2,638
Views
8
CrossRef citations to date
0
Altmetric
Theory and Methods

High-Dimensional Spatial Quantile Function-on-Scalar Regression

, , &
Pages 1563-1578 | Received 30 Aug 2018, Accepted 18 Dec 2020, Published online: 07 Mar 2021

References

  • Ardekani, B. A., Bachman, A. H., Figarsky, K., and Sidtis, J. J. (2014), “Corpus Callosum Shape Changes in Early Alzheimer’s Disease: An MRI Study Using the OASIS Brain Database,” Brain Structure and Function, 219, 343–352. DOI: 10.1007/s00429-013-0503-0.
  • Bárdossy, A. (2006), “Copula-Based Geostatistical Models for Groundwater Quality Parameters,” Water Resources Research, 42, W11416. DOI: 10.1029/2005WR004754.
  • Biegon, A., Eberling, J., Richardson, B., Roos, M., Wong, S., Reed, B. R., and Jagust, W. (1994), “Human Corpus Callosum in Aging and Alzheimer’s Disease: A Magnetic Resonance Imaging Study,” Neurobiology of Aging, 15, 393–397. DOI: 10.1016/0197-4580(94)90070-1.
  • Bouyé, E., and Salmon, M. (2013), “Dynamic Copula Quantile Regressions and Tail Area Dynamic Dependence in Forex Markets,” in Copulae and Multivariate Probability Distributions in Finance, eds. A. Dias, M. Salmon, and C. Adcock, London: Routledge, pp. 125–154.
  • Bowman, A. (2010), “Functional Data Analysis With R and MATLAB,” Journal of Statistical Software, 34, 1–2. DOI: 10.18637/jss.v034.b03.
  • Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J. (2011), “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations and Trends[textregistered] in Machine Learning, 3, 1–122.
  • Cai, T. T., and Hall, P. (2006), “Prediction in Functional Linear Regression,” The Annals of Statistics, 34, 2159–2179. DOI: 10.1214/009053606000000830.
  • Cai, T. T., and Yuan, M. (2012), “Minimax and Adaptive Prediction for Functional Linear Regression,” Journal of the American Statistical Association, 107, 1201–1216. DOI: 10.1080/01621459.2012.716337.
  • Cai, Z., and Xu, X. (2008), “Nonparametric Quantile Estimations for Dynamic Smooth Coefficient Models,” Journal of the American Statistical Association, 103, 1595–1608. DOI: 10.1198/016214508000000977.
  • Chen, X., Koenker, R., and Xiao, Z. (2009), “Copula Based Nonlinear Quantile Autoregression,” The Econometrics Journal, 12, S50–S67. DOI: 10.1111/j.1368-423X.2008.00274.x.
  • Colom, R., Stein, J. L., Rajagopalan, P., Martínez, K., Hermel, D., Wang, Y., Álvarez-Linera, J., Burgaleta, M., Quiroga, M. Á., Shih, P. C., and Thompson, P. M. (2013), “Hippocampal Structure and Human Cognition: Key Role of Spatial Processing and Evidence Supporting the Efficiency Hypothesis in Females,” Intelligence, 41, 129–140. DOI: 10.1016/j.intell.2013.01.002.
  • Crambes, C., Kneip, A., and Sarda, P. (2009), “Smoothing Splines Estimators for Functional Linear Regression,” The Annals of Statistics, 37, 35–72. DOI: 10.1214/07-AOS563.
  • Cressie, N. (1985), “Fitting Variogram Models by Weighted Least Squares,” Mathematical Geology, 17, 563–586. DOI: 10.1007/BF01032109.
  • ——— (2015), Statistics for Spatial Data (Revised Edition), New York: Wiley.
  • De Angelis, P. L., Pardalos, P. M., and Toraldo, G. (1997), “Quadratic Programming With Box Constraints,” in Developments in Global Optimization, eds. I. M. Bomze, T. Csendes, R. Horst, and P. M. Pardalos, Boston, MA: Springer, pp. 73–93.
  • De Backer, M., Ghouch, A. E., and Van Keilegom, I. (2017), “Semiparametric Copula Quantile Regression for Complete or Censored Data,” Electronic Journal of Statistics, 11, 1660–1698. DOI: 10.1214/17-EJS1273.
