Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-07T10:36:35.576Z Has data issue: false hasContentIssue false
  • English
  • Français

A Unified Approach to Functional Principal Component Analysis and Functional Multiple-Set Canonical Correlation

Published online by Cambridge University Press:  01 January 2025

Ji Yeh Choi ,
Heungsun Hwang ,
Michio Yamamoto ,
Kwanghee Jung  and
Todd S. Woodward
Ji Yeh Choi*
Affiliation:
McGill University
Heungsun Hwang
Affiliation:
McGill University
Michio Yamamoto
Affiliation:
Kyoto University
Kwanghee Jung
Affiliation:
University of Texas Health Science Center
Todd S. Woodward
Affiliation:
University of British Columbia and British Columbia Mental Health and Addiction Research Institute
*
Correspondence should be made to Ji Yeh Choi, Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, QC H3A 1B1 Canada. Email: ji.yeh.choi@mail.mcgill.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Functional principal component analysis (FPCA) and functional multiple-set canonical correlation analysis (FMCCA) are data reduction techniques for functional data that are collected in the form of smooth curves or functions over a continuum such as time or space. In FPCA, low-dimensional components are extracted from a single functional dataset such that they explain the most variance of the dataset, whereas in FMCCA, low-dimensional components are obtained from each of multiple functional datasets in such a way that the associations among the components are maximized across the different sets. In this paper, we propose a unified approach to FPCA and FMCCA. The proposed approach subsumes both techniques as special cases. Furthermore, it permits a compromise between the techniques, such that components are obtained from each set of functional data to maximize their associations across different datasets, while accounting for the variance of the data well. We propose a single optimization criterion for the proposed approach, and develop an alternating regularized least squares algorithm to minimize the criterion in combination with basis function approximations to functions. We conduct a simulation study to investigate the performance of the proposed approach based on synthetic data. We also apply the approach for the analysis of multiple-subject functional magnetic resonance imaging data to obtain low-dimensional components of blood-oxygen level-dependent signal changes of the brain over time, which are highly correlated across the subjects as well as representative of the data. The extracted components are used to identify networks of neural activity that are commonly activated across the subjects while carrying out a working memory task.

Keywords

functional datafunctional principal component analysisfunctional multiple-set canonical correlation analysisalternating regularized least squares algorithm
Type
Original Paper
Information
Psychometrika , Volume 82 , Issue 2 , June 2017 , pp. 427 - 441
Copyright
Copyright © 2016 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

