1932

Abstract

Modern studies from a variety of fields record multiple functional observations according to either multivariate, longitudinal, spatial, or time series designs. We refer to such data as second-generation functional data because their analysis—unlike typical functional data analysis, which assumes independence of the functions—accounts for the complex dependence between the functional observations and requires more advanced methods. In this article, we provide an overview of the techniques for analyzing second-generation functional data with a focus on highlighting the key methodological intricacies that stem from the need for modeling complex dependence, compared with independent functional data. For each of the four types of second-generation functional data presented—multivariate functional data, longitudinal functional data, functional time series and spatially functional data—we discuss how the widely popular functional principal component analysis can be extended to these settings to define, identify main directions of variation, and describe dependence among the functions. In addition to modeling, we also discuss prediction, statistical inference, and application to clustering. We close by discussing future directions in this area.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-statistics-032921-033726
2023-03-10
2024-06-20
Loading full text...

Full text loading...

/deliver/fulltext/statistics/10/1/annurev-statistics-032921-033726.html?itemId=/content/journals/10.1146/annurev-statistics-032921-033726&mimeType=html&fmt=ahah

Literature Cited

  1. Antoniadis A, Paparoditis E, Sapatinas T. 2006. A functional wavelet–kernel approach for time series prediction. J. R. Stat. Soc. Ser. B 68:5837–57
    [Google Scholar]
  2. Arnone E, Azzimonti L, Nobile F, Sangalli LM. 2019. Modeling spatially dependent functional data via regression with differential regularization. J. Multivar. Anal. 170:275–95
    [Google Scholar]
  3. Arnone E, Sangalli LM, Lila E, Ramsay J, Formaggia L 2022. fdapde: Functional data analysis and partial differential equations (PDE); statistical analysis of functional and spatial data, based on regression with PDE regularization. R Package version 1.1-8. https://CRAN.R-project.org/package=fdaPDE
    [Google Scholar]
  4. Aston JA, Kirch C. 2012. Evaluating stationarity via change-point alternatives with applications to fMRI data. Ann. Appl. Stat. 6:41906–48
    [Google Scholar]
  5. Aue A, Horváth L, Pellatt DF. 2017. Functional generalized autoregressive conditional heteroskedasticity. J. Time Ser. Anal. 38:13–21
    [Google Scholar]
  6. Aue A, Norinho DD, Hörmann S. 2015. On the prediction of stationary functional time series. J. Am. Stat. Assoc. 110:509378–92
    [Google Scholar]
  7. Aue A, Rice G, Sönmez O. 2018. Detecting and dating structural breaks in functional data without dimension reduction. J. R. Stat. Soc. Ser. B 80:3509–29
    [Google Scholar]
  8. Aue A, Van Delft A. 2020. Testing for stationarity of functional time series in the frequency domain. Ann. Stat. 48:52505–47
    [Google Scholar]
  9. Berkes I, Gabrys R, Horváth L, Kokoszka P. 2009. Detecting changes in the mean of functional observations. J. R. Stat. Soc. Ser. B 71:5927–46
    [Google Scholar]
  10. Bernardi MS, Sangalli LM, Mazza G, Ramsay JO. 2017. A penalized regression model for spatial functional data with application to the analysis of the production of waste in Venice province. Stochastic Environ. Res. Risk Assess. 31:123–38
    [Google Scholar]
  11. Berrendero JR, Justel A, Svarc M. 2011. Principal components for multivariate functional data. Comput. Stat. Data Anal. 55:92619–34
    [Google Scholar]
  12. Besse PC, Cardot H. 1996. Approximation spline de la prévision d'un processus fonctionnel autorégressif d'ordre 1. Can. J. Stat. 24:4467–87
    [Google Scholar]
  13. Besse PC, Cardot H, Stephenson DB. 2000. Autoregressive forecasting of some functional climatic variations. Scand. J. Stat. 27:4673–87
    [Google Scholar]
  14. Biernacki C, Celeux G, Govaert G. 2000. Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Trans. Pattern Anal. Mach. Intel. 22:7719–25
    [Google Scholar]
  15. Bohorquez M, Giraldo R, Mateu J 2016. Optimal sampling for spatial prediction of functional data. Stat. Methods Appl. 25:139–54
    [Google Scholar]
  16. Bosq D. 1991. Modelization, nonparametric estimation and prediction for continuous time processes. Nonparametric Functional Estimation and Related Topics G Roussas 509–29 New York: Springer
    [Google Scholar]
  17. Bosq D. 2000. Linear Processes in Function Spaces: Theory and Applications New York: Springer
    [Google Scholar]
  18. Bosq D. 2014. Computing the best linear predictor in a Hilbert space. Applications to general ARMAH processes. J. Multivar. Anal. 124:436–50
    [Google Scholar]