  • Demarta, S., and McNeil, A. J. (2005), “The t Copula and Related Copulas,” International Statistical Review, 73, 111–129. DOI: 10.1111/j.1751-5823.2005.tb00254.x.
  • Du, P., and Wang, X. (2014), “Penalized Likelihood Functional Regression,” Statistica Sinica, 24, 1017–1041. DOI: 10.5705/ss.2012.235.
  • Efron, B. (1986), “How Biased Is the Apparent Error Rate of a Prediction Rule?,” Journal of the American Statistical Association, 81, 461–470. DOI: 10.1080/01621459.1986.10478291.
  • Ferraty, F., and Vieu, P. (2006), Nonparametric Functional Data Analysis: Methods, Theory, Applications and Implementations, New York: Springer.
  • Frisoni, G. B., Ganzola, R., Canu, E., Rüb, U., Pizzini, F. B., Alessandrini, F., Zoccatelli, G., Beltramello, A., Caltagirone, C., and Thompson, P. M. (2008), “Mapping Local Hippocampal Changes in Alzheimer’s Disease and Normal Ageing With MRI at 3 Tesla,” Brain, 131, 3266–3276. DOI: 10.1093/brain/awn280.
  • Genton, M. G. (1998), “Variogram Fitting by Generalized Least Squares Using an Explicit Formula for the Covariance Structure,” Mathematical Geology, 30, 323–345.
  • Goldsmith, J., and Kitago, T. (2016), “Assessing Systematic Effects of Stroke on Motor Control by Using Hierarchical Function-on-Scalar Regression,” Journal of the Royal Statistical Society, Series C, 65, 215–236. DOI: 10.1111/rssc.12115.
  • Greven, S., and Scheipl, F. (2017), “A General Framework for Functional Regression Modelling,” Statistical Modelling, 17, 1–35. DOI: 10.1177/1471082X16681317.
  • Gutenbrunner, C., and Jurecková, J. (1992), “Regression Rank Scores and Regression Quantiles,” The Annals of Statistics, 20, 305–330. DOI: 10.1214/aos/1176348524.
  • Gutenbrunner, C., Jurečková, J., Koenker, R., and Portnoy, S. (1993), “Tests of Linear Hypotheses Based on Regression Rank Scores,” Journal of Nonparametric Statistics, 2, 307–331. DOI: 10.1080/10485259308832561.
  • Guttorp, P., and Gneiting, T. (2006), “Studies in the History of Probability and Statistics XLIX on the Matérn Correlation Family,” Biometrika, 93, 989–995. DOI: 10.1093/biomet/93.4.989.
  • Hall, P., and Horowitz, J. L. (2007), “Methodology and Convergence Rates for Functional Linear Regression,” The Annals of Statistics, 35, 70–91. DOI: 10.1214/009053606000000957.
  • Hallin, M., Lu, Z. L., and Yu, K. (2009), “Local Linear Spatial Quantile Regression,” Bernoulli, 15, 659–686. DOI: 10.3150/08-BEJ168.
  • Inano, S., Takao, H., Hayashi, N., Abe, O., and Ohtomo, K. (2011), “Effects of Age and Gender on White Matter Integrity,” American Journal of Neuroradiology, 32, 2103–2109. DOI: 10.3174/ajnr.A2785.
  • Ivanescu, A. E., Staicu, A.-M., Scheipl, F., and Greven, S. (2015), “Penalized Function-on-Function Regression,” Computational Statistics, 30, 539–568. DOI: 10.1007/s00180-014-0548-4.
  • Kato, K. (2012), “Estimation in Functional Linear Quantile Regression,” The Annals of Statistics, 40, 3108–3136. DOI: 10.1214/12-AOS1066.
  • Kazianka, H., and Pilz, J. (2010), “Copula-Based Geostatistical Modeling of Continuous and Discrete Data Including Covariates,” Stochastic Environmental Research and Risk Assessment, 24, 661–673. DOI: 10.1007/s00477-009-0353-8.
