Gaussian kernels for density estimation with compositional data

Common simplifications of the bandwidth matrix cannot be applied to existing kernels for density estimation with compositional data. In this paper, kernel density estimation methods are modified on the basis of recent developments in compositional data analysis and bandwidth matrix selection theory. The isometric log-ratio normal kernel is used to define a new estimator in which the smoothing parameter is chosen from the most general class of bandwidth matrices on the basis of a recently proposed plug-in algorithm. Both simulated and real examples are presented in which the behaviour of our approach is illustrated, which shows the advantage of the new estimator over existing proposed methods.     

Nonparametric kernel density estimation near the boundary

Standard fixed symmetric kernel-type density estimators are known to encounter problems for positive random variables with a large probability mass close to zero. It is shown that, in such settings, alternatives of asymmetric gamma kernel estimators are superior, but also differ in asymptotic and finite sample performance conditionally on the shape of the density near zero and the exact form of the chosen kernel. Therefore, a refined version of the gamma kernel with an additional tuning parameter adjusted according to the shape of the density close to the boundary is suggested. A data-driven method for the appropriate choice of the modified gamma kernel estimator is also provided. An extensive simulation study compares the performance of this refined estimator to those of standard gamma kernel estimates and standard boundary corrected and adjusted fixed kernels. It is found that the finite sample performance of the proposed new estimator is superior in all settings. Two empirical applications based on high-frequency stock trading volumes and realized volatility forecasts demonstrate the usefulness of the proposed methodology in practice.     

面向小目标图像的快速核密度估计图像阈值分割算法

针对当前小目标图像阈值分割研究工作面临的难题,提出了快速核密 度估计图像阈值分割新方法.首先给出了基于加权核密度估计器的概率计算模 型,通过引入二阶Renyi熵作为阈值选取准则,提出了基于核密度估计的图像阈 值分割算法 (Kernel density estimator based image thresholding algorithm, KDET), 然后通过引入快速压缩集密度估计 (Fast reduced set density estimator, FRSDE)技术,得到核密度估计的 稀疏权系数表示形式,提出快速核密度估计图像阈值分割算法fastKDET,并从 理论上对相关性质进行了深入探讨.实验表明,本文算法对小目标图像 阈值分割问题具有更广泛的适应性,并且对参数变化不敏感.     

Nonparametric density estimation for positive time series

The Gaussian kernel density estimator is known to have substantial problems for bounded random variables with high density at the boundaries. For independent and identically distributed data, several solutions have been put forward to solve this boundary problem. In this paper, we propose the gamma kernel estimator as a density estimator for positive time series data from a stationary α-mixing process. We derive the mean (integrated) squared error and asymptotic normality. In a Monte Carlo simulation, we generate data from an autoregressive conditional duration model and a stochastic volatility model. We study the local and global behavior of the estimator and we find that the gamma kernel estimator outperforms the local linear density estimator and the Gaussian kernel estimator based on log-transformed data. We also illustrate the good performance of the h-block cross-validation method as a bandwidth selection procedure. An application to data from financial transaction durations and realized volatility is provided.     

A geometric approach to non-parametric density estimation

A novel non-parametric density estimator is developed based on geometric principles. A penalised centroidal Voronoi tessellation forms the basis of the estimator, which allows the data to self-organise in order to minimise estimate bias and variance. This approach is a marked departure from usual methods based on local averaging, and has the advantage of being naturally adaptive to local sample density (scale-invariance). The estimator does not require the introduction of a plug-in kernel, thus avoiding assumptions of symmetricity and morphology. A numerical experiment is conducted to illustrate the behaviour of the estimator, and it's characteristics are discussed.     