  19. Brockwell PJ, Davis RA. 1991. Time Series: Theory and Methods New York: Springer. , 2nd ed..
    [Google Scholar]
  20. Caballero W, Giraldo R, Mateu J 2013. A universal kriging approach for spatial functional data. Stochastic Environ. Res. Risk Assess. 27:71553–63
    [Google Scholar]
  21. Cerovecki C, Francq C, Hörmann S, Zakoian JM. 2019. Functional garch models: the quasi-likelihood approach and its applications. J. Econom. 209:2353–75
    [Google Scholar]
  22. Chen K, Delicado P, Müller HG. 2017. Modelling function-valued stochastic processes, with applications to fertility dynamics. J. R. Stat. Soc. Ser. B 79:1177–96
    [Google Scholar]
  23. Chen K, Müller HG. 2012. Modeling repeated functional observations. J. Am. Stat. Assoc. 107:5001599–609
    [Google Scholar]
  24. Chen W, Genton MG, Sun Y. 2021. Space-time covariance structures and models. Annu. Rev. Stat. Appl. 8:191–215
    [Google Scholar]
  25. Chiou JM. 2012. Dynamical functional prediction and classification, with application to traffic flow prediction. Ann. Appl. Stat. 6:41588–614
    [Google Scholar]
  26. Chiou JM, Chen YT, Yang YF. 2014. Multivariate functional principal component analysis: a normalization approach. Stat. Sin. 24:1571–96
    [Google Scholar]
  27. Chiou JM, Li PL. 2007. Functional clustering and identifying substructures of longitudinal data. J. R. Stat. Soc. Ser. B 69:4679–99
    [Google Scholar]
  28. Chiou JM, Müller HG. 2014. Linear manifold modelling of multivariate functional data. J. R. Stat. Soc. Ser. B 76:3605–26
    [Google Scholar]
  29. Chiou JM, Müller HG. 2016. A pairwise interaction model for multivariate functional and longitudinal data. Biometrika 103:2377–96
    [Google Scholar]
  30. Claeskens G, Hubert M, Slaets L, Vakili K. 2014. Multivariate functional halfspace depth. J. Am. Stat. Assoc. 109:505411–23
    [Google Scholar]
  31. Crainiceanu CM, Staicu AM, Ray S, Punjabi N. 2012. Bootstrap-based inference on the difference in the means of two correlated functional processes. Stat. Med. 31:263223–40
    [Google Scholar]
  32. Cressie N. 2015. Statistics for Spatial Data New York: Wiley
    [Google Scholar]
  33. Dai W, Genton MG. 2018a. An outlyingness matrix for multivariate functional data classification. Stat. Sin. 28:42435–54
    [Google Scholar]
  34. Dai W, Genton MG. 2018b. Functional boxplots for multivariate curves. Stat 7:1e190
    [Google Scholar]
  35. Dai W, Genton MG. 2019. Directional outlyingness for multivariate functional data. Comput. Stat. Data Anal. 131:50–65
    [Google Scholar]
  36. Damon J, Guillas S. 2002. The inclusion of exogenous variables in functional autoregressive ozone forecasting. Environmetrics 13:7759–74
    [Google Scholar]
  37. Delicado P, Giraldo R, Comas C, Mateu J. 2010. Statistics for spatial functional data: some recent contributions. Environmetrics 21:3-4224–39
    [Google Scholar]
  38. Didericksen D, Kokoszka P, Zhang X. 2012. Empirical properties of forecasts with the functional autoregressive model. Comput. Stat. 27:2285–98
    [Google Scholar]
  39. Ferraty F, Vieu P. 2006. Nonparametric Functional Data Analysis: Theory and Practice New York: Springer
    [Google Scholar]
  40. Giraldo R, Delicado P, Mateu J. 2010. Continuous time-varying kriging for spatial prediction of functional data: an environmental application. J. Agric. Biol. Environ. Stat. 15:166–82
    [Google Scholar]
  41. Giraldo R, Delicado P, Mateu J. 2011. Ordinary kriging for function-valued spatial data. Environ. Ecol. Stat. 18:3411–26
    [Google Scholar]
