  We present a variant of the hyper-quadtree that divides a multidimensional
space according to the hyperplanes associated to the principal components of
the data in each hyperquadrant. Each of the $2^\lambda$ hyper-quadrants is a
data partition in a $\lambda$-dimension subspace, whose intrinsic
dimensionality $\lambda\leq d$ is reduced from the root dimensionality $d$ by
the principal components analysis, which discards the irrelevant eigenvalues of
the local covariance matrix. In the present method a component is irrelevant if
its length is smaller than, or comparable to, the local inter-data spacing.
Thus, the covariance hyper-quadtree is fully adaptive to the local
dimensionality. The proposed data-structure is used to compute the anisotropic
K nearest neighbors (kNN), supported by the Mahalanobis metric. As an
application, we used the present k nearest neighbors method to perform density
estimation over a noisy data distribution. Such estimation method can be
further incorporated to the smoothed particle hydrodynamics, allowing computer
simulations of anisotropic fluid flows.
