Summary We propose a novel class of dynamic shrinkage processes for Bayesian time series and regression analysis. Building on a global–local framework of prior construction, in which continuous scale mixtures of Gaussian distributions are employed for both desirable shrinkage properties and computational tractability, we model dependence between the local scale parameters. The resulting processes inherit the desirable shrinkage behaviour of popular global–local priors, such as the horseshoe prior, but provide additional localized adaptivity, which is important for modelling time series data or regression functions with local features. We construct a computationally efficient Gibbs sampling algorithm based on a Pólya–gamma scale mixture representation of the process proposed. Using dynamic shrinkage processes, we develop a Bayesian trend filtering model that produces more accurate estimates and tighter posterior credible intervals than do competing methods, and we apply the model for irregular curve fitting of minute‐by‐minute Twitter central processor unit usage data. In addition, we develop an adaptive time varying parameter regression model to assess the efficacy of the Fama–French five‐factor asset pricing model with momentum added as a sixth factor. Our dynamic analysis of manufacturing and healthcare industry data shows that, with the exception of the market risk, no other risk factors are significant except for brief periods.
We present a Bayesian approach for modeling multivariate, dependent functional data. To account for the three dominant structural features in the data-functional, time dependent, and multivariate components-we extend hierarchical dynamic linear models for multivariate time series to the functional data setting. We also develop Bayesian spline theory in a more general constrained optimization framework.The proposed methods identify a time-invariant functional basis for the functional observations, which is smooth and interpretable, and can be made common across multivariate observations for additional information sharing. The Bayesian framework permits joint estimation of the model parameters, provides exact inference (up to MCMC error) on specific parameters, and allows generalized dependence structures. Sampling from the posterior distribution is accomplished with an efficient Gibbs sampling algorithm. We illustrate the proposed framework with two applications: (1) multi-economy yield curve data from the recent global recession, and (2) local field potential brain signals in rats, for which we develop a multivariate functional time series approach for multivariate time-frequency analysis. Supplementary materials, including R code and the multi-economy yield curve data, are available online.
Arterial stiffening occurs with age and is associated with lack of exercise. Notably both age and lack of exercise are major cardiovascular risk factors. While it is well established that bulk arterial stiffness increases with age, more recent data suggest that the intima, the innermost arterial layer, also stiffens during aging. Micro-scale mechanical characterization of individual layers is important because cells primarily sense the matrix that they are in contact with and not necessarily the bulk stiffness of the vessel wall. To investigate the relationship between age, exercise, and subendothelial matrix stiffening, atomic force microscopy was utilized here to indent the subendothelial matrix of the thoracic aorta from young, aged-sedentary, and aged-exercised mice, and elastic modulus values were compared to conventional pulse wave velocity measurements. The subendothelial matrix elastic modulus was elevated in aged-sedentary mice compared to young or aged-exercised mice, and the macro-scale stiffness of the artery was found to linearly correlate with the subendothelial matrix elastic modulus. Notably, we also found that with age, there exists an increase in the point-to-point variations in modulus across the subendothelial matrix, indicating non-uniform stiffening. Importantly, this heterogeneity is reversible with exercise. Given that vessel stiffening is known to cause aberrant endothelial cell behavior, and the spatial heterogeneities we find exist on a length scale much smaller than the size of a cell, these data suggest that further investigation in the heterogeneity of the subendothelial matrix elastic modulus is necessary to fully understand the effects of physiological matrix stiffening on cell function.
We develop a fully Bayesian framework for function-on-scalars regression with many predictors. The functional data response is modeled nonparametrically using unknown basis functions, which produces a flexible and data-adaptive functional basis. We incorporate shrinkage priors that effectively remove unimportant scalar covariates from the model and reduce sensitivity to the number of (unknown) basis functions. For variable selection in functional regression, we propose a decision theoretic posterior summarization technique, which identifies a subset of covariates that retains nearly the predictive accuracy of the full model. Our approach is broadly applicable for Bayesian functional regression models, and unlike existing methods provides joint rather than marginal selection of important predictor variables.Computationally scalable posterior inference is achieved using a Gibbs sampler with linear time complexity in the number of predictors. The resulting algorithm is empirically faster than existing frequentist and Bayesian techniques, and provides joint estimation of model parameters, prediction and imputation of functional trajectories, and uncertainty quantification via the posterior distribution. A simulation study demonstrates improvements in estimation accuracy, uncertainty quantification, and variable selection relative to existing alternatives. The methodology is applied to actigraphy data to investigate the association between intraday physical activity and responses to a sleep questionnaire.
For time-ordered functional data, an important yet challenging task is to forecast functional observations with uncertainty quantification. Scalar predictors are often observed concurrently with functional data and provide valuable information about the dynamics of the functional time series. We develop a fully Bayesian framework for dynamic functional regression, which employs scalar predictors to model the time-evolution of functional data. Functional within-curve dependence is modeled using unknown basis functions, which are learned from the data. The unknown basis provides substantial dimension reduction, which is essential for scalable computing, and may incorporate prior knowledge such as smoothness or periodicity. The dynamics of the time-ordered functional data are specified using a time-varying parameter regression model in which the effects of the scalar predictors evolve over time. To guard against overfitting, we design shrinkage priors that regularize irrelevant predictors and shrink toward timeinvariance. Simulation studies decisively confirm the utility of these modeling and prior choices. Posterior inference is available via a customized Gibbs sampler, which offers unrivaled scalability for Bayesian dynamic functional regression. The methodology is applied to model and forecast yield curves using macroeconomic predictors, and demonstrates exceptional forecasting accuracy and uncertainty quantification over the span of four decades.
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