  42. Giraldo R, Mateu J, Delicado P. 2012. geofd: An R package for function-valued geostatistical prediction. Rev. Colomb. Estad. 35:3385–407
    [Google Scholar]
  43. Goldsmith J, Scheipl F, Huang L, Wrobel J, Di C et al. 2021. refund: Regression with functional data. R Package version 0.1-24. https://CRAN.R-project.org/package=refund
    [Google Scholar]
  44. Goulard M, Voltz M. 1993. Geostatistical interpolation of curves: a case study in soil science. Geostatistics Tróia '92 A Soares 805–16 New York: Springer
    [Google Scholar]
  45. Greven S, Crainiceanu C, Caffo B, Reich D 2011. Longitudinal functional principal component analysis. Recent Advances in Functional Data Analysis and Related Topics F Ferraty 149–54 New York: Springer
    [Google Scholar]
  46. Gromenko O, Kokoszka P, Zhu L, Sojka J. 2012. Estimation and testing for spatially indexed curves with application to ionospheric and magnetic field trends. Ann. Appl. Stat. 6:669–96
    [Google Scholar]
  47. Guo W. 2002. Functional mixed effects models. Biometrics 58:1121–28
    [Google Scholar]
  48. Guo W. 2004. Functional data analysis in longitudinal settings using smoothing splines. Stat. Methods Med. Res. 13:149–62
    [Google Scholar]
  49. Han K, Hadjipantelis PZ, Wang JL, Kramer MS, Yang S et al. 2018. Functional principal component analysis for identifying multivariate patterns and archetypes of growth, and their association with long-term cognitive development. PLOS ONE 13:11e0207073
    [Google Scholar]
  50. Happ C, Greven S. 2018. Multivariate functional principal component analysis for data observed on different (dimensional) domains. J. Am. Stat. Assoc. 113:649–59
    [Google Scholar]
  51. Happ-Kurz C. 2021. mfpca: Multivariate functional principal component analysis for data observed on different dimensional domains. R Package version 1.3-9. https://CRAN.R-project.org/package=MFPCA
    [Google Scholar]
  52. Holmes EE, Ward EJ, Wills K 2012. MARSS: multivariate autoregressive state-space models for analyzing time-series data. R J. 4:111–19
    [Google Scholar]
  53. Hörmann S, Horváth L, Reeder R. 2013. A functional version of the arch model. Econom. Theory 29:2267–88
    [Google Scholar]
  54. Hörmann S, Kidziński Ł, Hallin M. 2015. Dynamic functional principal components. J. R. Stat. Soc. Ser. B 77:2319–48
    [Google Scholar]
  55. Hörmann S, Kokoszka P. 2010. Weakly dependent functional data. Ann. Stat. 38:31845–84
    [Google Scholar]
  56. Hörmann S, Kokoszka P 2011. Consistency of the mean and the principal components of spatially distributed functional data. Recent Advances in Functional Data Analysis and Related Topics F Ferraty 169–75 New York: Springer
    [Google Scholar]
  57. Horváth L, Hušková M, Kokoszka P. 2010. Testing the stability of the functional autoregressive process. J. Multivar. Anal. 101:2352–67
    [Google Scholar]
  58. Horváth L, Kokoszka P. 2012. Inference for Functional Data with Applications New York: Springer
    [Google Scholar]
  59. Horváth L, Kokoszka P, Rice G. 2014. Testing stationarity of functional time series. J. Econom. 179:166–82
    [Google Scholar]
  60. Horváth L, Rice G, Whipple S. 2016. Adaptive bandwidth selection in the long run covariance estimator of functional time series. Comput. Stat. Data Anal. 100:676–93
    [Google Scholar]
  61. Huang L, Reiss PT, Xiao L, Zipunnikov V, Lindquist MA, Crainiceanu CM. 2017. Two-way principal component analysis for matrix-variate data, with an application to functional magnetic resonance imaging data. Biostatistics 18:2214–29
    [Google Scholar]
