Regularization methods for the solution of a nonlinear least-squares problem in tomography

In this work we study a polyenergetic and multimaterial model for the breast image reconstruction in Digital Tomosynthesis, taking into consideration the variety of the materials forming the object and the polyenergetic nature of the X-rays beam. The modelling of the problem leads to the resolution of a high-dimensional nonlinear least-squares problem that, due to its nature of inverse ill-posed problem, needs some kind of regularization. We test two main classes of methods: the Levenberg-Marquardt method (together with the Conjugate Gradient method for the computation of the descent direction) and two limited-memory BFGS-like methods (L-BFGS). We perform some experiments for different values of the regularization parameter (constant or varying at each iteration), tolerances and stop conditions. Finally, we analyse the performance of the several methods comparing relative errors, iterations number, times and the qualities of the reconstructed images.

Comparison of partial least squares regression and principal component regression for pelvic shape prediction.

This paper studied two different regression techniques for pelvic shape prediction, i.e., the partial least square regression (PLSR) and the principal component regression (PCR). Three different predictors such as surface landmarks, morphological parameters, or surface models of neighboring structures were used in a cross-validation study to predict the pelvic shape. Results obtained from applying these two different regression techniques were compared to the population mean model. In almost all the prediction experiments, both regression techniques unanimously generated better results than the population mean model, while the difference on prediction accuracy between these two regression methods is not statistically significant (α=0.01).

A laser induced breakdown spectroscopy quantitative analysis method based on the robust least squares support vector machine regression model

Data fluctuation in multiple measurements of Laser Induced Breakdown Spectroscopy (LIBS) greatly affects the accuracy of quantitative analysis. A new LIBS quantitative analysis method based on the Robust Least Squares Support Vector Machine (RLS-SVM) regression model is proposed. The usual way to enhance the analysis accuracy is to improve the quality and consistency of the emission signal, such as by averaging the spectral signals or spectrum standardization over a number of laser shots. The proposed method focuses more on how to enhance the robustness of the quantitative analysis regression model. The proposed RLS-SVM regression model originates from the Weighted Least Squares Support Vector Machine (WLS-SVM) but has an improved segmented weighting function and residual error calculation according to the statistical distribution of measured spectral data. Through the improved segmented weighting function, the information on the spectral data in the normal distribution will be retained in the regression model while the information on the outliers will be restrained or removed. Copper elemental concentration analysis experiments of 16 certified standard brass samples were carried out. The average value of relative standard deviation obtained from the RLS-SVM model was 3.06% and the root mean square error was 1.537%. The experimental results showed that the proposed method achieved better prediction accuracy and better modeling robustness compared with the quantitative analysis methods based on Partial Least Squares (PLS) regression, standard Support Vector Machine (SVM) and WLS-SVM. It was also demonstrated that the improved weighting function had better comprehensive performance in model robustness and convergence speed, compared with the four known weighting functions.

High-Throughput Prediction of Acacia and Eucalypt Lignin Syringyl/Guaiacyl Content Using FT-Raman Spectroscopy and Partial Least Squares Modeling

High-throughput techniques are necessary to efficiently screen potential lignocellulosic feedstocks for the production of renewable fuels, chemicals, and bio-based materials, thereby reducing experimental time and expense while supplanting tedious, destructive methods. The ratio of lignin syringyl (S) to guaiacyl (G) monomers has been routinely quantified as a way to probe biomass recalcitrance. Mid-infrared and Raman spectroscopy have been demonstrated to produce robust partial least squares models for the prediction of lignin S/G ratios in a diverse group of Acacia and eucalypt trees. The most accurate Raman model has now been used to predict the S/G ratio from 269 unknown Acacia and eucalypt feedstocks. This study demonstrates the application of a partial least squares model composed of Raman spectral data and lignin S/G ratios measured using pyrolysis/molecular beam mass spectrometry (pyMBMS) for the prediction of S/G ratios in an unknown data set. The predicted S/G ratios calculated by the model were averaged according to plant species, and the means were not found to differ from the pyMBMS ratios when evaluating the mean values of each method within the 95 % confidence interval. Pairwise comparisons within each data set were employed to assess statistical differences between each biomass species. While some pairwise appraisals failed to differentiate between species, Acacias, in both data sets, clearly display significant differences in their S/G composition which distinguish them from eucalypts. This research shows the power of using Raman spectroscopy to supplant tedious, destructive methods for the evaluation of the lignin S/G ratio of diverse plant biomass materials. © 2015, The Author(s).

