about stylistic differences can be obtained by comparing the performances of different artists across common musical pieces. In this article, we develop a statistical methodology for identifying and quantifying systematic stylistic differences among artists that are consistent across audio recordings of a common set of pieces, in terms of several musical features. Our focus is on a comparison of ten different orchestras, based on data from audio recordings of the nine Beethoven symphonies. As generative or fully parametric models of raw audio data can be highly complex, and more complex than necessary for our goal of identifying differences between orchestras, we propose to reduce the data from a set of audio recordings down to pairwise distances between orchestras, based on different musical characteristics of the recordings, such as tempo, dynamics, and timbre. For each of these characteristics, we obtain multiple pairwise distance matrices, one for each movement of each symphony. We develop a hierarchical multidimensional scaling (HMDS) model to identify and quantify systematic differences between orchestras in terms of these three musical characteristics, and interpret the results in the context of known qualitative information about the orchestras. This methodology is able to recover several expected systematic similarities between orchestras, as well as to identify some more novel results. For example, we find that modern recordings exhibit a high degree of similarity to each other, as compared to older recordings., Comment: Published in the Annals of Applied Statistics (https://projecteuclid.org/euclid.aoas/1608346888)

Medical research institutions have generated massive amounts of biological data by genetically profiling hundreds of cancer cell lines. In parallel, academic biology labs have conducted genetic screens on small numbers of cancer cell lines under custom experimental conditions. In order to share information between these two approaches to scientific discovery, this article proposes a "frequentist assisted by Bayes" (FAB) procedure for hypothesis testing that allows historical information from massive genomics datasets to increase the power of hypothesis tests in specialized studies. The exchange of information takes place through a novel probability model for multimodal genomics data, which distills historical information pertaining to cancer cell lines and genes across a wide variety of experimental contexts. If the relevance of the historical information for a given study is high, then the resulting FAB tests can be more powerful than the corresponding classical tests. If the relevance is low, then the FAB tests yield as many discoveries as the classical tests. Simulations and practical investigations demonstrate that the FAB testing procedure can increase the number of effects discovered in genomics studies while still maintaining strict control of type I error and false discovery rates.

This article develops $p$-values for evaluating means of normal populations that make use of indirect or prior information. A $p$-value of this type is based on a biased test statistic that is optimal on average with respect to a probability distribution that encodes indirect information about the mean parameter, resulting in a smaller $p$-value if the indirect information is accurate. In a variety of multiparameter settings, we show how to adaptively estimate the indirect information for each mean parameter while still maintaining uniformity of the $p$-values under their null hypotheses. This is done using a linking model through which indirect information about the mean of one population may be obtained from the data of other populations. Importantly, the linking model does not need to be correct to maintain the uniformity of the $p$-values under their null hypotheses. This methodology is illustrated in several data analysis scenarios, including small area inference, spatially arranged populations, interactions in linear regression, and generalized linear models.

Motivated by applications to Bayesian inference for statistical models with orthogonal matrix parameters, we present $\textit{polar expansion},$ a general approach to Monte Carlo simulation from probability distributions on the Stiefel manifold. To bypass many of the well-established challenges of simulating from the distribution of a random orthogonal matrix $\boldsymbol{Q},$ we construct a distribution for an unconstrained random matrix $\boldsymbol{X}$ such that $\boldsymbol{Q}_X,$ the orthogonal component of the polar decomposition of $\boldsymbol{X},$ is equal in distribution to $\boldsymbol{Q}.$ The distribution of $\boldsymbol{X}$ is amenable to Markov chain Monte Carlo (MCMC) simulation using standard methods, and an approximation to the distribution of $\boldsymbol{Q}$ can be recovered from a Markov chain on the unconstrained space. When combined with modern MCMC software, polar expansion allows for routine and flexible posterior inference in models with orthogonal matrix parameters. We find that polar expansion with adaptive Hamiltonian Monte Carlo is an order of magnitude more efficient than competing MCMC approaches in a benchmark protein interaction network application. We also propose a new approach to Bayesian functional principal components analysis which we illustrate in a meteorological time series application., Comment: 24 pages, 4 figures, 1 table

In many regression settings the unknown coefficients may have some known structure, for instance they may be ordered in space or correspond to a vectorized matrix or tensor. At the same time, the unknown coefficients may be sparse, with many nearly or exactly equal to zero. However, many commonly used priors and corresponding penalties for coefficients do not encourage simultaneously structured and sparse estimates. In this paper we develop structured shrinkage priors that generalize multivariate normal, Laplace, exponential power and normal-gamma priors. These priors allow the regression coefficients to be correlated a priori without sacrificing elementwise sparsity or shrinkage. The primary challenges in working with these structured shrinkage priors are computational, as the corresponding penalties are intractable integrals and the full conditional distributions that are needed to approximate the posterior mode or simulate from the posterior distribution may be non-standard. We overcome these issues using a flexible elliptical slice sampling procedure, and demonstrate that these priors can be used to introduce structure while preserving sparsity.