  • Koay, C. G., Chang, L.-C., Carew, J. D., Pierpaoli, C., and Basser, P. J. (2006), “A Unifying Theoretical and Algorithmic Framework for Least Squares Methods of Estimation in Diffusion Tensor Imaging,” Journal of Magnetic Resonance, 182, 115–125. DOI: 10.1016/j.jmr.2006.06.020.
  • Kochunov, P., Thompson, P. M., Lancaster, J. L., Bartzokis, G., Smith, S., Coyle, T., Royall, D. R., Laird, A., and Fox, P. T. (2007), “Relationship Between White Matter Fractional Anisotropy and Other Indices of Cerebral Health in Normal Aging: Tract-Based Spatial Statistics Study of Aging,” Neuroimage, 35, 478–487. DOI: 10.1016/j.neuroimage.2006.12.021.
  • Koenker, R. (2004), “Quantile Regression for Longitudinal Data,” Journal of Multivariate Analysis, 91, 74–89. DOI: 10.1016/j.jmva.2004.05.006.
  • ——— (2005), Quantile Regression, New York: Cambridge University Press.
  • Koenker, R., and Bassett, G., Jr.(1978), “Regression Quantiles,” Econometrica, 46, 33–50. DOI: 10.2307/1913643.
  • Koenker, R., Chernozhukov, V., He, X., and Peng, L. (2017), Handbook of Quantile Regression, New York: Chapman and Hall/CRC.
  • Koenker, R., Ng, P., and Portnoy, S. (1994), “Quantile Smoothing Splines,” Biometrika, 81, 673–680. DOI: 10.1093/biomet/81.4.673.
  • Koenker, R., and Park, B. J. (1996), “An Interior Point Algorithm for Nonlinear Quantile Regression,” Journal of Econometrics, 71, 265–283. DOI: 10.1016/0304-4076(96)84507-6.
  • Kostov, P. (2009), “A Spatial Quantile Regression Hedonic Model of Agricultural Land Prices,” Spatial Economic Analysis, 4, 53–72. DOI: 10.1080/17421770802625957.
  • Kraus, D., and Czado, C. (2017), “D-Vine Copula Based Quantile Regression,” Computational Statistics & Data Analysis, 110, 1–18.
  • Lahiri, S. N., Lee, Y., and Cressie, N. (2002), “On Asymptotic Distribution and Asymptotic Efficiency of Least Squares Estimators of Spatial Variogram Parameters,” Journal of Statistical Planning and Inference, 103, 65–85. DOI: 10.1016/S0378-3758(01)00198-7.
  • Li, Y., Liu, Y., and Zhu, J. (2007), “Quantile Regression in Reproducing Kernel Hilbert Spaces,” Journal of the American Statistical Association, 102, 255–268. DOI: 10.1198/016214506000000979.
  • Lorensen, W. E., and Cline, H. E. (1987), “Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” in ACM Siggraph Computer Graphics (Vol. 21), pp. 163–169. DOI: 10.1145/37402.37422.
  • Lu, Z., Tang, Q., and Cheng, L. (2014, 02), “Estimating Spatial Quantile Regression With Functional Coefficients: A Robust Semiparametric Framework,” Bernoulli, 20, 164–189. DOI: 10.3150/12-BEJ480.
  • Luders, E., Thompson, P. M., Kurth, F., Hong, J.-Y., Phillips, O. R., Wang, Y., Gutman, B. A., Chou, Y.-Y., Narr, K. L., and Toga, A. W. (2013), “Global and Regional Alterations of Hippocampal Anatomy in Long-Term Meditation Practitioners,” Human Brain Mapping, 34, 3369–3375. DOI: 10.1002/hbm.22153.
  • Matérn, B. (2013), Spatial Variation (Vol. 36), New York: Springer-Verlag.
  • Matheron, G. (1963), “Principles of Geostatistics,” Economic Geology, 58, 1246–1266. DOI: 10.2113/gsecongeo.58.8.1246.
  • Meyer, M., and Woodroofe, M. (2000), “On the Degrees of Freedom in Shape-Restricted Regression,” The Annals of Statistics, 28, 1083–1104. DOI: 10.1214/aos/1015956708.
