Annals of Statistics

Functional data analysis for density functions by transformation to a Hilbert space

Alexander Petersen and Hans-Georg Müller

Full-text: Open access

Enhanced PDF (626 KB)

Abstract

Functional data that are nonnegative and have a constrained integral can be considered as samples of one-dimensional density functions. Such data are ubiquitous. Due to the inherent constraints, densities do not live in a vector space and, therefore, commonly used Hilbert space based methods of functional data analysis are not applicable. To address this problem, we introduce a transformation approach, mapping probability densities to a Hilbert space of functions through a continuous and invertible map. Basic methods of functional data analysis, such as the construction of functional modes of variation, functional regression or classification, are then implemented by using representations of the densities in this linear space. Representations of the densities themselves are obtained by applying the inverse map from the linear functional space to the density space. Transformations of interest include log quantile density and log hazard transformations, among others. Rates of convergence are derived for the representations that are obtained for a general class of transformations under certain structural properties. If the subject-specific densities need to be estimated from data, these rates correspond to the optimal rates of convergence for density estimation. The proposed methods are illustrated through simulations and applications in brain imaging.

Article information

Source
Ann. Statist., Volume 44, Number 1 (2016), 183-218.

Dates
Received: December 2014
Revised: July 2015
First available in Project Euclid: 10 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1449755961

Digital Object Identifier
doi:10.1214/15-AOS1363

Mathematical Reviews number (MathSciNet)
MR3449766

Zentralblatt MATH identifier
1331.62203

Subjects
Primary: 62G05: Estimation
Secondary: 62G07: Density estimation 62G20: Asymptotic properties

Keywords
Basis representation kernel estimation log hazard prediction quantiles samples of density functions rate of convergence Wasserstein metric

Citation

Petersen, Alexander; Müller, Hans-Georg. Functional data analysis for density functions by transformation to a Hilbert space. Ann. Statist. 44 (2016), no. 1, 183--218. doi:10.1214/15-AOS1363. https://projecteuclid.org/euclid.aos/1449755961


