Penalized multivariate Whittle likelihood for power spectrum estimation

Graves

et al. 2019

Stat Biosci

The twenty-four hour sleep-wake pattern known as the rest-activity rhythm (RAR) is associated with many aspects of health and well-being. Researchers have utilized a number of interpretable, person-specific RAR measures that can be estimated from actigraphy. Actigraphs are wearable devices that dynamically record acceleration and provide indirect measures of physical activity over time. One class of useful RAR measures are those that quantify variability around a mean circadian pattern. However, current parametric and nonparametric RAR measures used by applied researchers can only quantify variability from a limited or undefined number of rhythmic sources. The primary goal of this article is to consider a new measure of RAR variability: the log-power spectrum of stochastic error around a circadian mean. This functional measure quantifies the relative contributions of variability about a circadian mean from all possibly frequencies, including weekly, daily, and high-frequency sources of variation. It can be estimated through a two-stage procedure that smooths the log-periodogram of residuals after estimating a circadian mean. The development of this measure was motivated by a study of depression in older adults and revealed that slow, rhythmic variations in activity from a circadian pattern are correlated with depression symptoms.

Section: Estimationmentioning

confidence: 91%

Measuring Variability in Rest-Activity Rhythms from Actigraphy with Application to Characterizing Symptoms of Depression

Graves

et al. 2019

Stat Biosci

“…While in the univariate setting the spectrum is smoothed on the logarithmic scale to preserve positivity, Cholesky components of spectral matrices can be smoothed to preserve positive-definiteness in the multivariate setting (Dai and Guo, 2004; Rosen and Stoffer, 2007; Krafty and Collinge, 2013). The modified Cholesky decomposition assures that, for a spectral matrix f ( ω ), there exists a unique P × P lower triangular complex matrix Θ( ω ) with ones on the diagonal and a unique P × P positive diagonal matrix Ψ ( ω ) such that…”

Section: Methodological Background: Spectral Domain Analysismentioning

confidence: 99%

“…Efficient nonparametric methods that preserve the positive-definite Hermitian structure of spectral matrices have been developed for the simpler, classical problem of estimating the power spectrum of a multivariate time series from a single subject by modeling Cholesky components of spectral matrices as functions of frequency (Dai and Guo, 2004; Rosen and Stoffer, 2007; Krafty and Collinge, 2013). In this article, we extend this framework to develop a new approach to analyzing data from multiple subjects that models Cholesky components as functions of both frequency and outcome.…”

Section: Introductionmentioning

confidence: 99%

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

Journal of the American Statistical Association

Rosen

Stoffer

et al. 2017

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 exibly 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.

“…Prior distributions for the coefficients are selected to regularize integrated squared first derivatives and formulate Bayesian linear penalized splines. As noted by Krafty and Collinge (2013) and by Zhang (2016), this Bayesian linear penalized spline model for local spectra differs somewhat from that used by Rosen and Stoffer (2007) for stationary time series. Rosen and Stoffer (2007) uses a model that is periodic, but not restricted to be odd or even.…”

Section: The Modelmentioning

confidence: 99%

Adaptive Bayesian Time–Frequency Analysis of Multivariate Time Series

Journal of the American Statistical Association

2018

This article introduces a nonparametric approach to multivariate time-varying power spectrum analysis. The procedure adaptively partitions a time series into an unknown number of approximately stationary segments, where some spectral components may remain unchanged across segments, allowing components to evolve differently over time. Local spectra within segments are fit through Whittle likelihood based penalized spline models of modified Cholesky components, which provide flexible nonparametric estimates that preserve positive definite structures of spectral matrices. The approach is formulated in a Bayesian framework, in which the number and location of partitions are random, and relies on reversible jump Markov chain and Hamiltonian Monte Carlo methods that can adapt to the unknown number of segments and parameters. By averaging over the distribution of partitions, the approach can approximate both abrupt and slow-varying changes in spectral matrices. Empirical performance is evaluated in simulation studies and illustrated through analyses of electroencephalography during sleep and of the El Niño-Southern Oscillation.