  62. Hubert M, Rousseeuw PJ, Segaert P. 2015. Multivariate functional outlier detection. Stat. Methods Appl. 24:2177–202
    [Google Scholar]
  63. Hyndman RJ, Shang HL. 2021. ftsa: Functional time series analysis. R Package version 6.1. https://CRAN.R-project.org/package=ftsa
    [Google Scholar]
  64. Hyndman RJ, Ullah MS. 2007. Robust forecasting of mortality and fertility rates: a functional data approach. Comput. Stat. Data Anal. 51:104942–56
    [Google Scholar]
  65. Ieva F, Paganoni AM. 2013. Depth measures for multivariate functional data. Commun. Stat. Theory Methods 42:71265–76
    [Google Scholar]
  66. Ieva F, Paganoni AM, Pigoli D, Vitelli V. 2013. Multivariate functional clustering for the morphological analysis of electrocardiograph curves. J. R. Stat. Soc. Ser. C 62:3401–18
    [Google Scholar]
  67. Jacques J, Preda C. 2014. Model-based clustering for multivariate functional data. Comput. Stat. Data Anal. 71:92–106
    [Google Scholar]
  68. Julien D, Serge G 2010. far: Modelization for functional autoregressive processes. R Package version 0.6-3. https://CRAN.R-project.org/package=far
    [Google Scholar]
  69. Kayano M, Dozono K, Konishi S. 2010. Functional cluster analysis via orthonormalized Gaussian basis expansions and its application. J. Classif. 27:2211–30
    [Google Scholar]
  70. King MC, Staicu AM, Davis JM, Reich BJ, Eder B. 2018. A functional data analysis of spatiotemporal trends and variation in fine particulate matter. Atmos. Environ. 184:233–43
    [Google Scholar]
  71. Klepsch J, Klüppelberg C, Wei T. 2017. Prediction of functional ARMA processes with an application to traffic data. Econom. Stat. 1:128–49
    [Google Scholar]
  72. Kokoszka P, Reimherr M. 2013a. Asymptotic normality of the principal components of functional time series. Stochastic Proc. Appl. 123:51546–62
    [Google Scholar]
  73. Kokoszka P, Reimherr M. 2013b. Determining the order of the functional autoregressive model. J. Time Ser. Anal. 34:1116–29
    [Google Scholar]
  74. Koner S, Park SY, Staicu AM. 2021. PROFIT: projection-based test in longitudinal functional data. arXiv:2104.11355 [stat.ME]
  75. Koner S, Staicu AM, Maity A. 2022. PROLIFIC: projection-based test for lack of importance of smooth functional effect in crossover design. arXiv:2205.08577 [stat.ME]
  76. Kowal DR, Matteson DS, Ruppert D. 2019. Functional autoregression for sparsely sampled data. J. Bus. Econ. Stat. 37:197–109
    [Google Scholar]
  77. Li C, Xiao L 2021. mfaces: Fast covariance estimation for multivariate sparse functional data. R Package version 0.1-3. https://CRAN.R-project.org/package=mfaces
    [Google Scholar]
  78. Li C, Xiao L, Luo S 2020. Fast covariance estimation for multivariate sparse functional data. Stat 9:1e245
    [Google Scholar]
  79. Liu C, Ray S, Hooker G. 2017. Functional principal component analysis of spatially correlated data. Stat. Comput. 27:61639–54
    [Google Scholar]
  80. Liu X, Ma S, Chen K 2022. Multivariate functional regression via nested reduced-rank regularization. J. Comput. Graph. Stat. 31:1231–40
    [Google Scholar]
  81. Martínez-Hernández I, Genton MG 2020. Recent developments in complex and spatially correlated functional data. Braz. J. Probab. Stat. 34:2204–29
    [Google Scholar]
  82. Martínez-Hernández I, Genton MG, González-Farías G. 2019. Robust depth-based estimation of the functional autoregressive model. Comput. Stat. Data Anal. 131:66–79
    [Google Scholar]
  83. Mas A. 2002. Weak convergence for the covariance operators of a Hilbertian linear process. Stochastic Proc. Appl. 99:1117–35
    [Google Scholar]
