Regression models for functional data by reproducing kernel Hilbert spaces methods. (English) Zbl 1104.62043
Summary: Nonparametric regression models are developed when the predictor is a function-valued random variable \(X=\{X_{t}\}_{t\in T}\). Based on a representation of the regression function \(f(X)\) in a reproducing kernel Hilbert space, such models generalize the classical setting used in statistical learning theory. Two applications corresponding to scalar and categorical response random variables are performed on stock-exchange and medical data. The results of different regression models are compared.
MSC:
| 62G08 | Nonparametric regression and quantile regression |
| 46N30 | Applications of functional analysis in probability theory and statistics |
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
Keywords:
functional data; stochastic process; RKHS; statistical learning; simulations; uniform convergence; consistency; curve distriminationSoftware:
fda (R)References:
| [1] | Aguilera, A. M.; Gutiérrez, R.; Ocãna, F.; Valderama, M. J., Approximation of estimators in the PCA of stochastic process using B-splines, Commun. Statist. Comput. Simulation, 25, 3, 671-690 (1996) · Zbl 0937.62602 |
| [2] | Aguilera, A. M.; Ocãna, F.; Valderama, M. J., An approximated principal component prediction model for continuous-time stochastic process, Appl. Stochastic Models Data Anal., 13, 61-72 (1997) · Zbl 0885.62109 |
| [3] | Aguilera, A. M.; Ocana, F.; Valderama, M. J., Stochastic modeling for evolution of stock-prices by means of principal component analysis, Appl. Stochastic Models Business Ind., 15, 4, 227-234 (1999) · Zbl 0961.91009 |
| [4] | Aronszajn, N., Theory of reproducing kernels, Amer. Math. Soc. Trans., 63, 337-404 (1950) · Zbl 0037.20701 |
| [5] | Berlinet, A.; Thomas-Agnan, C., Reproducing Kernel Hilbert spaces in Probability and Statistics (2004), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 1145.62002 |
| [6] | Cardot, H.; Sarda, P., Estimation in generalized linear models for functional data via penalized likelihood, J. Multivariate Anal., 92, 24-41 (2005) · Zbl 1065.62127 |
| [7] | Cardot, H.; Ferraty, F.; Sarda, P., Functional linear model, Statist. Probab. Lett., 45, 11-22 (1999) · Zbl 0962.62081 |
| [8] | Craven, P.; Wahba, G., Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation, Numer. Math., 31, 377-403 (1979) · Zbl 0377.65007 |
| [9] | Cucker, F.; Smale, S., On the mathematical foundations of learning, Bull. Amer. Math. Soc., 39, 1, 1-49 (2001) · Zbl 0983.68162 |
| [10] | Duhamel, A.; Bourriez, J. L.; Devos, P.; Krystkowiak, P.; Destee, A.; Derambure, P.; Defebvre, L., Statistical tools for clinical gait analysis, Gait Posture, 20, 204-212 (2004) |
| [11] | Escabias, M.; Aguilera, A. M.; Valderama, M. J., Principal component estimation of functional logistic regression: discussion of two different approaches, J. Nonparametric Statist., 16, 3-4, 365-384 (2004) · Zbl 1065.62114 |
| [12] | Escabias, M.; Aguilera, A. M.; Valderama, M. J., Modelling environmental data by functional principal component logistic regression, Environmetrics, 16, 1, 95-107 (2005) |
| [13] | Ferraty, F.; Vieu, P., Curves discrimination: a nonparametric approach, Comput. Statist. Data Anal., 44, 161-173 (2003) · Zbl 1429.62241 |
| [14] | Ferraty, F.; Vieu, P., Nonparametric models for functional data with application in regression, time series prediction and curve discrimination, J. Nonparametric Statist., 16, 1-2, 111-125 (2004) · Zbl 1049.62039 |
| [15] | Girard, D., A fast Monte-Carlo cross-validation procedure for large least squares problems with noisy data, Numer. Math., 56, 1-23 (1989) · Zbl 0665.65010 |
| [16] | Golub, G. H.; Heath, M.; Wahba, G., Generalized cross validation as a method for choosing a good ridge parameter, Technometrics, 21, 215-224 (1979) · Zbl 0461.62059 |
| [17] | James, G. M., Generalized linear models with functional predictors, J. Roy. Statist. Soc. Ser. B, 64, 3, 411-432 (2002) · Zbl 1090.62070 |
| [18] | Kimeldorf, G. S.; Wahba, G., Some results on Tchebycheffian spline functions, J. Math. Anal. Appl., 33, 82-95 (1971) · Zbl 0201.39702 |
| [19] | Mendelson, S., Learnability in Hilbert spaces with reproducing Kernels, J. Complexity, 18, 152-170 (2002) · Zbl 1004.68131 |
| [20] | Parzen, E., An approach to time series anlysis, Ann. Math. Statist., 32, 951-989 (1961) · Zbl 0107.13801 |
| [21] | Preda, C.; Saporta, G., PLS regression on a sochastic process, Comput. Statist. Data Anal., 48, 1, 149-158 (2005) · Zbl 1429.62224 |
| [22] | Ramsay, J. O.; Silverman, B. W., Functional data analysis. Springer Series in Statistics (1997), Springer: Springer New York · Zbl 0882.62002 |
| [23] | Ramsay, J. O.; Silverman, B. W., Applied Functional Data Analysis: Methods and Case Studies (2002), Springer: Springer Berlin · Zbl 1011.62002 |
| [24] | Ratcliffe, S. J.; Leader, L. R.; Heller, G. Z., Functional data analysis with application to periodically stimulated foetal heart rate data. I: functional regression, Statist. Med., 21, 1103-1114 (2002) |
| [25] | Ratcliffe, S. J.; Leader, L. R.; Heller, G. Z., Functional data analysis with application to periodically stimulated foetal heart rate data II: functional logistic regression, Statist. Med., 21, 1115-1127 (2002) |
| [26] | Schölkopf, B.; Smola, A., Learning with Kernes (2002), MIT Press: MIT Press Cambridge |
| [27] | Vapnik, V., Statistical Learning Theory (1998), Wiley: Wiley NY · Zbl 0935.62007 |
| [28] | Wahba, G., 1990. Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, vol. 59.; Wahba, G., 1990. Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, vol. 59. · Zbl 0813.62001 |
| [29] | Wahba, G., 2002. Soft and Hard Classification by Reproducing Kernel Hilbert Space Methods. Department of Statistics, University of Wisconsin. Technical Report 1067.; Wahba, G., 2002. Soft and Hard Classification by Reproducing Kernel Hilbert Space Methods. Department of Statistics, University of Wisconsin. Technical Report 1067. · Zbl 1106.62338 |
| [30] | Wahba, G.; Lin, X.; Gao, F.; Xiang, D.; Klein, R.; Klein, B., The biais-variance tradeoff and the randomized GACV, (Kearns, M.; Solla, S.; Cohn, D., Advances in Information Processing Systems, vol. 11 (1999), MIT Press: MIT Press Cambridge), 620-626 |
| [31] | Xiang, D.; Wahba, G., A generalized approximate cross validation for smoothing splines with non-Gaussian data, Statist. Sinica, 6, 675-692 (1996) · Zbl 0854.62044 |
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