Quantile regression for functional partially linear model in ultra-high dimensions

Quantile regression for functional partially linear model in ultra-high dimensions is proposed and studied. By focusing on the conditional quantiles, where conditioning is on both multiple random processes and high-dimensional scalar covariates, the proposed model can lead to a comprehensive description of the scalar response. To select and estimate important variables, a double penalized functional quantile objective function with two nonconvex penalties is developed, and the optimal tuning parameters involved can be chosen by a two-step technique. Based on the difference convex analysis (DCA), the asymptotic properties of the resulting estimators are established, and the convergence rate of the prediction of the conditional quantile function can be obtained. Simulation studies demonstrate a competitive performance against the existing approach. A real application to Alzheimer’s Disease Neuroimaging Initiative (ADNI) data is used to illustrate the practicality of the proposed model.

[1]  T. Tony Cai,et al.  Prediction in functional linear regression , 2006 .

[2]  Kehui Chen,et al.  Conditional quantile analysis when covariates are functions, with application to growth data , 2012 .

[3]  H. Müller,et al.  Functional Data Analysis for Sparse Longitudinal Data , 2005 .

[4]  T. P. Dinh,et al.  Convex analysis approach to d.c. programming: Theory, Algorithm and Applications , 1997 .

[5]  Mohan Giri,et al.  Genes associated with Alzheimer’s disease: an overview and current status , 2016, Clinical interventions in aging.

[6]  Linglong Kong,et al.  Partial functional linear quantile regression for neuroimaging data analysis , 2015, Neurocomputing.

[7]  Arno Klein,et al.  101 Labeled Brain Images and a Consistent Human Cortical Labeling Protocol , 2012, Front. Neurosci..

[8]  Fan Wang,et al.  Regularized partially functional quantile regression , 2017, J. Multivar. Anal..

[9]  Bo Peng,et al.  An Iterative Coordinate Descent Algorithm for High-Dimensional Nonconvex Penalized Quantile Regression , 2015 .

[10]  Tolga Ertekin,et al.  Total intracranial and lateral ventricle volumes measurement in Alzheimer’s disease: A methodological study , 2016, Journal of Clinical Neuroscience.

[11]  Sudha Seshadri,et al.  Genome-wide analysis of genetic loci associated with Alzheimer disease. , 2010, JAMA.

[12]  Arno Klein,et al.  Large-scale evaluation of ANTs and FreeSurfer cortical thickness measurements , 2014, NeuroImage.

[13]  Pascal Sarda,et al.  Quantile regression when the covariates are functions , 2005, math/0703056.

[14]  Qingguo Tang,et al.  Partial functional linear quantile regression , 2014 .

[15]  Runze Li,et al.  Quantile Regression for Analyzing Heterogeneity in Ultra-High Dimension , 2012, Journal of the American Statistical Association.

[16]  Rudolph E Tanzi,et al.  Diabetes-Associated SorCS1 Regulates Alzheimer's Amyloid-β Metabolism: Evidence for Involvement of SorL1 and the Retromer Complex , 2010, The Journal of Neuroscience.

[17]  Yufeng Liu,et al.  VARIABLE SELECTION IN QUANTILE REGRESSION , 2009 .

[18]  Fang Yao,et al.  Partially functional linear regression in high dimensions , 2016 .

[19]  Kengo Kato,et al.  Estimation in functional linear quantile regression , 2012, 1202.4850.

[20]  P. Hall,et al.  On properties of functional principal components analysis , 2006 .

[21]  Hongtu Zhu,et al.  Bayesian Generalized Low Rank Regression Models for Neuroimaging Phenotypes and Genetic Markers , 2014, Journal of the American Statistical Association.

[22]  Jean-François Dartigues,et al.  Education and Risk for Alzheimer's Disease: Sex Makes a Difference EURODEM Pooled Analyses , 2000 .

[23]  Jianqing Fan,et al.  Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties , 2001 .