Nonlinear principal components analysis: Introduction and application.

The authors provide a didactic treatment of nonlinear (categorical) principal components analysis (PCA). This method is the nonlinear equivalent of standard PCA and reduces the observed variables to a number of uncorrelated principal components. The most important advantages of nonlinear over linear PCA are that it incorporates nominal and ordinal variables and that it can handle and discover nonlinear relationships between variables. Also, nonlinear PCA can deal with variables at their appropriate measurement level; for example, it can treat Likert-type scales ordinally instead of numerically. Every observed value of a variable can be referred to as a category. While performing PCA, nonlinear PCA converts every category to a numeric value, in accordance with the variable's analysis level, using optimal quantification. The authors discuss how optimal quantification is carried out, what analysis levels are, which decisions have to be made when applying nonlinear PCA, and how the results can be interpreted. The strengths and limitations of the method are discussed. An example applying nonlinear PCA to empirical data using the program CATPCA (J. J. Meulman, W. J. Heiser, & SPSS, 2004) is provided. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Bootstrap estimates of standard errors in validity generalization.

Bootstrapping is introduced as a method for approximating the standard errors of validity generalization (VG) estimates. A Monte Carlo study was conducted to evaluate the accuracy of bootstrap validity-distribution parameter estimates, bootstrap standard error estimates, and nonparametric bootstrap confidence intervals. In the simulation study the authors manipulated the sample sizes per correlation coefficient, the number of coefficients per VG analysis, and the variance of the distribution of true correlation coefficients. The results indicate that the standard error estimates produced by the bootstrapping procedure were very accurate. It is recommended that the bootstrap standard-error estimates and confidence intervals be used in the interpretation of the results of VG analyses. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Assessing program effects in the presence of treatment—baseline interactions: A latent curve approach.

Methods for assessing treatment effects of longitudinal randomized intervention are considered. The focus is on modeling the nonlinear relationship between treatment effects and baseline often observed in prevention programs designed for at-risk populations. Piecewise linear growth modeling was used to study treatment effects during the different periods of development. A multistep multiple-group analysis procedure is proposed for assessing treatment effects in the presence of nonlinear treatment-baseline interactions. Standard errors of the estimates from this multistep procedure were obtained using a bootstrap approach. The methods are illustrated using data from the Johns Hopkins Prevention Research Center involving an intervention aimed at improving classroom behavior. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Dimensionality assessment of ordered polytomous items with parallel analysis.

Parallel analysis (PA) is an often-recommended approach for assessment of the dimensionality of a variable set. PA is known in different variants, which may yield different dimensionality indications. In this article, the authors considered the most appropriate PA procedure to assess the number of common factors underlying ordered polytomously scored variables. They proposed minimum rank factor analysis (MRFA) as an extraction method, rather than the currently applied principal component analysis (PCA) and principal axes factoring. A simulation study, based on data with major and minor factors, showed that all procedures consistently point at the number of major common factors. A polychoric-based PA slightly outperformed a Pearson-based PA, but convergence problems may hamper its empirical application. In empirical practice, PA-MRFA with a 95% threshold based on polychoric correlations or, in case of nonconvergence, Pearson correlations with mean thresholds appear to be a good choice for identification of the number of common factors. PA-MRFA is a common-factor-based method and performed best in the simulation experiment. PA based on PCA with a 95% threshold is second best, as this method showed good performances in the empirically relevant conditions of the simulation experiment. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Nonlinear Controller Synthesis for Single-Input Multiple-Output Systems with Application to Cruise Missile

The purpose of this paper is to present a new systematic procedure for synthesis of nonlinear controllers for single-input multiple-output systems. The procedure is based on several describing function models of the plant, and the nonlinear gains are obtained via an inverse describing function technique. The procedure and associated software are applied to control the bank angle of a cruise missile. The results are compared with a linear controller and another nonlinear controller that was previously reported in the open literature; it is shown that the developed procedure has resulted in a less conservative design (i.e., performance is not sacrificed to assure absolute stability).     