Export citation

References

  • [1] Achard, S., Salvador, R., Whitcher, B., Suckling, J. and Bullmore, E. (2006). A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs. J. Neurosci. 26 63–72.
  • [2] Allen, E. A., Damaraju, E., Plis, S. M., Erhardt, E. B., Eichele, T. and Calhoun, V. D. (2012). Tracking whole-brain connectivity dynamics in the resting state. Cerebral Cortex bhs352.
  • [3] Ash, R. B. and Gardner, M. F. (1975). Topics in Stochastic Processes. Academic Press, New York.
    Mathematical Reviews (MathSciNet): MR448463
    Zentralblatt MATH: 0317.60014
  • [4] Bali, J. L., Boente, G., Tyler, D. E. and Wang, J.-L. (2011). Robust functional principal components: A projection-pursuit approach. Ann. Statist. 39 2852–2882.
    Mathematical Reviews (MathSciNet): MR3012394
    Zentralblatt MATH: 1246.62145
    Digital Object Identifier: doi:10.1214/11-AOS923
    Project Euclid: euclid.aos/1327413771
  • [5] Bassett, D. S. and Bullmore, E. (2006). Small-world brain networks. Neuroscientist 12 512–523.
  • [6] Benko, M., Härdle, W. and Kneip, A. (2009). Common functional principal components. Ann. Statist. 37 1–34.
    Mathematical Reviews (MathSciNet): MR2488343
    Zentralblatt MATH: 1169.62057
    Digital Object Identifier: doi:10.1214/07-AOS516
    Project Euclid: euclid.aos/1232115926
  • [7] Besse, P. and Ramsay, J. O. (1986). Principal components analysis of sampled functions. Psychometrika 51 285–311.
    Mathematical Reviews (MathSciNet): MR848110
    Digital Object Identifier: doi:10.1007/BF02293986
  • [8] Bolstad, B. M., Irizarry, R. A., Åstrand, M. and Speed, T. P. (2003). A comparison of normalization methods for high density oligonucleotide array data based on variance and bias. Bioinformatics 19 185–193.
  • [9] Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Applications. Lecture Notes in Statistics 149. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1783138
    Zentralblatt MATH: 0962.60004
  • [10] Bouezmarni, T. and Rolin, J.-M. (2003). Consistency of the beta kernel density function estimator. Canad. J. Statist. 31 89–98.
    Mathematical Reviews (MathSciNet): MR1985506
    Digital Object Identifier: doi:10.2307/3315905
  • [11] Buckner, R. L., Sepulcre, J., Talukdar, T., Krienen, F. M., Liu, H., Hedden, T., Andrews-Hanna, J. R., Sperling, R. A. and Johnson, K. A. (2009). Cortical hubs revealed by intrinsic functional connectivity: Mapping, assessment of stability, and relation to Alzheimer’s disease. J. Neurosci. 29 1860–1873.
  • [12] Cai, T. T. and Hall, P. (2006). Prediction in functional linear regression. Ann. Statist. 34 2159–2179.
    Mathematical Reviews (MathSciNet): MR2291496
    Zentralblatt MATH: 1106.62036
    Digital Object Identifier: doi:10.1214/009053606000000830
    Project Euclid: euclid.aos/1169571793
  • [13] Castro, P. E., Lawton, W. H. and Sylvestre, E. A. (1986). Principal modes of variation for processes with continuous sample curves. Technometrics 28 329–337.
  • [14] Chen, S. X. (1999). Beta kernel estimators for density functions. Comput. Statist. Data Anal. 31 131–145.
    Mathematical Reviews (MathSciNet): MR1718494
  • [15] Cowling, A. and Hall, P. (1996). On pseudodata methods for removing boundary effects in kernel density estimation. J. R. Stat. Soc. Ser. B. Stat. Methodol. 58 551–563.
    Mathematical Reviews (MathSciNet): MR1394366
    Zentralblatt MATH: 06166829
  • [16] Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136–154.
    Mathematical Reviews (MathSciNet): MR650934
    Zentralblatt MATH: 0539.62064
    Digital Object Identifier: doi:10.1016/0047-259X(82)90088-4
  • [17] Delicado, P. (2011). Dimensionality reduction when data are density functions. Comput. Statist. Data Anal. 55 401–420.
    Mathematical Reviews (MathSciNet): MR2736564
  • [18] Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The $L_{1}$ View. Wiley, New York.
  • [19] Dubin, J. A. and Müller, H.-G. (2005). Dynamical correlation for multivariate longitudinal data. J. Amer. Statist. Assoc. 100 872–881.
    Mathematical Reviews (MathSciNet): MR2201015
    Zentralblatt MATH: 1117.62320
    Digital Object Identifier: doi:10.1198/016214504000001989
  • [20] Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Statist. 27 642–669.
    Mathematical Reviews (MathSciNet): MR83864
    Zentralblatt MATH: 0073.14603
    Digital Object Identifier: doi:10.1214/aoms/1177728174
    Project Euclid: euclid.aoms/1177728174
  • [21] Egozcue, J. J., Diaz-Barrero, J. L. and Pawlowsky-Glahn, V. (2006). Hilbert space of probability density functions based on Aitchison geometry. Acta Math. Sin. (Engl. Ser.) 22 1175–1182.
    Mathematical Reviews (MathSciNet): MR2245249
    Zentralblatt MATH: 1113.46016
    Digital Object Identifier: doi:10.1007/s10114-005-0678-2
  • [22] Ferreira, L. K. and Busatto, G. F. (2013). Resting-state functional connectivity in normal brain aging. Neuroscience & Biobehavioral Reviews 37 384–400.
  • [23] Fletcher, P. T., Lu, C., Pizer, S. M. and Joshi, S. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions on Medical Imaging 23 995–1005.
  • [24] Gajek, L. (1986). On improving density estimators which are not Bona fide functions. Ann. Statist. 14 1612–1618.
    Mathematical Reviews (MathSciNet): MR868324
    Zentralblatt MATH: 0623.62034
    Digital Object Identifier: doi:10.1214/aos/1176350182
    Project Euclid: euclid.aos/1176350182
  • [25] Hall, P. and Horowitz, J. L. (2007). Methodology and convergence rates for functional linear regression. Ann. Statist. 35 70–91.
    Mathematical Reviews (MathSciNet): MR2332269
    Zentralblatt MATH: 1114.62048
    Digital Object Identifier: doi:10.1214/009053606000000957
    Project Euclid: euclid.aos/1181100181
  • [26] Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 109–126.
    Mathematical Reviews (MathSciNet): MR2212577
    Zentralblatt MATH: 1141.62048
    Digital Object Identifier: doi:10.1111/j.1467-9868.2005.00535.x