  84. Mateu J, Romano E. 2017. Advances in spatial functional statistics. Stochastic Environ. Res. Risk Assess. 31:1–6
    [Google Scholar]
  85. Menafoglio A, Secchi P, Dalla Rosa M 2013. A universal kriging predictor for spatially dependent functional data of a Hilbert space. Electron. J. Stat. 7:2209–40
    [Google Scholar]
  86. Monestiez P, Nerini D 2008. A cokriging method for spatial functional data with applications in oceanology. Functional and Operatorial Statistics S Dabo-Niang, F Ferraty 237–42 New York: Springer
    [Google Scholar]
  87. Morris JS. 2015. Functional regression. Annu. Rev. Stat. Appl. 2:321–59
    [Google Scholar]
  88. Morris JS, Baladandayuthapani V, Herrick RC, Sanna P, Gutstein H. 2011. Automated analysis of quantitative image data using isomorphic functional mixed models, with application to proteomics data. Ann. Appl. Stat. 5:2A894
    [Google Scholar]
  89. Morris JS, Carroll RJ. 2006. Wavelet-based functional mixed models. J. R. Stat. Soc. Ser. B 68:2179–99
    [Google Scholar]
  90. Panaretos VM, Tavakoli S. 2013. Fourier analysis of stationary time series in function space. Ann. Stat. 41:2568–603
    [Google Scholar]
  91. Park J, Ahn J. 2017. Clustering multivariate functional data with phase variation. Biometrics 73:1324–33
    [Google Scholar]
  92. Park SY, Staicu AM 2015. Longitudinal functional data analysis. Stat 4:1212–26
    [Google Scholar]
  93. Park SY, Staicu AM, Xiao L, Crainiceanu CM. 2017. Simple fixed-effects inference for complex functional models. Biostatistics 19:2137–52
    [Google Scholar]
  94. Pebesma EJ. 2004. Multivariable geostatistics in S: the gstat package. Comput. Geosci. 30:7683–91
    [Google Scholar]
  95. Peng J, Müller HG. 2008. Distance-based clustering of sparsely observed stochastic processes, with applications to online auctions. Ann. Appl. Stat. 2:31056–77
    [Google Scholar]
  96. Qiao X, Guo S, James GM. 2019. Functional graphical models. J. Am. Stat. Assoc. 114:525211–22
    [Google Scholar]
  97. Ramsay J, Silverman B 2005. Principal components analysis for functional data. Functional Data Analysis JO Ramsay, BW Silverman 147–72 New York: Springer. , 2nd ed..
    [Google Scholar]
  98. Ramsay JO, Graves S, Hooker G. 2021. fda: Functional data analysis. R Package version 5.5.1. https://CRAN.R-project.org/package=fda
  99. Rubín T, Panaretos VM. 2020. Sparsely observed functional time series: estimation and prediction. Electron. J. Stat. 14:11137–210
    [Google Scholar]
  100. Schabenberger O, Gotway CA. 2017. Statistical Methods for Spatial Data Analysis Boca Raton, FL: Chapman and Hall/CRC
    [Google Scholar]
  101. Scheffler A, Telesca D, Li Q, Sugar CA, Distefano C et al. 2020. Hybrid principal components analysis for region-referenced longitudinal functional EEG data. Biostatistics 21:1139–57
    [Google Scholar]
  102. Schlather M, Malinowski A, Menck PJ, Oesting M, Strokorb K. 2015. Analysis, simulation and prediction of multivariate random fields with package RandomFields. J. Stat. Softw. 63:81–25
    [Google Scholar]
  103. Scott MA, Simonoff JS, Marx BD. 2013. The SAGE handbook of multilevel modeling London: SAGE
    [Google Scholar]
  104. Serban N, Staicu AM, Carroll RJ. 2013. Multilevel cross-dependent binary longitudinal data. Biometrics 69:4903–13
    [Google Scholar]
  105. Shamshoian J, Şentürk D, Jeste S, Telesca D. 2022. Bayesian analysis of longitudinal and multidimensional functional data. Biostatistics 23:2558–73
    [Google Scholar]
  106. Staicu AM, Crainiceanu CM, Carroll RJ. 2010. Fast methods for spatially correlated multilevel functional data. Biostatistics 11:2177–94
