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Nonparametric estimation of mutual information is used in a wide range ofscientific problems to quantify dependence between variables. The k-nearestneighbor (knn) methods are consistent, and therefore expected to work well forlarge sample size. These methods use geometrically regular local volumeelements. This practice allows maximum localization of the volume elements, butcan also induce a bias due to a poor description of the local geometry of theunderlying probability measure. We introduce a new class of knn estimators thatwe call geometric knn estimators (g-knn), which use more complex local volumeelements to better model the local geometry of the probability measures. As anexample of this class of estimators, we develop a g-knn estimator of entropyand mutual information based on elliptical volume elements, capturing the localstretching and compression common to a wide range of dynamical systemsattractors. A series of numerical examples in which the thickness of theunderlying distribution and the sample sizes are varied suggest that localgeometry is a source of problems for knn methods such as theKraskov-St\"{o}gbauer-Grassberger (KSG) estimator when local geometric effectscannot be removed by global preprocessing of the data. The g-knn methodperforms well despite the manipulation of the local geometry. In addition, theexamples suggest that the g-knn estimators can be of particular relevance toapplications in which the system is large, but data size is limited.

Geometric k-nearest neighbor estimation of entropy and mutual information