Data clustering is one area of data mining that falls into the class unsupervised learning. Cluster analysis divides different classes by discovering internal structure set objects and their relationship. This paper presented a new density method based on modified inversion formula estimation. should allow to improve performance robustness k-means, Gaussian mixture model, other methods. The pri...

Faà di Bruno's formula gives an expression for the derivatives of composition two real-valued functions. In this paper we prove a multivariate and synthesised version in higher dimensions, providing combinatorial chain compositions F ( 1 ) ∘ … m functions l : R N → terms sums over labelled trees. We give several applications formula, including new involution inversion power series. use framewor...

We give a one-parameter deformation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder-Frabetti-Krattenthaler for the antipode of the noncommutative Faá di Bruno algebra. Namely, we obtain a closed formula for the antipode of the one-parameter deformation of this Hopf algebra discovered by Foissy.

A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. The inversion statistic, which is the more intricate one, suffices for specializing a closely related formula to one for the type A Hall-Littlewood polynomials (spher...