Abstract: We present a method to numerically estimate the densities of a discretely sampled data based on a binary space partitioning tree. We start with a root node containing all the particles and then recursively divide each node into two nodes each containing roughly equal number of particles, until each of the nodes contains only one particle. The volume of such a leaf node provides an estimate of the local density and its shape provides an estimate of the variance. We implement an entropy-based node splitting criterion that results in a significant improvement in the estimation of densities compared to earlier work. The method is completely metric free and can be applied to arbitrary number of dimensions. We use this method to determine the appropriate metric at each point in space and then use kernel-based methods for calculating the density. The kernel-smoothed estimates were found to be more accurate and have lower dispersion. We apply this method to determine the phase-space densities of dark matter haloes obtained from cosmological N-body simulations. We find that contrary to earlier studies, the volume distribution function v(f) of phase-space density f does not have a constant slope but rather a small hump at high phase-space densities. We demonstrate that a model in which a halo is made up by a superposition of Hernquist spheres is not capable in explaining the shape of v(f) versus f relation, whereas a model which takes into account the contribution of the main halo separately roughly reproduces the behaviour as seen in simulations. The use of the presented method is not limited to calculation of phase-space densities, but can be used as a general purpose data-mining tool and due to its speed and accuracy it is ideally suited for analysis of large multidimensional data sets.
Credit: Sharma, Sanjib; Steinmetz, Matthias