Bayesian functional data modeling for heterogeneous volatility. (English) Zbl 1384.62122

Zhu, Bin; Dunson, David B.

Bayesian Anal. 12, No. 2, 335-350 (2017).

Summary: Although there are many methods for functional data analysis, less emphasis is put on characterizing variability among volatilities of individual functions. In particular, certain individuals exhibit erratic swings in their trajectory while other individuals have more stable trajectories. There is evidence of such volatility heterogeneity in blood pressure trajectories during pregnancy, for example, and reason to suspect that volatility is a biologically important feature. Most functional data analysis models implicitly assume similar or identical smoothness of the individual functions, and hence can lead to misleading inferences on volatility and an inadequate representation of the functions. We propose a novel class of functional data analysis models characterized using hierarchical stochastic differential equations. We model the derivatives of a mean function and deviation functions using Gaussian processes, while also allowing covariate dependence including on the volatilities of the deviation functions. Following a Bayesian approach to inference, a Markov chain Monte Carlo algorithm is used for posterior computation. The methods are tested on simulated data and applied to blood pressure trajectories during pregnancy.

MSC:

62G05	Nonparametric estimation
62F15	Bayesian inference
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
62P10	Applications of statistics to biology and medical sciences; meta analysis

Keywords:

Bayesian functional data analysis; Gaussian process; state space model; stochastic differential equation; volatility heterogeneity