On the optimality of the max-depth and max-rank classifiers for spherical data

Ondřej Vencálek; Houyem Demni; Amor Messaoud; Giovanni C. Porzio

Applications of Mathematics (2020)

  • Volume: 65, Issue: 3, page 331-342
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Vencálek, Ondřej, et al. "On the optimality of the max-depth and max-rank classifiers for spherical data." Applications of Mathematics 65.3 (2020): 331-342. <http://eudml.org/doc/297301>.

@article{Vencálek2020,
abstract = {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.},
author = {Vencálek, Ondřej, Demni, Houyem, Messaoud, Amor, Porzio, Giovanni C.},
journal = {Applications of Mathematics},
keywords = {depth-based classifier; von Mises-Fisher distribution; directional data; cosine depth},
language = {eng},
number = {3},
pages = {331-342},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the optimality of the max-depth and max-rank classifiers for spherical data},
url = {http://eudml.org/doc/297301},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Vencálek, Ondřej
AU - Demni, Houyem
AU - Messaoud, Amor
AU - Porzio, Giovanni C.
TI - On the optimality of the max-depth and max-rank classifiers for spherical data
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 3
SP - 331
EP - 342
AB - 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.
LA - eng
KW - depth-based classifier; von Mises-Fisher distribution; directional data; cosine depth
UR - http://eudml.org/doc/297301
ER -

References

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  1. Agostinelli, C., Romanazzi, M., 10.1007/s10651-012-0218-z, Environ. Ecol. Stat. 20 (2013), 253-270. (2013) MR3068658DOI10.1007/s10651-012-0218-z
  2. Batschelet, E., Circular Statistics in Biology, Mathematics in Biology. Academic Press, London (1981). (1981) Zbl0524.62104MR0659065
  3. Bowers, J. A., Morton, I. D., Mould, G. I., 10.1016/S0141-1187(99)00025-5, Appl. Ocean Research 22 (2000), 13-30. (2000) DOI10.1016/S0141-1187(99)00025-5
  4. Chang, T., 10.2307/1403630, Int. Stat. Rev. 61 (1993), 299-316. (1993) DOI10.2307/1403630
  5. Demni, H., Messaoud, A., Porzio, G. C., 10.1007/978-3-030-25147-5_4, Applications in Statistical Computing Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Cham (2019), 49-60. (2019) MR3970229DOI10.1007/978-3-030-25147-5_4
  6. Fisher, N. I., 10.1016/0191-8141(89)90012-6, J. Struct. Geol. 11 (1989), 775-778. (1989) DOI10.1016/0191-8141(89)90012-6
  7. Ghosh, A. K., Chaudhuri, P., 10.1111/j.1467-9469.2005.00423.x, Scand. J. Stat. 32 (2005), 327-350. (2005) Zbl1089.62075MR2188677DOI10.1111/j.1467-9469.2005.00423.x
  8. Hubert, M., Rousseeuw, P., Segaert, P., 10.1007/s11634-016-0269-3, Adv. Data Anal. Classif., ADAC 11 (2017), 445-466. (2017) Zbl1414.62247MR3688976DOI10.1007/s11634-016-0269-3
  9. James, G., Witten, D., Hastie, T., Tibshirani, R., 10.1007/978-1-4614-7138-7, Springer Texts in Statistics 103. Springer, New York (2013). (2013) Zbl1281.62147MR3100153DOI10.1007/978-1-4614-7138-7
  10. Kirschstein, T., Liebscher, S., Pandolfo, G., Porzio, G. C., Ragozini, G., 10.1016/j.csda.2018.08.028, Comput. Stat. Data Anal. 133 (2019), 53-75. (2019) Zbl07027245MR3926466DOI10.1016/j.csda.2018.08.028
  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. (2018) 
  12. Kosiorowski, D., About phase transitions in Kendall's shape space, Acta Univ. Lodz., Folia Oeconomica 206 (2007), 137-155. (2007) 
  13. Leong, P., Carlile, S., 10.1016/S0165-0270(97)00201-X, J. Neurosci. Methods 80 (1998), 191-200. (1998) DOI10.1016/S0165-0270(97)00201-X
  14. Ley, C., Sabbah, C., Verdebout, T., 10.1214/14-EJS904, Electron. J. Stat. 8 (2014), 795-816. (2014) Zbl1349.62197MR3217789DOI10.1214/14-EJS904
  15. Liu, R. Y., 10.1214/aos/1176347507, Ann. Stat. 18 (1990), 405-414. (1990) Zbl0701.62063MR1041400DOI10.1214/aos/1176347507
  16. Liu, R. Y., Singh, K., 10.1214/aos/1176348779, Ann. Stat. 20 (1992), 1468-1484. (1992) Zbl0766.62027MR1186260DOI10.1214/aos/1176348779
  17. Makinde, O. S., Fasoranbaku, O. A., 10.1080/02664763.2017.1342783, J. Appl. Stat. 45 (2018), 1106-1117. (2018) MR3774534DOI10.1080/02664763.2017.1342783
  18. Mardia, K. V., Jupp, P. E., 10.1002/9780470316979, Wiley Series in Probability and Statistics. John Wiley & Sons, Chichester (2000). (2000) Zbl0935.62065MR1828667DOI10.1002/9780470316979
  19. Paindaveine, D., Verdebout, T., 10.1007/978-3-319-12442-1_14, Mathematical Statistics and Limit Theorems Springer, Cham (2015), 249-269. (2015) Zbl1320.62131MR3380740DOI10.1007/978-3-319-12442-1_14
  20. Pandolfo, G., D'Ambrosio, A., Porzio, G. C., 10.1285/i20705948v11n2p447, Electron. J. Appl. Stat. Anal. 11 (2018), 447-462. (2018) MR3887392DOI10.1285/i20705948v11n2p447
  21. Pandolfo, G., Paindaveine, D., Porzio, G. C., 10.1002/cjs.11479, Can. J. Stat. 46 (2018), 593-609. (2018) Zbl07193349MR3902616DOI10.1002/cjs.11479
  22. Saw, J. G., 10.1093/biomet/65.1.69, Biometrika 65 (1978), 69-73. (1978) Zbl0379.62035MR0497510DOI10.1093/biomet/65.1.69
  23. Small, C. G., 10.2307/3314859, Can. J. Stat. 15 (1987), 31-39. (1987) Zbl0622.62054MR0887986DOI10.2307/3314859
  24. Tukey, J. W., Mathematics and the picturing of data, Proceedings of the International Congress of Mathematicians Canad. Math. Congress, Montreal (1975), 523-531. (1975) Zbl0347.62002MR0426989
  25. Vencálek, O., 10.17713/ajs.v46i3-4.677, Austrian J. Stat. 46 (2017), 117-128. (2017) DOI10.17713/ajs.v46i3-4.677

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