A step-by-step guide to non-linear regression analysis of experimental data using a Microsoft Excel spreadsheet

The objective of this present study was to introduce a simple, easily understood method for carrying out non-linear regression analysis based on user input functions. While it is relatively straightforward to fit data with simple functions such as linear or logarithmic functions, fitting data with more complicated non-linear functions is more difficult. Commercial specialist programmes are available that will carry out this analysis, but these programmes are expensive and are not intuitive to learn. An alternative method described here is to use the SOLVER function of the ubiquitous spreadsheet programme Microsoft Excel, which employs an iterative least squares fitting routine to produce the optimal goodness of fit between data and function. The intent of this paper is to lead the reader through an easily understood step-by-step guide to implementing this method, which can be applied to any function in the form y=f(x), and is well suited to fast, reliable analysis of data in all fields of biology.     

BJ: an S-Plus program to fit linear regression models to censored data using the Buckley-James method

Most researchers are familiar with ordinary multiple regression models, most commonly fitted using the method of least squares. The method of Buckley and James (J. Buckley, I. James, Linear regression with censored data, Biometrika 66 (1979) 429-436.) is an extension of least squares for fitting multiple regression models when the response variable is right-censored as in the analysis of survival time data. The Buckley-James method has been shown to have good statistical properties under usual regularity conditions (T.L. Lai, Z. Ying, Large sample theory of a modified Buckley-James estimator for regression analysis with censored data, Ann. Stat. 19 (1991) 1370-1402.). Nevertheless, even after 20 years of its existence, it is almost never used in practice. We believe that this is mainly due to lack of software and we describe here an S-Plus program that through its inclusion in a public domain function library fully exploits the power of the S-Plus programming environment. This environment provides multiple facilities for model specification, diagnostics, statistical inference, and graphical depiction of the model fit.     

乙基橙偶氮苯聚合物的全光开关特性曲线拟合中的应用

BP神经网络通过调节连接权重可以实现以任意精度逼近非线性函数,利用这一特点可以对非线性函数关系进行拟合。偶氮苯聚合物的全开关特性曲线是非线性,很难用数学函数表达式来描述。因而本文首先介绍神经网络的基本原理和BP算法神经网络,然后BP神经网络应用于的全光开关特性曲线拟合,在MATLAB环境下,利用实验数据进行了实验测试,结果表明该方法处理数据精度高,拟合效果好。     

一种基于样本加权的多尺度核支持向量机方法

多核学习方法是机器学习领域中的一个新的热点。核方法通过将数据映射到高维空间来增加线性分类器的计算能力,是目前解决非线性模式分析与分类问题的一种有效途径。但是在一些复杂的情况下,单个核函数构成的核学习方法并不能完全满足如数据异构或者不规则、样本规模大、样本分布不平坦等实际应用中的需求问题,因此将多个核函数进行组合以期获得更好的结果,是一种必然的发展趋势。因此提出一种基于样本加权的多尺度核支持向量机方法,通过不同尺度核函数对样本的拟合能力进行加权,从而得到基于样本加权的多尺度核支持向量机决策函数。通过在多个数据集上的实验分析可以得出所提方法对于各个数据集都获得了很高的分类准确率。     

基于差分进化算法的B 样条曲线曲面拟合

应用B 样条曲线曲面拟合内在形状带有间断或者尖点的数据时,最小二乘法得到的 拟合结果往往在间断和尖点处误差较大,原因在于最小二乘法将拟合函数B 样条的节点固定。本 文在利用3 次B 样条曲线和曲面拟合数据时,应用差分进化算法设计出一种能够自适应地设置B 样条节点的方法,同时对节点的数量和位置进行优化,使得B 样条拟合曲线曲面在间断和尖点处 产生拟多重节点,实现高精度地拟合采样于带有间断或尖点的曲线和曲面数据。     

Learning local objective functions for robust face model fitting

Model-based techniques have proven to be successful in interpreting the large amount of information contained in images. Associated fitting algorithms search for the global optimum of an objective function, which should correspond to the best model fit in a given image. Although fitting algorithms have been the subject of intensive research and evaluation, the objective function is usually designed ad hoc, based on implicit and domain-dependent knowledge. In this article, we address the root of the problem by learning more robust objective functions. First, we formulate a set of desirable properties for objective functions and give a concrete example function that has these properties. Then, we propose a novel approach that learns an objective function from training data generated by manual image annotations and this ideal objective function. In this approach, critical decisions such as feature selection are automated, and the remaining manual steps hardly require domain-dependent knowledge. Furthermore, an extensive empirical evaluation demonstrates that the obtained objective functions yield more robustness. Learned objective functions enable fitting algorithms to determine the best model fit more accurately than with designed objective functions.     

A Unified Gradient-Based Approach for Combining ASM into AAM

Active Appearance Model (AAM) framework is a very useful method that can fit the shape and appearance model to the input image for various image analysis and synthesis problems. However, since the goal of the AAM fitting algorithm is to minimize the residual error between the model appearance and the input image, it often fails to accurately converge to the landmark points of the input image. To alleviate this weakness, we have combined Active Shape Models (ASM) into AAMs, in which ASMs try to find correct landmark points using the local profile model. Since the original objective function of the ASM search is not appropriate for combining these methods, we derive a gradient based iterative method by modifying the objective function of the ASM search. Then, we propose a new fitting method that combines the objective functions of both ASM and AAM into a single objective function in a gradient based optimization framework. Experimental results show that the proposed fitting method reduces the average fitting error when compared with existing fitting methods such as ASM, AAM, and Texture Constrained-ASM (TC-ASM) and improves the performance of facial expression recognition significantly.     

振型函数拟合方法求解移动荷载作用下连续梁的响应

利用ANSYS对连续梁桥进行模态分析,得到各阶频率及振型数据.利用MATLAB编制程序对振型函数数据进行函数拟合,拟合函数采用等截面直梁的振型函数通式,相比样条插值函数和振型函数修正方法,该方法拟合得到的振型函数更为简单,更加和实际振型吻合,便于工程应用.利用拟合的振型函数,对移动荷载作用下连续梁的响应进行分析,应用方便,且得到相当好的结果.该方法具有较好的工程应用价值.     

Matlab在化工数据及模型参数拟合中的应用

在化工研究中,经常需要将实验数据拟合为数学模型,以定量描述过程或现象自变量与因变量之间的关系,模型参数的求取,一般采用最小二乘法。本文介绍了Matlab语言中利用最小二乘法以求取模型参数的几种功能函数及使用方法,通过举例对单变量曲线拟合、多变量曲线拟合及微分形式拟合进行了介绍,说明Matlab语言可拟合大部分实验数据,达到节约数据处理的时间,提高模型参数精度的目的。     

Implicit Fitting Using Radial Basis Functions with Ellipsoid Constraint

Implicit planar curve and surface fitting to a set of scattered points plays an important role in solving a wide variety of problems occurring in computer graphics modelling, computer graphics animation, and computer assisted surgery. The fitted implicit surfaces can be either algebraic or non‐algebraic. The main problem with most algebraic surface fitting algorithms is that the surface fitted to a given data set is often unbounded, multiple sheeted, and disconnected when a high degree polynomial is used, whereas a low degree polynomial is too simple to represent general shapes. Recently, there has been increasing interest in non‐algebraic implicit surface fitting. In these techniques, one popular way of representing an implicit surface has been the use of radial basis functions. This type of implicit surface can represent various shapes to a high level of accuracy. In this paper, we present an implicit surface fitting algorithm using radial basis functions with an ellipsoid constraint. This method does not need to build interior and exterior layers for the given data set or to use information on surface normal but still can fit the data accurately. Furthermore, the fitted shape can still capture the main features of the object when the data sets are extremely sparse. The algorithm involves solving a simple general eigen‐system and a computation of the inverse or psedo‐inverse of a matrix, which is straightforward to implement.     

