Nonlinear and additive principal component analysis for functional data. (English) Zbl 1461.62241
Summary: We introduce a nonlinear additive functional principal component analysis (NAFPCA) for vector-valued functional data. This is a generalization of functional principal component analysis and allows the relations among the random functions involved to be nonlinear. The method is constructed via two additively nested Hilbert spaces of functions, in which the first space characterizes the functional nature of the data, and the second space captures the nonlinear dependence. In the meantime, additivity is imposed so that we can avoid high-dimensional kernels in the functional space, which causes the curse of dimensionality. Along with the NAFPCA, we also develop a method of selection of the number of principal components and the tuning parameters that determines the degree of nonlinearity, as well as the asymptotic results for both the fully observed and the incompletely observed functional data. Simulation results show that the new method performs better than functional principal component analysis when the relations among random functions are nonlinear. We apply the new method to online handwritten digits and electroencephalogram (EEG) data sets.
MSC:
| 62R10 | Functional data analysis |
| 62H25 | Factor analysis and principal components; correspondence analysis |
| 62H12 | Estimation in multivariate analysis |
| 62H35 | Image analysis in multivariate analysis |
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
Keywords:
functional data analysis; handwritten digit classification; nonlinear dimension reduction; principal component analysis; reproducing kernel Hilbert spaceSoftware:
fda (R)References:
| [1] | Ahn, J., A stable hyperparameter selection for the Gaussian RBF kernel for discrimination, Stat. Anal. Data Min.: ASA Data Sci. J., 3, 3, 142-148 (2010) · Zbl 07260239 |
| [2] | Berlinet, A.; Thomas-Agnan, C., Reproducing Kernel Hilbert Spaces in Probability and Statistics (2004), Springer-Verlag: Springer-Verlag New York · Zbl 1145.62002 |
| [3] | Carmeli, C.; De Vito, E.; Toigo, A.; Umanitá, V., Vector valued reproducing kernel Hilbert spaces and universality, Anal. Appl., 8, 01, 19-61 (2010) · Zbl 1195.46025 |
| [4] | Christmann, A.; Steinwart, I., Support Vector Machines (2008), Springer-Verlag: Springer-Verlag New York · Zbl 1203.68171 |
| [5] | Conway, J. B., A Course in Functional Analysis (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0706.46003 |
| [6] | Dinuzzo, F.; Schölkopf, B., The representer theorem for Hilbert spaces: a necessary and sufficient condition, Adv. Neural Inf. Process. Syst., 189-196 (2012) |
| [7] | Fukumizu, K.; Bach, F. R.; Jordan, M. I., Kernel dimension reduction in regression, Ann. Statist., 37, 4, 1871-1905 (2009) · Zbl 1168.62049 |
| [8] | Golub, G. H.; Heath, M.; Wahba, G., Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21, 2, 215-223 (1979) · Zbl 0461.62059 |
| [9] | Hall, P.; Müller, H.; Wang, J., Properties of principal component methods for functional and longitudinal data analysis, Ann. Statist., 34, 3, 1493-1517 (2006) · Zbl 1113.62073 |
| [10] | Happ, C.; Greven, S., Multivariate functional principal component analysis for data observed on different (dimensional) domains, J. Amer. Statist. Assoc., 113, 522, 649-659 (2018) · Zbl 1398.62154 |
| [11] | Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0576.15001 |
| [12] | Horváth, L.; Kokoszka, P., Inference for Functional Data with Applications (2012), Springer-Verlag: Springer-Verlag New York · Zbl 1279.62017 |
| [13] | Hsing, T.; Eubank, R., Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators (2015), John Wiley & Sons: John Wiley & Sons West Sussex, UK · Zbl 1338.62009 |
| [14] | Jacques, J.; Preda, C., Model-based clustering for multivariate functional data, Comput. Statist. Data Anal., 71, 92-106 (2014) · Zbl 1471.62096 |
| [15] | Jolliffe, I., Principal Component Analysis (2011), Springer-Verlag: Springer-Verlag New York · Zbl 1011.62064 |
| [16] | Kokoszka, P.; Reimherr, M., Introduction to Functional Data Analysis (2017), CRC Press: CRC Press Boca Raton, FL · Zbl 1411.62004 |
| [17] | Lee, K.-Y.; Li, B.; Chiaromonte, F., A general theory for nonlinear sufficient dimension reduction: Formulation and estimation, Ann. Statist., 41, 1, 221-249 (2013) · Zbl 1347.62018 |
| [18] | Li, B.; Artemiou, A.; Li, L., Principal support vector machines for linear and nonlinear sufficient dimension reduction, Ann. Statist., 39, 6, 3182-3210 (2011) · Zbl 1246.62153 |
| [19] | Li, B.; Chun, H.; Zhao, H., Sparse estimation of conditional graphical models with application to gene networks, J. Amer. Statist. Assoc., 107, 497, 152-167 (2012) · Zbl 1261.62049 |
| [20] | Li, B.; Chun, H.; Zhao, H., On an additive semigraphoid model for statistical networks with application to pathway analysis, J. Amer. Statist. Assoc., 109, 507, 1188-1204 (2014) · Zbl 1368.62180 |
| [21] | Li, Y.; Guan, Y., Functional principal component analysis of spatiotemporal point process with applications in disease surveillance, J. Amer. Statist. Assoc., 109, 507, 1205-1215 (2014) · Zbl 1368.62161 |
| [22] | Li, B.; Song, J., Nonlinear sufficient dimension reduction for functional data, Ann. Statist., 45, 3, 1059-1095 (2017) · Zbl 1371.62003 |
| [23] | Minh, H. Q., Some properties of gaussian reproducing kernel hilbert space and their implications for function approximation and learning theory, Constr. Approx., 32, 307-338 (2010) · Zbl 1204.68157 |
| [24] | Ramsay, J.; Li, X., Curve registration, J. R. Stat. Soc. Ser. B, 60, 351-363 (1998) · Zbl 0909.62033 |
| [25] | Ramsay, J.; Silverman, B., Functional Data Analysis, 430 (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1079.62006 |
| [26] | Rasmussen, C. E.; Williams, C. K., Gaussian Processes for Machine Learning, 79-104 (2006), MIT Press: MIT Press Cambridge, MA · Zbl 1177.68165 |
| [27] | Schölkopf, B.; Herbrich, R.; Smola, A. J., A generalized representer theorem, Comput. Learn. Theory, 416-426 (2001) · Zbl 0992.68088 |
| [28] | Schölkopf, B.; Smola, A.; Müller, K., Nonlinear component analysis as a kernel eigenvalue problem, Neural Comput., 10, 1299-1319 (1998) |
| [29] | Sriperumbudur, B. K.; Gretton, A.; Fukumizu, K.; Schölkopf, B.; Lanckriet, G. R., Hilbert space embeddings and metrics on probability measures, J. Mach. Learn. Res., 11, Apr, 1517-1561 (2010) · Zbl 1242.60005 |
| [30] | Wang, J.-L.; Chiou, J.-M.; Müller, H.-G., Functional data analysis, Annu. Rev. Stat. Appl., 3, 257-295 (2016) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
