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. 2018 Mar;105(1):165-184.
doi: 10.1093/biomet/asx075. Epub 2018 Jan 19.

Scalar-on-Image Regression via the Soft-Thresholded Gaussian Process

Jian Kang  1 , Brian J Reich  2 , Ana-Maria Staicu  3
Affiliations

Scalar-on-Image Regression via the Soft-Thresholded Gaussian Process

Jian Kang et al. Biometrika. 2018 Mar.

Abstract

This work concerns spatial variable selection for scalar-on-image regression. We propose a new class of Bayesian nonparametric models and develop an efficient posterior computational aigorithm. The proposed soft-thresholded Gaussian process provides large prior support over the class of piecewise-smooth, sparse, and continuous spatially-varying regression coefficient functions. In addition, under some mild regularity conditions the soft-thresholded Gaussian proess prior leads to the posterior consistency for parameter estimation and variable selection for scalar-on-image regression, even when the number of predictors is larger than the sample size. The proposed method is compared to alternatives via simulation and applied to an electroen-cephalography study of alcoholism.

Keywords: Electroencephalography; Gaussian processes; Posterior consistency; Spatial variable selection.

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Figures

Fig. 1.
Fig. 1.
Standard electrode position nomenclature for 10–10 system
Fig. 2.
Fig. 2.
True βv images used in the simulation study.
Fig. 3.
Fig. 3.
Receiving operating characteristic curves with the area under the curve, AUC, for the leave-one-out cross validation of the EEG data by six different methods: lasso (black solid, AUC = 0.789), functional principal component analysis using the leading eigen-vectors that explain 80% (red solid, AUC = 0.775), 90% (red dashes, AUC = 0.789), 95% (red dots, AUC = 0.777) of variations, Gaussian process approach (green solid, AUC = 0.770) and soft-thresholded Gaussian process approach (navy solid, AUC = 0.818)
Fig. 4.
Fig. 4.
Estimated spatial-temporal effects of the EEG image predictors by four different methods: Lasso, functional principal component analysis, Gaussian process and soft-thresholded Gaussian process. The Gaussian process and soft-thresholded Gaussian process estimates are posterior means.
Fig. 5.
Fig. 5.
Summary of analysis of the EEG data by the soft-thresholded Gaussian process. Panel (a) plots the posterior probability of a nonzero β(s, t); each electrode is a line plotted over time t. The remaining panels map either the posterior probability of a nonzero β(s, t) or the posterior mean of β(s, t) at individual time points
See this image and copyright information in PMC

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