Modeling Covariance Matrices via Partial Autocorrelations

We study the role of partial autocorrelations in the reparameterization and parsimonious modeling of a covariance matrix. The work is motivated by and tries to mimic the phenomenal success of the partial autocorrelations function (PACF) in model formulation, removing the positive-definiteness constraint on the autocorrelation function of a stationary time series and in reparameterizing the stationarity-invertibility domain of ARMA models. It turns out that once an order is fixed among the variables of a general random vector, then the above properties continue to hold and follows from establishing a one-to-one correspondence between a correlation matrix and its associated matrix of partial autocorrelations. Connections between the latter and the parameters of the modified Cholesky decomposition of a covariance matrix are discussed. Graphical tools similar to partial correlograms for model formulation and various priors based on the partial autocorrelations are proposed. We develop frequentist/Bayesian procedures for modelling correlation matrices, illustrate them using a real dataset, and explore their properties via simulations.

Dynamic conditionally linear mixed models for longitudinal data

We develop a new class of models, dynamic conditionally linear mixed models, for longitudinal data by decomposing the within-subject covariance matrix using a special Cholesky decomposition. Here 'dynamic' means using past responses as covariates and 'conditional linearity' means that parameters entering the model linearly may be random, but nonlinear parameters are nonrandom. This setup offers several advantages and is surprisingly similar to models obtained from the first-order linearization method applied to nonlinear mixed models. First, it allows for flexible and computationally tractable models that include a wide array of covariance structures; these structures may depend on covariates and hence may differ across subjects. This class of models includes, e.g., all standard linear mixed models, antedependence models, and Vonesh-Carter models. Second, it guarantees the fitted marginal covariance matrix of the data is positive definite. We develop methods for Bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited.