Iterative Beam Hardening Correction for Multi-Material Objects

In this paper, we propose an iterative beam hardening correction method that is applicable for the case with multiple materials. By assuming that the materials composing scanned object are known and that they are distinguishable by their linear attenuation coefficients at some given energy, the beam hardening correction problem is converted into a nonlinear system problem, which is then solved iteratively. The reconstructed image is the distribution of linear attenuation coefficient of the scanned object at a given energy. So there are no beam hardening artifacts in the image theoretically. The proposed iterative scheme combines an accurate polychromatic forward projection with a linearized backprojection. Both forward projection and backprojection have high degree of parallelism, and are suitable for acceleration on parallel systems. Numerical experiments with both simulated data and real data verifies the validity of the proposed method. The beam hardening artifacts are alleviated effectively. In addition, the proposed method has a good tolerance on the error of the estimated x-ray spectrum.     

Robust and reliable estimation via recursive nonlinear dynamic data reconciliation based on cubature Kalman filter

Since measurements of process variables are subject to measurements errors as well as process variability, data reconciliation is the procedure of optimally adjusting measured date so that the adjusted values obey the conservation laws and constraints. Thus, data reconciliation for dynamic systems is fundamental and important for control, fault detection, and system optimization. Attempts to successfully implement estimators are often hindered by serve process nonlinearities, complicated state constraints, and un-measurable perturbations. As a constrained minimization problem, the dynamic data reconciliation is dynamically carried out to product smoothed estimates with variances from the original data. Many algorithms are proposed to solve such state estimation such as the extended Kalman filter (EKF), the unscented Kalman filter, and the cubature Kalman filter (CKF). In this paper, we investigate the use of CKF algorithm in comparative with the EKF to solve the nonlinear dynamic data reconciliation problem. First we give a broad overview of the recursive nonlinear data dynamic reconciliation (RNDDR) scheme, then present an extension to the CKF algorithm, and finally address the issue of how to solve the constraints in the CKF approach. The CCRNDDR method is proposed by applying the RNDDR in the CKF algorithm to handle nonlinearity and algebraic constraints and bounds. As the sampling idea is incorporated into the RNDDR framework, more accurate estimates can obtained via the recursive nature of the estimation procedure. The performance of the CKF approach is compared with EKF and RNDDR on nonlinear process systems with constraints. The conclusion is that with an error optimization solution of the correction step, the reformulated CKF shows high performance on the selection of nonlinear constrained process systems. Simulation results show the CCRNDDR is an efficient, accurate and stable method for real-time state estimation for nonlinear dynamic processes.     

Two paradigms of mathematical population biology. Attempting to synthesise

Whatever popular the slogan of nonlinear ecological interactions has been in theory, practical ecology professes the projection matrix paradigm, which is essentially linear, i.e., the linear matrix model paradigm for discrete-structured population dynamics. The dominant eigenvalue lamda1, of the projection matrix L is considered as a growth potential of the population. It provides for a quantitative measure of the fitness at which the species is adapted to the given environment, the measure being adequate and accurate, given the data of"identified individuals" type. The case of"identified individuals with unknown parents" bears uncertainty in the status-specific reproduction rates, which eliminates in a unique way (for a broad class of structures and life cycle graphs) by maximizing lamda1(L) under the constraints ensuing from the data and knowledge of species biology. The paradigm of linearity gives way to nonlinear models when modeled are species interactions, such as competition for shared resources, and where the outcome of interaction depends on the population structure of the competitors. This circumstance dictates a need for the synthesis of two paradigms, which is achieved in nonlinear matrix operators as models of interaction between the species whose populations are discrete-structured.     

Viscoelastic behavior of erythrocyte membrane.

A nonlinear viscoelastic relation is developed to describe the viscoelastic properties of erythrocyte membrane. This constitutive equation is used in the analysis of the time-dependent aspiration of an erythrocyte membrane into a micropipette. Equations governing this motion are reduced to a nonlinear integral equation of the Volterra type. A numerical procedure based on a finite difference scheme is used to solve the integral equation and to match the experimental data. The data, aspiration length vs. time, is used to determine the relaxation function at each time step. The inverse problem of obtaining the time dependence of the aspiration length from a given relaxation function is also solved. Analytical results obtained are applied to the experimental data of Chien et al. 1978. Biophys. J. 24:463-487. A relaxation function similar to that of a four-parameter solid with a shear-thinning viscous term is proposed.     

