Frequent problems in calculating integrals and optimizing objective functions: a case study in density deconvolution

Many statistical procedures involve calculation of integrals or optimization (minimization or maximization) of some objective function. In practical implementation of these, the user often has to face specific problems such as seemingly numerical instability of the integral calculation, choices of grid points, appearance of several local minima or maxima, etc. In this paper we provide insights into these problems (why and when are they happening?), and give some guidelines of how to deal with them. Such problems are not new, neither are the ways to deal with them, but it is worthwhile to devote serious considerations to them. For a transparant and clear discussion of these issues, we focus on a particular statistical problem: nonparametric estimation of a density from a sample that contains measurement errors. The discussions and guidelines remain valid though in other contexts. In the density deconvolution setting, a kernel density estimator has been studied in detail in the literature. The estimator is consistent and fully data-driven procedures have been proposed. When implemented in practice however, the estimator can turn out to be very inaccurate if no adequate numerical procedures are used. We review the steps leading to the calculation of the estimator and in selecting parameters of the method, and discuss the various problems encountered in doing so.     

东京都与日本近代资本主义的都市建设——兼谈日本近代化变革的非彻底性

明治维新固然是日本近代资本主义的开端性标志,但明治政府拉开这一序幕的第一步骤,却是东京迁都。这意味着,具有资本主义转型内涵的近代化功能首先在都城建制中得到落实。当这一落实逐步构成了对地方的强辐射和绝对影响时,日本的资本主义改造才得到了初步的实现。     

灰色成分数据模型在中国产业结构分析预测中的应用

针对成分数据这种特殊类型的统计数据,提出一种新的预测建模方法:对于一列按照时间顺序收集的成分数据,先运用对数变换使成分数据降维,然后对降维后的数据运用GM(1,1)模型进行预测,最后再将预测值进行反对数变换,从而得到了各成分的预测值.根据提出的方法,建立了中国产业结构的预测模型,并分析了中国产业结构的发展趋势和未来状况.经检验,运用该方法预测出的数据与实际值十分吻合.     

基于小波变换的图像数据压缩方法

小波变换继承了傅立叶分析优点，主要使用在静态和动态图像压缩领域，已经成为某些图像压缩的国际标准。二维小波变换是在第一代图像压缩基础上，采用小波变换为主的多尺度分析算法，使处理后的图像可以达到30—60倍的高压缩比率，也更适合在目前低宽带的网络中传输。     

Matching pursuit by undecimated discrete wavelet transform for non-stationary time series of arbitrary length

We describe how to formulate a matching pursuit algorithm which successively approximates a periodic non-stationary time series with orthogonal projections onto elements of a suitable dictionary. We discuss how to construct such dictionaries derived from the maximal overlap (undecimated) discrete wavelet transform (MODWT). Unlike the standard discrete wavelet transform (DWT), the MODWT is equivariant under circular shifts and may be computed for an arbitrary length time series, not necessarily a multiple of a power of 2. We point out that when using the MODWT and continuing past the level where the filters are wrapped, the norms of the dictionary elements may, depending on N, deviate from the required value of unity and require renormalization.We analyse a time series of subtidal sea levels from Crescent City, California. The matching pursuit shows in an iterative fashion how localized dictionary elements (scale and position) account for residual variation, and in particular emphasizes differences in construction for varying parts of the series.     

规则区域上Helmholtz方程的一种快速算法

采用有限差分法对Ｈｅｌｍｈｏｌｔｚ 方程进行五点差分离散，在规则区域上引入快速傅里叶变换（ＦＦＴ） ， 将差分方程变换成一组三对角方程， 使求解规则子区域上 Ｈｅｌｍｈｏｌｔｚ 方程的计算量降为Ｏ（ Ｐｌｇ Ｐ） ，最后的数值结果证明了文中的算法是一种快速算法     

Modelling cell generation times by using the tempered stable distribution

Summary.  We show that the family of tempered stable distributions has considerable potential for modelling cell generation time data. Several real examples illustrate how these distributions can improve on currently assumed models, including the gamma and inverse Gaussian distributions which arise as special cases. Our applications concentrate on the generation times of oligodendrocyte progenitor cells and the yeast Saccharomyces cerevisiae . Numerical inversion of the Laplace transform of the probability density function provides fast and accurate approximations to the tempered stable density, for which no closed form generally exists. We also show how the asymptotic population growth rate is easily calculated under a tempered stable model.     

Mixed inverse problems arising in the estimation of calibration factors in positron emission tomography

A penalized likelihood approach to the estimation of calibration factors in positron emission tomography (PET) is considered, in particular the problem of estimating the efficiency of PET detectors. Varying efficiencies among the detectors create a non-uniform performance and failure to account for the non-uniformities would lead to streaks in the image, so efficient estimation of the non-uniformities is desirable to reduce the propagation of noise to the final image. The relevant data set is provided by a blank scan, where a model may be derived that depends only on the sources affecting non-uniformities: inherent variation among the detector crystals and geometric effects. Physical considerations suggest a novel mixed inverse model with random crystal effects and smooth geometric effects. Using appropriate penalty terms, the penalized maximum likelihood estimates are derived and an efficient computational algorithm utilizing the fast Fourier transform is developed. Data-driven shrinkage and smoothing parameters are chosen to minimize an estimate of the predictive loss function. Various examples indicate that the approach proposed works well computationally and compares well with the standard method.     

浅谈马克思对思维方式的变革

实践思维方式是马克思哲学把握属于人的世界,理解、诠释和评价一切哲学问题的根本思维方式,是对形而上学哲学思维方式的根本变革。马克思哲学从人的实践活动出发,从现实生活出发,去改造现实世界,以实践作为根据,去理解社会生活的本质和破解形而上学哲学对历史神秘主义的理解。     

A general family of NBU classes of life distributions

In this paper we introduce and study a family of new better than used ageing notions parameterized by a function h. The new better than used , the new better than used of a specified age and the new better than used in total time on test transform order , are special cases of this new family. We study some properties of the new family, and we give some applications of it in actuarial science, reliability theory and statistics.