The interpoint depth for directional data


Abstract


The notion of the interpoint depth is applied to spherical spaces by us-ing an appropriate angular distance function for data lying on the surfaceof the unit hypersphere. The traditional multivariate methods, indeed, arenot suitable for the analysis of directional data and this holds true also fordistance measures and related depth based methods. The interpoint depthfor directional data possesses some nice properties and can be used for highdimensional data analysis. This notion of depth is particularly useful toinvestigate local features of distribution such as multimodality and can beexploited to deal with many statistical problems. The behavior of the pro-posed depth is investigated by means of simulated data. In addition threeinteresting applications are presented.

DOI Code: 10.1285/i20705948v13n2p358

Keywords: Data depth; Spherical distance; Spherical variables; Uniformity

References


Banerjee, A., Dhillon, I. S., Ghosh, J., and Sra, S. (2005). Clustering on the unit

hypersphere using von Mises-Fisher distributions. J. Mach. Learn. Res., 6:1345–1382.

Bingham, C. (1974). An antipodally symmetric distribution on the sphere. Ann. Statist.,2(6):1201–1225.

Buchta, C., Kober, M., Feinerer, I., and Hornik, K. (2012). Spherical k-means clustering. Journal of Statistical Software, 50(10):1–22.

Cai, T., Fan, J., and Jiang, T. (2013). Distributions of angles in random packing on spheres. The Journal of Machine Learning Research, 14(1):1837–1864.

Chavan, A. R. and Shirke, D. T. (2016). Nonparametric tests for testing equality of location parameters of two multivariate distributions. Electronic Journal of Applied Statistical Analysis, 9(2):417–432.

Dong, Y. and Lee, S. M. (2014). Depth functions as measures of representativeness. Statistical Papers, 55(4):1079–1105.

Dutta, S., Ghosh, A. K., Chaudhuri, P., et al. (2011). Some intriguing properties of tukey’s half-space depth. Bernoulli, 17(4):1420–1434.

Fisher, N. (1985). Spherical medians. Journal of the Royal Statistical Society. Series B (Methodological), pages 342–348.

Ley, C. and Verdebout, T. (2017). Modern directional statistics. Chapman and Hall/CRC.

Li, J., Cuesta-Albertos, J. A., and Liu, R. Y. (2012). Dd-classifier: Nonparametric classification procedure based on dd-plot. Journal of the American Statistical Association, 107(498):737–753.

Li, J. and Liu, R. Y. (2004). New nonparametric tests of multivariate locations and scales using data depth. Statistical Science, pages 686–696.

Liu, R. Y., Parelius, J. M., Singh, K., et al. (1999). Multivariate analysis by data depth: descriptive statistics, graphics and inference,(with discussion and a rejoinder by liu and singh). The annals of statistics, 27(3):783–858.

Liu, R. Y. and Singh, K. (1992). Ordering directional data: concepts of data depth on circles and spheres. The Annals of Statistics, pages 1468–1484.

Liu, Z. and Modarres, R. (2011). Lens data depth and median. Journal of Nonparametric Statistics, 23(4):1063–1074.

Lok, W. and Lee, S. M. (2011). A new statistical depth function with applications to multimodal data. Journal of Nonparametric Statistics, 23(3):617–631.

Mardia, K. V. and Jupp, P. E. (2009). Directional statistics, volume 494. John Wiley & Sons.

Paindaveine, D. and Van Bever, G. (2013). From depth to local depth: a focus on centrality. Journal of the American Statistical Association, 108(503):1105–1119.

Pandolfo, G., D’Ambrosio, A., and Porzio, G. C. (2018). A note on depth-based classification of circular data. Electronic Journal of Applied Statistical Analysis, 11(2):447–462.

Pandolfo, G. and Porzio, G. (2018). Dd-classifier for angular data with an application to protein structures. In Book of Abstracts, page 66.

Wilson, R. C., Hancock, E. R., Pekalska, E., and Duin, R. P. (2014). Spherical and hyperbolic embeddings of data. IEEE transactions on pattern analysis and machine intelligence, 36(11):2255–2269.

Zuo, Y. and Serfling, R. (2000). General notions of statistical depth function. Annals of statistics, pages 461–482.


Full Text: pdf
کاغذ a4

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