A nonlinear regression program for small computers

A BASIC computer program for performing weighted nonlinear regression is described and a listing of the program is given. The program, which is small and simple to use, has been designed to be run by users with little knowledge of mathematics or computers. Robust methods of analysis are described which may be applied to data in which experimental errors are not normally distributed, and the program incorporates one such method. It is shown that the program is useful for the analysis of data conforming to the Michaelis-Menten equation, a single exponential, and to binding equations, and other applications are discussed.     

Application of a new method of nonlinear dynamical system identification to biochemical problems

The system identification method for a variety of nonlinear dynamic models is elaborated. The problem of identification of an original nonlinear model presented as a system of ordinary differential equations in the Cauchy explicit form with a polynomial right part reduces to the solution of the system of linear equations for the constants of the dynamical model. In other words, to construct an integral model of the complex system (phenomenon), it is enough to collect some data array characterizing the time-course of dynamical parameters of the system. Collection of such a data array has always been a problem. However difficulties emerging are, as a rule, not principal and may be overcome almost without exception. The potentialities of the method under discussion are demonstrated by the example of the test problem of multiparametric nonlinear oscillator identification. The identification method proposed may be applied to the study of different biological systems and in particular the enzyme kinetics of complex biochemical reactions.     

Parametric analysis of the biomechanical response of head subjected to the primary blast loading – a data mining approach

Traumatic brain injury due to primary blast loading has become a signature injury in recent military conflicts and terrorist activities. Extensive experimental and computational investigations have been conducted to study the interrelationships between intracranial pressure response and intrinsic or ‘input’ parameters such as the head geometry and loading conditions. However, these relationships are very complicated and are usually implicit and ‘hidden’ in a large amount of simulation/test data. In this study, a data mining method is proposed to explore such underlying information from the numerical simulation results. The heads of different species are described as a highly simplified two-part (skull and brain) finite element model with varying geometric parameters. The parameters considered include peak incident pressure, skull thickness, brain radius and snout length. Their interrelationship and coupling effect are discovered by developing a decision tree based on the large simulation data-set. The results show that the proposed data-driven method is superior to the conventional linear regression method and is comparable to the nonlinear regression method. Considering its capability of exploring implicit information and the relatively simple relationships between response and input variables, the data mining method is considered to be a good tool for an in-depth understanding of the mechanisms of blast-induced brain injury. As a general method, this approach can also be applied to other nonlinear complex biomechanical systems.     

Analysis of Schizophrenia Data Using A Nonlinear Threshold Index Logistic Model

Genetic information, such as single nucleotide polymorphism (SNP) data, has been widely recognized as useful in prediction of disease risk. However, how to model the genetic data that is often categorical in disease class prediction is complex and challenging. In this paper, we propose a novel class of nonlinear threshold index logistic models to deal with the complex, nonlinear effects of categorical/discrete SNP covariates for Schizophrenia class prediction. A maximum likelihood methodology is suggested to estimate the unknown parameters in the models. Simulation studies demonstrate that the proposed methodology works viably well for moderate-size samples. The suggested approach is therefore applied to the analysis of the Schizophrenia classification by using a real set of SNP data from Western Australian Family Study of Schizophrenia (WAFSS). Our empirical findings provide evidence that the proposed nonlinear models well outperform the widely used linear and tree based logistic regression models in class prediction of schizophrenia risk with SNP data in terms of both Types I/II error rates and ROC curves.     

Semiparametric analysis of zero-inflated count data

Medical and public health research often involve the analysis of count data that exhibit a substantially large proportion of zeros, such as the number of heart attacks and the number of days of missed primary activities in a given period. A zero-inflated Poisson regression model, which hypothesizes a two-point heterogeneity in the population characterized by a binary random effect, is generally used to model such data. Subjects are broadly categorized into the low-risk group leading to structural zero counts and high-risk (or normal) group so that the counts can be modeled by a Poisson regression model. The main aim is to identify the explanatory variables that have significant effects on (i) the probability that the subject is from the low-risk group by means of a logistic regression formulation; and (ii) the magnitude of the counts, given that the subject is from the high-risk group by means of a Poisson regression where the effects of the covariates are assumed to be linearly related to the natural logarithm of the mean of the counts. In this article we consider a semiparametric zero-inflated Poisson regression model that postulates a possibly nonlinear relationship between the natural logarithm of the mean of the counts and a particular covariate. A sieve maximum likelihood estimation method is proposed. Asymptotic properties of the proposed sieve maximum likelihood estimators are discussed. Under some mild conditions, the estimators are shown to be asymptotically efficient and normally distributed. Simulation studies were carried out to investigate the performance of the proposed method. For illustration purpose, the method is applied to a data set from a public health survey conducted in Indonesia where the variable of interest is the number of days of missed primary activities due to illness in a 4-week period.     