Stable Boundary Element Method/Finite Element Method Procedure for Dynamic Fluid–Structure Interactions

The stability problem has prevented the application of the boundary element method/finite element method (BEM/FEM) coupling procedure in dynamic fluid–structure interaction problems for the last 2 decades. It has been proved that the linear θ method can make a significant stability improvement for the time domain BEM scheme. With the use of the linear θ method, the BEM/FEM coupling procedure is applied to two-dimensional time domain fluid–structure interaction problems. The fluid domain is acoustic and modeled by taking advantage of the BEM scheme that is suitable to either finite or infinite domains. An internal source has been considered in BEM formulations, and no artificial boundary needs to be introduced for the infinite domain. The structure is modeled by finite elements that can be either linear or nonlinear. Two classical examples are given to show the validity of the coupling procedure in fluid–structure interaction problems and the significant stability improvement given by the linear θ method to the BEM/FEM coupling procedure.     

基于PCA-BP神经网络的尾矿库坝体稳定性分析

基于尾矿库坝体稳定性影响因素具有复杂多变性、随机不确定性和非线性的特点,为了确保尾矿库稳定并且有效预防尾矿库事故的发生,建立了基于PCA和BP神经网络的尾矿库坝体稳定性研究模型,该模型通过应用SPSS软件对来自多个尾矿库失稳实例的原始数据进行标准化,找出影响尾矿库坝体稳定的主成分,然后把主成分作为BP输入样本,运用MATLAB软件进行训练仿真。结果表明：在BP训练前,利用PCA算法对原始样本进行预处理,能有效地提高训练的速度和精度,得出了PCA-BP神经网络模型在尾矿库坝体稳定性分析中是可行的。     

Distribution Networks with Linear and Nonlinear Loads: Cost-Based Method for Planning Dispersed Generation and Capacitor Units under Uncertainty

This paper discusses the optimal allocation of capacitors and dispersed generation units in a medium-voltage distribution system with linear and nonlinear loads. An optimization problem is formulated and solved for optimal siting and sizing, with the objective of minimizing costs sustained by the distributor for reactive power service, even containing the impact of the waveform distortions. A three-step procedure, based on genetic algorithm and decision theory, is applied to obtain a solution for the optimization problem that takes into account the unavoidable uncertainties that the dispersed generation production and the loads introduce. The numeric applications performed on an 18-bus test distribution system demonstrate of the effectiveness of the proposed procedure.     

Linearized Stability and Accuracy for Step-by-Step Solutions of Certain Nonlinear Systems

In this work, stability and accuracy of the Newmark method for nonlinear systems are obtained from a linearized analysis. This analysis reveals that an unconditionally stable integration method for linear elastic systems is unconditionally stable for nonlinear systems and a conditionally stable integration method for linear elastic systems remains conditionally stable for nonlinear systems except that its upper stability limit might vary with the step degree of nonlinearity and step degree of convergence. A sufficient condition to have a stable computation for nonlinear systems in a whole step-by-step integration procedure is also developed in this study. Furthermore, it is also found that numerical accuracy in the solution of nonlinear systems is closely related to the step degree of nonlinearity and step degree of convergence although its characteristics are similar to those of the preceding works for linear elastic systems. Since these results are obtained from a linearized analysis, they can be applicable to the nonlinear systems that satisfied the simplifications for the analysis but may not be applicable to general nonlinear systems.     

Stability Analysis of Direct Integration Algorithms Applied to MDOF Nonlinear Structural Dynamics

Direct integration algorithms are typically used to solve temporally discretized equations of motion to evaluate the performance of structures under dynamic loading. The stability of these direct integration algorithms are usually investigated for linear elastic structures. However integration algorithms are often applied to structures with nonlinear behavior. This paper presents a procedure based on discrete control theory to investigate the stability of direct integration algorithms applied to multidegree-of-freedom (MDOF) nonlinear structures. The discrete root locus approach is used to investigate properties of the poles of the discrete transfer function matrix representing the nonlinear structural dynamics and to assess the stability of the integration algorithm. The procedure is illustrated using a nonlinear shear building MDOF system to investigate the stability of popular direct integration algorithms, including the Newmark family of integration algorithms, the Hilber-Hughes-Taylor α-method, and two newly developed explicit integration algorithms. Stability limits are derived for the direct integration algorithms that are found to be conditionally stable.     

