A Cheaper Way to Compute Generalized Cross-Validation as a Stopping Rule for Linear Stationary Iterative Methods

We apply generalized cross-validation (GCV) as a stopping rule for general linear stationary iterative methods for solving very large-scale, ill-conditioned problems. We present a new general formula for the influence operator for these methods and, using this formula and a Monte Carlo approach, we show how to compute the GCV function at a cheaper cost. Then we apply our approach to a well known iterative method (ART) with simulated data in positron emission tomography (PET).     

The identification problem for the exponential Radon transform

The exponential Radon transform, which arises in single photon emission computed tomography, is defined by ? ?(μ:ω,s) = ∫_R?(sω + tomega;^?) e^μt dt?. Here ? is a compactly supported distribution in the plane which represents the location and intensity of a radio-pharmaceutical in a body of constant, but unknown, attenuation μ, and ω is a direction. The identification problem is to determine the attenuation μ from the data ?? with ? unknown. We will show that μ can be determined from the data if and only if ? is not a radial distribution and give formulae for computing μ when ? is not radial.     

Estimating Precision in Functional Images

Abstract

Functional imaging of biologic parameters like in vivo tissue metabolism is made possible by Positron Emission Tomography (PET). Many techniques have been suggested for extracting such images from dynamic time-course sequences of reconstructed PET scans. Quantitating the precision of these estimates is important for drawing inferences on the biologic parameters. Analytic variance formulas are not immediate owing to the nonlinear methods used in extraction. The usual resampling approach is infeasible because each image reconstruction in PET is a computationally demanding solution to a high-dimensional linear inverse problem. We suggest an alternative simulation approach that approximates the distribution of reconstructed PET scans and performs a parametric bootstrap in the imaging domain. Results on a simplified model chosen to match the characteristics of PET reconstruction are very encouraging. Mixture analysis is used to estimate functional images; however, the suggested approach is general enough to extend to other techniques or imaging methods.     

On projected gradient methods for regularizing reconstruction of synchrotron radiation spectra distribution function

We study the numerical methods for the reconstruction of the spectral distribution function of SR by measurement of the attenuation of the SR energy spectrum. The reconstruction of the spectral distribution function of SR is an ill-posed integral operator equation of the first kind. Therefore, robust method to overcome the ill-posedness and to improve the computational efficiency is a major task in numerical computation. Because of the physical meaning of the problem, we study projected gradient methods in this paper. Numerical simulations are given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A method of iterative data refinement and its applications

Iterative data refinement (IDR) is a general procedure for producing a sequence of estimates of the data that would be collected by a measuring device which is idealized to a certain extent, starting from the data that are collected by an actual measuring device. Following a discussion of the fundamentals of IDR, we present a number of previously published procedures which are special cases of it. We concentrate on examples from medical imaging. In particular, we discuss beam hardening correction in x-ray computerized tomography, attenuation correction in emission computerized tomography, and compensation for missing data in reconstruction from projections. We also show that a standard method of numerical mathematics (the parallel chord method) as well as a whole family of constrained iterative restoration algorithms are special cases of IDR. Thus IDR provides a common framework within which a number of originally different looking procedures are presented and discussed. We also present a result of theoretical nature concerning the initial behavior of IDR.     

Positron Emission Tomography and Random Coefficients Regression

We further explore the relation between random coefficients regression (RCR) and computerized tomography. Recently, Beran et al. (1996, Ann. Statist., 24, 2569–2592) explored this connection to derive an estimation method for the non-parametric RCR problem which is closely related to image reconstruction methods in X-ray computerized tomography. In this paper we emphasize the close connection of the RCR problem with positron emission tomography (PET). Specifically, we show that the RCR problem can be viewed as an idealized (continuous) version of a PET experiment, by demonstrating that the nonparametric likelihood of the RCR problem is equivalent to that of a specific PET experiment. Consequently, methods independently developed for either of the two problems can be adapted from one problem to the other. To demonstrate the close relation between the two problems we use the estimation method of Beran, Feuerverger and Hall for image reconstruction in PET.     