Random orthogonal matrices play an important role in probability and statistics, arising in multivariate analysis, directional statistics, and models of physical systems, among other areas. Calculations involving random orthogonal matrices are complicated by their constrained support. Accordingly, we parametrize the Stiefel and Grassmann manifolds, represented as subsets of orthogonal matrices, in terms of Euclidean parameters using the Cayley transform. We derive the necessary Jacobian terms for change of variables formulas. Given a density defined on the Stiefel or Grassmann manifold, these allow us to specify the corresponding density for the Euclidean parameters, and vice versa. As an application, we describe and illustrate through examples a Markov chain Monte Carlo approach to simulating from distributions on the Stiefel and Grassmann manifolds. Finally, we establish an asymptotic independent normal approximation for the distribution of the Euclidean parameters which corresponds to the uniform distribution on the Stiefel manifold. This result contributes to the growing literature on normal approximations to the entries of random orthogonal matrices or transformations thereof., Comment: 34 pages, 2 figures

Network datasets typically exhibit certain types of statistical dependencies, such as within-dyad correlation, row and column heterogeneity, and third-order dependence patterns such as transitivity and clustering. The first two of these can be well-represented statistically with a social relations model, a type of additive random effects model originally developed for continuous dyadic data. Third-order patterns can be represented with multiplicative random effects models, which are related to matrix decompositions commonly used for matrix-variate data analysis. Additionally, these multiplicative random effects models generalize other popular latent variable network models, such as the stochastic blockmodel and the latent space model. In this article we review a general regression framework for the analysis of network data that combines these two types of random effects and accommodates a variety of network data types, including continuous, binary and ordinal network relations.

In multiple testing scenarios, typically the sign of a parameter is inferred when its estimate exceeds some significance threshold in absolute value. Typically, the significance threshold is chosen to control the experimentwise type I error rate, family-wise type I error rate or the false discovery rate. However, controlling these error rates does not explicitly control the sign error rate. In this paper, we propose two procedures for adaptively selecting an experimentwise significance threshold in order to control the sign error rate. The first controls the sign error rate conservatively, without any distributional assumptions on the parameters of interest. The second is an empirical Bayes procedure, and achieves optimal performance asymptotically when a model for the distribution of the parameters is correctly specified. We also discuss an adaptive procedure to minimize the sign error rate when the experimentwise type I error rate is held fixed., Comment: 19 pages, 4 figures

Many penalized maximum likelihood estimators correspond to posterior mode estimators under specific prior distributions. Appropriateness of a particular class of penalty functions can therefore be interpreted as the appropriateness of a prior for the parameters. For example, the appropriateness of a lasso penalty for regression coefficients depends on the extent to which the empirical distribution of the regression coefficients resembles a Laplace distribution. We give a testing procedure of whether or not a Laplace prior is appropriate and accordingly, whether or not using a lasso penalized estimate is appropriate. This testing procedure is designed to have power against exponential power priors which correspond to $\ell_q$ penalties. Via simulations, we show that this testing procedure achieves the desired level and has enough power to detect violations of the Laplace assumption when the numbers of observations and unknown regression coefficients are large. We then introduce an adaptive procedure that chooses a more appropriate prior and corresponding penalty from the class of exponential power priors when the null hypothesis is rejected. We show that this can improve estimation of the regression coefficients both when they are drawn from an exponential power distribution and when they are drawn from a spike-and-slab distribution.

We introduce a simple and extendable coevolution model for the analysis of longitudinal network and nodal attribute data. The model features parameters that describe three phenomena: homophily, contagion and autocorrelation of the network and nodal attribute process. Homophily here describes how changes to the network may be associated with between-node similarities in terms of their nodal attributes. Contagion refers to how node-level attributes may change depending on the network. The model we present is based upon a pair of intertwined autoregressive processes. We obtain least-squares parameter estimates for continuous-valued fully-observed network and attribute data. We also provide methods for Bayesian inference in several other cases, including ordinal network and attribute data, and models involving latent nodal attributes. These model extensions are applied to an analysis of international relations data and to data from a study of teen delinquency and friendship networks., Comment: 20 pages

