A nonparametric classification scheme with mean squared error criterion

A statistical classification scheme for a given set of data requires knowledge of the probability distribution of the observations. Traditional approaches to this problem have revolved around chosen various parametric forms for the probability distribution and evaluating these by goodness of fit methods. Among the difficulties with this method are that it is time consuming, it may not lead to satisfactory results and it may lie beyond the statistical expertise of many practitioners. In this paper, the author's consider the use of a recently developed nonparametric probability density estimator in classification schemes with mean squared error loss criterion. Classical parametric approaches are compared to the nonparametric method on simulated data on the basis of the misclassification probability. Real data from the medical and biological sciences are also used to illustrate the usefulness of the nonparametric method.     

一种基于均值偏移的自动运动分割方法

运动分割是计算机视觉领域研究的重要内容。提出一种基于均值偏移的自动运动分割算法。该方法首先用特征点匹配关系获得特征点的运动轨迹,并以轨迹的运动向量作为特征,再用均值偏移算法对轨迹的运动向量进行聚类。均值偏移缩小相似的运动向量之间的差别,同时扩大不同运动的运动向量之间的差距。为了自动获得运动分类数,还提出了一种基于非参数核密度的自动分类方法,该方法通过估计运动向量的密度分布,用核密度图自动确定运动分类数。实验结果表明,该算法分割精度高、鲁棒性好,能够自动确定运动分类数。     

Control-loop diagnosis using continuous evidence through kernel density estimation

While most previous work in the subject of Bayesian Fault diagnosis and control loop diagnosis use discretized evidence for performing diagnosis (an example of evidence being a monitor reading), discretizing continuous evidence can result in information loss. This paper proposes the use of kernel density estimation, a non-parametric technique for estimating the density functions of continuous random variables. Kernel density estimation requires the selection of a bandwidth parameter, used to specify the degree of smoothing, and a number of bandwidth selection techniques (optimal Gaussian, sample-point adaptive, and smoothed cross-validation) are discussed and compared. Because kernel density estimation is known to have reduced performance in high dimensions, this paper also discusses a number of existing preprocessing methods that can be used to reduce the dimensionality (grouping according to dependence, and independent component analysis). Bandwidth selection and dimensionality reduction techniques are tested on a simulation and an industrial process.     

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.     

Efficient On-Line Nonparametric Kernel Density Estimation

Nonparametric density estimation has broad applications in computational finance especially in cases where high frequency data are available. However, the technique is often intractable, given the run times necessary to evaluate a density. We present a new and efficient algorithm based on multipole techniques. Given the n kernels that estimate the density, current methods take O(n) time directly to sum the kernels to perform a single density query. In an on-line algorithm where points are continually added to the density, the cumulative O(n ² ) running time for n queries makes it very costly, if not impractical, to compute the density for large n . Our new Multipole-accelerated On-line Density Estimation (MODE) algorithm is general in that it can be applied to any kernel (in arbitrary dimensions) that admits a Taylor series expansion. The running time for a density query reduces to O (logn) or even constant time, depending on the kernel chosen, and, hence, the cumulative running time is reduced to O (n logn) or O(n) , respectively. Our results show that the MODE algorithm provides dramatic advantages over the direct approach to density evaluation. For example, we show using a modest computing platform that on-line density updates and queries for 1 million points and two dimensions take 8 days to compute using the direct approach versus 40 seconds with the MODE approach. Received June 2, 1997; revised March 3, 1998.     

A semiparametric density estimation approach to pattern classification

A new multivariate density estimator suitable for pattern classifier design is proposed. The data are first transformed so that the pattern vector components with the most non-Gaussian structure are separated from the Gaussian components. Nonparametric density estimation is then used to capture the non-Gaussian structure of the data while parametric Gaussian conditional density estimation is applied to the rest of the components. Both simulated and real data sets are used to demonstrate the potential usefulness of the proposed approach.     