Regression Model for Daily Maximum Stream Temperature

An empirical model is developed to predict daily maximum stream temperatures for the summer period. The model is created using a stepwise linear regression procedure to select significant predictors. The predictive model includes a prediction confidence interval to quantify the uncertainty. The methodology is applied to the Truckee River in California and Nevada. The stepwise procedure selects daily maximum air temperature and average daily flow as the variables to predict maximum daily stream temperature at Reno, Nev. The model is shown to work in a predictive mode by validation using three years of historical data. Using the uncertainty quantification, the amount of required additional flow to meet a target stream temperature with a desired level of confidence is determined.     

A Monte Carlo study of the inferential properties of three methods of shape comparison

Three inferential morphometric methods, Euclidean distance matrix analysis (EDMA), Bookstein's edge-matching method (EMM), and the Procrustes method, were applied to facial landmark data. A Monte Carlo simulation was conducted with three sample sizes, ranging from n = 10 to 50, to assess type I error rates and the power of the tests to detect group differences for two- and three-dimensional representations of forms. Type I error rates for EMM were at or below nominal levels in both two and three dimensions. Procrustes in 2D and EDMA in 2D and 3D produced inflated type I error rates in all conditions, but approached acceptable levels with moderate cell sizes. Procrustes maintained error rates below the nominal levels in 2D. The power of EMM was high compared with the other methods in both 2D and 3D, but, conflicting EMM decisions were provided depending on which pair (2D) or triad (3D) of landmarks were selected as reference points. EDMA and Procrustes were more powerful in 2D data than for 3D data. Interpretation of these results must take into account that the data used in this simulation were selected because they represent real data that might have been collected during a study or experiment. These data had characteristics which violated assumptions central to the methods here with unequal variances about landmarks, correlated errors, and correlated landmark locations; therefore these results may not generalize to all conditions, such as cases with no violations of assumptions. This simulation demonstrates, however, limitations of each procedure that should be considered when making inferences about shape comparisons.     

Perforin expression in human cell-mediated luteolysis

For a variety of reasons, small vessels have low signal intensity in magnetic resonance angiography. When the vessel signal intensity is lower than the signal intensity of background tissues, these vessels tend not to be visible on maximum-intensity-projection images. The authors developed a nonlinear second-difference spatial filtering technique that enhances the details of small vessels while suppressing both noise and uniform background tissue. Two similar nonlinear second-difference filters are presented and compared with the linear Laplacian second-difference filter. To evaluate the performance of these filters, they were applied to intracranial three-dimensional time-of-flight MR angiographic data and the results compared with the vessel enhancement obtained with a simple second-difference Laplacian filter and with magnetization transfer contrast (MTC) techniques. The comparisons demonstrated that nonlinear filtering and MTC techniques result in similar improvement in small-vessel visibility and apparent continuity. A quantitative comparison demonstrated that the improvement in the contrast-to-noise ratio is much greater with the nonlinear filters than the Laplacian filter.     

Estimating the nonlinear and interactive effects of latent variables.

Describes a procedure that enables researchers to estimate nonlinear and interactive effects of latent variables in structural equation models. Given that the latent variables are normally distributed, the parameters of such models can be estimated. To do this, products of the measured variables are used as indicators of latent product variables. Estimation must be done using a procedure that allows nonlinear constraints on parameters. The procedure is demonstrated in 3 examples. The 1st 2 examples use artificial data with known parameter values. These parameters are successfully recovered by the procedure. The final complex example uses national election survey data. (14 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Advances in testing the statistical significance of mediation effects.

P. A. Frazier, A. P. Tix, and K. E. Barron (2004) highlighted a normal theory method popularized by R. M. Baron and D. A. Kenny (1986) for testing the statistical significance of indirect effects (i.e., mediator variables) in multiple regression contexts. However, simulation studies suggest that this method lacks statistical power relative to some other approaches. The authors describe an alternative developed by P. E. Shrout and N. Bolger (2002) based on bootstrap resampling methods. An example and step-by-step guide for performing bootstrap mediation analyses are provided. The test of joint significance is also briefly described as an alternative to both the normal theory and bootstrap methods. The relative advantages and disadvantages of each approach in terms of precision in estimating confidence intervals of indirect effects, Type I error, and Type II error are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Discriminant analysis: Statistical inference using the jackknife and bootstrap procedures.