Measuring influence and determining what drives it are persistent questions in political science and in network analysis more generally. Herein we focus on the domain of international relations. Our major substantive question is: How can we determine what characteristics make an actor influential? To address the topic of influence, we build on a multilinear tensor regression framework (MLTR) that captures influence relationships using a tensor generalization of a vector autoregression model. Influence relationships in that approach are captured in a pair of n x n matrices and provide measurements of how the network actions of one actor may influence the future actions of another. A limitation of the MLTR and earlier latent space approaches is that there are no direct mechanisms through which to explain why a certain actor is more or less influential than others. Our new framework, social influence regression, provides a way to statistically model the influence of one actor on another as a function of characteristics of the actors. Thus we can move beyond just estimating that an actor influences another to understanding why. To highlight the utility of this approach, we apply it to studying monthly-level conflictual events between countries as measured through the Integrated Crisis Early Warning System (ICEWS) event data project., Comment: 23 pages, 8 figures

We propose an adaptive confidence interval procedure (CIP) for the coefficients in the normal linear regression model. This procedure has a frequentist coverage rate that is constant as a function of the model parameters, yet provides smaller intervals than the usual interval procedure, on average across regression coefficients. The proposed procedure is obtained by defining a class of CIPs that all have exact $1-\alpha$ frequentist coverage, and then selecting from this class the procedure that minimizes a prior expected interval width. Such a procedure may be described as "frequentist, assisted by Bayes" or FAB. We describe an adaptive approach for estimating the prior distribution from the data so that exact non-asymptotic $1-\alpha$ coverage is maintained. Additionally, in a "$p$ growing with $n$" asymptotic scenario, this adaptive FAB procedure is asymptotically Bayes-optimal among $1-\alpha$ frequentist CIPs.

Consider the problem of estimating the entries of an unknown mean matrix or tensor given a single noisy realization. In the matrix case, this problem can be addressed by decomposing the mean matrix into a component that is additive in the rows and columns, i.e.\ the additive ANOVA decomposition of the mean matrix, plus a matrix of elementwise effects, and assuming that the elementwise effects may be sparse. Accordingly, the mean matrix can be estimated by solving a penalized regression problem, applying a lasso penalty to the elementwise effects. Although solving this penalized regression problem is straightforward, specifying appropriate values of the penalty parameters is not. Leveraging the posterior mode interpretation of the penalized regression problem, moment-based empirical Bayes estimators of the penalty parameters can be defined. Estimation of the mean matrix using these these moment-based empirical Bayes estimators can be called LANOVA penalization, and the corresponding estimate of the mean matrix can be called the LANOVA estimate. The empirical Bayes estimators are shown to be consistent. Additionally, LANOVA penalization is extended to accommodate sparsity of row and column effects and to estimate an unknown mean tensor. The behavior of the LANOVA estimate is examined under misspecification of the distribution of the elementwise effects, and LANOVA penalization is applied to several datasets, including a matrix of microarray data, a three-way tensor of fMRI data and a three-way tensor of wheat infection data.

Confidence intervals for the means of multiple normal populations are often based on a hierarchical normal model. While commonly used interval procedures based on such a model have the nominal coverage rate on average across a population of groups, their actual coverage rate for a given group will be above or below the nominal rate, depending on the value of the group mean. Alternatively, a coverage rate that is constant as a function of a group's mean can be simply achieved by using a standard $t$-interval, based on data only from that group. The standard $t$-interval, however, fails to share information across the groups and is therefore not adaptive to easily obtained information about the distribution of group-specific means. In this article we construct confidence intervals that have a constant frequentist coverage rate and that make use of information about across-group heterogeneity, resulting in constant-coverage intervals that are narrower than standard $t$-intervals on average across groups. Such intervals are constructed by inverting biased tests for the mean of a normal population. Given a prior distribution on the mean, Bayes-optimal biased tests can be inverted to form Bayes-optimal confidence intervals with frequentist coverage that is constant as a function of the mean. In the context of multiple groups, the prior distribution is replaced by a model of across-group heterogeneity. The parameters for this model can be estimated using data from all of the groups, and used to obtain confidence intervals with constant group-specific coverage that adapt to information about the distribution of group means.

We introduce a Bayesian approach to conduct inferential analyses on dyadic data while accounting for interdependencies between observations through a set of additive and multiplicative effects (AME). The AME model is built on a generalized linear modeling framework and is thus flexible enough to be applied to a variety of contexts. We contrast the AME model to two prominent approaches in the literature: the latent space model (LSM) and the exponential random graph model (ERGM). Relative to these approaches, we show that the AME approach is a) to be easy to implement; b) interpretable in a general linear model framework; c) computationally straightforward; d) not prone to degeneracy; e) captures 1st, 2nd, and 3rd order network dependencies; and f) notably outperforms ERGMs and LSMs on a variety of metrics and in an out-of-sample context. In summary, AME offers a straightforward way to undertake nuanced, principled inferential network analysis for a wide range of social science questions.