Bezdek, J. C. (1974). Numerical taxonomy with fuzzy sets. Journal of Mathematical Biology, 1, (1), 57–71CrossRefGoogle Scholar
Cairo, T. A., & Woodward, T. S., & Ngan, E. T. C. (2006). Decreased encoding efficiency in schizophrenia. Biological Psychiatry, 59, (8), 740–746CrossRefGoogle ScholarPubMed
Di, C-Z, & Crainiceanu, C. M., & Caffo, B. S., & Punjabi, N. M. (2009). Multilevel functional principal component analysis. The Annals of Applied Statistics, 3, (1), 458CrossRefGoogle ScholarPubMed
Dunn, J. C. (1974). A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters. Journal of Cybernetics, 3, 32–57CrossRefGoogle Scholar
Escabias, M., & Aguilera, A. M., & Valderrama, M. J. (2004). Principal component estimation of functional logistic regression: Discussion of two different approaches. Journal of Nonparametric Statistics, 16, (3–4), 365–384CrossRefGoogle Scholar
Fox, M. D., & Snyder, A. Z., & Vincent, J. L., & Corbetta, M., & Van Essen, D. C., & Raichle, M. E. (2005). The human brain is intrinsically organized into dynamic, anticorrelated functional networks. Proceedings of the National Academy of Sciences of the United States of America, 102, 9673–9678CrossRefGoogle ScholarPubMed
Giraldo, R., & Delicado, P., & Mateu, J. (2011). Ordinary kriging for function-valued spatial data. Environmental and Ecological Statistics, 18, (3), 411–426CrossRefGoogle Scholar
Goutte, C., & Nielsen, F. A., & Hansen, L. K. (2000). Modeling the hemodynamic response in fMRI using smooth FIR filters. IEEE Transactions on Medical Imaging, 19, 1188–1201CrossRefGoogle ScholarPubMed
Hancock, P. A., & Hutchinson, M. F. (2006). Spatial interpolation of large climate data sets using bivariate thin plate smoothing splines. Environmental Modelling and Software, 21, (12), 1684–1694CrossRefGoogle Scholar
Hartigan, J. A., & Wong, M. A. (1979). Algorithm AS 136: A k-means clustering algorithm. Applied Statistics, 28, (1), 100–108CrossRefGoogle Scholar
Hastie, T., & Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning: Data mining, inference, and prediction, 2New York: SpringerCrossRefGoogle Scholar
He, G., & Müller, H. G., & Wang, J. L. (2003). Functional canonical analysis for square integrable stochastic processes. Journal of Multivariate Analysis, 85, (1), 54–77CrossRefGoogle Scholar
Huettel, S. A., & Song, A. W., & McCarthy, G. (2009). Functional magnetic resonance imaging, Massachusetts: Sinauer.Google Scholar
Hunter, M. A., & Takane, Y. (2002). Constrained principal component analysis: Various applications. Journal of Educational and Behavioral Statistics, 27, (2), 105–145CrossRefGoogle Scholar
Hwang, H., & Jung, K., & Takane, Y., & Woodward, T. S. (2012). Functional multiple-set canonical correlation analysis. Psychometrika, 77, (1), 48–64CrossRefGoogle Scholar
Hwang, H., & Jung, K., & Takane, Y., & Woodward, T. S. (2013). A unified approach to multiple-set canonical correlation analysis and principal components analysis. British Journal of Mathematical and Statistical Psychology, 66, (2), 308–321CrossRefGoogle ScholarPubMed
Hwang, H., & Suk, H. W., & Lee, J-H, & Moskowitz, D. S., & Lim, J. (2012). Functional extended redundancy analysis. Psychometrika, 77, (3), 524–542CrossRefGoogle ScholarPubMed
Jackson, I., & Sirois, S. (2009). Infant cognition: going full factorial with pupil dilation. Developmental Science, 12, (4), 670–679CrossRefGoogle ScholarPubMed
Kneip, A., & Utikal, K. J. (2001). Inference for density families using functional principal component analysis. Journal of the American Statistical Association, 96, (454), 519–542CrossRefGoogle Scholar
Leurgans, S. E., & Moyeed, R. A., & Silverman, B. W. (1993). Canonical correlation analysis when the data are curves. Journal of the Royal Statistical Society Series B (Methodological), 55, 725–740.CrossRefGoogle Scholar
MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the fifth Berkeley symposium on mathematical statistics and probability (pp. 281–297).Google Scholar
Marra, G., & Miller, D. L., & Zanin, L. (2012). Modelling the spatiotemporal distribution of the incidence of resident foreign population. Statistica Neerlandica, 66, (2), 133–160CrossRefGoogle Scholar
Metzak, P. D., & Feredoes, E., & Takane, Y., & Wang, L., & Weinstein, S., & Cairo, T. A., & Ngan, E. T. C., & Woodward, T. S. (2011). Constrained principal component analysis reveals functionally connected load-dependent networks involved in multiple stages of working memory. Human Brain Mapping, 32, (6), 856–871CrossRefGoogle ScholarPubMed
Metzak, P. D., & Riley, J. D., & Wang, L., & Whitman, J. C., & Ngan, E. T. C., & Woodward, T. S. (2011). Decreased efficiency of task-positive and task-negative networks during working memory in schizophrenia. Schizophrenia Bulletin, 38, (4), 803–813CrossRefGoogle ScholarPubMed
Mishra, S. K. (2009). Representation-constrained canonical corrleation analysis: A hybridization of canonical correlation and principal component analyses. Journal of Applied Economic Sciences (JAES), 7, 115–124.Google Scholar
Ramsay, J. O., & Dalzell, C. J. (1991). Some tools for functional data analysis. Journal of the Royal Statistical Society. Series B (Methodological), 53, 539–572.CrossRefGoogle Scholar
Ramsay, J. O., & Silverman, B. W. (2005). Functional data analysis, 2New York: Springer.CrossRefGoogle Scholar
Ramsay, T. (2002). Spline smoothing over difficult regions. Journal of the Royal Statistical Society, 64, (2), 307–319CrossRefGoogle Scholar
Rice, J. A., & Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society, 53, 233–243.CrossRefGoogle Scholar
Ruiz-Medina, M. D. (2012). New challenges in spatial and spatiotemporal functional statistics for high-dimensional data. Spatial Statistics, 1, 82–91CrossRefGoogle Scholar
Sibson, R., & Stone, G. (1991). Computation of thin-plate splines. SIAM Journal on Scientific and Statistical Computing, 12, (6), 1304–1313CrossRefGoogle Scholar
Tan, T., & Choi, J. Y., & Hwang, H. (2015). Fuzzy clusterwise functional extended redundancy analysis. Behaviormetrika, 42, (1), 37–62CrossRefGoogle Scholar
Tian, T. S. (2010). Functional data analysis in brain imaging studies. Frontiers in Psychology, 1, 1–11.CrossRefGoogle ScholarPubMed
Vines, B. W., Nuzzo, R. L., & Levitin, D. J. (2005). Analyzing temporal dynamics in music: Differential calculus, physics, and functional data analysis techniques. Music Perception, 23, 137–152.CrossRefGoogle Scholar
Vines, B. W., Wanderley, M. M., Krumhansl, C. L., Nuzzo, R. L., & Levitin, D. J. (2004). Performance gestures of musicians: What structural and emotional information do they convey? In Gesture-based communication in human-computer interaction (pp. 468–478). Berlin: Springer.Google Scholar
Viviani, R., & Grön, G., & Spitzer, M. (2005). Functional principal component analysis of fMRI data. Human Brain Mapping, 24, (2), 109–129CrossRefGoogle ScholarPubMed
Yao, F., & Müller, H-G, & Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association, 100, (470), 577–590CrossRefGoogle Scholar