LedoitWolfCovariance¶

Online Ledoit-Wolf shrinkage covariance.

Shrinks the exponentially weighted sample covariance towards a scaled identity (a sphere) by a data-driven intensity, following Ledoit & Wolf (2004). The shrinkage intensity is estimated online from the dispersion of the per-observation scatter around the running covariance, and is recomputed on read; the per-step update stays O(d^2) with no stored window.

When to use it. The raw sample covariance is a poor estimate when the number of variables is large relative to the (effective) number of observations: it is noisy and often ill-conditioned or singular, which wrecks anything that inverts it (portfolio optimisation, Mahalanobis distances, Gaussian likelihoods). Shrinkage trades a little bias for a large reduction in variance, producing a well-conditioned, invertible matrix. Ledoit-Wolf picks the shrinkage intensity for you, so it is a strong default when there are many assets relative to clean history.

Parameters¶

• fading_factor

  Type → float

  Default → 0.5

  Recency weight of the underlying exponentially weighted covariance. The effective sample size is roughly 1 / fading_factor.

Attributes¶

• matrix

Examples¶

On the ten stocks of datasets.SP500Stocks with a short effective memory, the raw covariance is poorly conditioned; Ledoit-Wolf shrinkage tames it.

import numpy as np
from river import covariance, datasets

ewa = covariance.EwaCovariance(fading_factor=0.05)
lw = covariance.LedoitWolfCovariance(fading_factor=0.05)
for x, _ in datasets.SP500Stocks():
    ewa.update(x)
    lw.update(x)

def condition_number(cov):
    names = sorted({i for i, _ in cov.matrix})
    M = np.array([[cov[i, j] for j in names] for i in names])
    return np.linalg.cond(M)

condition_number(ewa) > condition_number(lw)

np.True_

Methods¶

References¶

• Ledoit, O. and Wolf, M., 2004. A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2), pp.365-411.