On a Method of Multivariate Density Estimate Basedon Nearest Neighbours Graphs

A method of multivariate density estimation based on the reweighted nearest neighbours,mimicking the natural neighbours techniques, is presented. Estimation of multivariate densityis important for machine learning, astronomy, biology, physics and econometrics. A 2-additivefuzzy measure is constructed based on proxies for pairwise interaction indices. The neighboursof a point lying in nearly the same direction are treated as redundant, and the contributionof the farthest neighbour is transferred to the nearer neighbour. The calculation of the localpoint density estimate is performed by the discrete Choquet integral, so that the contributionsof the neighbours all around that point are accounted for. This way an approximation to theSibson’s natural neighbours is computed. The method relieves the computational burden of theDelaunay tessellation-based natural neighbours approach in higher dimensions, whose complexityis exponential in the dimension of the data. This method is suitable for density estimates ofstructured data (possibly lying on lower dimensional manifolds), as the nearest neighbours differsignificantly from the natural neighbours in this case.