Nonparametric density estimation using Copula Transform, Bayesian sequential partitioning and diffusion-based Kernel estimator

Non-parametric density estimation methods are more flexible than parametric methods, due to the fact that they do not assume any specific shape or structure for the data. Most non-parametric methods, like Kernel estimation, require tuning of parameters to achieve good data smoothing, a non-trivial task, even in low dimensions. In higher dimensions, sparsity of data in local neighborhoods becomes a challenge even for non-parametric methods. In this paper we use the copula transform and two efficient non-parametric methods to develop a new method for improved non-parametric density estimation in multivariate domain. After separation of marginal and joint densities using copula transform, a diffusion-based kernel estimator is employed to estimate the marginals. Next, Bayesian sequential partitioning (BSP) is used in the joint density estimation

Offline and Online Density Estimation for Large High-Dimensional Data

Density estimation has wide applications in machine learning and data analysis techniques including clustering, classification, multimodality analysis, bump hunting and anomaly detection. In high-dimensional space, sparsity of data in local neighborhood makes many of parametric and nonparametric density estimation methods mostly inefficient. This work presents development of computationally efficient algorithms for high-dimensional density estimation, based on Bayesian sequential partitioning (BSP). Copula transform is used to separate the estimation of marginal and joint densities, with the purpose of reducing the computational complexity and estimation error. Using this separation, a parallel implementation of the density estimation algorithm on a 4-core CPU is presented. Also, some example applications of the high-dimensional density estimation in density-based classification and clustering are presented. Another challenge in the area of density estimation rises in dealing with online sources of data, where data is arriving over an open-ended and non-stationary stream. This calls for efficient algorithms for online density estimation. An online density estimator needs to be capable of providing up-to-date estimates of the density, bound to the available computing resources and requirements of the application. In response to this, BBSP method for online density estimation is introduced. It works based on collecting and processing the data in blocks of fixed size, followed by a weighted averaging over block-wise estimates of the density. Proper choice of block size is discussed via simulations for streams of synthetic and real datasets. Further, with the purpose of efficiency improvement in offline and online density estimation, progressive update of the binary partitions in BBSP is proposed, which as simulation results show, leads into improved accuracy as well as speed-up, for various block sizes