Quantification of chlorpheniramine maleate enantiomers by ultraviolet spectroscopy and chemometric methods

Chlorpheniramine maleate (CLOR) enantiomers were quantified by ultraviolet spectroscopy and partial least squares regression. The CLOR enantiomers were prepared as inclusion complexes with beta-cyclodextrin and 1-butanol with mole fractions in the range from 50 to 100%. For the multivariate calibration the outliers were detected and excluded and variable selection was performed by interval partial least squares and a genetic algorithm. Figures of merit showed results for accuracy of 3.63 and 2.83% (S)-CLOR for root mean square errors of calibration and prediction, respectively. The ellipse confidence region included the point for the intercept and the slope of 1 and 0, respectively. Precision and analytical sensitivity were 0.57 and 0.50% (S)-CLOR, respectively. The sensitivity, selectivity, adjustment, and signal-to-noise ratio were also determined. The model was validated by a paired t test with the results obtained by high-performance liquid chromatography proposed by the European pharmacopoeia and circular dichroism spectroscopy. The results showed there was no significant difference between the methods at the 95% confidence level, indicating that the proposed method can be used as an alternative to standard procedures for chiral analysis.

Rock fracture mean trace length estimation and confidence interval calculation using maximum likelihood methods

There has been a resurgence of interest in the mean trace length estimator of Pahl for window sampling of traces. The estimator has been dealt with by Mauldon and Zhang and Einstein in recent publications. The estimator is a very useful one in that it is non-parametric. However, despite some discussion regarding the statistical distribution of the estimator, none of the recent works or the original work by Pahl provide a rigorous basis for the determination a confidence interval for the estimator or a confidence region for the estimator and the corresponding estimator of trace spatial intensity in the sampling window. This paper shows, by consideration of a simplified version of the problem but without loss of generality, that the estimator is in fact the maximum likelihood estimator (MLE) and that it can be considered essentially unbiased. As the MLE, it possesses the least variance of all estimators and confidence intervals or regions should therefore be available through application of classical ML theory. It is shown that valid confidence intervals can in fact be determined. The results of the work and the calculations of the confidence intervals are illustrated by example. (C) 2003 Elsevier Science Ltd. All rights reserved.

Time domain design of fractional differintegrators using least-squares

In this paper we propose the use of the least-squares based methods for obtaining digital rational approximations (IIR filters) to fractional-order integrators and differentiators of type sα, α∈R. Adoption of the Padé, Prony and Shanks techniques is suggested. These techniques are usually applied in the signal modeling of deterministic signals. These methods yield suboptimal solutions to the problem which only requires finding the solution of a set of linear equations. The results reveal that the least-squares approach gives similar or superior approximations in comparison with other widely used methods. Their effectiveness is illustrated, both in the time and frequency domains, as well in the fractional differintegration of some standard time domain functions.

Design of Digital Fractional-Order Integrators and Differentiators by Least Squares

First IFAC Workshop on Fractional Differentiation and Its Application - 19-21 July 2004, Enseirb, Bordeaux, France - FDA'04

Diffusional kurtosis imaging using a fast heuristic constrained linear least squares algorithm: a plugin for OsiriX

Diffusion Kurtosis Imaging (DKI) is a fairly new magnetic resonance imag-ing (MRI) technique that tackles the non-gaussian motion of water in biological tissues by taking into account the restrictions imposed by tissue microstructure, which are not considered in Diffusion Tensor Imaging (DTI), where the water diffusion is considered purely gaussian. As a result DKI provides more accurate information on biological structures and is able to detect important abnormalities which are not visible in standard DTI analysis. This work regards the development of a tool for DKI computation to be implemented as an OsiriX plugin. Thus, as OsiriX runs under Mac OS X, the pro-gram is written in Objective-C and also makes use of Apple’s Cocoa framework. The whole program is developed in the Xcode integrated development environ-ment (IDE). The plugin implements a fast heuristic constrained linear least squares al-gorithm (CLLS-H) for estimating the diffusion and kurtosis tensors, and offers the user the possibility to choose which maps are to be generated for not only standard DTI quantities such as Mean Diffusion (MD), Radial Diffusion (RD), Axial Diffusion (AD) and Fractional Anisotropy (FA), but also DKI metrics, Mean Kurtosis (MK), Radial Kurtosis (RK) and Axial Kurtosis (AK).The plugin was subjected to both a qualitative and a semi-quantitative analysis which yielded convincing results. A more accurate validation pro-cess is still being developed, after which, and with some few minor adjust-ments the plugin shall become a valid option for DKI computation

Reflections on Partial Least Squares Path Modeling

The purpose of the present article is to take stock of a recent exchange in Organizational Research Methods between critics (Rönkkö & Evermann, 2013) and proponents (Henseler et al., 2014) of partial least squares path modeling (PLS-PM). The two target articles were centered around six principal issues, namely whether PLS-PM: (1) can be truly characterized as a technique for structural equation modeling (SEM); (2) is able to correct for measurement error; (3) can be used to validate measurement models; (4) accommodates small sample sizes; (5) is able to provide null hypothesis tests for path coefficients; and (6) can be employed in an exploratory, model-building fashion. We summarize and elaborate further on the key arguments underlying the exchange, drawing from the broader methodological and statistical literature in order to offer additional thoughts concerning the utility of PLS-PM and ways in which the technique might be improved. We conclude with recommendations as to whether and how PLS-PM serves as a viable contender to SEM approaches for estimating and evaluating theoretical models.

