Rank based alternatives for sample covariance/correlation matrix

[mnarayan]

Description

Enables implementation of simple nonparametric variant of graphical lasso or any other future estimators for the precision matrix.

Overview of nonparametric alternatives

The class of nonparanormal (monotone transformations + gaussian graphical models) and transelliptical (monotone transformations + elliptical graphical models) are covered by the use of

• spearman
• kendall's tau
• winsorized or trim-mean correlation estimates
• symmetric rank covariances includes Hoeffding's D, Bergma-Dassios sign covariance
• k-root of sample covariance substitutes sample covariance or correlation with k-root. Use k=2 to mirror benefits of the sqrt-lasso. Cannot support negative eigenvalues.

Implementation

Task involves creating alternative functions for the rank-based sample correlation/covariance and offering this an alternative to current empirical covariance or correlation options.

Key Steps:

1. Take rows of observations and transform into ranks
2. Unbiased spearman's rankk correlation (eq. 8 and eq. 9) for all pairs of features

The Kendall's tau concordance variant is another alternative, that has better nicer variations for handling ties, weighting higher ranks differently from lower ranks. However, scipy.stats.kendalltau does not support 2D arrays so it is likely to be slow for large dimensions.

Furthermore, the bias correction for correlation via kendalltau amounts using sin(pi / 2 * kendalltau)

Note 1: These rank correlation estimates can violate positive semi-definite requirements (all though the biased spearman and kendall are always P.S.D). Thus, not all graphical model estimators will be compatible with these rank correlation estimates. **However regularized Dantzig-CLIME, CONCORD, and some pseudolikelihood type estimators should be able to handle rank_correlation matrices with negative eigenvalues. **

Note 2: Take advantage of RobustScaler and RankScaler if possible, as well as other BaseEstimators from sklearn.

References

Liu, Han, John Lafferty, and Larry Wasserman.
"The nonparanormal: Semiparametric estimation of high dimensional undirected graphs."
Journal of Machine Learning Research 10.Oct (2009): 2295-2328.

Wilcox, R. R. (1993), Some results on a Winsorized correlation coefficient. British Journal of Mathematical and Statistical Psychology, 46: 339–349.
doi:10.1111/j.2044-8317.1993.tb01020.x

Xue, Lingzhou; Zou, Hui.
Regularized rank-based estimation of high-dimensional nonparanormal graphical models.
Ann. Statist. 40 (2012), no. 5, 2541--2571. doi:10.1214/12-AOS1041.
http://projecteuclid.org/download/pdfview_1/euclid.aos/1359987530

Liu, Han; Han, Fang; Yuan, Ming; Lafferty, John; Wasserman, Larry.
High-dimensional semiparametric Gaussian copula graphical models.
Ann. Statist. 40 (2012), no. 4, 2293--2326. doi:10.1214/12-AOS1037 https://projecteuclid.org/euclid.aos/1358951383

Rina Foygel Barber, Mladen Kolar
"ROCKET: Robust Confidence Intervals via Kendall's Tau for Transelliptical Graphical Models"
https://arxiv.org/abs/1502.07641

Vahe Avagyan, Andrés M. Alonso & Francisco J. Nogales (2017)
"Improving the Graphical Lasso Estimation for the Precision Matrix Through Roots of the Sample Covariance Matrix", Journal of Computational and Graphical Statistics, 26:4, 865-872, DOI: 10.1080/10618600.2017.1340890