arbitragelab.codependence.information

Implementations of mutual information (I) and variation of information (VI) codependence measures from Cornell lecture slides: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3512994&download=yes

Module Contents

Functions

get_optimal_number_of_bins(→ int)

Calculates optimal number of bins for discretization based on number of observations

get_mutual_info(→ float)

Returns mutual information (MI) between two vectors.

variation_of_information_score(→ float)

Returns variantion of information (VI) between two vectors.
get_optimal_number_of_bins(num_obs: int, corr_coef: float = None) → int

Calculates optimal number of bins for discretization based on number of observations and correlation coefficient (univariate case).

Algorithms used in this function were originally proposed in the works of Hacine-Gharbi et al. (2012) and Hacine-Gharbi and Ravier (2018). They are described in the Cornell lecture notes: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3512994&download=yes (p.26)

Parameters:

  • num_obs – (int) Number of observations.

  • corr_coef – (float) Correlation coefficient, used to estimate the number of bins for univariate case.

Returns:

(int) Optimal number of bins.

get_mutual_info(x: numpy.array, y: numpy.array, n_bins: int = None, normalize: bool = False, estimator: str = 'standard') → float

Returns mutual information (MI) between two vectors.

This function uses the discretization with the optimal bins algorithm proposed in the works of Hacine-Gharbi et al. (2012) and Hacine-Gharbi and Ravier (2018).

Read Cornell lecture notes for more information about the mutual information: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3512994&download=yes.

This function supports multiple ways the mutual information can be estimated:

  1. standard - the standard way of estimation - binning observations according to a given number of bins and applying the MI formula.

  2. standard_copula - estimating the copula (as a normalized ranking of the observations) and applying the standard mutual information estimator on it.

  3. copula_entropy - estimating the copula (as a normalized ranking of the observations) and calculating its entropy. Then MI estimator = (-1) * copula entropy.

The last two estimators’ implementation is taken from the blog post by Dr. Gautier Marti. Read this blog post for more information about the differences in the estimators: https://gmarti.gitlab.io/qfin/2020/07/01/mutual-information-is-copula-entropy.html

Parameters:

  • x – (np.array) X vector.

  • y – (np.array) Y vector.

  • n_bins – (int) Number of bins for discretization, if None the optimal number will be calculated. (None by default)

  • normalize – (bool) Flag used to normalize the result to [0, 1]. (False by default)

  • estimator – (str) Estimator to be used for calculation. [standard, standard_copula, copula_entropy] (standard by default)

Returns:

(float) Mutual information score.

variation_of_information_score(x: numpy.array, y: numpy.array, n_bins: int = None, normalize: bool = False) → float

Returns variantion of information (VI) between two vectors.

This function uses the discretization using optimal bins algorithm proposed in the works of Hacine-Gharbi et al. (2012) and Hacine-Gharbi and Ravier (2018).

Read Cornell lecture notes for more information about the variation of information: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3512994&download=yes.

Parameters:

  • x – (np.array) X vector.

  • y – (np.array) Y vector.

  • n_bins – (int) Number of bins for discretization, if None the optimal number will be calculated. (None by default)

  • normalize – (bool) True to normalize the result to [0, 1]. (False by default)

Returns:

(float) Variation of information score.