Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Liu, Chong; Ray, Surajit; Hooker, Giles (2017)
Publisher: Springer
Languages: English
Types: Article
Subjects: HA
This paper focuses on the analysis of spatially correlated functional data. We propose a parametric model for spatial correlation and the between-curve correlation is modeled by correlating functional principal component scores of the functional data. Additionally, in the sparse observation framework, we propose a novel approach of spatial principal analysis by conditional expectation to explicitly estimate spatial correlations and reconstruct individual curves. Assuming spatial stationarity, empirical spatial correlations are calculated as the ratio of eigenvalues of the smoothed covariance surface Cov (Xi(s),Xi(t))(Xi(s),Xi(t)) and cross-covariance surface Cov (Xi(s),Xj(t))(Xi(s),Xj(t)) at locations indexed by i and j. Then a anisotropy Matérn spatial correlation model is fitted to empirical correlations. Finally, principal component scores are estimated to reconstruct the sparsely observed curves. This framework can naturally accommodate arbitrary covariance structures, but there is an enormous reduction in computation if one can assume the separability of temporal and spatial components. We demonstrate the consistency of our estimates and propose hypothesis tests to examine the separability as well as the isotropy effect of spatial correlation. Using simulation studies, we show that these methods have some clear advantages over existing methods of curve reconstruction and estimation of model parameters.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1970)
    • Abramowitz, M., Stegun, I.A., et al.: Handbook of mathematical functions. Appl. Math. Ser. 55, 62 (1966)
    • Aston, J.A., Pigoli, D., Tavakoli, S.: Tests for separability in nonparametric covariance operators of random surfaces. arXiv preprint arXiv:1505.02023 (2015)
    • Avriel, M.: Nonlinear Programming: analysis and Methods. Courier Corporation, New York (2003)
    • Baladandayuthapani, V., Mallick, B.K.: Young Hong, M., Lupton, J.R., Turner, N.D., Carroll, R.J.: Bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis. Biometrics 64(1), 64-73 (2008)
    • Banerjee, S., Johnson, G.A.: Coregionalized single-and multiresolution spatially varying growth curve modeling with application to weed growth. Biometrics 62(3), 864-876 (2006)
    • Bowman, A. W. and Azzalini, A.: R package sm: nonparametric smoothing methods (version 2.2-5), University of Glasgow, UK and Universita` di Padova, Italia. http://www.stats.gla.ac.uk/ adrian/sm, http://azzalini.stat.unipd.it/Book_sm (2013)
    • Cressie, N.: Statistics for Spatial Data. Wiley, New York (2015)
    • Diggle, P.J., Ribeiro Jr., P.J., Christensen, O.F.: An Introduction to Model-Based Geostatistics, Spatial Statistics and Computational Methods, pp. 43-86. Springer, New York (2003)
    • Gromenko, O., Kokoszka, P., Zhu, L., Sojka, J.: Estimation and testing for spatially indexed curves with application to ionospheric and magnetic field trends. Ann. Appl. Stat. 6(2), 669-696 (2012). doi:10.1214/11-AOAS524
    • Hall, P., Müller, H., Wang, J.: Properties of principal component methods for functional and longitudinal data analysis. Ann. Stat. 34(3), 1493-1517 (2006)
    • Hurvich, C.M., Simonoff, J.S., Tsai, C.-L.: Smoothing parameter selection in nonparametric regression using an improved akaike information criterion. J. R. Stat. Soc. 60(2), 271-293 (1998)
    • James, G., Sugar, C.: Clustering for sparsely sampled functional data. J. Am. Stat. Assoc. 98(462), 397-408 (2003)
    • Li, P.-L., Chiou, J.-M.: Identifying cluster number for subspace projected functional data clustering. Comput. Stat. Data Anal. 55(6), 2090-2103 (2011)
    • Li, Y., Hsing, T.: Uniform convergence rates for nonparametric regression and principal component analysis in functional and longitudinal data. Ann. Stat. 38(6), 3321-3351 (2010)
    • Li, Y., Wang, N., Hong, M., Turner, N.D., Lupton, J.R., Carroll, R.J.: Nonparametric estimation of correlation functions in longitudinal and spatial data, with application to colon carcinogenesis experiments. Ann. Stat. 35(4), 1608-1643 (2007)
    • Liu, C., Ray, S., Hooker, G., Friedl, M.: Functional factor analysis for periodic remote sensing data. Ann. Appl. Stat. 6(2), 601-624 (2012)
    • Mateu, J.: Spatially correlated functional data, in 'Spatial2 conference: spatial data methods for environmental and ecological processes, Foggia (IT), 1-2 September 2011', IT (2011)
    • Menafoglio, A., Petris, G.: Kriging for hilbert-space valued random fields: the operatorial point of view. J. Multivar. Anal. 146, 84-94 (2016). doi:10.1016/j.jmva.2015.06.012
    • 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)
    • 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(4), 995-1015 (2009)
    • R Development Core Team.: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2010)
    • Ramsay, J.O., Silverman, B.W.: Functional Data Analysis, 2nd edn. Springer, New York (2005)
    • Rice, J., Silverman, B.: Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. Ser. B (Methodol.) 53(1), 233-243 (1991)
    • Ruppert, D., Carroll, R.J., Gill, R., Ripley, B.D., Ross, S., Stein, M., Wand, M.P., Williams, D.: Semiparametric Regression. Cambridge University Press, Cambridge. https://cds.cern.ch/record/997686 (2003)
    • Staicu, A., Crainiceanu, C.M., Reich, D.S., Ruppert, D.: Modeling functional data with spatially heterogeneous shape characteristics. Biometrics 68(2), 331-343 (2012)
    • Staicu, A.M., Crainiceanu, C.M. and Carroll, R.J. (2010) Fast methods for spatially correlated multilevel functional data. Biostatistics 11(2), 177-194. http://www.ncbi.nlm.nih.gov/pmc/articles/ PMC2830578/pdf/kxp058.pdf
    • Yao, F., Lee, T.: Penalized spline models for functional principal component analysis. J. R. Stat. Soc. Ser. B 68(1), 3 (2006)
    • Yao, F., Müller, H., Clifford, A., Dueker, S., Follett, J., Lin, Y., Buchholz, B., Vogel, J.: Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate. Biometrics 59(3), 676-685 (2003)
    • Yao, F., Müller, H., Wang, J.: Functional data analysis for sparse longitudinal data. J. Am. Stat. Assoc. 100(470), 577-590 (2005)
    • Zhou, L., Huang, J.Z., Martinez, J.G., Maity, A., Baladandayuthapani, V., Carroll, R.J.: Reduced rank mixed effects models for spatially correlated hierarchical functional data. J. Am. Stat. Assoc 105(489), 390-400. http://EconPapers.repec.org/RePEc: bes:jnlasa:v:105:i:489:y:2010:p:390-400 (2010)
  • No related research data.
  • No similar publications.

Share - Bookmark

Download from

Funded by projects

  • NSF | CMG: Functional Data Modeli...

Cite this article