Mahalanobis depth#

mahalanobis(x, data, exact=True, mah_estimate='moment', mah_parMcd=0.75, 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 Mahalanobis 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=False, which leads to approx- imate computation of the Mahalanobis depth using the method defined by the argument solver. If exact=True, the Mahalanobis depth is computed exactly, using the closed-form expression.
mah_estimate: A character string specifying which estimates to use when calculating the Mahalanobis depth; can be “‘moment’” or 'MCD', determining whether traditional moment or Minimum Covariance Determinant (MCD) estimates for mean and covariance are used. By default 'moment' is used.
mah_parMcd: is the value of the argument alpha for the function covMcd; is used when mah.estimate = 'MCD'.
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

Mahalanobis, P. C. (1936). On the generalized distance in statistics. Proceedings of the National Institute of Sciences of India, 12, 49–55.
Mosler, K. and Mozharovskyi, P. (2022). Choosing among notions of multivariate depth statistics. Statistical Science, 37(3), 348-368.

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)
>>> mahalanobis(x, data)
[0.17849871 0.10412453 0.1331417  0.13578021 0.3154836  0.29103769
    0.13398989 0.13913017 0.59339051 0.10556139]
>>> mahalanobis(x, data, exact="True", mah_estimate="MCD", mah_parMcd = 0.75)
[0.17758703 0.10367974 0.131705   0.13575221 0.31847867 0.29034948
    0.13291613 0.13792774 0.59094958 0.10491694]