Importance sampling of products from illumination and BRDF using spherical radial basis functions

In this paper, a new approach for the importance sampling of products from a complex high dynamic range (HDR) environment map and measured bidirectional reflectance distribution function (BRDF) data using spherical radial basis functions (SRBFs) is presented. In the pre-process, a complex HDR environment map and measured BRDF data are transformed into a scattered SRBF representation by using a non-uniform and non-negative SRBF fitting algorithm. An initial guess is determined for the fitting operation. In the run-time rendering process, after the product of the two SRBFs is evaluated, this is used to guide the number of samples. The sampling is done by mixing samples from the various “product” SRBFs using multiple importance sampling. Hence, the proposed approach efficiently renders images with multiple HDR environment maps and measured BRDFs.     

Compact implicit surface reconstruction via low-rank tensor approximation

Implicit representations have gained an increasing popularity in geometric modeling and computer graphics due to their ability to represent shapes with complicated geometry and topology. However, the storage requirement, e.g. memory or disk usage, for implicit representations of complex models is relatively large. In this paper, we propose a compact representation for multilevel rational algebraic spline (MRAS) surfaces using low-rank tensor approximation technique, and exploit its applications in surface reconstruction. Given a set of 3D points equipped with oriented normals, we first fit them with an algebraic spline surface defined on a box that bounds the point cloud. We split the bounding box into eight sub-cells if the fitting error is greater than a given threshold. Then for each sub-cell over which the fitting error is greater than the threshold, an offset function represented by an algebraic spline function of low rank is computed by locally solving a convex optimization problem. An algorithm is presented to solve the optimization problem based on the alternating direction method of multipliers (ADMM) and the CANDECOMP/PARAFAC (CP) decomposition of tensors. The procedure is recursively performed until a certain accuracy is achieved. To ensure the global continuity of the MRAS surface, quadratic B-spline weight functions are used to blend the offset functions. Numerous experiments show that our approach can greatly reduce the storage of the reconstructed implicit surface while preserve the fitting accuracy compared with the state-of-the-art methods. Furthermore, our method has good adaptability and is able to produce reconstruction results with high quality.     

基于椭球约束的径向基函数隐式曲面重建

对三维点云进行隐式曲面重建是解决虚拟现实等方面所存在问题的关键。本文提出 了一种基于椭球约束的径向基函数隐式曲面建模的算法,该方法在仅有点云信息的前提下仍能够 非常精确地拟合点云数据。当点云稀疏时拟合后的模型可以非常好地保证模型的主要特征,但对 于拟合大规模数据点集时,模型会出现冗余现象,保特征效果不理想且效率低下。需将点云进行 适当分割,然后并行拟合被分割点云并将它们进行光滑拼接处理。实验效果表明该算法保特征效 果非常好且效率明显提高。     

Robust Fitting on Poorly Sampled Data for Surface Light Field Rendering and Image Relighting

Two‐dimensional (2D) parametric colour functions are widely used in Image‐Based Rendering and Image Relighting. They make it possible to express the colour of a point depending on a continuous directional parameter: the viewing or the incident light direction. Producing such functions from acquired data is promising but difficult. Indeed, an intensive acquisition process resulting in dense and uniform sampling is not always possible. Conversely, a simpler acquisition process results in sparse, scattered and noisy data on which parametric functions can hardly be fitted without introducing artefacts. Within this context, we present two contributions. The first one is a robust least‐squares‐based method for fitting 2D parametric colour functions on sparse and scattered data. Our method works for any amount and distribution of acquired data, as well as for any function expressed as a linear combination of basis functions. We tested our fitting for both image‐based rendering (surface light fields) and image relighting using polynomials and spherical harmonics. The second one is a statistical analysis to measure the robustness of any fitting method. This measure assesses a trade‐off between precision of the fitting and stability with respect to input sampling conditions. This analysis along with visual results confirm that our fitting method is robust and reduces reconstruction artefacts for poorly sampled data while preserving the precision for a dense and uniform sampling.     

