Abstract
We enlarge the number of available functional depths by introducing the kernelized functional spatial depth (KFSD). KFSD is a local-oriented and kernel-based version of the recently proposed functional spatial depth (FSD) that may be useful for studying functional samples that require an analysis at a local level. In addition, we consider supervised functional classification problems, focusing on cases in which the differences between groups are not extremely clear-cut or the data may contain outlying curves. We perform classification by means of some available robust methods that involve the use of a given functional depth, including FSD and KFSD, among others. We use the functional k-nearest neighbor classifier as a benchmark procedure. The results of a simulation study indicate that the KFSD-based classification approach leads to good results. Finally, we consider two real classification problems, obtaining results that are consistent with the findings observed with simulated curves.
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Notes
If \(Y\) is not concentrated on a straight line and is not strongly concentrated around single points, the spatial median \(m\) of \(Y\) is the unique solution of (5) for \(u\) equal to the zero element in \(\mathbb {H}\).
For FMD, HMD, RTD and IDD we have used the corresponding R functions that are available in the R package fda.usc on CRAN (Febrero and Oviedo de la Fuente 2012); for MBD we have followed the guidelines contained in Sun et al. (2012); for FSD and KFSD we have built some functions for R, which are available upon request. Features of the workstation: Intel Core i7-3.40 GHz and 16GB of RAM.
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Acknowledgments
The authors would like to thank the associate editor and four anonymous referees for their helpful comments. This research was partially supported by Spanish Ministry of Education and Science Grant 2007/04438/001, by Spanish Ministry of Science and Innovation Grant 2012/00084/001, and by MCI Grant MTM2008-03010.
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Sguera, C., Galeano, P. & Lillo, R. Spatial depth-based classification for functional data. TEST 23, 725–750 (2014). https://doi.org/10.1007/s11749-014-0379-1
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DOI: https://doi.org/10.1007/s11749-014-0379-1
Keywords
- Functional depths
- Functional outliers
- Functional spatial depth
- Kernelized functional spatial depth
- Supervised functional classification