Relative performance of mutual information estimation methods for quantifying the dependence among short and noisy data

Abstract

Commonly used dependence measures, such as linear correlation, cross-correlogram, or Kendall's , cannot capture the complete dependence structure in data unless the structure is restricted to linear, periodic, or monotonic. Mutual information ͑MI͒ has been frequently utilized for capturing the complete dependence structure including nonlinear dependence. Recently, several methods have been proposed for the MI estimation, such as kernel density estimators ͑KDEs͒, k-nearest neighbors ͑KNNs͒, Edgeworth approximation of differential entropy, and adaptive partitioning of the XY plane. However, outstanding gaps in the current literature have precluded the ability to effectively automate these methods, which, in turn, have caused limited adoptions by the application communities. This study attempts to address a key gap in the literaturespecifically, the evaluation of the above methods to choose the best method, particularly in terms of their robustness for short and noisy data, based on comparisons with the theoretical MI estimates, which can be computed analytically, as well with linear correlation and Kendall's . Here we consider smaller data sizes, such as 50, 100, and 1000, and within this study we characterize 50 and 100 data points as very short and 1000 as short. We consider a broader class of functions, specifically linear, quadratic, periodic, and chaotic, contaminated with artificial noise with varying noise-to-signal ratios. Our results indicate KDEs as the best choice for very short data at relatively high noise-to-signal levels whereas the performance of KNNs is the best for very short data at relatively low noise levels as well as for short data consistently across noise levels. In addition, the optimal smoothing parameter of a Gaussian kernel appears to be the best choice for KDEs while three nearest neighbors appear optimal for KNNs. Thus, in situations where the approximate data sizes are known in advance and exploratory data analysis and/or domain knowledge can be used to provide a priori insights into the noise-to-signal ratios, the results in the paper point to a way forward for automating the process of MI estimation.

Minimum Mutual Information and Non-Gaussianity through the Maximum Entropy Method: Estimation from Finite Samples

“…This paper is the natural follow-up of PP12 [12], studying now the statistics (mean or bias, variance and distribution) of the MinMI estimation errors: min, H is the ME estimation issued from N-sized samples of iid outcomes. Those errors are roughly similar to those of MI and entropy generic estimator's errors (see [13,14] for a thorough review and performance comparisons between MI estimators). Their mean (bias), variance and higher-order moments are written in terms of 1 N  powers, thus covering intermediate and asymptotic N ranges [15], with specific applications in neurophysiology [16,17,18].…”

Towards Inferring Communication Patterns in Online Social Networks

“…, R k }, e.g., Both entropy and conditional entropy are related to mutual information through the following expression: I(R 1 ; R 2 ) = H(R 1 ) − H(R 1 |R 2 ). We have favoured mutual information over other measures of statistical dependence, such as correlation coefficients, for its equitability, i.e., its ability to detect general, not only linear or monotonic, dependence Khan et al (2007); Kinney and Atwal (2014). We have chosen however to present results in this paper in terms of conditional entropy, which is trivial to obtain from the corresponding mutual information.…”

Objective reduction based on nonlinear correlation information entropy

“…We find NCIE (Wang et al 2005) to be a very robust measure for both linearly and nonlinearly correlated datasets, which has been applied to the analysis of neurophysiological signals (Pereda et al 2005), the quantification of the dependence among noisy data (Khan et al 2007), etc. Therefore, we adopt NCIE as a correlation measure in objective reduction and study its impact on online objective reduction approaches.…”