    [Google Scholar]
  107. Staicu AM, Lahiri SN, Carroll RJ. 2015. Significance tests for functional data with complex dependence structure. J. Stat. Plan. Inference 156:1–13
    [Google Scholar]
  108. Tidemann-Miller B, Reich B, Staicu AM. 2016. Modeling multivariate mixed-response functional data. arXiv:1601.02461 [stat.ME]
  109. Tokushige S, Yadohisa H, Inada K. 2007. Crisp and fuzzy k-means clustering algorithms for multivariate functional data. Comput. Stat. 22:11–16
    [Google Scholar]
  110. Volkmann A. 2021. multifamm: Multivariate functional additive mixed models. R Package version 0.1.1. https://CRAN.R-project.org/package=multifamm
    [Google Scholar]
  111. Volkmann A, Stöcker A, Scheipl F, Greven S. 2021. Multivariate functional additive mixed models. arXiv:2103.06606 [stat.ME]
  112. Wang JL, Chiou JM, Müller HG. 2016. Functional data analysis. Annu. Rev. Stat. Appl. 3:257–95
    [Google Scholar]
  113. Wang K, Tsung F. 2021. Hierarchical sparse functional principal component analysis for multistage multivariate profile data. IISE Trans. 53:158–73
    [Google Scholar]
  114. Wood SN. 2006. Generalized Additive Models: An Introduction with R Boca Raton, FL: Chapman and Hall/CRC
    [Google Scholar]
  115. Wrobel J, Park SY, Staicu AM, Goldsmith J. 2016. Interactive graphics for functional data analyses. Stat 5:1108–18
    [Google Scholar]
  116. Xiao L. 2020. Asymptotic properties of penalized splines for functional data. Bernoulli 26:42847–75
    [Google Scholar]
  117. Yuan Y, Gilmore JH, Geng X, Martin S, Chen K et al. 2014. FMEM: Functional mixed effects modeling for the analysis of longitudinal white matter Tract data. NeuroImage 84:753–64
    [Google Scholar]
  118. Zhang H, Li Y. 2021. Unified principal component analysis for sparse and dense functional data under spatial dependency. J. Bus. Econ. Stat. https://doi.org/10.1080/07350015.2021.1938085
    [Crossref] [Google Scholar]
  119. Zhang JT, Chen J. 2007. Statistical inferences for functional data. Ann. Stat. 35:31052–79
    [Google Scholar]
  120. Zhang X, Shao X, Hayhoe K, Wuebbles DJ. 2011. Testing the structural stability of temporally dependent functional observations and application to climate projections. Electron. J. Stat. 5:1765–96
    [Google Scholar]
  121. Zhu H, Chen K, Luo X, Yuan Y, Wang JL. 2019. FMEM: functional mixed effects models for longitudinal functional responses. Stat. Sin. 29:42007
    [Google Scholar]
  122. Zhu H, Li R, Kong L. 2012. Multivariate varying coefficient model for functional responses. Ann. Stat. 40:52634
    [Google Scholar]
  123. Zhu H, Morris JS, Wei F, Cox DD. 2017. Multivariate functional response regression, with application to fluorescence spectroscopy in a cervical pre-cancer study. Comput. Stat. Data Anal. 111:88–101
    [Google Scholar]
  124. Zhu H, Strawn N, Dunson DB. 2016. Bayesian graphical models for multivariate functional data. J. Mach. Learn. Res. 17:2041–27
    [Google Scholar]
  125. Zipunnikov V, Greven S, Shou H, Caffo B, Reich DS, Crainiceanu C. 2014. Longitudinal high-dimensional principal components analysis with application to diffusion tensor imaging of multiple sclerosis. Ann. Appl. Stat. 8:42175
    [Google Scholar]
  126. Zuo Y, Serfling R. 2000. General notions of statistical depth function. Ann. Stat. 28:2461–82
    [Google Scholar]
/content/journals/10.1146/annurev-statistics-032921-033726
Loading
/content/journals/10.1146/annurev-statistics-032921-033726
Loading

Data & Media loading...

Supplementary Data

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error