Integrated data depth for smooth functions and its application in supervised classification
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This paper concerns depth functions suitable for smooth functional data. We suggest a modification of the integrated data depth that takes into account the shape properties of the functions. This is achieved by including a derivative(s) into the definition of the suggested depth measures. We then further investigate the use of integrated data depths in supervised classification problems. The performances of classification rules based on different data depths are investigated, both in simulated and real data sets. As the proposed depth function provides a natural alternative to the depth function based on random projections, the difference in the performances of these two methods are discussed in more detail.
KeywordsData depths Functional data Integrated data depths Supervised classification
The authors thank the Editor and three reviewers for their valuable comments which led to a considerable improvement of the paper. This research was supported by the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy). The work of the D. Hlubinka was supported by the Grant GACR P402/14-07234S. The I. Gijbels gratefully acknowledges support from the GOA/12/014—project of the Research Fund KU Leuven. The work of the fourth author was partially supported by the Czech Science Foundation Project No. P402/12/G097 “DYME–Dynamic Models in Economics”. Currently he is a Research Assistant of the Research Foundation—Flanders, and acknowledges support from this foundation.
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