"""Gerber Covariance Estimators."""
# Copyright (c) 2023
# Author: Hugo Delatte <delatte.hugo@gmail.com>
# License: BSD 3 clause
# Implementation derived from:
# scikit-learn, Copyright (c) 2007-2010 David Cournapeau, Fabian Pedregosa, Olivier
# Grisel Licensed under BSD 3 clause.
import numpy as np
import numpy.typing as npt
from skfolio.moments.covariance._base import BaseCovariance
from skfolio.utils.stats import corr_to_cov
[docs]
class GerberCovariance(BaseCovariance):
"""Gerber Covariance estimator.
Robust co-movement measure which ignores fluctuations below a certain threshold
while simultaneously limiting the effects of extreme movements.
The Gerber statistic extends Kendall's Tau by counting the proportion of
simultaneous co-movements in series when their amplitudes exceed data-dependent
thresholds.
Three variant has been published:
* Gerber et al. (2015): tend to produce matrices that are non-PSD.
* Gerber et al. (2019): alteration of the denominator of the above statistic.
* Gerber et al. (2022): final alteration to ensure PSD matrix.
The last two variants are implemented.
Parameters
----------
window_size : int, optional
Window size. The model is fitted on the last `window_size` observations.
The default (`None`) is to use all the data.
threshold : float, default=0.5
Gerber threshold. The default value is `0.5`.
psd_variant : bool, default=True
If this is set to True, the Gerber et al. (2022) variant is used to ensure a
positive semi-definite matrix.
Otherwise, the Gerber et al. (2019) variant is used.
The default is `True`.
nearest : bool, default=True
If this is set to True, the covariance is replaced by the nearest covariance
matrix that is positive definite and with a Cholesky decomposition than can be
computed. The variance is left unchanged.
A covariance matrix that is not positive definite often occurs in high
dimensional problems. It can be due to multicollinearity, floating-point
inaccuracies, or when the number of observations is smaller than the number of
assets. For more details, see :func:`~skfolio.utils.stats.cov_nearest`.
The default is `True`.
higham : bool, default=False
If this is set to True, the Higham & Nick (2002) algorithm is used to find the
nearest PD covariance, otherwise the eigenvalues are clipped to a threshold
above zeros (1e-13). The default is `False` and use the clipping method as the
Higham & Nick algorithm can be slow for large datasets.
higham_max_iteration : int, default=100
Maximum number of iteration of the Higham & Nick (2002) algorithm.
The default value is `100`.
Attributes
----------
covariance_ : ndarray of shape (n_assets, n_assets)
Estimated covariance.
n_features_in_ : int
Number of assets seen during `fit`.
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of assets seen during `fit`. Defined only when `X`
has assets names that are all strings.
References
----------
.. [1] "The gerber statistic: A robust co-movement measure for portfolio
optimization".
The Journal of Portfolio Management.
Gerber, S., B. Javid, H. Markowitz, P. Sargen, and D. Starer (2022).
.. [2] "The gerber statistic: A robust measure of correlation".
Gerber, S., B. Javid, H. Markowitz, P. Sargen, and D. Starer (2019).
.. [3] "Enhancing multi-asset portfolio construction under modern portfolio theory
with a robust co-movement measure".
Social Science Research network Working Paper Series.
Gerber, S., H. Markowitz, and P. Pujara (2015).
.. [4] "Deconstructing the Gerber Statistic".
Flint & Polakow, 2023.
"""
def __init__(
self,
window_size: int | None = None,
threshold: float = 0.5,
psd_variant: bool = True,
nearest: bool = True,
higham: bool = False,
higham_max_iteration: int = 100,
):
super().__init__(
nearest=nearest,
higham=higham,
higham_max_iteration=higham_max_iteration,
)
self.window_size = window_size
self.threshold = threshold
self.psd_variant = psd_variant
[docs]
def fit(self, X: npt.ArrayLike, y=None) -> "GerberCovariance":
"""Fit the Gerber covariance estimator.
Parameters
----------
X : array-like of shape (n_observations, n_assets)
Price returns of the assets.
y : Ignored
Not used, present for API consistency by convention.
Returns
-------
self : GerberCovariance
Fitted estimator.
"""
X = self._validate_data(X)
if self.window_size is not None:
X = X[-self.window_size :]
if not (1 > self.threshold > 0):
raise ValueError("The threshold must be between 0 and 1")
n_observations = X.shape[0]
std = X.std(axis=0).reshape((-1, 1))
u = X >= std.T * self.threshold
d = X <= -std.T * self.threshold
n = np.invert(u) & np.invert(d) # np.invert preferred that ~ for type hint
n = n.astype(int)
u = u.astype(int)
d = d.astype(int)
concordant = u.T @ u + d.T @ d
discordant = u.T @ d + d.T @ u
h = concordant - discordant
if self.psd_variant:
corr = h / (n_observations - n.T @ n)
else:
h_sqrt = np.sqrt(np.diag(h)).reshape((-1, 1))
corr = h / (h_sqrt @ h_sqrt.T)
covariance = corr_to_cov(corr, std.reshape(-1))
self._set_covariance(covariance)
return self