Constrained least-squares

The method of least-squares is intended to fit a function to a set of data points closely, so as to satisfy a particular criterion of closeness. The approximating function can be taken to be a linear combination of linearly independent functions of an independent variable.If the number of data points is the same as the number of linearly independent primitive functions, then the compound function formed by the linear combination of the primitive functions will fit the data exactly.The method of least-squares is designed to fit an approximant to the data when the order of the approximant is incapable of yielding an exact fit to all the data. For example, given three data points, a linear approximant cannot fit the data points, usually. Likewise, given five data points (or more) a cubic approximant cannot fit the data exactly.In the classical method, all data points are considered to be equally important (or equally doubtful). If some of the data points are more reliable than others, it is easy to attach weights to these points, and with this modification to proceed with the classical method.But sometimes, some of the data points are absolutely reliable. This might suggest that we attach infinite weights to such points, somehow. But this presents arithmetical difficulties.We call “absolutely reliable” data points, “hard points”, and all the other points “soft points”, a terminology borrowed from the automobile, aircraft, and ship-building industries.The following paper shows how we can generate a compound approximant that fits the hard points exactly, and satisfies the least-squares criterion for all the soft points. Then the classical method exhibits itself as a special case of this more general solution.     

基于BP神经网络的隐式曲线构造方法

隐式曲线与曲面是当前计算机图形学研究的热点之一。通过把BP神经网络与隐式曲线构造原理相结合，提出了一种构造隐式曲线的新方法，即首先由约束点构造神经网络的输入与输出，把描述物体边界曲线的隐式函数转化为显式函数；然后用BP神经网络对此显式函数进行逼近；最后由仿真曲面得到物体边界的拟合曲线。该新方法不同于传统的对显式函数的逼近方法，因为传统方法无法描述封闭的曲线；也不同于基于优化的拟合隐式曲线方法，因为它无须考虑函数的形式或多项式的次数。实验表明，该新方法有很强的物体边界描述能力和缺损修复能力，因而在物体边界重建、缺损图像复原等领域有一定的应用前景。     

一种基于文法演化自动拟合非线性数据的蜂群算法

将文法演化方法引入蜂群算法, 基于上下文文法定义多种数学函数, 提出了自动拟合非线性数据的蜂群算法BCGE, 并给出了通过基因截断、基因增补及利用领域知识加速BCGE的方法。基于文法演化的BCGE比基于文法规划的其他算法更为简洁, 且存储基因型所需的空间也远比其他算法存储语法树所需的空间少。通过五个测试函数的实验表明, BCGE能够有效地拟合非常复杂的非线性数据。     

Data fitting with a spline using a real-coded genetic algorithm

To obtain a good approximation for data fitting with a spline, frequently we have to deal with knots as variables. The problem to be solved then becomes a continuous nonlinear and multivariate optimization problem with many local optima. Therefore, it is difficult to obtain the global optimum. In this paper, we propose a method for solving this problem by using a real-coded genetic algorithm. Our method can treat not only data with a smooth underlying function, but also data with an underlying function having discontinuous points and/or cusps. We search for the best model among candidate models by using the Bayes Information Criterion (BIC). With this, we can appropriately determine the number and locations of knots automatically and simultaneously. Five examples of data fitting are given to show the performance of our method.     

frailtypack: a computer program for the analysis of correlated failure time data using penalized likelihood estimation

Correlated survival outcomes occur quite frequently in the biomedical research. Available software is limited, particularly if we wish to obtain smoothed estimate of the baseline hazard function in the context of random effects model for correlated data. The main objective of this paper is to describe an R package called frailtypack that can be used for estimating the parameters in a shared gamma frailty model with possibly right-censored, left-truncated stratified survival data using penalized likelihood estimation. Time-dependent structure for the explanatory variables and/or extension of the Cox regression model to recurrent events are also allowed. This program can also be used simply to obtain directly a smooth estimate of the baseline hazard function. To illustrate the program we used two data sets, one with clustered survival times, the other one with recurrent events, i.e., the rehospitalizations of patients diagnosed with colorectal cancer. We show how to fit the model with recurrent events and time-dependent covariates using Andersen-Gill approach.     

基于模糊理论的随机需求系统

需求量是供销系统成本优化中的重要指标.目前常见的研究方法是采用泊松分布来拟合.通过数据统计和分析,我们发现采用模糊三角去拟合随机需求量有更好的拟合度和科学性.采用模糊理论对随机需求进行了建模分析,设计了随机需求系统的算法并最终对其进行了实现.