Efficient Multidimensional Functional Data Analysis Using Marginal Product Basis Systems

References

• Allen, G. I. (2013), “Multi-Way Functional Principal Components Analysis,” in 2013 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), pp. 220–223.
   Google Scholar
• Allen, G. I., and Weylandt, M. (2019), “Sparse and Functional Principal Components Analysis,” 2019 IEEE Data Science Workshop (DSW). DOI: 10.1109/DSW.2019.8755778.
   Google Scholar
• Bartels, R. H., and Stewart, G. W. (1972), “Solution of the Matrix Equation AX + XB = C,” Communications of the ACM, 15, 820–826. DOI: 10.1145/361573.361582.
   Web of Science ®Google Scholar
• Benjamini, Y., and Hochberg, Y. (1995), “Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing,” Journal of the Royal Statistical Society, Series B, 57, 289–300. DOI: 10.1111/j.2517-6161.1995.tb02031.x.
   Web of Science ®Google Scholar
• Bertsekas, D. P. (1997), “Nonlinear Programming,” Journal of the Operational Research Society, 48, 334–334. DOI: 10.1057/palgrave.jors.2600425.
   Web of Science ®Google Scholar
• 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 in Machine Learning, 3, 1–122. DOI: 10.1561/2200000016.
   Google Scholar
• Chen, K., Delicado, P., and Müller, H.-G. (2017), “Modelling Function-Valued Stochastic Processes, with Applications to Fertility Dynamics,” Journal of the Royal Statistical Society, Series B, 79, 177–196. DOI: 10.1111/rssb.12160.
   Google Scholar
• Chen, L.-H., and Jiang, C.-R. (2017), “Multi-Dimensional Functional Principal Component Analysis,” Statistics and Computing, 27, 1181–1192. DOI: 10.1007/s11222-016-9679-5.
   Web of Science ®Google Scholar
• Happ, C., and Greven, S. (2018), “Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains,” Journal of the American Statistical Association, 113, 649–659. DOI: 10.1080/01621459.2016.1273115.
   Web of Science ®Google Scholar
• Hillar, C. J., and Lim, L.-H. (2013), “Most Tensor Problems Are NP-Hard,” Journal of the ACM, 60, 1–39. DOI: 10.1145/2512329.
   Web of Science ®Google Scholar
• Hsing, T., and Eubank, R. (2015), Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators, Chichester: Wiley.
   Google Scholar
• Huang, J. Z., Shen, H., and Buja, A. (2009), “The Analysis of Two-Way Functional Data using Two-Way Regularized Singular Value Decompositions,” Journal of the American Statistical Association, 104, 1609–1620. DOI: 10.1198/jasa.2009.tm08024.
   Web of Science ®Google Scholar
• Huang, K., Sidiropoulos, N. D., and Liavas, A. P. (2016), “A Flexible and Efficient Algorithmic Framework for Constrained Matrix and Tensor Factorization,” IEEE Transactions on Signal Processing, 64, 5052–5065. DOI: 10.1109/TSP.2016.2576427.
   Web of Science ®Google Scholar
• Khong, E., Odenwald, N., Hashim, E., and Cusimano, M. D. (2016), “Diffusion Tensor Imaging Findings in Post-Concussion Syndrome Patients After Mild Traumatic Brain Injury: A Systematic Review,” Frontiers in Neurology, 7, 156. DOI: 10.3389/fneur.2016.00156.
   PubMed Web of Science ®Google Scholar
• Kraus, M. F., Susmaras, T., Caughlin, B. P., Walker, C. J., Sweeney, J. A., and Little, D. M. (2007), “White Matter Integrity and Cognition in Chronic Traumatic Brain Injury: A Diffusion Tensor Imaging Study,” Brain, 130, 2508–2519. DOI: 10.1093/brain/awm216.