空气中PM2.5的扩散和衰减模型

PM2.5的时空分布及其演变规律十分复杂.为刻画PM2.5的发生、扩散和衰减规律,提出点源、线源和面源叠加的多源模型描述区域内的多污染源对某一监测点的影响.考虑风力风速、太阳辐射强度、湿度等天气和季节因素以及重力、湍流扩散、分子扩散等对源强强度的影响,提出源衰减、湿沉积、化学迁移叠加衰减模型,用监测点的PM2.5浓度数据对污染源强度和衰减系数进行反演求解.针对西安市某些监测点处的PM2.5浓度突然增至数倍且延续数小时,建立污染扩散预测与评估方法,对提升前后污染源源强进行分析,给出重度污染区域,并用数据的人工统计定性验证模型的合理性.     

On multivariate attenuation factors

In this paper, we introduce the method of discrete attenuation factors for the approximate computation of multivariate discrete Fourier transforms. We consider attenuation factors related with multivariate discrete Bernoulli functions and deduce a best approximation property of the corresponding method of attenuation factors. Choosing a unique approach to the discrete and non-discrete settings, we emphasize the close relation between both cases and interpret results in the literature from a more general point of view.     

Micromechanics-Supported Conversion of Computer Tomographic (CT) Images into Anisotropic and Inhomogeneous Finite Element Models of Organs: the Case of a Human Mandible

We here present a strategy for reliable prediction of elastic properties from X-ray attenuation coefficients visualized in Computer Tomographic images, as basis for Finite Element models. By example, we show the distribution of the axial normal stress throughout a human mandible, due to a bite on the leftmost premolar. Remarkably, this distribution is not heavily altered if we replace the inhomogeneous material distribution by one discerning merely cortical and trabecular bone, but it is strongly affected by the consideration of material anisotropy. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic Perturbations of Line Solitons of KP

We consider the properties of localized solutions of the KP equation coupled to a stochastic noise. Corresponding to white noise, we find that the traveling waves are destroyed asymptotically, and we determine the distribution of the wave position and the arrival time. For generalized Ornstein–Uhlenbeck processes, we show that the only effect of noise is to render the asymptotic position random; in particular, when the noise has a sufficiently strong attenuation mechanism, the random wave coincides asymptotically with the unperturbed one. We also consider linearization of the corresponding Cauchy problem in the plane corresponding to this kind of initial data.     

导化积合与幂弦指数分布:金融危机后的经济行为

运用导化积合方法,构造出一类随机变量的分布:ω-幂弦指数分布.此分布具有周期性衰减震荡特征,可以很好地描述金融危机发生时,政府主动调整后的经济运行规律,也可以作为地震余震的数学模型.     

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Solar cell is the basic component of satellite photovoltaic panels with complicated redundant system structure. Its reliability plays an important role in the system, and its performance shows a degradation trend over time. In this paper, study is conducted for the solar cell degradation modeling and reliability analysis based on practical testing results. Specifically, we illustrate an accelerated test for the attenuation ratio character test under different accumulative irradiation levels, focusing on the heteroscedasticity of the collected testing data. A heteroscedastic linear model is proposed, and the life distribution of the photovoltaic panel is obtained by using Fiducial method. A numerical example is shown for the purpose of illustration.     

Bias reduction in the two-stage method for degradation data analysis

Degradation data are usually collected for assessing the reliability of the product. We propose a new two-stage method to analyze degradation data. The degradation path is fitted by the nonlinear mixed effects model in the first stage, and the parameters in lifetime distribution are estimated by maximizing the asymptotic marginal distribution of pseudo lifetimes in the second stage. The new method has many advantages: (i) it does not require the distributions on random effects, (ii) the historical information about lifetime distribution of the product can be incorporated easily, and thus the estimated lifetime distribution has a closed form, (iii) bias correction term is automatically embedded into the asymptotic marginal distribution of pseudo lifetime. Finally, simulation studies and real data analysis are performed for illustration.     