  • Nikoloulopoulos, A. K., Joe, H., and Li, H. (2012), “Vine Copulas With Asymmetric Tail Dependence and Applications to Financial Return Data,” Computational Statistics & Data Analysis, 56, 3659–3673.
  • Patenaude, B., Smith, S. M., Kennedy, D. N., and Jenkinson, M. (2011), “A Bayesian Model of Shape and Appearance for Subcortical Brain Segmentation,” Neuroimage, 56, 907–922. DOI: 10.1016/j.neuroimage.2011.02.046.
  • Pizer, S. M., Fritsch, D. S., Yushkevich, P. A., Johnson, V. E., and Chaney, E. L. (1999), “Segmentation, Registration, and Measurement of Shape Variation via Image Object Shape,” IEEE Transactions on Medical Imaging, 18, 851–865. DOI: 10.1109/42.811263.
  • Portnoy, S. (1997), “On Computation of Regression Quantiles: Making the Laplacian Tortoise Faster,” in Lecture Notes-Monograph Series, pp. 187–200.
  • Portnoy, S., and Koenker, R. (1997), “The Gaussian Hare and the Laplacian Tortoise: Computability of Squared-Error Versus Absolute-Error Estimators,” Statistical Science, 12, 279–300. DOI: 10.1214/ss/1030037960.
  • Ramsay, J. O., and Silverman, B. W. (2005), Functional Data Analysis, New York: Springer-Verlag.
  • ——— (2007), Applied Functional Data Analysis: Methods and Case Studies, New York: Springer-Verlag.
  • Reich, B. J. (2012), “Spatiotemporal Quantile Regression for Detecting Distributional Changes in Environmental Processes,” Journal of the Royal Statistical Society, Series C, 61, 535–553. DOI: 10.1111/j.1467-9876.2011.01025.x.
  • Reich, B. J., Fuentes, M., and Dunson, D. B. (2011), “Bayesian Spatial Quantile Regression,” Journal of the American Statistical Association, 106, 6–20. DOI: 10.1198/jasa.2010.ap09237.
  • Reiss, P. T., Huang, L., and Mennes, M. (2010), “Fast Function-on-Scalar Regression With Penalized Basis Expansions,” The International Journal of Biostatistics, 6, 1–27. DOI: 10.2202/1557-4679.1246.
  • Rosen, W. G., Mohs, R. C., and Davis, K. L. (1984), “A New Rating Scale for Alzheimer’s Disease,” The American Journal of Psychiatry, 141, 1356–1364.
  • Scher, A., Xu, Y., Korf, E., White, L., Scheltens, P., Toga, A., Thompson, P., Hartley, S., Witter, M., Valentino, D., and Launer, L. (2007), “Hippocampal Shape Analysis in Alzheimer’s Disease: A Population-Based Study,” Neuroimage, 36, 8–18. DOI: 10.1016/j.neuroimage.2006.12.036.
  • Schwarz, G. (1978), “Estimating the Dimension of a Model,” The Annals of Statistics, 6, 461–464. DOI: 10.1214/aos/1176344136.
  • Shi, J., Lepore, N., Gutman, B. A., Thompson, P. M., Baxter, L. C., Caselli, R. J., and Wang, Y. (2014), “Genetic Influence of Apolipoprotein E4 Genotype on Hippocampal Morphometry: An N = 725 Surface-Based Alzheimer’s Disease Neuroimaging Initiative Study,” Human Brain Mapping, 35, 3903–3918. DOI: 10.1002/hbm.22447.
  • Shi, J., Thompson, P. M., Gutman, B., and Wang, Y. (2013), “Surface Fluid Registration of Conformal Representation: Application to Detect Disease Burden and Genetic Influence on Hippocampus,” NeuroImage, 78, 111–134. DOI: 10.1016/j.neuroimage.2013.04.018.
  • Smith, S. M., Jenkinson, M., Johansen-Berg, H., Rueckert, D., Nichols, T. E., Mackay, C. E., Watkins, K. E., Ciccarelli, O., Cader, M. Z., Matthews, P. M., and Behrens, T. E. (2006), “Tract-Based Spatial Statistics: Voxelwise Analysis of Multi-Subject Diffusion Data,” NeuroImage, 31, 1487–1505. DOI: 10.1016/j.neuroimage.2006.02.024.
