A boundary corrected expansion of the moments of nearest neighbor distributions

In this paper, the moments of nearest neighbor distance distributions are examined. While the asymptotic form of such moments is well-known, the boundary effect has this far resisted a rigorous analysis. Our goal is to develop a new technique that allows a closed-form high order expansion, where the boundaries are taken into account up to the first order. The resulting theoretical predictions are tested via simulations and found to be much more accurate than the first order approximation obtained by neglecting the boundaries. While our results are of theoretical interest, they definitely also have important applications in statistics and physics. As a concrete example, we mention estimating Renyi entropies of probability distributions. Moreover, the algebraic technique developed may turn out to be useful in other, related problems including estimation of the Shannon differential entropy

On the bounds of the expected nearest neighbor distance

In this paper, we give some contributions for special distributions having unbounded support    for which we derive upper and lower bounds on the expected nearest neighbor distance of the extreme value (Gumbel) distribution as typical

The Hellinger Correlation

In this paper, the defining properties of a valid measure of the dependence between two random variables are reviewed and complemented with two original ones, shown to be more fundamental than other usual postulates. While other popular choices are proved to violate some of these requirements, a class of dependence measures satisfying all of them is identified. One particular measure, that we call the Hellinger correlation, appears as a natural choice within that class due to both its theoretical and intuitive appeal. A simple and efficient nonparametric estimator for that quantity is proposed. Synthetic and real-data examples finally illustrate the descriptive ability of the measure, which can also be used as test statistic for exact independence testing

k-Nearest Neighbor Curves in Imaging Data Classification

Background: Lung disease quantification via medical image analysis is classically difficult. We propose a method based on normalized nearest neighborhood distance classifications for comparing individual CT scan air-trapping distributions (representing 3D segmented parenchyma). Previously, between-image comparisons were precluded by the variation inherent to parenchyma segmentations, the dimensions of which are patient- and image-specific by nature.Method: Nearest neighbor distance estimations are normalized by a theoretical distance according to the uniform distribution of air trapping. This normalization renders images (of different sizes, shapes, and/or densities) comparable. The estimated distances for the k-nearest neighbor describe the proximity of point patterns over the image. Our approach assumes and requires a defined homogeneous space; therefore, a completion pretreatment is applied beforehand.Results: Model robustness is characterized via simulation in order to verify that the required initial transformations do not bias uniformly sampled results. Additional simulations were performed to assess the discriminant power of the method for different point pattern profiles. Simulation results demonstrate that the method robustly recognizes pattern dissimilarity. Finally, the model is applied on real data for illustrative purposes.Conclusion: We demonstrate that a parenchyma-cuboid completion method provides the means of characterizing air-trapping patterns in a chosen segmentation and, importantly, comparing such patterns between patients and images