Structure coefficients from discriminant analyses assist the substantive interpretation with rules of thumb indicating whether variables load on discriminant functions. A variable's structure coefficient should at least be statistically different from zero before it is interpreted. Unfortunately, standard errors of structure coefficients are not available. However, the bootstrap and the jackknife procedures provide statistical tests in such circumstances. Bootstrap and jackknife analyses of an example data set obtained different interpretations from those using the usual rules of thumb. Results from Monte Carlo studies, in terms of Type I error rates and confidence interval coverage, did not support the usual rules of thumb and clearly showed the superiority of the standard bootstrap test and the bootstrap percentile confidence interval. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Confidence intervals for the receiver operating characteristic area in studies with small samples

RATIONALE AND OBJECTIVES: The authors performed this study to address two practical questions. First, how large does the sample size need to be for confidence intervals (CIs) based on the usual asymptotic methods to be appropriate? Second, when the sample size is smaller than this threshold, what alternative method of CI construction should be used? MATERIALS AND METHODS: The authors performed a Monte Carlo simulation study where 95% CIs were constructed for the receiver operating characteristic (ROC) area and for the difference between two ROC areas for rating and continuous test results--for ROC areas of moderate and high accuracy--by using both parametric and nonparametric estimation methods. Alternative methods evaluated included several bootstrap CIs and CIs with the Student t distribution. RESULTS: For the difference between two ROC areas, CIs based on the asymptotic theory provided adequate coverage even when the sample size was very small (20 patients). In contrast, for a single ROC area, the asymptotic methods do not provide adequate CI coverage for small samples; for ROC areas of high accuracy, the sample size must be large (more than 200 patients) for the asymptotic methods to be applicable. The recommended alternative (bootstrap percentile, bootstrap t, or bootstrap bias-corrected accelerated method) depends on the estimation approach, format of the test results, and ROC area. CONCLUSION: Currently, there is not a single best alternative for constructing CIs for a single ROC area for small samples.     

Determining the Number of Clusters by Sampling With Replacement.

A split-sample replication criterion originally proposed by J. E. Overall and K. N. Magee (1992) as a stopping rule for hierarchical cluster analysis is applied to multiple data sets generated by sampling with replacement from an original simulated primary data set. An investigation of the validity of this bootstrap procedure was undertaken using different combinations of the true number of latent populations, degrees of overlap, and sample sizes. The bootstrap procedure enhanced the accuracy of identifying the true number of latent populations under virtually all conditions. Increasing the size of the resampled data sets relative to the size of the primary data set further increased accuracy. A computer program to implement the bootstrap stopping rule is made available via a referenced Web site. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Kinetic parameter estimation by numerical algorithms and multiple linear regression: theoretical

A new method is presented for the determination of kinetic parameters based on a functional relationship among experimental data derived from the postulated model. The data, even though containing errors, are manifestations of this relationship, which should be satisfied by parameters fitted to the system. The procedure involves the use of numerical integration and/or differentiation of the data, followed by multiple linear regression. It does not require initial estimates or repetitive iteration for linear systems and can be applied to nonlinear models. The accuracy of estimated parameter depends on the goodness of the particular numerical approximation method used.     

A tetrad test for causal indicators.

The authors propose a confirmatory tetrad analysis test to distinguish causal from effect indicators in structural equation models. The test uses "nested" vanishing tetrads that are often implied when comparing causal and effect indicator models. The authors present typical models that researchers can use to determine the vanishing tetrads for 4 or more variables. They also provide the vanishing tetrads for mixtures of causal and effect indicators, for models with fewer than 4 indicators per latent variable, or for cases with correlated errors. The authors illustrate the test results for several simulation and empirical examples and emphasize that their technique is a theory-testing rather than a model-generating approach. They also review limitations of the procedure including the indistinguishable tetrad equivalent models, the largely unknown finite sample behavior of the test statistic, and the inability of any procedure to fully validate a model specification. (PsycINFO Database Record (c) 2010 APA, all rights reserved)