Scalar‐on‐image regression via the soft‐thresholded Gaussian process

Summary 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 algorithm. 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 process 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 electroencephalography study of alcoholism.

[1]  R. Tibshirani,et al.  The solution path of the generalized lasso , 2010, 1005.1971.

[2]  Jian Kang,et al.  Thresholded Multiscale Gaussian Processes with Application to Bayesian Feature Selection for Massive Neuroimaging Data , 2015, 1504.06074.

[3]  P. Reiss,et al.  Functional Generalized Linear Models with Images as Predictors , 2010, Biometrics.

[4]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[5]  Ciprian M. Crainiceanu,et al.  refund: Regression with Functional Data , 2013 .

[6]  David Higdon,et al.  Non-Stationary Spatial Modeling , 2022, 2212.08043.

[7]  J. Møller,et al.  Handbook of Spatial Statistics , 2008 .

[8]  J. Ghosh,et al.  Posterior consistency of logistic Gaussian process priors in density estimation , 2007 .

[9]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[10]  J. Møller,et al.  An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants , 2006 .

[11]  Sudipto Banerjee,et al.  Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geostatistical Datasets , 2014, Journal of the American Statistical Association.

[12]  Robert Haining,et al.  Statistics for spatial data: by Noel Cressie, 1991, John Wiley & Sons, New York, 900 p., ISBN 0-471-84336-9, US $89.95 , 1993 .

[13]  Luo Xiao,et al.  Fast bivariate P‐splines: the sandwich smoother , 2013 .

[14]  Hung Hung,et al.  Matrix variate logistic regression model with application to EEG data. , 2011, Biostatistics.

[15]  N. Altman,et al.  On dimension folding of matrix- or array-valued statistical objects , 2010, 1002.4789.

[16]  Xiao Wang,et al.  Generalized Scalar-on-Image Regression Models via Total Variation , 2017, Journal of the American Statistical Association.

[17]  C. Kelly,et al.  WAVELET-DOMAIN REGRESSION AND PREDICTIVE INFERENCE IN PSYCHIATRIC NEUROIMAGING. , 2015, The annals of applied statistics.

[18]  Lexin Li,et al.  Regularized matrix regression , 2012, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[19]  S. Ghosal,et al.  Posterior consistency of Gaussian process prior for nonparametric binary regression , 2006, math/0702686.

[20]  Russell T. Shinohara,et al.  A Spatio-Temporal Model for Longitudinal Image-on-Image Regression , 2019, Statistics in biosciences.

[21]  James A. Coan,et al.  Spatial Bayesian variable selection and grouping for high-dimensional scalar-on-image regression , 2015, 1509.04069.

[22]  Montserrat Fuentes,et al.  Spatial variable selection methods for investigating acute health effects of fine particulate matter components , 2015, Biometrics.

[23]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[24]  R. Nelsen.  An Introduction to Copulas , 1998 .

[25]  S. Ghosal,et al.  Bayesian Estimation of the Spectral Density of a Time Series , 2004 .

[26]  Brian J. Reich,et al.  Spatial Signal Detection Using Continuous Shrinkage Priors , 2019, Technometrics.

[27]  B. Carlin,et al.  Spatial Analyses of Periodontal Data Using Conditionally Autoregressive Priors Having Two Classes of Neighbor Relations , 2007 .

[28]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[29]  Ciprian M Crainiceanu,et al.  Smooth Scalar-on-Image Regression via Spatial Bayesian Variable Selection , 2014, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[30]  H. Lüders,et al.  American Electroencephalographic Society Guidelines for Standard Electrode Position Nomenclature , 1991, Journal of clinical neurophysiology : official publication of the American Electroencephalographic Society.

[31]  R. Tibshirani,et al.  Sparsity and smoothness via the fused lasso , 2005 .

[32]  D. Nychka,et al.  A Multiresolution Gaussian Process Model for the Analysis of Large Spatial Datasets , 2015 .

[33]  Sw. Banerjee,et al.  Hierarchical Modeling and Analysis for Spatial Data , 2003 .

[34]  L. Fahrmeir,et al.  Spatial Bayesian Variable Selection With Application to Functional Magnetic Resonance Imaging , 2007 .