A self‐guided search for good local minima of the sum‐of‐squared‐error in nonlinear least squares regression

Hard modeling of nonlinear chemical or biological systems is highly relevant as a model function together with values for model parameters provides insights in the systems' functionalities. Deriving values for said model parameters via nonlinear regression, however, is challenging as usually one of the numerous local minima of the sum‐of‐squared errors (SSEs) is determined; furthermore, for different starting points, different minima may be found. Thus, nonlinear regression is prone to low accuracy and low reproducibility. Therefore, there is a need for a generally applicable, automated initialization of nonlinear least squares algorithms, which reaches a good, reproducible solution after spending a reasonable computation time probing the SSE‐hypersurface. For this purpose, a three‐step methodology is presented in this study. First, the SSE‐hypersurface is randomly probed in order to estimate probability density functions of initial model parameter that generally lead to an accurate fit solution. Second, these probability density functions then guide a high‐resolution sampling of the SSE‐hypersurface. This second probing focuses on those model parameter ranges that are likely to produce a low SSE. As the probing continues, the most appropriate initial guess is retained and eventually utilized in a subsequent nonlinear regression. It is shown that this “guided random search” derives considerably better regression solutions than linearization of model functions, which has so far been considered the best‐case scenario. Examples from infrared spectroscopy, cell culture monitoring, reaction kinetics, and image analyses demonstrate the broad and successful applicability of this novel method. Copyright © 2014 John Wiley & Sons, Ltd.     

Evaluation of preprocessing steps to compensate for magnetic field distortions due to body movements in BOLD fMRI

Blood oxygenation level-dependent (BOLD) functional magnetic resonance imaging (fMRI) is currently the dominant technique for non-invasive investigation of brain functions. One of the challenges with BOLD fMRI, particularly at high fields, is compensation for the effects of spatiotemporally varying magnetic field inhomogeneities (ΔB₀) caused by normal subject respiration and, in some studies, movement of the subject during the scan to perform tasks related to the functional paradigm. The presence of ΔB₀ during data acquisition distorts reconstructed images and introduces extraneous fluctuations in the fMRI time series that decrease the BOLD contrast-to-noise ratio. Optimization of the fMRI data-processing pipeline to compensate for geometric distortions is of paramount importance to ensure high quality of fMRI data. To investigate ΔB₀ caused by subject movement, echo-planar imaging scans were collected with and without concurrent motion of a phantom arm. The phantom arm was constructed and moved by the experimenter to emulate forearm motions while subjects remained still and observed a visual stimulation paradigm. These data were then subjected to eight different combinations of preprocessing steps. The best preprocessing pipeline included navigator correction, a complex phase regressor and spatial smoothing. The synergy between navigator correction and phase regression reduced geometric distortions better than either step in isolation and preconditioned the data to make them more amenable to the benefits of spatial smoothing. The combination of these steps provided a 10% increase in t-statistics compared to only navigator correction and spatial smoothing and reduced the noise and false activations in regions where no legitimate effects would occur.     

On the selection of the most adequate radial basis function

Radial basis function (RBF) methods can provide excellent interpolants for a large number of poorly distributed data points. For any finite data set in any Euclidean space, one can construct an interpolation of the data by using RBFs. However, RBF interpolant trends between and beyond the data points depend on the RBF used and may exhibit undesirable trends using some RBFs while the trends may be desirable using other RBFs. The fact that a certain RBF is commonly used for the class of problems at hand, previous good behavior in that (or other) class of problems, and bibliography, are just some of the many valid reasons given to justify a priori selection of RBF. Even assuming that the justified choice of the RBF is most likely the correct choice, one should nonetheless confirm numerically that, in fact, the most adequate RBF for the problem at hand is the RBF chosen a priori. The main goal of this paper is to alert the analyst as to the danger of a priori selection of RBF and to present a strategy to numerically choose the most adequate RBF that better captures the trends of the given data set. The wing weight data fitting problem is used to illustrate the benefits of an adequate choice of RBF for each given data set.     

非参数固定效应Panel Data模型的统计推断

本文主要讨论了非参数固定效应Panel data模型的估计与检验问题,首先我们利用Profile最小二乘方法得到了固定效应与非参数部分的估计;接着基于比较原假设与备择假设下模型拟合的残差平方和的思想针对固定效应的检验问题构造了检验统计量,并给出了计算检验p-值的F分布逼近法。     

半参数回归模型独立情形的分离法估计

对半参数回归模型,用L2最佳逼近加矩估计的方法,推出其非参数部分的依L2与强相合联合收敛意义下的估计,及参数部分的强相合与相合渐近正态估计,并设计实行了一个模拟实验.     

Khmaladze transformation of integrated variance processes with applications to goodness-of-fit testing

In the common nonparametric regression model the problem of testing for a specific parametric form of the variance function is considered. Recently Dette and Hetzler [8] proposed a test statistic which is based on an empirical process of pseudo residuals. The process converges weakly to a Gaussian process with a complicated covariance kernel depending on the data generating process. In the present paper we consider a standardized version of this process and propose an application of the Khmaladze transformation to obtain asymptotically distribution-free tests for the corresponding Kolmogorov-Smirnov and Cramér-von Mises functionals. The finite-sample properties of the proposed tests are investigated by means of a simulation study.      

A new regression method based on independent component analysis

Based on independent component analysis (ICA), a new regression method, independent component regression (ICR), was developed to build the model of NIR spectra and the routine components of plant samples. It is found that ICR and principal component regression (PCR) are completely equivalent when they are applied in quantitative prediction. However, independent components (ICs) can give more chemical explanation than principal components (PCs) because independence is a high-order statistic that is a much stronger condition than orthogonality. Three ICs are obtained by ICA from the NIR spectra of plant samples; it is found that they are strongly correlated to the NIR spectra of water, hydrocarbons and organonitrogen compounds, respectively. Therefore, ICA may be a promising tool to retrieve both quantitative and qualitative information from complex chemical data sets.