  • [27] Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493–1517.
    Mathematical Reviews (MathSciNet): MR2278365
    Zentralblatt MATH: 1113.62073
    Digital Object Identifier: doi:10.1214/009053606000000272
    Project Euclid: euclid.aos/1152540756
  • [28] Höffding, W. (1940). Maszstabinvariante Korrelationstheorie. Schr. Math. Inst. U. Inst. Angew. Math. Univ. Berlin 5 181–233.
    Mathematical Reviews (MathSciNet): MR4426
  • [29] Hron, K., Menafoglio, A., Templ, M., Hruzova, K. and Filzmoser, P. (2014). Simplicial principal component analysis for density functions in Bayes spaces. MOX-report 25 2014.
  • [30] Jones, M. C. (1992). Estimating densities, quantiles, quantile densities and density quantiles. Ann. Inst. Statist. Math. 44 721–727.
  • [31] Jones, M. C. and Rice, J. A. (1992). Displaying the important features of large collections of similar curves. Amer. Statist. 46 140–145.
    Mathematical Reviews (MathSciNet): MR1199044
    Zentralblatt MATH: 0763.47002
    Digital Object Identifier: doi:10.1080/03081089208818125
  • [32] Kneip, A. and Utikal, K. J. (2001). Inference for density families using functional principal component analysis. J. Amer. Statist. Assoc. 96 519–542.
    Mathematical Reviews (MathSciNet): MR1946423
    Zentralblatt MATH: 1019.62060
    Digital Object Identifier: doi:10.1198/016214501753168235
  • [33] Li, Y. and Hsing, T. (2010). Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Statist. 38 3321–3351.
    Mathematical Reviews (MathSciNet): MR2766854
    Digital Object Identifier: doi:10.1214/10-AOS813
    Project Euclid: euclid.aos/1284988408
  • [34] Mallows, C. L. (1972). A note on asymptotic joint normality. Ann. Math. Statist. 43 508–515.
    Mathematical Reviews (MathSciNet): MR298812
    Zentralblatt MATH: 0238.60017
    Digital Object Identifier: doi:10.1214/aoms/1177692631
    Project Euclid: euclid.aoms/1177692631
  • [35] Müller, H. G. and Stadtmüller, U. (1999). Multivariate boundary kernels and a continuous least squares principle. J. R. Stat. Soc. Ser. B. Stat. Methodol. 61 439–458.
    Mathematical Reviews (MathSciNet): MR1680306
    Zentralblatt MATH: 0927.62035
    Digital Object Identifier: doi:10.1111/1467-9868.00186
  • [36] Müller, H.-G. and Yao, F. (2008). Functional additive models. J. Amer. Statist. Assoc. 103 1534–1544.
    Mathematical Reviews (MathSciNet): MR2504202
    Zentralblatt MATH: 1286.62040
    Digital Object Identifier: doi:10.1198/016214508000000751
  • [37] Parzen, E. (1979). Nonparametric statistical modeling. J. Amer. Statist. Assoc. 74 105–121.
    Mathematical Reviews (MathSciNet): MR529528
    Zentralblatt MATH: 0407.62001
    Digital Object Identifier: doi:10.1080/01621459.1979.10481621
  • [38] Petersen, A. and Müller, H.-G. (2015). Supplement to “Functional data analysis for density functions by transformation to a Hilbert space.” DOI:10.1214/15-AOS1363SUPP.
  • [39] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2168993
  • [40] Rosenblatt, M. (1952). Remarks on a multivariate transformation. Ann. Math. Statist. 23 470–472.
    Mathematical Reviews (MathSciNet): MR49525
    Zentralblatt MATH: 0047.13104
    Digital Object Identifier: doi:10.1214/aoms/1177729394
    Project Euclid: euclid.aoms/1177729394
  • [41] Sheline, Y. I. and Raichle, M. E. (2013). Resting state functional connectivity in preclinical Alzheimer’s disease. Biol. Psychiatry 74 340–347.
  • [42] Srivastava, A., Jermyn, I. and Joshi, S. (2007). Riemannian analysis of probability density functions with applications in vision. Proceedings from IEEE Conference on Computer Vision and Pattern Recognition 25 1–8.
  • [43] Srivastava, A., Klassen, E., Joshi, S. H. and Jermyn, I. H. (2011a). Shape analysis of elastic curves in Euclidean spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 33 1415–1428.
  • [44] Srivastava, A., Wu, W., Kurtek, S., Klassen, E. and Marron, J. S. (2011b). Registration of functional data using Fisher–Rao metric. Available at arXiv:1103.3817v2 [math.ST].
    arXiv: 1103.3817v2
  • [45] Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2724359
  • [46] Tukey, J. W. (1965). Which part of the sample contains the information? Proc. Natl. Acad. Sci. USA 53 127–134.
    Mathematical Reviews (MathSciNet): MR172387
    Digital Object Identifier: doi:10.1073/pnas.53.1.127
  • [47] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.
    Mathematical Reviews (MathSciNet): MR1964483
    Zentralblatt MATH: 1106.90001
  • [48] Wand, M. P., Marron, J. S. and Ruppert, D. (1991). Transformations in density estimation. J. Amer. Statist. Assoc. 86 343–361.
    Mathematical Reviews (MathSciNet): MR1137118
    Zentralblatt MATH: 0742.62046
    Digital Object Identifier: doi:10.1080/01621459.1991.10475041
  • [49] Worsley, K. J., Chen, J.-I., Lerch, J. and Evans, A. C. (2005). Comparing functional connectivity via thresholding correlations and singular value decomposition. Philosophical Transactions of the Royal Society B: Biological Sciences 360 913–920.
  • [50] Zhang, Z. and Müller, H.-G. (2011). Functional density synchronization. Comput. Statist. Data Anal. 55 2234–2249.
    Mathematical Reviews (MathSciNet): MR2786984

Supplemental materials

  • The Wasserstein metric, Wasserstein–Fréchet mean, simulation results and additional proofs. The supplementary material includes additional discussion on the Wasserstein distance and the rate of convergence of the Wasserstein–Fréchet mean is derived. Additional simulation results are presented for FVE values using the Wasserstein metric, similar to the boxplots in Figure 2, which correspond to FVE values using the $L^{2}$ metric. All assumptions are listed in one place. Lastly, additional proofs of auxiliary results are provided.
    Digital Object Identifier: doi:10.1214/15-AOS1363SUPP
    Supplemental PDF (233 KB)

The Institute of Mathematical Statistics

Annals of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more