Welcome to covmats’s documentation!#
covmats#
🐍 Covariance matrices representation.
The complete and up to date documentation can be found here: https://covmats.readthedocs.io.
🎯 Motivations#
Calculations involving covariance matrices (e.g. linear algebra, data whitening, multivariate normal function evaluation) are often performed more efficiently using a decomposition of the covariance matrix instead of the covariance matrix itself. For large scale application, a dense covariance matrix would not even fit in memory and one must rely on low-rank approximations. This package allows the user to construct an object representing a covariance matrix using any of several decompositions/approximations and perform calculations using a common interface.
The common interface CovarianceMatrix can be seen as a extension of the class scipy.stats.Covariance as it inherits from it (thus making it compatible with all scipy.stats functions and classes) and dope it with LinearOperator capabilities.
The package is used in large-scale inversion packages such as pypcga, pyesmda and pyrtid.
🚀 Quick start#
To install covmats, the easiest way is through pip:
pip install covmats
Or alternatively using conda
conda install covmats
You might also clone the repository and install from source
pip install -e .
Once the installation is done, covmats is straightforward to use and proposes the following full-rank covariance representations:
It also provides low-rank approximations suitable for large scale problems:
as well as kernel-based, matrix-free linear operators for point-cloud data:
and a small hierarchy of prior/drift terms for geostatistical regularization:
The two companion tutorial notebooks, examples_covariances.py and examples_priors.py, walk through every one of these classes in detail. The rest of this section gives a condensed overview.
Let’s start by importing numpy, scipy and covmats for the tests and define a random number generator seed for reproducibility:
import numpy as np
import scipy as sp
import covmats
rng_seed = 2026
First example with a diagonal covariance matrix#
In the following, we define a (3 x 3) covariance matrix defining only its diagonal (i.e., all elements of the random vectors are independent). Using scipy, it is possible to compute the pdf.
d = [1, 2, 3]
A33 = np.diag(d) # a diagonal covariance matrix
x = [4, -2, 5] # a point of interest
dist = sp.stats.multivariate_normal(mean=[0, 0, 0], cov=A33)
dist.pdf(x)
np.float64(4.9595685102808205e-08)
It is possible to obtain a dense representation in a straightforward manner:
cov_diag33 = covmats.CovViaDiagonal(d)
dist = sp.stats.multivariate_normal(mean=[0, 0, 0], cov=cov_diag33)
dist.pdf(x)
np.float64(4.9595685102808205e-08)
It is compatible with the stats API from scipy since the base class inherit from Covariance.
Every representation also exposes todense, solve, precision and
log_pdet, as well as sample_mvnormal, whiten and colorize for
fast Monte-Carlo sampling and data whitening:
samples = cov_diag33.sample_mvnormal(shape=[10000], random_state=rng_seed)
np.round(np.cov(samples, rowvar=False), 1)
array([[ 1., 0., -0.],
[ 0., 2., -0.],
[-0., -0., 3.]])
Cholesky and precision-based representations#
CovViaCholesky wraps a dense Cholesky factor L such that A = L @ L.T, while CovViaPrecisionCholesky does the same for the precision matrix Q = A^-1 (the natural representation for Gaussian Markov Random Fields, where Q is sparse). Sparse counterparts, CovViaSparseCholesky and CovViaSparsePrecisionCholesky, build on a SparseCholeskyFactor (an LDL’ factorization) for large, sparse problems:
rng = np.random.default_rng(rng_seed)
n = 4
B = rng.random((n, n))
A = B @ B.T + n * np.eye(n) # a random SPD matrix
cov_cho = covmats.CovViaCholesky(np.linalg.cholesky(A))
np.allclose(cov_cho.todense(), A)
True
Low-rank representations#
For very large problems, CovViaEigenFactorization (a truncated eigen decomposition) and CovViaEnsemble (an ensemble of anomalies, as used in ensemble Kalman filtering) never require forming the dense covariance matrix:
ensemble = rng.multivariate_normal(np.zeros(6), np.eye(6) * 2 + 0.3, size=200)
cov_ens = covmats.CovViaEnsemble(ensemble)
cov_ens.shape
(6, 6)
get_linop_eigen_factorization and eigen_factorize_cov_mat build a randomized low-rank eigen factorization directly from any CovarianceMatrix or CovKernelAsLinop instance, and get_explained_var reports how much variance each retained mode captures – useful to pick a truncation rank.
Kernel-based linear operators#
When the covariance is defined analytically through a kernel evaluated on a point cloud, CovKernelAsLinop (dense evaluation, any point cloud) and CovKernelAsLinopViaFFT (FFT-based, regular grids only) expose a LinearOperator without requiring the covariance matrix to be assembled up front:
pts = covmats.get_pts_coords_regular_grid(mesh_dim=1.0, shape=(6, 6))
cov_kernel = covmats.CovKernelAsLinop(
pts, lambda d: np.exp(-d), len_scale=np.array([2.0, 2.0])
)
cov_kernel.shape
(36, 36)
Priors and drift matrices#
Alongside covariance representations, covmats provides PriorTerm subclasses to describe the deterministic mean/trend of a field, from a simple NullPriorTerm to a LinearDriftMatrix expressing a spatially-varying trend as m = X @ beta:
drift = covmats.LinearDriftMatrix(pts)
drift.beta = np.array([1.0, 0.5, -0.5])
drift.get_values(np.zeros(pts.shape[0])).shape
(36,)
Conditional and unconditional simulations#
🏗️ Complete example with supporting paper coming Q1 2027.
🔑 License#
This project is released under the BSD 3-Clause License.
Copyright (c) 2026, Antoine COLLET. All rights reserved.
For more details, see the LICENSE file included in this repository.
⚠️ Disclaimer#
This software is provided “as is”, without warranty of any kind, express or implied, including but not limited to the warranties of merchantability, fitness for a particular purpose, or non-infringement. In no event shall the authors or copyright holders be liable for any claim, damages, or other liability, whether in an action of contract, tort, or otherwise, arising from, out of, or in connection with the software or the use or other dealings in the software.
By using this software, you agree to accept full responsibility for any consequences, and you waive any claims against the authors or contributors.
📧 Contact#
For questions, suggestions, or contributions, you can reach out via:
Email: antoinecollet5@gmail.com
GitHub: antoinecollet5/covmats
We welcome contributions!
📚 References#
TODO
Free software: SPDX-License-Identifier: BSD-3-Clause