Abstract
We provide a MATLAB toolbox, BFDA, that implements a Bayesian hierarchical model to smooth multiple functional data with the assumptions of the same underlying Gaussian process distribution, a Gaussian process prior for the mean function, and an Inverse-Wishart process prior for the covariance function. This model-based approach can borrow strength from all functional data to increase the smoothing accuracy, as well as estimate the mean-covariance functions simultaneously. An option of approximating the Bayesian inference process using cubic B-spline basis functions is integrated in BFDA, which allows for efficiently dealing with high-dimensional functional data. Examples of using BFDA in various scenarios and conducting follow-up functional regression are provided. The advantages of BFDA include: (1) Simultaneously smooths multiple functional data and estimates the mean-covariance functions in a nonparametric way; (2) flexibly deals with sparse and high-dimensional functional data with stationary and nonstationary covariance functions, and without the requirement of common observation grids; (3) provides accurately smoothed functional data for follow-up analysis.
Highlights
Since Ramsay and Dalzell (1991) first coined the term “functional data analysis” (FDA) for analyzing data that are realizations of a continuous function, many statistical methods and tools have been proposed for FDA
BFDA is flexible for analyzing sparse and dense functional data without the requirement of common observation grids, suitable for analyzing functional data with both stationary and nonstationary covariance functions, and efficient for high-dimensional functional data using a Bayesian framework with Approximations by Basis Functions (BABF) (Yang et al 2015)
The MATLAB tool BFDA presented in this paper can simultaneously smooth multiple functional observations and estimate the mean-covariance functions, assuming the functional data are from the same Gaussian process (GP)
Summary
Since Ramsay and Dalzell (1991) first coined the term “functional data analysis” (FDA) for analyzing data that are realizations of a continuous function, many statistical methods and tools have been proposed for FDA. We provide a MATLAB toolbox BFDA for simultaneously smoothing multiple functional observations from the same distribution and estimating the underlying mean-covariance functions, using a nonparametric Bayesian Hierarchical Model (BHM) with Gaussian-Wishart processes (Yang et al 2016). This model-based approach borrows strength through modeling the shared mean-covariance functions, producing more accurate smoothing results than the individually smoothing methods (Yang et al 2016). BFDA provides options for implementing the standard Bayesian GP regression method, conducting Bayesian principal component analysis, and using the fdaM package for follow-up FDA
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