  • Stein, C. M. (1981), “Estimation of the Mean of a Multivariate Normal Distribution,” The Annals of Statistics, 9, 1135–1151. DOI: 10.1214/aos/1176345632.
  • Su, L., and Yang, Z. (2007), “Instrumental Variable Quantile Estimation of Spatial Autoregressive Models,” Development Economics Working Papers 22476, East Asian Bureau of Economic Research.
  • Sun, X., Du, P., Wang, X., and Ma, P. (2018), “Optimal Penalized Function-on-Function Regression Under a Reproducing Kernel Hilbert Space Framework,” Journal of the American Statistical Association, 113, 1601–1611. DOI: 10.1080/01621459.2017.1356320.
  • Wahba, G. (1990), Spline Models for Observational Data (Vol. 59), Philadelphia, PA: SIAM.
  • Wang, H. J., Feng, X., and Dong, C. (2019), “Copula-Based Quantile Regression for Longitudinal Data,” Statistica Sinica, 29, 245–264.
  • Wang, H. J., Zhu, Z., and Zhou, J. (2009), “Quantile Regression in Partially Linear Varying Coefficient Models,” The Annals of Statistics, 37, 3841–3866. DOI: 10.1214/09-AOS695.
  • Wang, X., Zhu, H., and Alzheimer’s Disease Neuroimaging Initiative (2017), “Generalized Scalar-on-Image Regression Models via Total Variation,” Journal of the American Statistical Association, 112, 1156–1168. DOI: 10.1080/01621459.2016.1194846.
  • Wang, Y., Song, Y., Rajagopalan, P., An, T., Liu, K., Chou, Y.-Y., Gutman, B., Toga, A. W., and Thompson, P. M. (2011), “Surface-Based TBM Boosts Power to Detect Disease Effects on the Brain: An N = 804 ADNI Study,” Neuroimage, 56, 1993–2010. DOI: 10.1016/j.neuroimage.2011.03.040.
  • Wang, Y., Yuan, L., Shi, J., Greve, A., Ye, J., Toga, A. W., Reiss, A. L., and Thompson, P. M. (2013), “Applying Tensor-Based Morphometry to Parametric Surfaces Can Improve MRI-Based Disease Diagnosis,” Neuroimage, 74, 209–230. DOI: 10.1016/j.neuroimage.2013.02.011.
  • Wyss-Coray, T. (2016), “Ageing, Neurodegeneration and Brain Rejuvenation,” Nature, 539, 180–186. DOI: 10.1038/nature20411.
  • Yang, Y., and He, X. (2015), “Quantile Regression for Spatially Correlated Data: An Empirical Likelihood Approach,” Statistica Sinica, 25, 261–274.
  • Yao, F., Müller, H.-G., and Wang, J.-L. (2005), “Functional Linear Regression Analysis for Longitudinal Data,” The Annals of Statistics, 33, 2873–2903. DOI: 10.1214/009053605000000660.
  • Yi, G. Y., and He, W. (2009), “Median Regression Models for Longitudinal Data With Dropouts,” Biometrics, 65, 618–625. DOI: 10.1111/j.1541-0420.2008.01105.x.
  • Yuan, M. (2006), “GACV for Quantile Smoothing Splines,” Computational Statistics & Data Analysis, 50, 813–829.
  • Yuan, M., and Cai, T. T. (2010), “A Reproducing Kernel Hilbert Space Approach to Functional Linear Regression,” The Annals of Statistics, 38, 3412–3444. DOI: 10.1214/09-AOS772.
  • Zhu, H., Kong, L., Li, R., Styner, M., Gerig, G., Lin, W., and Gilmore, J. H. (2011), “FADTTS: Functional Analysis of Diffusion Tensor Tract Statistics,” NeuroImage, 56, 1412–1425. DOI: 10.1016/j.neuroimage.2011.01.075.
  • Zhu, H., Zhang, H., Ibrahim, J. G., and Peterson, B. S. (2007), “Statistical Analysis of Diffusion Tensors in Diffusion-Weighted Magnetic Resonance Imaging Data,” Journal of the American Statistical Association, 102, 1085–1102. DOI: 10.1198/016214507000000581.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.