Probability density estimation from optimally condensed data samples

The requirement to reduce the computational cost of evaluating a point probability density estimate when employing a Parzen window estimator is a well-known problem. This paper presents the Reduced Set Density Estimator that provides a kernel-based density estimator which employs a small percentage of the available data sample and is optimal in the L/sub 2/ sense. While only requiring /spl Oscr/(N/sup 2/) optimization routines to estimate the required kernel weighting coefficients, the proposed method provides similar levels of performance accuracy and sparseness of representation as Support Vector Machine density estimation, which requires /spl Oscr/(N/sup 3/) optimization routines, and which has previously been shown to consistently outperform Gaussian Mixture Models. It is also demonstrated that the proposed density estimator consistently provides superior density estimates for similar levels of data reduction to that provided by the recently proposed Density-Based Multiscale Data Condensation algorithm and, in addition, has comparable computational scaling. The additional advantage of the proposed method is that no extra free parameters are introduced such as regularization, bin width, or condensation ratios, making this method a very simple and straightforward approach to providing a reduced set density estimator with comparable accuracy to that of the full sample Parzen density estimator.     

Entropy based robust estimator and its application to line-based mapping

This paper presents a robust mapping algorithm for an application in autonomous robots. The method is inspired by the notion of entropy from information theory. A kernel density estimator is adopted to estimate the appearance probability of samples directly from the data. An Entropy Based Robust (EBR) estimator is then designed that selects the most reliable inliers of the line segments. The inliers maintained by the entropy filter are those samples that carry more information. Hence, the parameters extracted from EBR estimator are accurate and robust to the outliers. The performance of the EBR estimator is illustrated by comparing the results with the performance of three other estimators via simulated and real data.     

基于核密度估计的入侵检测方法

通过分析现有入侵检测技术的不足,探讨基于孤立点挖掘的入侵检测技术的优势,提出一种基于核密度估计的入侵检测方法。该方法通过核密度估计求出孤立点的近似集,再通过筛选近似集获得最终的孤立点集合,从而检测入侵记录。阐述了具体实现方案,通过仿真实验验证了该方法的可行性。     

Estimating multidimensional probability fields using the Field Estimator for Arbitrary Spaces (FiEstAS) with applications to astrophysics

The Field Estimator for Arbitrary Spaces (FiEstAS) computes the continuous probability density field underlying a given discrete data sample in multiple, non-commensurate dimensions. The algorithm works by constructing a metric-independent tessellation of the data space based on a recursive binary splitting. Individual, data-driven bandwidths are assigned to each point, scaled so that a constant “mass” M₀ is enclosed. Kernel density estimation may then be performed for different kernel shapes, and a combination of balloon and sample point estimators is proposed as a compromise between resolution and variance. A bias correction is evaluated for the particular (yet common) case where the density is computed exactly at the locations of the data points rather than at an uncorrelated set of locations. By default, the algorithm combines a top-hat kernel with M₀=2.0 with the balloon estimator and applies the corresponding bias correction. These settings are shown to yield reasonable results for a simple test case, a two-dimensional ring, that illustrates the performance for oblique distributions, as well as for a six-dimensional Hernquist sphere, a fairly realistic model of the dynamical structure of stellar bulges in galaxies and dark matter haloes in cosmological N-body simulations. Results for different parameter settings are discussed in order to provide a guideline to select an optimal configuration in other cases. Source code is available upon request.     

Bayesian adaptive bandwidth kernel density estimation of irregular multivariate distributions

In this paper, we propose a new methodology for multivariate kernel density estimation in which data are categorized into low- and high-density regions as an underlying mechanism for assigning adaptive bandwidths. We derive the posterior density of the bandwidth parameters via the Kullback-Leibler divergence criterion and use a Markov chain Monte Carlo (MCMC) sampling algorithm to estimate the adaptive bandwidths. The resulting estimator is referred to as the tail-adaptive density estimator. Monte Carlo simulation results show that the tail-adaptive density estimator outperforms the global-bandwidth density estimators implemented using different global bandwidth selection rules. The inferential potential of the tail-adaptive density estimator is demonstrated by employing the estimator to estimate the bivariate density of daily index returns observed from the USA and Australian stock markets.