Abstract
We consider spectral decompositions of multiple time series that arise in studies where the interest lies in assessing the influence of two or more factors. We write the spectral density of each time series as a sum of the spectral densities associated to the different levels of the factors. We then use Whittle’s approximation to the likelihood function and follow a Bayesian non-parametric approach to obtain posterior inference on the spectral densities based on Bernstein–Dirichlet prior distributions. The prior is strategically important as it carries identifiability conditions for the models and allows us to quantify our degree of confidence in such conditions. A Markov chain Monte Carlo (MCMC) algorithm for posterior inference within this class of frequency-domain models is presented.We illustrate the approach by analyzing simulated and real data via spectral one-way and two-way models. In particular, we present an analysis of functional magnetic resonance imaging (fMRI) brain responses measured in individuals who participated in a designed experiment to study pain perception in humans.
Highlights
Time series data from several subjects that can be classified into groups according to a set of features are oftentimes recorded in clinical and non-clinical studies
Suppose that a collection of time series {yi1,i2,h(t)} are recorded during a particular experimental setting in which i1 indexes the level of some factor, say A, i2 indexes the level of some other factor, say B, and h indexes the time series within given levels of the A and B factors, where i1 = 1 : N1, i2 = 1 : N2, h = 1 : Hi1,i2, and t = 1 : T
Such data may arise in designed experiments where factor A is a given treatment or experimental condition— e.g., the type of stimulus in a neuroscience experiment or the disease level, factor B is another treatment or experimental condition, and h indexes the individuals who were assigned to levels i1 and i2 of the A and B factors, respectively
Summary
Time series data from several subjects that can be classified into groups according to a set of features are oftentimes recorded in clinical and non-clinical studies. The method is used to analyze individual (not multiple) time series data obtained from functional magnetic resonance imaging (fMRI) experiments In such context the method of DiMatteo et al (2001) is a time-domain approach that assumes that a given time series y(t) can be modeled as y(t) = f (t) + (t), with f (t) =. We extend the methods and algorithms of Macaro (2010) to provide a Bayesian non-parametric spectral analysis of multiple time series recorded during designed experiments that may involve several factors, various levels within each factor and several individuals.
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