Using a multiplicative reparametrization, I show that a subclass of $L_q$ penalties with $q\leq 1$ can be expressed as sums of $L_2$ penalties. It follows that the lasso and other norm-penalized regression estimates may be obtained using a very simple and intuitive alternating ridge regression algorithm. As compared to a similarly intuitive EM algorithm for $L_q$ optimization, the proposed algorithm avoids some numerical instability issues and is also competitive in terms of speed. Furthermore, the proposed algorithm can be extended to accommodate sparse high-dimensional scenarios, generalized linear models, and can be used to create structured sparsity via penalties derived from covariance models for the parameters. Such model-based penalties may be useful for sparse estimation of spatially or temporally structured parameters., Comment: This revision includes a comparison to cyclic coordinate descent and a new algorithm for sparse high-dimensional settings

Many inference techniques for multivariate data analysis assume that the rows of the data matrix are realizations of independent and identically distributed random vectors. Such an assumption will be met, for example, if the rows of the data matrix are multivariate measurements on a set of independently sampled units. In the absence of an independent random sample, a relevant question is whether or not a statistical model that assumes such row exchangeability is plausible. One method for assessing this plausibility is a statistical test of row covariation. Maintenance of a constant type I error rate regardless of the column covariance or matrix mean can be accomplished with a test that is invariant under an appropriate group of transformations. In the context of a class of elliptically contoured matrix regression models (such as matrix normal models), I show that there are no non-trivial invariant tests if the number of rows is not sufficiently larger than the number of columns. Furthermore, I show that even if the number of rows is large, there are no non-trivial invariant tests that have power to detect arbitrary row covariance in the presence of arbitrary column covariance. However, we can construct biased tests that have power to detect certain types of row covariance that may be encountered in practice.

The current standard for confidence interval construction in the context of a possibly misspecified model is to use an interval based on the sandwich estimate of variance. These intervals provide asymptotically correct coverage, but small-sample coverage is known to be poor. By eliminating a plug-in assumption, we derive a pivot-based method for confidence interval construction under possibly misspecified models. When compared against confidence intervals generated by the sandwich estimate of variance, this method provides more accurate coverage of the pseudo-true parameter at small sample sizes. This is shown in the results of several simulation studies. Asymptotic results show that our pivot-based intervals have large sample efficiency equal to that of intervals based on the sandwich estimate of variance.

Dyadic data on pairs of objects, such as relational or social network data, often exhibit strong statistical dependencies. Certain types of second-order dependencies, such as degree heterogeneity and reciprocity, can be well-represented with additive random effects models. Higher-order dependencies, such as transitivity and stochastic equivalence, can often be represented with multiplicative effects. The "amen" package for the R statistical computing environment provides estimation and inference for a class of additive and multiplicative random effects models for ordinal, continuous, binary and other types of dyadic data. The package also provides methods for missing, censored and fixed-rank nomination data, as well as longitudinal dyadic data. This tutorial illustrates the "amen" package via example statistical analyses of several of these different data types., Comment: This is a vignette for the R package "amen"

Previous models of international conflict have suffered two shortfalls. They tended not to embody dynamic changes, focusing rather on static slices of behavior over time. These models have also been empirically evaluated in ways that assumed the independence of each country, when in reality they are searching for the interdependence among all countries. We illustrate a solution to these two hurdles and evaluate this new, dynamic, network based approach to the dependencies among the ebb and flow of daily international interactions using a newly developed, and openly available, database of events among nations.

A fundamental aspect of relational data, such as from a social network, is the possibility of dependence among the relations. In particular, the relations between members of one pair of nodes may have an effect on the relations between members of another pair. This article develops a type of regression model to estimate such effects in the context of longitudinal and multivariate relational data, or other data that can be represented in the form of a tensor. The model is based on a general multilinear tensor regression model, a special case of which is a tensor autoregression model in which the tensor of relations at one time point are parsimoniously regressed on relations from previous time points. This is done via a separable, or Kronecker-structured, regression parameter along with a separable covariance model. In the context of an analysis of longitudinal multivariate relational data, it is shown how the multilinear tensor regression model can represent patterns that often appear in relational and network data, such as reciprocity and transitivity., Comment: Published at http://dx.doi.org/10.1214/15-AOAS839 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We develop a higher order generalization of the LQ decomposition and show that this decomposition plays an important role in likelihood-based estimation and testing for separable, or Kronecker structured, covariance models, such as the multilinear normal model. This role is analogous to that of the LQ decomposition in likelihood inference for the multivariate normal model. Additionally, this higher order LQ decomposition can be used to construct an alternative version of the popular higher order singular value decomposition for tensor-valued data. We also develop a novel generalization of the polar decomposition to tensor-valued data., Comment: 30 pages