Hostname: page-component-5f745c7db-f9j5r Total loading time: 0 Render date: 2025-01-06T06:47:50.332Z Has data issue: true hasContentIssue false
  • English
  • Français

Generalized Functional Extended Redundancy Analysis

Published online by Cambridge University Press:  01 January 2025

Heungsun Hwang ,
Hye Won Suk ,
Yoshio Takane ,
Jang-Han Lee  and
Jooseop Lim
Heungsun Hwang*
Affiliation:
McGill University
Hye Won Suk
Affiliation:
Arizona State University
Yoshio Takane
Affiliation:
McGill University
Jang-Han Lee
Affiliation:
Chung-Ang University
Jooseop Lim
Affiliation:
Concordia University
*
Requests for reprints should be sent to Heungsun Hwang, Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, Quebec, H3A 1B1, Canada. E-mail: heungsun.hwang@mcgill.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Functional extended redundancy analysis (FERA) was recently developed to integrate data reduction into functional linear models. This technique extracts a component from each of multiple sets of predictor data in such a way that the component accounts for the maximum variance of response data. Moreover, it permits predictor and/or response data to be functional. FERA can be of use in describing overall characteristics of each set of predictor data and in summarizing the relationships between predictor and response data. In this paper, we extend FERA into the framework of generalized linear models (GLM), so that it can deal with response data generated from a variety of distributions. Specifically, the proposed method reduces each set of predictor functions to a component and uses the component for explaining exponential-family responses. As in GLM, we specify the random, systematic, and link function parts of the proposed method. We develop an iterative algorithm to maximize a penalized log-likelihood criterion that is derived in combination with a basis function expansion approach. We conduct two simulation studies to investigate the performance of the proposed method based on synthetic data. In addition, we apply the proposed method to two examples to demonstrate its empirical usefulness.