Numerical Solution to Generalized Burgers'-Fisher Equation Using Exp-Function Method Hybridized with Heuristic Computation

In this paper, a new heuristic scheme for the approximate solution of the generalized Burgers''-Fisher equation is proposed. The scheme is based on the hybridization of Exp-function method with nature inspired algorithm. The given nonlinear partial differential equation (NPDE) through substitution is converted into a nonlinear ordinary differential equation (NODE). The travelling wave solution is approximated by the Exp-function method with unknown parameters. The unknown parameters are estimated by transforming the NODE into an equivalent global error minimization problem by using a fitness function. The popular genetic algorithm (GA) is used to solve the minimization problem, and to achieve the unknown parameters. The proposed scheme is successfully implemented to solve the generalized Burgers''-Fisher equation. The comparison of numerical results with the exact solutions, and the solutions obtained using some traditional methods, including adomian decomposition method (ADM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), show that the suggested scheme is fairly accurate and viable for solving such problems.     

Monte Carlo EM for missing covariates in parametric regression models

We propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates. We allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout. When the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). We extend this method to continuous or mixed categorical and continuous covariates, and for arbitrary parametric regression models, by adapting a Monte Carlo version of the EM algorithm as discussed by Wei and Tanner (1990, Journal of the American Statistical Association 85, 699-704). In addition, we discuss the Gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log-concave. The log-concavity property of the conditional distributions will facilitate a straightforward implementation of the Gibbs sampler via the adaptive rejection algorithm of Gilks and Wild (1992, Applied Statistics 41, 337-348). We assume the model for the response given the covariates is an arbitrary parametric regression model, such as a generalized linear model, a parametric survival model, or a nonlinear model. We model the marginal distribution of the covariates as a product of one-dimensional conditional distributions. This allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the E-step. We present examples involving both simulated and real data.     

Nonlinear viscoelastic, thermodynamically consistent, models for biological soft tissue

The mechanical behavior of most biological soft tissue is nonlinear viscoelastic rather than elastic. Many of the models previously proposed for soft tissue involve ad hoc systems of springs and dashpots or require measurement of time-dependent constitutive coefficient functions. The model proposed here is a system of evolution differential equations, which are determined by the long-term behavior of the material as represented by an energy function of the type used for elasticity. The necessary empirical data is time independent and therefore easier to obtain. These evolution equations, which represent non-equilibrium, transient responses such as creep, stress relaxation, or variable loading, are derived from a maximum energy dissipation principle, which supplements the second law of thermodynamics. The evolution model can represent both creep and stress relaxation, depending on the choice of control variables, because of the assumption that a unique long-term manifold exists for both processes. It succeeds, with one set of material constants, in reproducing the loading-unloading hysteresis for soft tissue. The models are thermodynamically consistent so that, given data, they may be extended to the temperature-dependent behavior of biological tissue, such as the change in temperature during uniaxial loading. The Holzapfel et al. three-dimensional two-layer elastic model for healthy artery tissue is shown to generate evolution equations by this construction for biaxial loading of a flat specimen. A simplified version of the Shah-Humphrey model for the elastodynamical behavior of a saccular aneurysm is extended to viscoelastic behavior.     

等位基因多态性群体遗传结构的多元非线性分析方法

长期以来,对于多维基因多态性数据的多元统计分析,如计算遗传距离时昕用的聚类分析、分析群体遗传结构时所用的主成分分析、因子分析和典型相关分析等,一直应用为无约束条件数据而设计的经典多元线性分析方法,并没有注意基因多态性数据的“闭合效应”所带来的问题。从分析基因多态性数据的分布和结构特征入手,文中指出了基因多态性分布具有“闭合数据”的特点,分析了由于“闭合效应”的影响,经典多元线性方法用于群体遗传结构分析昕面临的困难。根据成分数据统计分析的理论和方法,提出了基因多态性群体遗传结构的多元非线性分析基本方法。并以主成分分析为例,通过实例比较和分析了经典线性主成分分析和“对数比”非线性主成分分析的结果,证明“对数比”非线性主成分分析方法是研究基因多态性群体遗传结构的良好方法,具有特异、灵敏等优点,其结果符合群体遗传学规律。     

