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Estimating historic movement of a climatological variable from a pair of misaligned functional data sets

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Abstract

We consider the problem of estimating the mean function from a pair of paleoclimatic functional data sets after one of them has been registered with the other. We establish that registering one data set with respect to the other is the appropriate way to formulate this problem. This is in contrast with estimation of the mean function on a ‘central’ time scale that is preferred in the analysis of multiple sets of longitudinal growth data. We show that if a consistent estimator of the time transformation is used for registration, the Nadaraya–Watson estimator of the mean function based on the registered data would be consistent under a few additional conditions. We study the potential change in asymptotic mean squared error of the estimator because of the contribution of the time-transformed data set. We demonstrate through simulations that the additional data can lead to improved estimation despite estimation error in registration. Analysis of three pairs of paleoclimatic data sets reveals some interesting points.

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References

  • Anklin M, Schwander J, Stauffer B, Tschumi J, Fuchs A, Barnola JM, Raynaud D (1997) CO\(_2\) record between 40 and 8 kyr b.p. from the Greenland ice core project ice core. J Geophys Res Oceans 102(C12):26539–26545. https://doi.org/10.1029/97JC00182

  • Bhaumik D, Srivastava R, Sengupta D (2017) Feature sensitive curve registration by kernel matching, submitted; arXiv:1704.03127v2

  • Blunier T, Chappellaz J, Schwander J, Stauffer B, Raynaud D (1995) Variations in atmospheric methane concentration during the holocene epoch. Nature 374(6517):46–49. https://doi.org/10.1038/374046a0

    Article  CAS  Google Scholar 

  • Brook EJ, Sowers T, Orchardo J (1996) Rapid variations in atmospheric methane concentration during the past 110,000 years. Science 273(5287):1087–1091. https://doi.org/10.1126/science.273.5278.1087

    Article  CAS  PubMed  Google Scholar 

  • Brumback LC, Lindstrom MJ (2004) Self modeling with flexible, random time transformations. Biometrics 60(2):461–470

    Article  PubMed  Google Scholar 

  • Collomb G (1977a) Estimation non paramétrique de la régression par la méthode du noyau : propriété de convergence asymptotiquememt normale indépendante. Annales scientifiques de l’Université de Clermont Mathématiques 65(15):24–46

    Google Scholar 

  • Collomb G (1977b) Quelques propriétés de la méthode du noyau pour 1’estimation non-paramétrique de la régression en un point fixé. Comptes Rendus de l’Acad des Sciences, Paris, Série A 285:289–292

    Google Scholar 

  • Collomb G (1981) Estimation nonparamétrique de la régression: revue bibliographique. Inter Stat Rev 49:75–93

    Article  Google Scholar 

  • Collomb G (1985) Nonparametric regression an up-to-date bibliography. Statistics 2:309–324

    Article  Google Scholar 

  • Eubank RL (1999) Nonparametric regression and spline smoothing, 2nd edn. Marcel Dekker Inc, New York

    Book  Google Scholar 

  • Gervini D, Gasser T (2004) Self-modelling warping functions. J R Stat Soc Ser B Stat Methodol 66(4):959–971

    Article  Google Scholar 

  • Gervini D, Gasser T (2005) Nonparametric maximum likelihood estimation of the structural mean of a sample of curves. Biometrika 92(4):801–820

    Article  Google Scholar 

  • Härdle W (1990) Applied nonparametric regression. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Jouzel J, Masson-Delmotte V, Cattani O, Dreyfus G, Falourd S, Hoffmann G, Minster B, Nouet J, Barnola JM, Chappellaz J, Fischer H, Gallet JC, Johnsen S, Leuenberger M, Loulergue L, Luethi D, Oerter H, Parrenin F, Raisbeck G, Raynaud D, Schilt A, Schwander J, Selmo E, Souchez R, Spahni R, Stauffer B, Steffensen JP, Stenni B, Stocker TF, Tison JL, Werner M, Wolff EW (2007) Orbital and millennial Antarctic climate variability over the past 800,000 years. Science 317(5839):793–796. https://doi.org/10.1126/science.1141038

    Article  CAS  PubMed  Google Scholar 

  • Kneip A, Engel J (1995) Model estimation in nonlinear regression under shape invariance. Ann Stat 23(2):551–570

    Article  Google Scholar 

  • Kneip A, Gasser T (1992) Statistical tools to analyze data representing a sample of curves. Ann Stat 20(3):1266–1305

    Article  Google Scholar 

  • Kneip A, Ramsay JO (2008) Combining registration and fitting for functional models. J Am Stat Assoc 103(483):1155–1165

    Article  CAS  Google Scholar 

  • Lawton WH, Sylvestre EA, Maggio MS (1972) Self modeling nonlinear regression. Technometrics 14(3):513–532

