A note on depth-based classification of circular data


Abstract


A procedure is developed in order to deal with the classification problem of objects in circular statistics. It is fully non-parametric and based on depth functions for directional data. Using the so-called DD-plot, we apply the k-nearest neighbors method in order to discriminate between competing groups. Three different notions of data depth for directional data are considered: the angular simplicial, the angular Tukey and the arc distance. We investigate and compare their performances through the average misclassification rate with respect to different distributional settings by using simulated and real data sets. Results show that the use of the arc distance depth should be generally preferred, and in some cases it outperforms the classifier based both on the angular simplicial and Tukey depths.

DOI Code: 10.1285/i20705948v11n2p447

Keywords: Angular depths, Supervised circular classification, K-NN, Misclassification rate

References


Ackermann, H.: A note on circular nonparametrical classication. Biometrical J 5, 577{587 (1997)

Agostinelli, C., Romanazzi, M.: Nonparametric analysis of directional data based on data depth. Environ Ecol Stat 20, 253{270 (2013)

Batschelet, E.: Circular statistics in biology. Academic Press, London, 1981.

Christmann, A., Rousseeuw, P.: Measuring overlap in logistic regression. Comput Statist Data Anal 37, 65{75 (2001)

Christmann, A., Fischer, P., Joachims, T.: Comparison between various regression depth methods and the support vector machine to approximate the minimum number of misclassications . Comput Statist 17, 273{287 (2002)

Dutta, S. and Ghosh, A. K.: On classication based on Lp depth with an adaptive choice of p. Technical Report No. R5/2011, Statistics and Mathematics Unit. Indian Statistical Institute, Kolkata, India (2011). Available at http://www.isical.ac. in/~statmath/html/publication/l_p_24_02_2011.pdf

El Katthabi, S., Streit, F.: Identication analysis in directional statistics, Comput Statist Data Anal 23, 45{63 (1996)

Ghosh, A.K., Chaudhuri, P.: On maximum depth and related classiers. Scand J Stat 32, 327{350 (2005)

Hartikainen, A., Oja, H.: On some parametric, nonparametric and semiparametric discrimination rules. In Data depth: robust multivariate analysis, computational geometry and applications. Liu, R. Y., Serfing, R., and Souvaine, D. L., eds. 1st edition. New York: AMS, 61{70 (2006)

Hubert, M., Van der Veeken, S.: Robust classication for skewed data. Adv Data Anal Classif 4, 239{254 (2010)

James, G., Witten, D., Hastie, T. and Tibshirani, R.: An Introduction to Statistical Learning: with Applications in R. Springer, New York (2013)

Morris, J., Layccock, P.J.: Discriminant analysis of directional data. Biometrika 61, 335{341 (1974)

Jornsten, R.: Clustering and classication based on the L1 data depth. J Multivariate Anal 90, 67{89 (2004)

Li, J., Cuesta-Albertos, J.A., and Liu, R.: DD-Classier: Nonparametric Classification Procedure Based on DD-plot. J Am Statist Assoc 107, 737{753 (2012)

Liu, Z., Modarres, R.: Lens data depth and median. J Nonparametr Statist 23, 1063{1074 (2011)

Liu, R.Y., Singh, K.: Ordering directional data. Concepts of data depth on circles and spheres. Ann Stat 20, 1468{1484 (1992)

Liu, R.Y., Singh, K.: Multivariate analysis by data depth: descriptive statistics, graphics and inference. Ann Stat 27, 783{1117 (1999)

Lopez-Cruz, P., Bielza, C., Larranaga, P.: Directional naive Bayes classiers. Pattern Anal Appl DOI:10.1007/s10044-013-0340-z. Springer, London (2013)

Lopez-Pintado, S. and Romo, J.: Depth-based classication for functional data. DIMACS Ser. Math. Theo. Comput. Sci., (Liu, R., Sering, R. and Souvaine, D. L. ed.) 72, 103{119 (2006)

Mardia, K.V., Jupp, E.P.: Statistics of directional data. Academic Press, London (1972)

Mosler, I., Hoberg, R.: Data analysis and classication with the zonoid depth. In Data depth: robust multivariate analysis, computational geometry and applications. Liu, R. Y., Serfling, R., and Souvaine, D. L., eds. 1st edition. New York: American

Mathematical Society, 49{59 (2006)

Paindaveine, D., Van Bever, G.: Nonparametrically Consistent Depth-Based Classifiers. Paper submitted to the Bernoulli. (2014)

Ruts, I., Rousseeuw, P.: Computing depth contours of bivariate points clouds. Comput Stat Data An 94, 388{402 (1996)

SenGupta, A., Ugwuowo, F.I.: A classication method for directional data with application to the human skull. Commun Stat Theory Methods 40, 457-466 (2011)

Serfling, R.: Depth functions functions in nonparametric multivariate inference.

Data depth: robust multivariate analysis, computational geometry and applications,

(Liu, R., Serfling, R. and Souvaine, D., eds.), 1-16 (2006)

Stoller, D.S.: Univariate two-population distribution-free discrimination. J Am Statist Assoc 49, 770-777 (1954)

Zuo, Y., Serfling, R.: General notions of depth function. Ann Stat 28, 461-482 (2000)


Full Text: pdf
کاغذ a4
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.