Keywords

functional data analysisfunctional extended redundancy analysisgeneralized linear modelsdata reductionexponential family responsespenalized likelihood
Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 1 , March 2015 , pp. 101 - 125
Copyright
Copyright © 2013 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Dauxois, J., Pousse, A., & Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: some applications to statistical inference. Journal of Multivariate Analysis, 12, 136–154.CrossRefGoogle Scholar
Davidian, M., Lin, X., & Wang, J.-L. (2004). Introduction: emerging issues in longitudinal and functional data analysis. Statistica Sinica, 14, 613–614.Google Scholar
Febrero-Bande, M., & Oviedo de la Fuente, M. (2012). Statistical computing in functional data analysis: the R package fda.usc. Journal of Statistical Software, 51, 1–28.CrossRefGoogle Scholar
Ferraty, F. (2011). Recent advances in functional data analysis and related topics. New York: Springer.CrossRefGoogle Scholar
Ferraty, F., Laksaci, A., Tadj, A., & Vieu, P. (2011). Kernel regression with functional response. Electronic Journal of Statistics, 5, 159–171.CrossRefGoogle Scholar
Ferraty, F., Mas, A., & Vieu, P. (2007). Nonparametric regression on functional data: inference and practical. Australian & New Zealand Journal of Statistics, 49, 267–286.CrossRefGoogle Scholar
Ferraty, F., & Romain, Y. (2011). The Oxford handbook of functional data analysis. Oxford: University Press.Google Scholar
Ferraty, F., Van Keilegom, I., & Vieu, P. (2012). Regression when both response and predictor are functions. Journal of Multivariate Analysis, 109, 10–28.CrossRefGoogle Scholar
Ferraty, F., & Vieu, P. (2006). Nonparametric functional data analysis: theory and practice. New York: Springer.Google Scholar
González-Manteiga, W., & Vieu, P. (2007). Statistics for functional data. Computational Statistics & Data Analysis, 51, 4788–4792.CrossRefGoogle Scholar
Green, P.J. (1984). Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternative (with discussion). Journal of the Royal Statistical Society. Series B, 46, 149–192.CrossRefGoogle Scholar
Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning: data mining, inference, and prediction (2nd ed.). New York: Springer.CrossRefGoogle Scholar
Heyde, C.C. (1997). Quasi-likelihood and its applications. A general approach to optimal parameter estimation. New York: Springer.Google Scholar
Hoerl, A.E., & Kennard, R.W. (1970). Ridge regression: application to nonorthogonal problems. Technometrics, 12, 69–82.CrossRefGoogle Scholar
Hwang, H. (2009). Regularized generalized structured component analysis. Psychometrika, 74, 517–530.CrossRefGoogle Scholar
Hwang, H., Suk, H.W., Lee, J.-H., Moskowitz, D.S., & Lim, J. (2012). Functional extended redundancy analysis. Psychometrika, 77, 524–542.CrossRefGoogle ScholarPubMed
Hwang, H., & Tomiuk, M.A. (2010). Fuzzy clusterwise quasi-likelihood generalized linear models. Advances in Data Analysis and Classification, 4, 255–270.CrossRefGoogle Scholar
Kreibig, S.D. (2010). Autonomic nervous system activity in emotion: a review. Biological Psychology, 84, 394–421.CrossRefGoogle ScholarPubMed
Le Cessie, S., & Van Houwelingen, J.C. (1992). Ridge estimators in logistic regression. Applied Statistics, 41, 191–201.CrossRefGoogle Scholar
Lee, A., & Silvapulle, M. (1988). Ridge estimation in logistic regression. Communications in Statistics. Simulation and Computation, 17, 1231–1257.CrossRefGoogle Scholar
Lian, H. (2011). Convergence of functional k-nearest neighbor regression estimate with functional responses. Electronic Journal of Statistics, 5, 31–40.CrossRefGoogle Scholar
McCullagh, P. (1983). Quasi-likelihood functions. The Annals of Statistics, 11, 59–67.CrossRefGoogle Scholar
McCullagh, P., & Nelder, J.A. (1989). Generalized linear models (2nd ed.). London: Chapman & Hall/CRC.CrossRefGoogle Scholar
McLachlan, G., & Peel, D. (2000). Finite mixture models. New York: Wiley.CrossRefGoogle Scholar
Nelder, J.A., & Wedderburn, R.W.M. (1972). Generalized linear models. Journal of the Royal Statistical Society. Series A, 135, 370–384.CrossRefGoogle Scholar
Ramsay, J.O., & Dalzell, C.J. (1991). Some tools for functional data analysis (with discussion). Journal of the Royal Statistical Society. Series B, 53, 539–572.CrossRefGoogle Scholar
Ramsay, J.O., Hooker, G., & Graves, S. (2009). Functional data analysis with R and Matlab. New York: Springer.CrossRefGoogle Scholar
Ramsay, J.O., & Silverman, B.W. (1997). Functional data analysis (1st ed.). New York: Springer.CrossRefGoogle Scholar
Ramsay, J.O., & Silverman, B.W. (2002). Applied functional data analysis: methods and case studies. New York: Springer.CrossRefGoogle Scholar
Ramsay, J.O., & Silverman, B.W. (2005). Functional data analysis (2nd ed.). New York: Springer.CrossRefGoogle Scholar
Rice, J.A. (2004). Functional and longitudinal data analysis: perspectives on smoothing. Statistica Sinica, 14, 631–647.Google Scholar
Rice, J.A., & Silverman, B.W. (1991). Estimating the mean and covariance structure non-parametrically when the data are curves. Journal of the Royal Statistical Society. Series B, 53, 233–243.CrossRefGoogle Scholar
Richards, F.S.G. (1961). A method of maximum-likelihood estimation. Journal of the Royal Statistical Society. Series B, 23, 469–475.CrossRefGoogle Scholar
Takane, Y., & Hwang, H. (2005). An extended redundancy analysis and its applications to two practical examples. Computational Statistics & Data Analysis, 49, 785–808.CrossRefGoogle Scholar
Valderrama, M.J. (2007). An overview to modelling functional data. Computational Statistics, 22, 331–334.CrossRefGoogle Scholar
Wedderburn, R.W.M. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika, 61, 439–447.Google Scholar
Wedel, M., & DeSarbo, W.S. (1995). A mixture likelihood approach for generalized linear models. Journal of Classification, 12, 21–55.CrossRefGoogle Scholar
Yee, T.W., & Hastie, T.J. (2003). Reduced-rank vector generalized linear models. Statistical Modelling, 3, 15–41.CrossRefGoogle Scholar
Yee, T.W., & Wild, C.J. (1999). Vector generalized additive models. Journal of the Royal Statistical Society. Series B, 58, 481–493.CrossRefGoogle Scholar