Add Multivariate T likelihood cost

[johannvk]

This PR adds a new cost function to skchange, the robust cost MultivariateTCost.

As the maximum likelihood estimator for the scale matrix of a multivariate student's T distribution is now known analytically we solve a fixed point iteration equation to find the MLE scale matrix on each interval, before evaluating the log likelihood of the sample at the computed MLE scale matrix estimate. However, using the capbilities of Numba, the performance is greatly improved as compared to a purely numpy and scipy based implementation.
Currently the column medians are used as a quick and robust estimator of the distribution mean.

The fatness of the tails in the T distribution is controlled by the degrees of freedom (dof), and when using this cost one can either specify a fixed degrees of freedom for the cost object, or estimate the degrees of freedom when fitting the cost to samples.

References:

• Aeschliman, Chad, Johnny Park, and Avinash C. Kak. “A Novel Parameter Estimation Algorithm for the Multivariate T-Distribution and Its Application to Computer Vision.” In Computer Vision – ECCV 2010, edited by Kostas Daniilidis, Petros Maragos, and Nikos Paragios, 594–607. Berlin, Heidelberg: Springer, 2010. link
• Ollila, Esa, Daniel P. Palomar, and Frédéric Pascal. “Shrinking the Eigenvalues of M-Estimators of Covariance Matrix.” arXiv, October 28, 2020. link
• Pascal, Frédéric, Esa Ollila, and Daniel P. Palomar. “Improved Estimation of the Degree of Freedom Parameter of Multivariate T-Distribution.” In 2021 29th European Signal Processing Conference (EUSIPCO), 860–64, 2021. link

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 98.74%. Comparing base (74edbf9) to head (80e5c64).
Report is 35 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #58      +/-   ##
==========================================
+ Coverage   98.53%   98.74%   +0.21%     
==========================================
  Files          49       50       +1     
  Lines        1838     2159     +321     
==========================================
+ Hits         1811     2132     +321     
  Misses         27       27              

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[johannvk]

I experienced a problem when testing this PR on another project using skchange now, and the issue was fixed when I upgraded numba to version 0.61 and its dependency llvmlite was upgraded from 0.43.0 to 0.44.0. Is it possible that we have a dependency on these newest versions of numba and llvmlite?