Zonoid depth#

zonoid(x, data, seed=0, exact=True, solver='neldermead', NRandom=1000, option=1, n_refinements=10, sphcap_shrink=0.5, alpha_Dirichlet=1.25, cooling_factor=0.95, cap_size=1, start='mean', space='sphere', line_solver='goldensection', bound_gc=True)[source]#

Description

Calculates the zonoid 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. 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.
exact: The type of the used method. The default is exact=True, which leads to exact computation of the zonoid depth using the method described by Dyckerhoff et al. (1996). If exact=False, approximate computation of the zonoid depth is performed using the method defined by the argument solver.
solver: The type of solver used to approximate the depth. {'simplegrid', 'refinedgrid', 'simplerandom', 'refinedrandom', 'coordinatedescent', 'randomsimplices', 'neldermead', 'simulatedannealing'}
NRandom: The total number of iterations to compute the depth. Some solvers are converging faster so they are run several time to achieve NRandom iterations.
option: If option=1, only approximated depths are returned.

If option=2, best directions to approximate depths are also returned.

If option=3, depths calculated at every iteration are also returned.

If option=4, random directions used to project depths are also returned with indices of converging for the solver selected.

n_refinements Set the maximum of iteration for computing the depth of one point. For solver='refinedrandom' or 'refinedgrid'.
sphcap_shrink: It’s the shrinking of the spherical cap. For solver='refinedrandom' or 'refinedgrid'.
alpha_Dirichlet: It’s the parameter of the Dirichlet distribution. For solver='randomsimplices'.
cooling_factor: It’s the cooling factor. For solver='simulatedannealing'.
cap_size: It’s the size of the spherical cap. For solver='simulatedannealing' or 'neldermead'.
start: {‘mean’, ‘random’}. For solver='simulatedannealing' or 'neldermead', it’s the method used to compute the first depth.
space: {'sphere', 'euclidean'}. For solver='coordinatedescent' or 'neldermead', it’s the type of spacecin which the solver is running.
line_solver: {'uniform', 'goldensection'}. For solver='coordinatedescent', it’s the line searh strategy used by this solver.
bound_gc: For solver='neldermead', it’s True if the search is limited to the closed hemisphere.

References

Dyckerhoff, R., Koshevoy, G. and Mosler, K. (1996). Zonoid data depth: Theory and computation. In A. Pratt, (Ed.), COMPSTAT 1996, Proceedings in Computational Statistics, Physica-Verlag, Heidelberg, 235–240.
Koshevoy, G. and Mosler, K. (1997). Zonoid trimming for multivariate distributions. The Annals of Statistics, 25, 1998–2017.
Dyckerhoff, R., Mozharovskyi, P., and Nagy, S. (2021). Approximate computation of projection depths. Computational Statistics and Data Analysis, 157, 107166.

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)
>>> zonoid(x, data)
[0.         0.00769552 0.03087017 0.         0.30945453 0.0142515
    0.         0.01970896 0.02169483 0.        ]