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Keywords:
depth-based classifier; von Mises-Fisher distribution; directional data; cosine depth
Summary:
The main goal of supervised learning is to construct a function from labeled training data which assigns arbitrary new data points to one of the labels. Classification tasks may be solved by using some measures of data point centrality with respect to the labeled groups considered. Such a measure of centrality is called data depth. In this paper, we investigate conditions under which depth-based classifiers for directional data are optimal. We show that such classifiers are equivalent to the Bayes (optimal) classifier when the considered distributions are rotationally symmetric, unimodal, differ only in location and have equal priors. The necessity of such assumptions is also discussed.
References:
[1] Agostinelli, C., Romanazzi, M.: Nonparametric analysis of directional data based on data depth. Environ. Ecol. Stat. 20 (2013), 253-270. DOI 10.1007/s10651-012-0218-z | MR 3068658
[2] Batschelet, E.: Circular Statistics in Biology. Mathematics in Biology. Academic Press, London (1981). MR 0659065 | Zbl 0524.62104
[3] Bowers, J. A., Morton, I. D., Mould, G. I.: Directional statistics of the wind and waves. Appl. Ocean Research 22 (2000), 13-30. DOI 10.1016/S0141-1187(99)00025-5
[4] Chang, T.: Spherical regression and the statistics of tectonic plate reconstructions. Int. Stat. Rev. 61 (1993), 299-316. DOI 10.2307/1403630
[5] Demni, H., Messaoud, A., Porzio, G. C.: The cosine depth distribution classifier for directional data. Applications in Statistical Computing Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Cham (2019), 49-60. DOI 10.1007/978-3-030-25147-5_4 | MR 3970229
[6] Fisher, N. I.: Smoothing a sample of circular data. J. Struct. Geol. 11 (1989), 775-778. DOI 10.1016/0191-8141(89)90012-6
[7] Ghosh, A. K., Chaudhuri, P.: On maximum depth and related classifiers. Scand. J. Stat. 32 (2005), 327-350. DOI 10.1111/j.1467-9469.2005.00423.x | MR 2188677 | Zbl 1089.62075
[8] Hubert, M., Rousseeuw, P., Segaert, P.: Multivariate and functional classification using depth and distance. Adv. Data Anal. Classif., ADAC 11 (2017), 445-466. DOI 10.1007/s11634-016-0269-3 | MR 3688976 | Zbl 1414.62247
[9] James, G., Witten, D., Hastie, T., Tibshirani, R.: An Introduction to Statistical Learning: With applications in R. Springer Texts in Statistics 103. Springer, New York (2013). DOI 10.1007/978-1-4614-7138-7 | MR 3100153 | Zbl 1281.62147
[10] Kirschstein, T., Liebscher, S., Pandolfo, G., Porzio, G. C., Ragozini, G.: On finite-sample robustness of directional location estimators. Comput. Stat. Data Anal. 133 (2019), 53-75. DOI 10.1016/j.csda.2018.08.028 | MR 3926466 | Zbl 07027245
[11] Klecha, T., Kosiorowski, D., Mielczarek, D., Rydlewski, J. P.: New proposals of a stress measure in a capital and its robust estimator. Available at https://arxiv.org/abs/1802.03756 (2018), 24 pages.
[12] Kosiorowski, D.: About phase transitions in Kendall's shape space. Acta Univ. Lodz., Folia Oeconomica 206 (2007), 137-155.
[13] Leong, P., Carlile, S.: Methods for spherical data analysis and visualization. J. Neurosci. Methods 80 (1998), 191-200. DOI 10.1016/S0165-0270(97)00201-X
[14] Ley, C., Sabbah, C., Verdebout, T.: A new concept of quantiles for directional data and the angular Mahalanobis depth. Electron. J. Stat. 8 (2014), 795-816. DOI 10.1214/14-EJS904 | MR 3217789 | Zbl 1349.62197
[15] Liu, R. Y.: On a notion of data depth based on random simplices. Ann. Stat. 18 (1990), 405-414. DOI 10.1214/aos/1176347507 | MR 1041400 | Zbl 0701.62063
[16] Liu, R. Y., Singh, K.: Ordering directional data: Concepts of data depth on circles and spheres. Ann. Stat. 20 (1992), 1468-1484. DOI 10.1214/aos/1176348779 | MR 1186260 | Zbl 0766.62027
[17] Makinde, O. S., Fasoranbaku, O. A.: On maximum depth classifiers: Depth distribution approach. J. Appl. Stat. 45 (2018), 1106-1117. DOI 10.1080/02664763.2017.1342783 | MR 3774534
[18] Mardia, K. V., Jupp, P. E.: Directional Statistics. Wiley Series in Probability and Statistics. John Wiley & Sons, Chichester (2000). DOI 10.1002/9780470316979 | MR 1828667 | Zbl 0935.62065
[19] Paindaveine, D., Verdebout, T.: Optimal rank-based tests for the location parameter of a rotationally symmetric distribution on the hypersphere. Mathematical Statistics and Limit Theorems Springer, Cham (2015), 249-269. DOI 10.1007/978-3-319-12442-1_14 | MR 3380740 | Zbl 1320.62131
[20] Pandolfo, G., D'Ambrosio, A., Porzio, G. C.: A note on depth-based classification of circular data. Electron. J. Appl. Stat. Anal. 11 (2018), 447-462. DOI 10.1285/i20705948v11n2p447 | MR 3887392
[21] Pandolfo, G., Paindaveine, D., Porzio, G. C.: Distance-based depths for directional data. Can. J. Stat. 46 (2018), 593-609. DOI 10.1002/cjs.11479 | MR 3902616 | Zbl 07193349
[22] Saw, J. G.: A family of distributions on the $m$-sphere and some hypothesis tests. Biometrika 65 (1978), 69-73. DOI 10.1093/biomet/65.1.69 | MR 0497510 | Zbl 0379.62035
[23] Small, C. G.: Measures of centrality for multivariate and directional distributions. Can. J. Stat. 15 (1987), 31-39. DOI 10.2307/3314859 | MR 0887986 | Zbl 0622.62054
[24] Tukey, J. W.: Mathematics and the picturing of data. Proceedings of the International Congress of Mathematicians Canad. Math. Congress, Montreal (1975), 523-531. MR 0426989 | Zbl 0347.62002
[25] Vencálek, O.: Depth-based classification for multivariate data. Austrian J. Stat. 46 (2017), 117-128. DOI 10.17713/ajs.v46i3-4.677
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