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Statistics and Computing

, Volume 26, Issue 5, pp 945–950 | Cite as

Maximal autocorrelation functions in functional data analysis

  • Giles Hooker
  • Steven Roberts
Article

Abstract

This paper proposes a new factor rotation for the context of functional principal components analysis. This rotation seeks to re-express a functional subspace in terms of directions of decreasing smoothness as represented by a generalized smoothing metric. The rotation can be implemented simply and we show on two examples that this rotation can improve the interpretability of the leading components.

Keywords

Factor rotation Functional data  Interpretability  Principal components analysis 

Notes

Acknowledgments

Giles Hooker was supported in part by NSF Grants DMS-1053252 and DEB-1353039. Steven Roberts was supported in part by ARC Grant DP140100551.

References

  1. Chiou, J.-M., Müller, H.-G., Wang, J.-L.: Functional response models. Stat. Sin. 14, 659–677 (2004)MathSciNetzbMATHGoogle Scholar
  2. Cunningham, J.P., Ghahramani, Z.: Unifying linear dimensionality reduction. arXiv preprint arXiv:1406.0873 (2014)
  3. Gallagher, N.B., Shaver, J.M., Bishop, R., Roginski, R.T., Wise, B.M.: Decompositions using maximum signal factors. J. Chemom. 28, 663–671 (2014)CrossRefGoogle Scholar
  4. Goldsmith, J., Bobb, J., Crainiceanu, C.M., Caffo, B., Reich, D.: Penalized functional regression. J. Comput. Graph. Stat. 20, 830–851 (2011)MathSciNetCrossRefGoogle Scholar
  5. Hall, P., Müller, H., Wang, J.: Properties of principal component methods for functional and longitudinal data analysis. Ann. Stat. 34, 1493–1517 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Henderson, A., Fletcher, J.S., Vickerman, J.C.: A comparison of PCA and MAF for ToF-SIMS image interpretation. Surf. Interface Anal. 41, 666–674 (2009)CrossRefGoogle Scholar
  7. Kaiser, H.F.: The varimax criterion for analytic rotation in factor analysis. Psychometrika 23, 187–200 (1958)CrossRefzbMATHGoogle Scholar
  8. Liu, C., Ray, S., Hooker, G., Friedl, M.: Functional factor analysis for periodic remote sensing data. Ann. Appl. Stat. 6, 601–624 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Magnano, L., Boland, J.W.: Generation of synthetic sequences of electricity demand: application in South Australia. Energy 32, 2230–2243 (2007)CrossRefGoogle Scholar
  10. Magnano, L., Boland, J.W., Hyndman, R.: Generation of synthetic sequences of half-hourly temperature. Environmetrics 19, 818–835 (2008)MathSciNetCrossRefGoogle Scholar
  11. Müller, H.-G., Yao, F.: Functional additive models. J. Am. Stat. Assoc. 103, 1534–1544 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Paul, D., Peng, J.: Principal components analysis for sparsely observed correlated functional data using a kernel smoothing approach. Electron. J. Stat. 5, 1960–2003 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Peng, J., Paul, D.: A geometric approach to maximum likelihood estimation of the functional principal components from sparse longitudinal data. J. Comput. Graph. Stat. 18, 995–1015 (2009)MathSciNetCrossRefGoogle Scholar
  14. Ramsay, J.O., Hooker, G., Graves, S.: Functional data analysis in R and Matlab. Springer, New York (2009)CrossRefzbMATHGoogle Scholar
  15. Ramsay, J.O., Silverman, B.W.: Functional data analysis, 2nd edn. Springer, New York (2005)zbMATHGoogle Scholar
  16. Ramsay, J.O., Wickham, H., Graves, S., Hooker, G.: fda: functional data analysis, r package version 2.4.0. (2013)Google Scholar
  17. Rice, J., Silverman, B.: Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. Ser. B 53, 233–243 (1991)MathSciNetzbMATHGoogle Scholar
  18. Sentürk, D., Müller, H.-G.: Inference for covariate adjusted regression via varying coefficient models. Ann. Stat. 34, 654–679 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Shang, H.L., Hyndman, R.J.: fds: functional data sets, r package version 1.7. (2013)Google Scholar
  20. Shapiro, D., Switzer, P.: Extracting time trends from multiple monitoring sites: Dept. of Statistics, Tech. rep., Stanford University, Tech. Rep. 132. (1989)Google Scholar
  21. Silverman, B.W.: Smoothed functional principal components by choice of norm. Ann. Stat. 24, 1–24 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Switzer, P., Green, A.: Min/max autocorrelation factors for multivariate spatial imagery: Dept. of Statistics, Tech. rep., Stanford University, Tech. Rep. 6. (1984)Google Scholar
  23. Woillez, M., Rivoirard, J., Petitgas, P.: Using min/max autocorrelation factors of survey-based indicators to follow the evolution of fish stocks in time. Aquat. Living Resour. 22, 193–200 (2009)CrossRefGoogle Scholar
  24. Yao, F., Müller, H.-G., Wang, J.-L.: Functional data analysis for sparse longitudinal data. J. Am. Stat. Assoc. 100, 577–590 (2005a)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Yao, F., Müller, H.-G., Wang, J.-L.: Functional linear regression analysis for longitudinal data. Ann. Stat. 33, 2873–2903 (2005b)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Biological Statistics and Computational BiologyCornell UniversityIthacaUSA
  2. 2.Research School of Finance, Actuarial Studies and Applied StatisticsAustralian National UniversityCanberraAustralia

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