When working with noisy data coming from complex processes it can be helpful to apply Kalman filters, and use repeated measurements to estimate the level of noise and to improve the estimate of the underlying process that generated the data.
But for the initial setting up of the filter it is necessary to make statements to the level of noise in the various channels, and the covariance between them.
The autocovariance least-squares method has been shown to be effective at off-line noise covariance estimation. This module is a Python translation of the ALS Package, Release 5.0 written in Octave. The latest release can be found on the Rawlings Group website.
The initial translation was performed using SMOP and then refined by hand. All basic functions and most tests have been verified to be correct, but the re-implementation of the ltv_als function has been skipped, as I was unable to translate some of the underlying functions. For further details see the corresponding file.
When I first translated this package, some of the tests revealed severe and quite opaque issues with some of the matrices when the installed NumPy library was compiled with MKL, as was (and still is) the default. It was not entirely clear what the underlying cause was, but could be resolved by using NumPy compiled with OpenBLAS. For more information check out the docstrings in als_utils.py. Now, around one year (mid-2025) later, this issue appears to be resolved, as the default installations do not crash. The comments are still included in case I am mistaken.
A currently working Anaconda/Mamba environment.yml is included in this repository.
As all function have been reimplemented using Numba, speed should be at the same level as the original Octave package, and is actually a bit higher for some of the functions.
You can find some translated examples in this Jupyter notebook. There are also a small number of tests (timing and the aforementioned NumPy issue) in another notebook.