Clustering using kernel entropy principal component analysis and variable kernel estimator

Clustering as unsupervised learning method is the mission of dividing data objects into clusters with common characteristics. In the present paper, we introduce an enhanced technique of the existing EPCA data transformation method. Incorporating the kernel function into the EPCA, the input space can be mapped implicitly into a high-dimensional of feature space. Then, the Shannon’s entropy estimated via the inertia provided by the contribution of every mapped object in data is the key measure to determine the optimal extracted features space. Our proposed method performs very well the clustering algorithm of the fast search of clusters’ centers based on the local densities’ computing. Experimental results disclose that the approach is feasible and efficient on the performance query.

Variable bandwidth kernel regression estimation

In this paper we propose a variable bandwidth kernel regression estimator for $i.i.d.$ observations in $\mathbb{R}^2$ to improve the classical Nadaraya-Watson estimator. The bias is improved to the order of $O(h_n^4)$ under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kernel method over the classical kernel method.

L2L^{2} boundedness for commutators of fractional differential type Marcinkiewicz integral with rough variable kernel and BMO Sobolev spaces

In this paper, for {0<\gamma<1} and {b\in I_{\gamma}(\mathrm{BMO})}, the authors give the {L^{2}({\mathbb{R}}^{n})} boundedness of {\mu_{\gamma;b}}, the commutator of a fractional differential type Marcinkiewicz integral with rough variable kernel, which is an extension of some known results.