Least squares regression with errors in both variables: case studies

Analytical curves are normally obtained from discrete data by least squares regression. The least squares regression of data involving significant error in both x and y values should not be implemented by ordinary least squares (OLS). In this work, the use of orthogonal distance regression (ODR) is discussed as an alternative approach in order to take into account the error in the x variable. Four examples are presented to illustrate deviation between the results from both regression methods. The examples studied show that, in some situations, ODR coefficients must substitute for those of OLS, and, in other situations, the difference is not significant.

Non invasive classification system of scoliosis curve types using least-squares support vector machines

Objective To determine scoliosis curve types using non invasive surface acquisition, without prior knowledge from X-ray data. Methods Classification of scoliosis deformities according to curve type is used in the clinical management of scoliotic patients. In this work, we propose a robust system that can determine the scoliosis curve type from non invasive acquisition of the 3D back surface of the patients. The 3D image of the surface of the trunk is divided into patches and local geometric descriptors characterizing the back surface are computed from each patch and constitute the features. We reduce the dimensionality by using principal component analysis and retain 53 components using an overlap criterion combined with the total variance in the observed variables. In this work, a multi-class classifier is built with least-squares support vector machines (LS-SVM). The original LS-SVM formulation was modified by weighting the positive and negative samples differently and a new kernel was designed in order to achieve a robust classifier. The proposed system is validated using data from 165 patients with different scoliosis curve types. The results of our non invasive classification were compared with those obtained by an expert using X-ray images. Results The average rate of successful classification was computed using a leave-one-out cross-validation procedure. The overall accuracy of the system was 95%. As for the correct classification rates per class, we obtained 96%, 84% and 97% for the thoracic, double major and lumbar/thoracolumbar curve types, respectively. Conclusion This study shows that it is possible to find a relationship between the internal deformity and the back surface deformity in scoliosis with machine learning methods. The proposed system uses non invasive surface acquisition, which is safe for the patient as it involves no radiation. Also, the design of a specific kernel improved classification performance.

The method of predicate least squares refinement

Parameters to be determined in a least squares refinement calculation to fit a set of observed data may sometimes usefully be `predicated' to values obtained from some independent source, such as a theoretical calculation. An algorithm for achieving this in a least squares refinement calculation is described, which leaves the operator in full control of the weight that he may wish to attach to the predicate values of the parameters.

A modified PNN algorithm with optimal PD modeling using the orthogonal least squares method

In this paper a modified algorithm is suggested for developing polynomial neural network (PNN) models. Optimal partial description (PD) modeling is introduced at each layer of the PNN expansion, a task accomplished using the orthogonal least squares (OLS) method. Based on the initial PD models determined by the polynomial order and the number of PD inputs, OLS selects the most significant regressor terms reducing the output error variance. The method produces PNN models exhibiting a high level of accuracy and superior generalization capabilities. Additionally, parsimonious models are obtained comprising a considerably smaller number of parameters compared to the ones generated by means of the conventional PNN algorithm. Three benchmark examples are elaborated, including modeling of the gas furnace process as well as the iris and wine classification problems. Extensive simulation results and comparison with other methods in the literature, demonstrate the effectiveness of the suggested modeling approach.

A constrained recursive least squares algorithm for adaptive combination of multiple models

In this paper, we develop a novel constrained recursive least squares algorithm for adaptively combining a set of given multiple models. With data available in an online fashion, the linear combination coefficients of submodels are adapted via the proposed algorithm.We propose to minimize the mean square error with a forgetting factor, and apply the sum to one constraint to the combination parameters. Moreover an l1-norm constraint to the combination parameters is also applied with the aim to achieve sparsity of multiple models so that only a subset of models may be selected into the final model. Then a weighted l2-norm is applied as an approximation to the l1-norm term. As such at each time step, a closed solution of the model combination parameters is available. The contribution of this paper is to derive the proposed constrained recursive least squares algorithm that is computational efficient by exploiting matrix theory. The effectiveness of the approach has been demonstrated using both simulated and real time series examples.