Conditional Spectral Analysis of Replicated Multiple Time Series with Application to Nocturnal Physiology

This article considers the problem of analyzing associations between power spectra of multiple time series and cross-sectional outcomes when data are observed from multiple subjects. The motivating application comes from sleep medicine, where researchers are able to non-invasively record physiological time series signals during sleep. The frequency patterns of these signals, which can be quantified through the power spectrum, contain interpretable information about biological processes. An important problem in sleep research is drawing connections between power spectra of time series signals and clinical characteristics; these connections are key to understanding biological pathways through which sleep affects, and can be treated to improve, health. Such analyses are challenging as they must overcome the complicated structure of a power spectrum from multiple time series as a complex positive-definite matrix-valued function. This article proposes a new approach to such analyses based on a tensor-product spline model of Cholesky components of outcome-dependent power spectra. The approach flexibly models power spectra as nonparametric functions of frequency and outcome while preserving geometric constraints. Formulated in a fully Bayesian framework, a Whittle likelihood based Markov chain Monte Carlo (MCMC) algorithm is developed for automated model fitting and for conducting inference on associations between outcomes and spectral measures. The method is used to analyze data from a study of sleep in older adults and uncovers new insights into how stress and arousal are connected to the amount of time one spends in bed

Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory

Bohmian mechnaics is the most naively obvious embedding imaginable of Schr\"odingers's equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function $\psi$ its configuration is typically random, with probability density $\rho$ given by $|\psi|^2$, the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, naturally emerges in Bohmian mechanics from the analysis of ``measurements.'' This analysis reveals the status of operators as observables in the description of quantum phenomena, and facilitates a clear view of the range of applicability of the usual quantum mechanical formulas.Comment: 77 page

Mixed membership stochastic blockmodels

Observations consisting of measurements on relationships for pairs of objects arise in many settings, such as protein interaction and gene regulatory networks, collections of author-recipient email, and social networks. Analyzing such data with probabilisic models can be delicate because the simple exchangeability assumptions underlying many boilerplate models no longer hold. In this paper, we describe a latent variable model of such data called the mixed membership stochastic blockmodel. This model extends blockmodels for relational data to ones which capture mixed membership latent relational structure, thus providing an object-specific low-dimensional representation. We develop a general variational inference algorithm for fast approximate posterior inference. We explore applications to social and protein interaction networks.Comment: 46 pages, 14 figures, 3 table

Spectral gene set enrichment (SGSE)

Motivation: Gene set testing is typically performed in a supervised context to quantify the association between groups of genes and a clinical phenotype. In many cases, however, a gene set-based interpretation of genomic data is desired in the absence of a phenotype variable. Although methods exist for unsupervised gene set testing, they predominantly compute enrichment relative to clusters of the genomic variables with performance strongly dependent on the clustering algorithm and number of clusters. Results: We propose a novel method, spectral gene set enrichment (SGSE), for unsupervised competitive testing of the association between gene sets and empirical data sources. SGSE first computes the statistical association between gene sets and principal components (PCs) using our principal component gene set enrichment (PCGSE) method. The overall statistical association between each gene set and the spectral structure of the data is then computed by combining the PC-level p-values using the weighted Z-method with weights set to the PC variance scaled by Tracey-Widom test p-values. Using simulated data, we show that the SGSE algorithm can accurately recover spectral features from noisy data. To illustrate the utility of our method on real data, we demonstrate the superior performance of the SGSE method relative to standard cluster-based techniques for testing the association between MSigDB gene sets and the variance structure of microarray gene expression data. Availability: http://cran.r-project.org/web/packages/PCGSE/index.html Contact: [email protected] or [email protected]

Multinomial Inverse Regression for Text Analysis

Text data, including speeches, stories, and other document forms, are often connected to sentiment variables that are of interest for research in marketing, economics, and elsewhere. It is also very high dimensional and difficult to incorporate into statistical analyses. This article introduces a straightforward framework of sentiment-preserving dimension reduction for text data. Multinomial inverse regression is introduced as a general tool for simplifying predictor sets that can be represented as draws from a multinomial distribution, and we show that logistic regression of phrase counts onto document annotations can be used to obtain low dimension document representations that are rich in sentiment information. To facilitate this modeling, a novel estimation technique is developed for multinomial logistic regression with very high-dimension response. In particular, independent Laplace priors with unknown variance are assigned to each regression coefficient, and we detail an efficient routine for maximization of the joint posterior over coefficients and their prior scale. This "gamma-lasso" scheme yields stable and effective estimation for general high-dimension logistic regression, and we argue that it will be superior to current methods in many settings. Guidelines for prior specification are provided, algorithm convergence is detailed, and estimator properties are outlined from the perspective of the literature on non-concave likelihood penalization. Related work on sentiment analysis from statistics, econometrics, and machine learning is surveyed and connected. Finally, the methods are applied in two detailed examples and we provide out-of-sample prediction studies to illustrate their effectiveness.Comment: Published in the Journal of the American Statistical Association 108, 2013, with discussion (rejoinder is here: http://arxiv.org/abs/1304.4200). Software is available in the textir package for