Linear Mixed Effects Models under Inequality Constraints with Applications

Constraints arise naturally in many scientific experiments/studies such as in, epidemiology, biology, toxicology, etc. and often researchers ignore such information when analyzing their data and use standard methods such as the analysis of variance (ANOVA). Such methods may not only result in a loss of power and efficiency in costs of experimentation but also may result poor interpretation of the data. In this paper we discuss constrained statistical inference in the context of linear mixed effects models that arise naturally in many applications, such as in repeated measurements designs, familial studies and others. We introduce a novel methodology that is broadly applicable for a variety of constraints on the parameters. Since in many applications sample sizes are small and/or the data are not necessarily normally distributed and furthermore error variances need not be homoscedastic (i.e. heterogeneity in the data) we use an empirical best linear unbiased predictor (EBLUP) type residual based bootstrap methodology for deriving critical values of the proposed test. Our simulation studies suggest that the proposed procedure maintains the desired nominal Type I error while competing well with other tests in terms of power. We illustrate the proposed methodology by re-analyzing a clinical trial data on blood mercury level. The methodology introduced in this paper can be easily extended to other settings such as nonlinear and generalized regression models.     

Multivariate Association and Dimension Reduction: A Generalization of Canonical Correlation Analysis

Summary In this article, we propose a new generalized index to recover relationships between two sets of random vectors by finding the vector projections that minimize an L ₂ distance between each projected vector and an unknown function of the other. The unknown functions are estimated using the Nadaraya–Watson smoother. Extensions to multiple sets and groups of multiple sets are also discussed, and a bootstrap procedure is developed to detect the number of significant relationships. All the proposed methods are assessed through extensive simulations and real data analyses. In particular, for environmental data from Los Angeles County, we apply our multiple‐set methodology to study relationships between mortality, weather, and pollutants vectors. Here, we detect existence of both linear and nonlinear relationships between the dimension‐reduced vectors, which are then used to build nonlinear time‐series regression models for the dimension‐reduced mortality vector. These findings also illustrate potential use of our method in many other applications. A comprehensive assessment of our methodologies along with their theoretical properties are given in a Web Appendix.     

A theoretical framework to analyze bend testing of soft tissue

It has been hypothesized that repetitive flexural stresses contribute to the fatigue-induced failure of bioprosthetic heart valves. Although experimental apparatuses capable of measuring the bending properties of biomaterials have been described, a theoretical framework to analyze the resulting data is lacking. Given the large displacements present in these bending experiments and the nonlinear constitutive behavior of most biomaterials, such a formulation must be based on finite elasticity theory. We present such a theory in this work, which is capable of fitting bending moment versus radius of curvature experimental data to an arbitrary strain energy function. A simple finite element model was constructed to study the validity of the proposed method. To demonstrate the application of the proposed approach, bend testing data from the literature for gluteraldehyde-fixed bovine pericardium were fit to a nonlinear strain energy function, which showed good agreement to the data. This method may be used to integrate bending behavior in constitutive models for soft tissue.     

The numerical method for the optimal supporting position and related optimal control for the catalytic reaction system

This paper considers the numerical approximation for the optimal supporting position and related optimal control of a catalytic reaction system with some control and state constraints, which is governed by a nonlinear partial differential equations with given initial and boundary conditions. By the Galerkin finite element method, the original problem is projected into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then the control parameterization method is applied to approximate the control and reduce the original system to an optimal parameter selection problem, in which both the position and related control are taken as decision variables to be optimized. This problem can be solved as a nonlinear optimization problem by a particle swarm optimization algorithm. The numerical simulations are given to illustrate the effectiveness of the proposed numerical approximation method.     

Efficient measurement error correction with spatially misaligned data

Association studies in environmental statistics often involve exposure and outcome data that are misaligned in space. A common strategy is to employ a spatial model such as universal kriging to predict exposures at locations with outcome data and then estimate a regression parameter of interest using the predicted exposures. This results in measurement error because the predicted exposures do not correspond exactly to the true values. We characterize the measurement error by decomposing it into Berkson-like and classical-like components. One correction approach is the parametric bootstrap, which is effective but computationally intensive since it requires solving a nonlinear optimization problem for the exposure model parameters in each bootstrap sample. We propose a less computationally intensive alternative termed the "parameter bootstrap" that only requires solving one nonlinear optimization problem, and we also compare bootstrap methods to other recently proposed methods. We illustrate our methodology in simulations and with publicly available data from the Environmental Protection Agency.