Estimating historic movement of a climatological variable from a pair of misaligned functional data sets
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Estimating historic movement of a climatological variable from a pair of misaligned functional data sets Dibyendu Bhaumik1
· Debasis Sengupta2
Received: 19 March 2019 / Revised: 30 June 2020 / Accepted: 3 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Curve registration · Ice core data · Nadaraya–Watson estimator · Nonparametric regression · Structural averaging
Handling Editor: Bryan F. J. Manly. The views expressed in this paper are strictly personal, and not those of the Reserve Bank of India. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10651020-00463-3) contains supplementary material, which is available to authorized users.
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Dibyendu Bhaumik [email protected] Debasis Sengupta [email protected]
1
Reserve Bank of India, Central Office Building, Fort, Mumbai 400001, India
2
Applied Statistics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India
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Environmental and Ecological Statistics
1 Introduction
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Paleoclimatic data on movement of atmospheric concentration of carbon dioxide with time, derived from ice-cores drilled at Lake Vostok and EPICA (The European Project for Ice Coring in Antarctica) Dome C in Antarctica (Lüthi et al. 2008; Petit et al. 1999), show remarkable similarity (see Fig. 1). The ups and downs of these curves are linked with different phases of the Earth’s paleoclimatic history. A more precise description of this movement should be possible by pooling of the two data sets for a combined estimate. However, due to distortion of the time scales arising from errors in radio isotope dating, the two sets of data can not be combined as it is. Methods of estimating an underlying location function for a given set of misaligned functional data are available in the literature. It is generally presumed that in the absence of misalignment, these functions would have a common mean or location function, referred to as the structural mean. The task
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