Bayesian Analysis

Variational Bayes for Functional Data Registration, Smoothing, and Prediction

Cecilia Earls and Giles Hooker

Full-text: Open access

Abstract

We propose a model for functional data registration that extends current inferential capabilities for unregistered data by providing a flexible probabilistic framework that 1) allows for functional prediction in the context of registration and 2) can be adapted to include smoothing and registration in one model. The proposed inferential framework is a Bayesian hierarchical model where the registered functions are modeled as Gaussian processes. To address the computational demands of inference in high-dimensional Bayesian models, we propose an adapted form of the variational Bayes algorithm for approximate inference that performs similarly to Markov Chain Monte Carlo (MCMC) sampling methods for well-defined problems. The efficiency of the adapted variational Bayes (AVB) algorithm allows variability in a predicted registered, warping, and unregistered function to be depicted separately via bootstrapping. Temperature data related to the El-Niño phenomenon is used to demonstrate the unique inferential capabilities for prediction provided by this model.

Article information

Source
Bayesian Anal. Volume 12, Number 2 (2017), 557-582.

Dates
First available in Project Euclid: 26 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.ba/1469553352

Digital Object Identifier
doi:10.1214/16-BA1013

Keywords
Bayesian modeling functional data functional prediction registration smoothing variational Bayes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Earls, Cecilia; Hooker, Giles. Variational Bayes for Functional Data Registration, Smoothing, and Prediction. Bayesian Anal. 12 (2017), no. 2, 557--582. doi:10.1214/16-BA1013. https://projecteuclid.org/euclid.ba/1469553352.


Export citation

References

  • Beran, R. (1990). “Calibrating prediction regions.” Journal of the American Statistical Association, 85(411): 715–723.
  • Bishop, C. (2006). Pattern Recognition and Machine Learning. Springer, New York.
  • Brumback, C. and Lindstrom, J. (2004). “Self-modeling with flexible, random time transformations.” Biometrics, 60: 461–470.
  • Earls, C. and Hooker, G. (2014). “Bayesian covariance estimation and inference in latent Gaussian process models.” Statistical Methodology, 18: 79–100.
  • Earls, C. and Hooker, G. (2016). “Appendices for Variational Bayes for Functional Data Registration, Smoothing, and Prediction.” Bayesian Analysis.
  • Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer, New York.
  • Gelman, A. (2006). “Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper.” Bayesian Analysis, 1(3): 515–534.
  • Gervini, D. and Gasser, T. (2004). “Self-modeling warping functions.” Journal of the Royal Statistical Society, Series B, 66: 959–971.
  • Gervini, D. and Gasser, T. (2005). “Nonparametric maximum likelihood estimation of the structure of a sample of curves.” Biometrika, 92: 801–820.
  • Goldsmith, J., Wand, M., and Crainiceanu, C. (2011). “Functional regression via variational Bayes.” Electronic Journal of Statistics, 5(572).
  • James, G. (2007). “Curve alignment by moments.” The Annals of Applied Statistics, 1(2): 480–501.
  • Kneip, A. and Gasser, T. (1992). “Statistical tools to analyze data representing a sample of curves.” The Annals of Statistics, 1(2): 480–501.
  • Kneip, A. and Gasser, T. (1995). “Searching for structure in curve samples.” Journal of the American Statistical Association, 90: 1179–1188.
  • Kneip, A. and Ramsay, J. O. (2008). “Combining registration and fitting for functional models.” Journal of the American Statistical Association, 103(483): 1155–1165.
  • Lange (1989). “Robust statistical modeling using the t distribution.” Journal of the American Statistical Association, 84(408): 881–896.
  • Liu, X. and Muller, H. (2004). “Functional convex averaging and synchronization for time-warped random curves.” Journal of the American Statistical Association, 99: 687–699.
  • Liu, X. and Yang, M. (2009). “Simultaneous curve registration and clustering for functional data.” Computational Statistics and Data Analysis, 53: 1361–1376.
  • Omerod, J. and Wand, M. (2010). “Explaining variational approximations.” The American Statistician, 64: 140–153.
  • Raket, L., Sommer, S., and Markussen, B. (2014). “A nonlinear mixed-effects model for simultaneous smoothing and registration of functional data.” Pattern Recognition Letters, 38: 1–7.
  • Ramsay, J. O. and Li, X. (1998). “Curve registration.” Journal of the Royal Statistical Society. Series B (Statistical Methodology), 60(2): 351–363.
  • Ramsay, J. O. and Silverman, B. (2005). Functional Data Analysis. Springer, New York.
  • Rao, C. (1945). “Information and accuracy attainable in the estimation of statistical parameters.” Bulletin of Calcutta Mathematical Society, 37: 81–91.
  • Ronn, B. (2001). “Nonparametric maximum likelihood estimation of shifted curves.” Journal of the Royal Statistical Society, B, 63: 243–259.
  • Sakoe, H. and Chiba, S. (1978). “Dynamic programming algorithm optimization for spoken word recognition.” IEEE Transactions on Acoustics, Speech and Signal Processing, 26(1): 43–49.
  • Sangalli, L., Secchi, P., Vantini, S., and Vitelli, V. (2010). “k-mean alignment for curve clustering.” Computational Statistics and Data Analysis, 54: 1219–1233.
  • Silverman, B. (1995). “Incorporating parametric effects into functional principal components analysis.” Journal of the Royal Statistical Society, 57: 673–689.
  • Srivastava, A., Wu, W., Kurtek, S., Klassen, E., and Marron, J. (2011). “Registration of functional data using Fisher–Rao metric.” arXiv:1103.3817.
  • Tang, R. and Muller, H. (2008). “Pairwise curve synchronization for functional data.” Biometrika, 95(4): 875–889.
  • Telesca, D. and Inoue, L. (2007). “Bayesian hierarchical curve registration.” Journal of the American Statistical Association, 103(481): 328–339.
  • Tuddenham, R. and Snyder, M. (1954). “Physical growth of California boys and girls from birth to eighteen years.” University of California Publications in Child Development I, 183–364.
  • Tzikas, D. G., Likas, A. C., and Galatsanos, N. P. (2008). “The variational approximation for Bayesian inference.” Signal Processing Magazine, IEEE, 25(6): 131–146.
  • Wang, B. and Titterington, D. (2005). “Inadequacy of interval estimates corresponding to variational Bayesian approximations.” Proc. 10th Int. Wrkshp Artificial Intelligence and Statistics, 373–380.
  • Wang, K. and Gasser, T. (1997). “Alignment of curves by dynamic time warping.” The Annals of Statistics, 25(3): 1251–1276.
  • Zhang, Y. and Telesca, D. (2014). “Joint clustering and registration of functional data.” arXiv:1403.7134.

Supplemental materials