    Article  Google Scholar 

  • Liu X, Müller HG (2004) Functional convex averaging and synchronization for time-warped random curves. J Am Stat Assoc 99(467):687–699

    Article  Google Scholar 

  • Loulergue L, Schilt A, Spahni R, Masson-Delmotte V, Blunier T, Lemieux B, Barnola JM, Raynaud D, Stocker T, Chappellaz J (2008) Orbital and millennial-scale features of atmospheric CH\(_4\) over the past 800,000 years. Nature 453:383–386

    CAS  PubMed  Google Scholar 

  • Lüthi D, Floch ML, Bereiter B, Blunier T, Barnola JM, Siegenthaler U, Raynaud D, Jouzel J, Fischer H, Kawamura K, Stocker T (2008) High-resolution carbon dioxide concentration record 650,000–800,000 years before present. Nature 453:379–382

    Article  PubMed  Google Scholar 

  • Nadaraya EA (1964) On estimating regression. Theory Prob Appl 9(1):141–142

    Article  Google Scholar 

  • Panaretos VM, Zemel Y (2016) Amplitude and phase variation of point processes. Ann Stat 44(2):771–812. https://doi.org/10.1214/15-AOS1387

    Article  Google Scholar 

  • Petit JR, Jouzel J, Raynaud D, Barkov NI, Barnola JM, Basile I, Bender M, Chappellaz J, Davis M, Delaygue G, Delmotte M, Kotlyakov VM, Legrand M, Lipenkov VY, Lorius C, Pepin L, Ritz C, Saltzman E, Stievenard M (1999) Climate and atmospheric history of the past 420,000 years from the vostok ice core, Antarctica. Nature 399(3):429–436

    Article  CAS  Google Scholar 

  • Rakêt LL, Sommer S, Markussen B (2014) A nonlinear mixed-effects model for simultaneous smoothing and registration of functional data. Pattern Recogn Lett 38:1–7

    Article  Google Scholar 

  • Ramsay JO, Li X (1998) Curve registration. J R Stat Soc Ser B Stat Methodol 60(2):351–363

    Article  Google Scholar 

  • Rønn BB (2001) Nonparametric maximum likelihood estimation for shifted curves. J R Stat Soc Ser B Stat Methodol 63(2):243–259

    Article  Google Scholar 

  • Schimek MG (2000) Smooting and regression—approaches computation and application. John Wiley & Sons, New York

    Book  Google Scholar 

  • Smith HJ, Wahlen M, Mastroianni D, Taylor K, Mayewski P (1997) The CO\(_2\) concentration of air trapped in greenland ice sheet project 2 ice formed during periods of rapid climate change. J Geophys Res Oceans 102(C12):26577–26582. https://doi.org/10.1029/97JC00163

    CAS  Google Scholar 

  • Srivastava A, Wu W, Kurtek S, Klassen E, Marron JS (2011) Registration of functional data using Fisher-Rao metric, arXiv:1103.3817v2

  • Tang R, Müller HG (2008) Pairwise curve synchronization for functional data. Biometrika 95(4):875–889

    Article  Google Scholar 

  • Wang K, Gasser T (1997) Alignment of curves by dynamic time warping. Ann Stat 25(3):1251–1276

    Article  Google Scholar 

  • Wang K, Gasser T (1999) Synchronizing sample curves nonparametrically. Ann Statist 27(2):439–460

    Article  Google Scholar 

  • Watson GS (1964) Smooth regression analysis. Sankhya Indian J Stat Ser A 26(4):359–372

    Google Scholar 

  • Watts A (1992) International law and the Antarctic treaty system. Grotius Publications Ltd., Cambridge

    Google Scholar 

  • Wrobel J, Zipunnikov V, Schrack J, Goldsmith J (2019) Registration for exponential family functional data. Biometrics 75(1):48–57. https://doi.org/10.1111/biom.12963

    Article  PubMed  Google Scholar 

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Acknowledgements

We thank two anonymous referees for very useful suggestions that led to substantial improvement of the paper.

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Authors and Affiliations

  1. Reserve Bank of India, Central Office Building, Fort, Mumbai, 400001, India

    Dibyendu Bhaumik

  2. Applied Statistics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India

    Debasis Sengupta

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  1. Dibyendu Bhaumik
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  2. Debasis Sengupta
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Correspondence to Dibyendu Bhaumik.

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Handling Editor: Bryan F. J. Manly.

The views expressed in this paper are strictly personal, and not those of the Reserve Bank of India.

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Bhaumik, D., Sengupta, D. Estimating historic movement of a climatological variable from a pair of misaligned functional data sets. Environ Ecol Stat 27, 729–751 (2020). https://doi.org/10.1007/s10651-020-00463-3

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