A bivariate interval censorship model for partnership formation

We consider a statistical problem of estimating a bivariate age distribution of newly formed partnership. The study is motivated by a type of data that consist of uncensored, right-censored, left-censored, interval-censored and missing observations in the coordinates of a bivariate random vector. A model is proposed for formulating such type of data. A feasible algorithm to estimate the generalized MLE (GMLE) of the bivariate distribution function is also proposed. We establish asymptotic properties for the GMLE under a discrete assumption on the underlying distributions and apply the method to the data set.     

Application of the cross-entropy method to clustering and vector quantization

We apply the cross-entropy (CE) method to problems in clustering and vector quantization. The CE algorithm for clustering involves the following iterative steps: (a) generate random clusters according to a specified parametric probability distribution, (b) update the parameters of this distribution according to the Kullback–Leibler cross-entropy. Through various numerical experiments, we demonstrate the high accuracy of the CE algorithm and show that it can generate near-optimal clusters for fairly large data sets. We compare the CE method with well-known clustering and vector quantization methods such as K-means, fuzzy K-means and linear vector quantization, and apply each method to benchmark and image analysis data.     

The Baum–Welch algorithm with limiting distribution constraints

We present a method of incorporating limiting distribution information in the Baum–Welch algorithm for estimating parameters of discrete-time, finite-state, Hidden Markov Models. We find that having, even imperfect, limiting distribution information can dramatically improve transition probability estimates. Additionally, we find that when (1) the underlying process is weakly correlated with the observable signal, and (2) the length of the data sequence is short, the additional information provided by the limiting distribution is substantial.     

Computing Scan Statistic p Values Using Importance Sampling,With Applications to Genetics and Medical Image Analysis

We present an importance sampling method for deciding, based on an observed random field, if a scan statistic provides significant evidence of increased activity in some localized region of time or space. Our method allows consideration of scan statistics based simultaneously on multiple scan geometries. Our approach yields an unbiased p value estimate whose variance is typically smaller than that of the naive hit-or-miss Monte Carlo technique when the p value is small. Furthermore, our p value estimate is often accurate for critical values that are not far enough in the tails of the null distribution to allow for accurate approximations via extreme value theory. The importance sampling approach unifies the analysis of various random field models, from (spatial) point processes to Gaussian random fields. For a scan statistic M, the method produces a p value of the form P[M ≥ τ] = Bρ, where B is the Bonferroni upper bound and the correction factor ρ measures the conservativeness of this upper bound. We present the application of our importance sampling estimator to multinomial sequences (molecular genetics), spatial point processes (digital mammography), and Gaussian random fields (PET scan brain imagery).     

On Procedures for Testing the Equivalence of Distribution Tails

Doklady Mathematics - We propose a method for testing the hypothesis about the equivalence of the distribution tail of observed data and a certain distribution tail, which is the analogue of the...     

Cross-calibration of X-ray µCT and MRX for tissue analysis

Ferrofluids are being considered as an aid for local cancer treatments, such as Magnetic Drug Targeting (MDT) and Magnetic Hyperthermia (MHT). Both methods make use of the strong influence of a magnetic field on the nanoparticles, with the aim of treating the cancer locally while reducing, or even eliminating, the side effects that usually occur during conventional cancer treatments. Microcomputed tomography analysis has been performed on tumour tissue after MDT and MHT in order to examine the distribution of the magnetic nanoparticles within the tissue. The majority of the measurements has been performed in a laboratory based on a polychromatic X-ray source. The strong energy dependence of the attenuation coefficient and the occurrence of the so called beam hardening artefacts make the quantitative evaluation of data acquired with polychromatic tomography equipment very difficult. In this paper we present a cross-calibration method for magnetorelaxometry and polychromatic X-ray tomography for biological tissue samples enriched with magnetic nanoparticles. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一种产生广义高斯分布随机数的新算法

本文提出了一种生成广义高斯分布 (GGD)随机数的通用算法 .该算法针对GGD密度函数衰减性的特点 ,采用变步长的方法 ,综合运用了逆函数法、近似复合抽样法及变换抽样法 .通过调整分布参数的数值 ,就能产生具有任何形状参数和任何方差的GGD随机数 ,简单易于实现 .最后将仿真实验结果与已有算法的结果做比较 ,并用 χ2检验法和Kolmogorov Smirnov检验法 (K S检验法 )验证该方法的有效性 .