   PubMed Web of Science ®Google Scholar
• Li, C., Xiao, L., and Luo, S. (2020), “Fast Covariance Estimation for Multivariate Sparse Functional Data,” Stat, 9, e245. DOI: 10.1002/sta4.245.
   PubMed Web of Science ®Google Scholar
• Li, Y., Huang, C., and Härdle, W. K. (2019), “Spatial Functional Principal Component Analysis with Applications to Brain Image Data,” Journal of Multivariate Analysis, 170, 263–274. Special Issue on Functional Data Analysis and Related Topics. DOI: 10.1016/j.jmva.2018.11.004.
   Web of Science ®Google Scholar
• Lynch, B., and Chen, K. (2018), “A Test of Weak Separability for Multi-Way Functional Data, with Application to Brain Connectivity Studies,” Biometrika, 105, 815–831. DOI: 10.1093/biomet/asy048.
   Web of Science ®Google Scholar
• Phan, A., Tichavský, P., and Cichocki, A. (2013), “Fast Alternating ls Algorithms for High Order candecomp/parafac Tensor Factorizations,” IEEE Transactions on Signal Processing, 61, 4834–4846. DOI: 10.1109/TSP.2013.2269903.
   Web of Science ®Google Scholar
• Post, A., Oeur, A., Hoshizaki, B., and Gilchrist, M. D. (2013), “Examination of the Relationship between Peak Linear and Angular Accelerations to Brain Deformation Metrics in Hockey Helmet Impacts,” Computer Methods in Biomechanics and Biomedical Engineering, 16, 511–519. DOI: 10.1080/10255842.2011.627559.
   PubMed Web of Science ®Google Scholar
• Ramsay, J. O., and Silverman, B. W. (2005), Functional Data Analysis, New York: Springer-Verlag.
   Google Scholar
• Silverman, B. W. (1996), “Smoothed Functional Principal Components Analysis by Choice of Norm,” The Annals of Statistics, 24, 1–24. DOI: 10.1214/aos/1033066196.
   Web of Science ®Google Scholar
• 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.
   PubMed Web of Science ®Google Scholar
• Stone, C. J. (1980), “Optimal Rates of Convergence for Nonparametric Estimators,” The Annals of Statistics, 8, 1348–1360. DOI: 10.1214/aos/1176345206.
   Web of Science ®Google Scholar
• Wang, J., Wong, R. K. W., and Zhang, X. (2020), “Low-Rank Covariance Function Estimation for Multidimensional Functional Data,” Journal of the American Statistical Association, 117, 809–822. DOI: 10.1080/01621459.2020.1820344.
   Web of Science ®Google Scholar
• Wang, Z., Wu, W., Liu, Y., Wang, T., Chen, X., Zhang, J., Zhou, G., and Chen, R. (2016), “Altered Cerebellar White Matter Integrity in Patients with Mild Traumatic Brain Injury in the Acute Stage,” PLoS One, 11, e0151489. DOI: 10.1371/journal.pone.0151489.
   PubMed Web of Science ®Google Scholar
• Wasserman, L. (2010), All of Nonparametric Statistics, New York: Springer.
   Google Scholar
• Xiao, L., Li, Y., and Ruppert, D. (2013), “Fast Bivariate p-splines: The Sandwich Smoother,” Journal of the Royal Statistical Society, Series B, 75, 577–599. DOI: 10.1111/rssb.12007.
   Google Scholar
• Xiong, K., Zhu, Y., Zhang, Y., Yin, Z., Zhang, J., Qiu, M., and Zhang W. (2014), “White Matter Integrity and Cognition in Mild Traumatic Brain Injury Following Motor Vehicle Accident,” Brain Research, 1591, 86–92. DOI: 10.1016/j.brainres.2014.10.030.
   PubMed Web of Science ®Google Scholar
• Zhang, J.-T., and Chen, J. (2007), “Statistical Inferences for Functional Data,” The Annals of Statistics, 35, 1052–1079. DOI: 10.1214/009053606000001505.
   Web of Science ®Google Scholar