Mahalanobis depth#

DepthEucl.mahalanobis(x: ndarray | None = None, exact: bool = True, mah_estimate: Literal['moment', 'mcd'] = 'moment', mah_parMcd: float = 0.75, solver='neldermead', NRandom=1000, 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, output_option: Literal['lowest_depth', 'final_depht_dir', 'all_depth', 'all_depth_directions'] = 'final_depht_dir', evaluate_dataset: bool = False) → ndarray[source]

Calculates the Mahalanobis depth of points w.r.t. a multivariate data set.

Arguments
x: array-like or None, default=None

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.

exactbool, delfaut=True

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_estimatestr, {“moment”, “mcd”}, default=”moment”

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_parMcdfloat, default=0.75

is the value of the argument alpha for the function covMcd; is used when mah.estimate = 'MCD'.

solverstr, default=”neldermead”

The type of solver used to approximate the depth. {'simplegrid', 'refinedgrid', 'simplerandom', 'refinedrandom', 'coordinatedescent', 'randomsimplices', 'neldermead', 'simulatedannealing'}

NRandomint, default=1000

The total number of iterations to compute the depth. Some solvers are converging faster so they are run several time to achieve NRandom iterations.

n_refinementsint, default = 10

Set the maximum of iteration for computing the depth of one point. For solver='refinedrandom' or 'refinedgrid'.

sphcap_shrinkfloat, default = 0.5

It’s the shrinking of the spherical cap. For solver='refinedrandom' or 'refinedgrid'.

alpha_Dirichletfloat, default = 1.25

It’s the parameter of the Dirichlet distribution. For solver='randomsimplices'.

cooling_factorfloat, default = 0.95

It’s the cooling factor. For solver='simulatedannealing'.

cap_sizeint | float, default = 1

It’s the size of the spherical cap. For solver='simulatedannealing' or 'neldermead'.

startstr {‘mean’, ‘random’}, default = mean

For solver='simulatedannealing' or 'neldermead', it’s the method used to compute the first depth.

spacestr {‘sphere’, ‘euclidean’}, default = ‘sphere’

For solver='coordinatedescent' or 'neldermead', it’s the type of spacecin which the solver is running.

line_solverstr {‘uniform’, ‘goldensection’}, default = goldensection

For solver='coordinatedescent', it’s the line searh strategy used by this solver.

bound_gcbool, default = True

For solver='neldermead', it’s True if the search is limited to the closed hemisphere.

output_optionstr {“lowest_depth”,”final_depht_dir”,”all_depth”,”all_depth_directions}, default = final_depht_dir

Determines what will be computated alongside with the final depth | If output_option=lowest_depth, only approximated depths are returned. | If output_option=final_depht_dir, best directions to approximate depths are also returned. | If output_option=all_depth, depths calculated at every iteration are also returned. | If output_option=all_depth_directions, random directions used to project depths are also returned with indices of converging for the solver selected.

evaluate_datasetbool, default=False

Determines if dataset loaded will be evaluated. Automatically sets x to dataset

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.model import DepthEucl
>>> 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]]
>>> np.random.seed(0)
>>> x = np.random.multivariate_normal([1,1,1,1,1], mat2, 10)
>>> data = np.random.multivariate_normal([0,0,0,0,0], mat1, 1000)
>>> model = DepthEucl().load_dataset(data)
>>> model.mahalanobis(x)
[0.17849871 0.10412453 0.1331417  0.13578021 0.3154836  0.29103769
    0.13398989 0.13913017 0.59339051 0.10556139]
>>> model.mahalanobisDepth
[0.17849871 0.10412453 0.1331417  0.13578021 0.3154836  0.29103769
    0.13398989 0.13913017 0.59339051 0.10556139]
>>> model.mahalanobis(x, 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]

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