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

Commonly used dependence measures, such as linear correlation, cross-correlogram, or Kendall’s $\tau$, cannot capture the complete dependence structure in data unless the structure is restricted to linear, periodic, or monotonic. Mutual information (MI) 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 (KDEs), $k$-nearest neighbors (KNNs), 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 literature—specifically, 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 $\tau$. 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.

Figure 1

(Color Online) Plot of 100 points with different noise-to-signal ratios (shown by plus) and with zero noise level (shown by dots). Noise-to-signal ratios on the left and right figures are 0.1 and 0.5, respectively. (a) $X\sim N(0,1)$, $Y:y_i={x_i}^2+{\epsilon }_i$, where $\epsilon \sim N(0,{\sigma }_{\epsilon })$ is the Gaussian noise with zero mean and ${\sigma }_{\epsilon }$ standard deviation. (b) $X:x_i=H_{x_i}+\epsilon x_i$, $Y:y_i=H_{y_i}+\epsilon y_i$, where $H_X$ and $H_Y$ are the $X$ and $Y$ components of the Henon map, respectively. $\epsilon x\sim N(0,{\sigma }_{H_X})$ and $\epsilon y\sim N(0,{\sigma }_{H_Y})$, where ${\sigma }_{H_X}$ and ${\sigma }_{H_Y}$ are the standard deviations of $H_X$ and $H_Y$, respectively.

Figure 2

(Color online) Linear: comparisons between linear CCs from LR and nonlinear CCs from KDE, KNN, Edgeworth, Cellucci, and Kendall’s $\tau$, at different noise-to-signal ratios $({\sigma }_{ϵ}∕{\sigma }_s)$ for (a) 50 points, (b) 100 points, and (c) 1000 points.

Figure 3

(Color online) Quadratic: comparisons between linear CCs from LR and nonlinear CCs from KDE, KNN, Edgeworth, Cellucci, and Kendall’s $\tau$, at different noise-to-signal ratios $({\sigma }_{ϵ}∕{\sigma }_s)$ for (a) 50 points, (b) 100 points, and (c) 1000 points.

Figure 4

(Color online) Periodic: comparisons between linear CCs from LR and nonlinear CCs from KDE, KNN, Edgeworth, Cellucci, and Kendall’s $\tau$, at different noise-to-signal ratios $({\sigma }_{ϵ}∕{\sigma }_s)$ for (a) 50 points, (b) 100 points, and (c) 1000 points. In (c), LR overlaps exactly with Edgeworth.

Figure 5

(Color online) Chaotic: comparisons between linear CCs from LR and nonlinear CCs from KDE, KNN, Edgeworth, Cellucci, and Kendall’s $\tau$, at different noise-to-signal ratios $({\sigma }_{ϵ}∕{\sigma }_s)$ for (a) 50 points, (b) 100 points, and (c) 1000 points.

Figure 6

(Color online) Performance of KDE and KNN with different values of smoothing parameter $\left(h\right)$ and number of nearest neighbors $\left(k\right)$, respectively. The results from quadratic and periodic functions are presented in the left and right, respectively. (a) KDE with 100 points, (b) KNN with 100 points, and (c) KNN with 1000 points. In (a), $h_o$ is the optimal smoothing parameter for a Gaussian kernel given in Eq. (8).

Figure 7

(Color online) Linear: normal (left) and kernel (right) densities with different noise-to-signal ratios $({\sigma }_{ϵ}∕{\sigma }_s)$ with 100 points. For kernel density, a Gaussian kernel with optimal smoothing parameter $h_o$ given in Eq. (8) is used. (a) ${\sigma }_{ϵ}∕{\sigma }_s=0.2$. (b) ${\sigma }_{ϵ}∕{\sigma }_s=0.9$. The linear dependence structure can be seen clearly in (a) but cannot be readily identified in (b) based on an eye estimation.

Figure 8

(Color online) Quadratic: normal (left) and kernel (right) densities with different noise-to-signal ratios $({\sigma }_{ϵ}∕{\sigma }_s)$ with 100 points. For kernel density, a Gaussian kernel with optimal smoothing parameter $h_o$ given in Eq. (8) is used. (a) ${\sigma }_{ϵ}∕{\sigma }_s=0.2$. (b) ${\sigma }_{ϵ}∕{\sigma }_s=0.9$. At low noise, such as in (a), the nonlinear dependence can be clearly seen as shown by the kernel density. However, at high noise, such as in (b), the dependence structure is not readily discernible visually from the kernel density.

Figure 9

(Color online) Periodic: normal (left) and kernel (right) densities with different noise-to-signal ratios $({\sigma }_{ϵ}∕{\sigma }_s)$ with 100 points. For kernel density, a Gaussian kernel with optimal smoothing parameter $h_o$ given in Eq. (8) is used. (a) ${\sigma }_{ϵ}∕{\sigma }_s=0.2$. (b) ${\sigma }_{ϵ}∕{\sigma }_s=0.9$. With increasing noise levels, the nonlinear dependence structure cannot be identified visually as shown by the kernel density plots.

Figure 10

(Color online) Chaotic: normal (left) and kernel (right) densities with different noise-to-signal ratios $({\sigma }_{ϵ}∕{\sigma }_s)$ with 100 points. For kernel density, a Gaussian kernel with optimal smoothing parameter $h_o$ given in Eq. (8) is used. (a) ${\sigma }_{ϵ}∕{\sigma }_s=0.2$. (b) ${\sigma }_{ϵ}∕{\sigma }_s=0.9$. Kernel density plot shows the Henon attractor in (a). However, the Henon attractor cannot be readily distinguished visually in (b).