Halfspace depth#

halfspace(x, data, numDirections=1000, exact=True, method='recursive', 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 exact and approximated Tukey (=halfspace, location) depth (Tukey, 1975) 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=False, which leads to approx- imate computation of the Tukey depth. If exact=True, the Tukey depth is computed exactly, with method='recursive' by default.
method: For exact=True, the Tukey depth is calculated as the minimum over all combinations of k points from data (see Details below). In this case parameter method specifies k, with possible values 1 for method='recursive' (by default), d−2 for method='plane', d−1 for 'method=line'. The name of the method may be given as well as just parameter exact, in which case the default method will be used.
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

Tukey, J. W. (1975). Mathematics and the picturing of data. In R. James (Ed.), Proceedings of the International Congress of Mathematicians, Volume 2, Canadian Mathematical Congress, 523–531.
Donoho, D. L. and M. Gasko (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. The Annals of Statistics, 20(4), 1803–1827.
Dyckerhoff, R. and Mozharovskyi, P. (2016): Exact computation of the halfspace depth. Computational Statistics and Data Analysis, 98, 19–30.
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, 2, 0],[0, 0, 1]]
>>> mat2=[[1, 0, 0],[0, 1, 0],[0, 0, 1]]
>>> x = np.random.multivariate_normal([1,1,1], mat2, 10)
>>> data = np.random.multivariate_normal([0,0,0], mat1, 200)
>>> halfspace(x, data)
[0.    0.005 0.005 0.    0.04  0.01  0.    0.    0.04  0.01 ]
>>> halfspace(x, data, exact=True)
[0.    0.005 0.005 0.    0.04  0.01  0.    0.    0.04  0.01 ]