shrinkage¶

Linear and nonlinear shrinkage estimators for covariance matrices, following Ledoit and Wolf.

Sample covariance matrices are noisy when the number of variables is large relative to the number of observations. Shrinkage pulls the sample estimate toward a structured target (linear shrinkage) or reshapes its entire eigenvalue spectrum (nonlinear shrinkage), trading a little bias for a large reduction in variance and producing a better-conditioned, invertible estimate.

Installation¶

```bash +RHIZA_SKIP pip install git+https://github.com/Jebel-Quant/shrinkage.git

## Usage

Both estimators take a raw data matrix of shape `(N, p)` — `N` observations on
`p` variables — and return a `(p, p)` covariance estimator. By default the data
is demeaned and the effective sample size is reduced by one; pass `k=0` to skip
demeaning, or `k=1` if `Y` is already demeaned.

### Linear shrinkage

`cov1para` implements the Ledoit-Wolf (2004) one-parameter estimator, which
shrinks the sample covariance toward a scaled identity (equal variances, zero
covariances):

```python
import numpy as np

from shrinkage.linear import cov1para

rng = np.random.default_rng(0)
Y = rng.standard_normal((250, 3))  # 250 observations, 3 variables

estimator = cov1para(Y)
print(estimator.shape)
print(np.allclose(estimator, estimator.T))  # symmetric

(3, 3)
True

Nonlinear shrinkage¶

nonlinear_shrinkage implements the Ledoit-Wolf quadratic-inverse shrinkage (QIS) estimator. Instead of a single shrinkage intensity it reshapes the whole eigenvalue spectrum, which is especially effective in the high-dimensional regime where p is comparable to (or larger than) N:

import numpy as np

from shrinkage.nonlinear import nonlinear_shrinkage

rng = np.random.default_rng(0)
Y = rng.standard_normal((200, 50))  # 200 observations, 50 variables

estimator = nonlinear_shrinkage(Y)
print(estimator.shape)
print(np.allclose(estimator, estimator.T))  # symmetric

(50, 50)
True

Documentation¶

See the API reference for the full estimator documentation.

References and acknowledgements¶

The estimators follow the work of Olivier Ledoit and Michael Wolf, and are ported from their reference implementation at pald22/covShrinkage:

• Ledoit, O. and Wolf, M. (2004). "A well-conditioned estimator for large-dimensional covariance matrices." Journal of Multivariate Analysis (the one-parameter cov1para estimator).
• Ledoit, O. and Wolf, M. (2022). "Quadratic shrinkage for large covariance matrices." Bernoulli (the nonlinear_shrinkage / QIS estimator).