Parameter estimation and system identification for continuously-observed quantum systems

This paper gives an overview of parameter estimation and system identification for quantum input–output systems by continuous observation of the output field. We present recent results on the quantum Fisher information of the output with respect to unknown dynamical parameters. We discuss the structure of continuous-time measurements as solutions of the quantum Zakai equation, and their relationship to parameter estimation methods. Proceeding beyond parameter estimation, the paper also gives an overview of the emerging topic of quantum system identification for black-box modelling of quantum systems by continuous observation of a travelling wave probe, for the case of ergodic quantum input–output systems and linear quantum systems. Empirical methods for such black-box modelling are also discussed.     

An algorithm for nonparametric GARCH modelling

A simple iterative algorithm for nonparametric first-order GARCH modelling is proposed. This method offers an alternative to fitting one of the many different parametric GARCH specifications that have been proposed in the literature. A theoretical justification for the algorithm is provided and examples of its application to simulated data from various stationary processes showing stochastic volatility, as well as empirical financial return data, are given. The nonparametric procedure is found to often give better estimates of the unobserved latent volatility process than parametric modelling with the standard GARCH(1,1) model, particularly in the presence of asymmetry and other departures from the standard GARCH specification. Extensions of the basic iterative idea to more complex time series models combining ARMA or GARCH features of possibly higher order are suggested.     

利用空间信息的核模糊C均值聚类算法

模糊聚类,特别是模糊C均值聚类算法（FCM）广泛地运用到图像的分割中。但是传统的算法未对数据对特征进行优化,亦未考虑图像的空间信息,对噪声图像分割不理想。在FCM目标函数中引入核函数,用内核引导距离代替传统的欧式距离,同时考虑到邻近象素的影响,增加了空间约束项,提出了利用空间信息的核FCM算法。通过对模拟图和仿真脑部MR图像的分割实验证明,该算法可以有效的分割含有噪声图像。     

Claw finding algorithms using quantum walk

The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. Given two functions, f$f$ and g$g$, with domain sizes N$N$ and M$M$(N≤M)$(N\le M)$, respectively, and the same range, the goal of the problem is to find x$x$ and y$y$ such that f(x)=g(y)$f\left(x\right)=g\left(y\right)$. This problem has been considered in both quantum and classical settings in terms of query complexity. This paper describes an optimal algorithm that uses quantum walk to solve this problem. Our algorithm can be slightly modified to solve the more general problem of finding a tuple consisting of elements in the two function domains that has a prespecified property. It can also be generalized to find a claw of k$k$ functions for any constant integer k>1$k>1$, where the domain sizes of the functions may be different.     

A programming tool to generate multi-site daily rainfall using a two-stage semi parametric model

Many problems in hydrology and agricultural science require extensive records of rainfall from multiple locations. Temporal and/or spatial coverage of rainfall data is often limited, so that stochastic models may be required to generate long synthetic rainfall records. This study describes a multi-site rainfall simulator (MRS) to stochastically generate daily rainfall at multiple locations. The MRS is available as an open-source software package in the R statistical computing environment. The software includes statistical analysis and graphics functions, and can display statistics and graphs at multiple time scales, including from individual sites and areal averages. The MRS thus provides a detailed set of modelling functions to simulate and analyse daily rainfall. The capabilities of the package are demonstrated using 30 gauges located in Sydney, Australia, and the results show that the model preserves observed year-to-year variability, interannual persistence and various daily distributional and space–time dependence attributes.     

Classification procedures using multivariate variable kernel density estimate

Nonparametric classification procedures derived from the multivariate kernel density estimate are examined. Conditions for weak and strong Bayes risk consistencies are given.     

Unconditionally secure cryptosystems based on quantum cryptography

Most modern cryptographic studies design cryptosystems and algorithms using mathematical concepts. In designing and analyzing cryptosystems and protocols, mathematical concepts are critical in supporting the claim that the intended cryptosystem is secure. Most early cryptographic algorithms are based either on factorization or on discrete logarithm problem. Such systems generally adopt rather simple mathematics, and, therefore, need extensive secondary index computation. This study discusses quantum cryptosystems, protection of system security, and optimization of system efficiency. Quantum cryptography detects intrusion and wiretap. In quantum mechanics, a wiretap is neither external nor passive; rather it modifies its entity based on the internal component of the system. The status of the quantum system changes once a wiretap is detected. Hence, only the designer of the system can discover the quantum status of the system; an eavesdropper can neither determine the quantum state nor duplicate the system. The quantum cryptosystem can achieve unconditional security, and thus guarantees secure communication.     

