Quantile regression for functional partially linear model in ultra-high dimensions. (English) Zbl 1469.62114
Summary: 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.
MSC:
| 62-08 | Computational methods for problems pertaining to statistics |
| 62G08 | Nonparametric regression and quantile regression |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62H25 | Factor analysis and principal components; correspondence analysis |
| 62R10 | Functional data analysis |
Keywords:
asymptotic property; double penalized quantile method; functional partially linear quantile model; functional principal component analysis; variable selectionSoftware:
QICDReferences:
| [1] | Allende, L.; Pizarro, H., On properties of functional principal components, J. R. Stat. Soc. Ser. B Stat. Methodol., 68, 1, 109-126, (2006) · Zbl 1141.62048 |
| [2] | Cai, T.; Hall, P., Prediction in functional linear regression, Ann. Statist., 34, 5, 2159-2179, (2006) · Zbl 1106.62036 |
| [3] | Cardot, H.; Crambes, C.; Sarda, P., Quantile regression when the covariates are functions, Nonparametr. Statist., 17, 7, 841-856, (2005) · Zbl 1077.62026 |
| [4] | Chen, K.; Müller, H.-G., Conditional quantile analysis when covariates are functions, with application to growth data, J. R. Stat. Soc. Ser. B Stat. Methodol., 74, 1, 67-89, (2012) · Zbl 1411.62095 |
| [5] | Ertekin, T.; Acer, N.; Koseoglu, E.; Zararsiz, G.; Sonmez, A.; Gums, K.; Kurtoglu, E., Total intracranial and lateral ventricle volumes measurement in alzheimer’s disease: A methodological study, J. Clin. Neurosci., 34, 133-139, (2016) |
| [6] | Fan, J.; Li, R., Variable selection via nonconcave penalized likelihood and its oracle properties, J. Amer. Statist. Assoc., 96, 456, 1348-1360, (2002) · Zbl 1073.62547 |
| [7] | Giri, M.; Man, Z.; Yang, L., Genes associated with alzheimers disease: an overview and current status, Clin. Interventions Aging, 11, 2, 665-681, (2016) |
| [8] | Kato, K., Estimation in functional linear quantile regression, Ann. Statist., 40, 6, 3108-3136, (2012) · Zbl 1296.62104 |
| [9] | Klein, A.; Tourville, J., 101 labeled brain images and a consistent human cortical labeling protocol, Front. Neurosci., 6, 171, (2012) |
| [10] | Koenker, R., Quantile regression, (2005), Cambridge University Press · Zbl 1111.62037 |
| [11] | Koenker, R.; Bassett, G., Regression quantiles, Econometrica, 46, 33-50, (1978) · Zbl 0373.62038 |
| [12] | Kong, D.; Xue, K.; Yao, F.; Zhang, H. H., Partially functional linear regression in high dimensions, Biometrika, 103, 1, 147-159, (2016) · Zbl 1452.62500 |
| [13] | Lane, R. F.; Raines, S. M.; Steele, J. W.; Ehrlich, M. E.; Lah, J. A.; Small, S. A.; Tanzi, R. E.; Attie, A. D.; Gandy, S., Diabetes-associated sorcs1 regulates alzheimer’s amyloid-beta metabolism: evidence for involvement of sorl1 and the retromer complex, J. Neurosci. Off. J. Soc. Neurosci., 30, 39, 13110-13115, (2010) |
| [14] | Letenneur, L.; Launer, L. J.; Andersen, K.; Dewey, M. E.; Ott, A.; Copeland, J. R.; Dartigues, J. F.; Kragh-Sorensen, P.; Baldereschi, M.; Brayne, C., Education and the risk for alzheimer’s disease: sex makes a difference. eurodem pooled analyses, Am. J. Epidemiol., 151, 11, 1064-1071, (2000) |
| [15] | Peng, B.; Wang, L., An iterative coordinate descent algorithm for high-dimensional nonconvex penalized quantile regression, J. Comput. Graph. Statist., 24, 3, 676-694, (2015) |
| [16] | Seshadri, S.; Fitzpatrick, A. L.; Ikram, M. A.; Destefano, A. L.; Gudnason, V.; Boada, M.; Bis, J. C.; Smith, A. V.; Carrasquillo, M. M.; Lambert, J. C., Genome-wide analysis of genetic loci associated with alzheimer disease, JAMA, 303, 18, 1832-1840, (2010) |
| [17] | Tang, Q.; Cheng, L., Partial functional linear quantile regression, Sci. China Math., 57, 12, 2589-2608, (2014) · Zbl 1308.62086 |
| [18] | Tao, P. D.; An, L. T.H., Convex analysis approach to dc programming: theory, algorithms and applications, Acta Math. Vietnam., 22, 1, 289-355, (1997) · Zbl 0895.90152 |
| [19] | Tustison, N. J.; Cook, P. A.; Klein, A.; Song, G.; Das, S. R.; Duda, J. T.; Kandel, B. M.; van Strien, N.; Stone, J. R.; Gee, J. C., Large-scale evaluation of ants and freesurfer cortical thickness measurements, Neuroimage, 99, 166-179, (2014) |
| [20] | Wang, L.; Wu, Y.; Li, R., Quantile regression for analyzing heterogeneity in ultra-high dimension, J. Amer. Statist. Assoc., 107, 497, 214-222, (2012) · Zbl 1328.62468 |
| [21] | Wu, Y.; Liu, Y., Variable selection in quantile regression, Statist. Sinica, 19, 2, 801-817, (2009) · Zbl 1166.62012 |
| [22] | Yao, F.; Müller, H.-G.; Wang, J.-L., Functional data analysis for sparse longitudinal data, J. Amer. Statist. Assoc., 100, 470, 577-590, (2005) · Zbl 1117.62451 |
| [23] | Yao, F.; Sue-Chee, S.; Wang, F., Regularized partially functional quantile regression, J. Multivariate Anal., 156, 39-56, (2017) · Zbl 1369.62083 |
| [24] | Yu, D.; Kong, L.; Mizera, I., Partial functional linear quantile regression for neuroimaging data analysis, Neurocomputing, 195, 74-87, (2016) |
| [25] | Zhu, H.; Khondker, Z.; Lu, Z.; Ibrahim, J. G., Bayesian generalized low rank regression models for neuroimaging phenotypes and genetic markers, J. Amer. Statist. Assoc., 109, 507, 977-990, (2014) · Zbl 1368.62298 |
| [26] | Zou, H.; Li, R., One-step sparse estimates in nonconcave penalized likelihood models, Ann. Statist., 36, 4, 1509-1533, (2008) · Zbl 1142.62027 |
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