Efficient Functional ANOVA Through Wavelet-Domain Markov Groves

ABSTRACT

We introduce a wavelet-domain method for functional analysis of variance (fANOVA). It is based on a Bayesian hierarchical model that employs a graphical hyperprior in the form of a Markov grove (MG)—that is, a collection of Markov trees—for linking the presence/absence of factor effects at all location-scale combinations, thereby incorporating the natural clustering of factor effects in the wavelet-domain across locations and scales. Inference under the model enjoys both analytical simplicity and computational efficiency. Specifically, the posterior of the full hierarchical model is available in closed form through a pyramid algorithm operationally similar to Mallat’s pyramid algorithm for discrete wavelet transform (DWT), achieving for exact Bayesian inference the same computational efficiency—linear in both the number of observations and the number of locations—as for carrying out the DWT. In particular, posterior probabilities of the presence of factor contributions to functional variation are directly available from the pyramid algorithm, while posterior samples for the factor effects can be drawn directly from the exact posterior through standard (not Markov chain) Monte Carlo. We investigate the performance of our method through extensive simulation and show that it substantially outperforms existing wavelet-domain fANOVA methods in a variety of common settings. We illustrate the method through analyzing the orthosis data. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian inference
• Functional data analysis
• Graphical models
• Hierarchical models
• Multi-scale inference
• Scalable inference

Supplementary Materials

The supplementary materials contain all proofs, additional technical details for the case with multiple factors, additional numerical results, and a sensitivity analysis.

Additional information

Funding

This research is partly supported by NSF grants DMS-1309057, DMS-1612889, and a Google Faculty Research Award.