Simple and effective boundary correction for kernel densities and regression with an application to the world income and Engel curve estimation

In both nonparametric density estimation and regression, the so-called boundary effects, i.e. the bias and variance increase due to one sided data information, can be quite serious. For estimation performed on transformed variables this problem can easily get boosted and may distort substantially the final estimates, and consequently the conclusions. After a brief review of some existing methods a new, straightforward and very simple boundary correction is proposed, applying local bandwidth variation at the boundaries. The statistical behavior is discussed and the performance for density and regression estimation is studied for small and moderate sample sizes. In a simulation study this method is shown to perform very well. Furthermore, it appears to be excellent for estimating the world income distribution, and Engel curves in economics.     

量子系统中状态估计方法的综述

从广泛用于实验量子领域的典型状态估计方法,到基于系统论观点、可用于量子反馈控制的状态估计方法,详细综述了4种测量方式下的相应量子状态估计方法及其适用背景.通过其发展历程的叙述,从本质上阐述了估计的基本原理,从技术上对各种方法进行了相应的分析和比较.同时,对量子状态估计和经典状态估计进行了相应的比较,并对量子系统中的状态估计方法作了总结.     

基于量子进化规划核聚类算法的图像分割

基于量子计算的并行性、进化计算简单、通用性好等优点,采用量子编码构造进化算法的染色体种群,再将二者引入到核聚类中来,提出了一种基于量子进化规划的核聚类算法.该算法充分利用了量子态的叠加性以及量子比特的概率表示,能够表示出许多可能的线性叠加状态,具有更好的种群多样性,因此将其用于解决核聚类算法中目标函数的优化问题,可以有效克服传统进化算法收敛速度慢以及早熟等问题.对Brodatz纹理图像及SAR图像进行分割,仿真实验结果表明该算法可以较好地改善图像分割效果.     

Enhancing principal direction divisive clustering

While data clustering has a long history and a large amount of research has been devoted to the development of numerous clustering techniques, significant challenges still remain. One of the most important of them is associated with high data dimensionality. A particular class of clustering algorithms has been very successful in dealing with such datasets, utilising information driven by the principal component analysis. In this work, we try to deepen our understanding on what can be achieved by this kind of approaches. We attempt to theoretically discover the relationship between true clusters in the data and the distribution of their projection onto the principal components. Based on such findings, we propose appropriate criteria for the various steps involved in hierarchical divisive clustering and develop compilations of them into new algorithms. The proposed algorithms require minimal user-defined parameters and have the desirable feature of being able to provide approximations for the number of clusters present in the data. The experimental results indicate that the proposed techniques are effective in simulated as well as real data scenarios.     

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.     

A kernel-based parametric method for conditional density estimation

A conditional density function, which describes the relationship between response and explanatory variables, plays an important role in many analysis problems. In this paper, we propose a new kernel-based parametric method to estimate conditional density. An exponential function is employed to approximate the unknown density, and its parameters are computed from the given explanatory variable via a nonlinear mapping using kernel principal component analysis (KPCA). We develop a new kernel function, which is a variant to polynomial kernels, to be used in KPCA. The proposed method is compared with the Nadaraya-Watson estimator through numerical simulation and practical data. Experimental results show that the proposed method outperforms the Nadaraya-Watson estimator in terms of revised mean integrated squared error (RMISE). Therefore, the proposed method is an effective method for estimating the conditional densities.     

An efficient and accurate quantum lattice-gas model for the many-body Schrödinger wave equation

Presented is quantum lattice-gas model for simulating the time-dependent evolution of a many-body quantum mechanical system of particles governed by the non-relativistic Schrödinger wave equation with an external scalar potential. A variety of computational demonstrations are given where the numerical predictions are compared with exact analytical solutions. In all cases, the model results accurately agree with the analytical predictions and we show that the model's error is second order in the temporal discretization and fourth order in the spatial discretization. The difficult problem of simulating a system of fermionic particles is also treated and a general computational formulation of this problem is given. For pedagogical purposes, the two-particle case is presented and the numerical dispersion of the simulated wave packets is compared with the analytical solutions.