Beta-skeleton depth#

betaSkeleton(x, data, beta=2, distance='Lp', Lp_p=2, mah_estimate='moment', mah_parMcd=0.75)[source]#
Description

Calculates the beta-skeleton depth of points w.r.t. a multivariate data set.

Arguments
x

Matrix of objects (numerical vector as one object) whose depth is to be calculated. Each row contains a d-variate point and should have the same dimension as data.

data

Matrix of data where each row contains a d-variate point, w.r.t. which the depth is to be calculated.

beta

The parameter defining the positionning of the balls’ centers, see Yang and Modarres (2017) for details. By default (together with other arguments) equals 2, which corresponds to the lens depth, see Liu and Modarres (2011).

distance

A character string defining the distance to be used for determining inclusion of a point into the lens (influence region), see Yang and Modarres (2017) for details. Possibilities are 'Lp' for the Lp-metric (default) or 'Mahalanobis' for the Mahalanobis distance adjustment.

Lp_p

A non-negative number defining the distance’s power equal 2 by default (Euclidean distance) is used only when distance='Lp'.

mah_estimate

A character string specifying which estimates to use when calculating sample covariance matrix; can be 'none', 'moment' or 'MCD', determining whether traditional moment or Minimum Covariance Determinant (MCD) estimates for mean and covariance are used. By default 'moment' is used. Is used only when distance='Mahalanobis'.

mah_parMcd

The value of the argument alpha for Minimum Covariance Determinant (MCD); is used when distance='Mahalanobis' and mah.estimate='MCD'.

References

  • Elmore, R. T., Hettmansperger, T. P. and Xuan, F. (2006). Spherical data depth and a multivariate median. In R. Y. Lui, R. Serfling, and D. L. Souvaine, (Eds.), Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications, DIMACS Series Discrete Mathematics and Theoretical Computer Science, 72, American Mathematical Society, Providence, RI, 87–101.

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

  • Kleindessner, M. and Von Luxburg, U. (2017). Lens depth function and k-relative neighborhood graph: Versatile tools for ordinal data analysis. Journal of Machine Learning Research, 18, 58, 52.

  • Yang, M. and Modarres, R. (2018). \({\beta}\)-skeleton depth functions and medians. Communications in Statistics - Theory and Methods, 47, 5127–5143.

Examples
>>> import numpy as np
>>> from depth.multivariate import *
>>> mat1=[[1, 0, 0, 0, 0],[0, 2, 0, 0, 0],[0, 0, 3, 0, 0],[0, 0, 0, 2, 0],[0, 0, 0, 0, 1]]
>>> mat2=[[1, 0, 0, 0, 0],[0, 1, 0, 0, 0],[0, 0, 1, 0, 0],[0, 0, 0, 1, 0],[0, 0, 0, 0, 1]]
>>> x = np.random.multivariate_normal([1,1,1,1,1], mat2, 10)
>>> data = np.random.multivariate_normal([0,0,0,0,0], mat1, 1000)
>>> BetaSkeleton(x, data)
[0.16467668 0.336002   0.43702102 0.25827828 0.4204044  0.46894895
0.27825225 0.11572372 0.4663003  0.18778579]