Local Least-squares Research Articles

Identifying 1-rectifiable measures in Carnot groups

Abstract We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of M. Badger and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable curves and the part of the measure that is singular to rectifiable curves. Our main result is entwined with an extension of analyst’s traveling salesman theorem, which characterizes the subsets of rectifiable curves in R 2 {{\mathbb{R}}}^{2} (P. W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1–15), in R n {{\mathbb{R}}}^{n} (K. Okikiolu, Characterization of subsets of rectifiable curves in R n {{\bf{R}}}^{n} , J. London Math. Soc. (2) 46 (1992), no. 2, 336–348), or in an arbitrary Carnot group (S. Li) in terms of local geometric least-squares data called Jones’ β \beta -numbers. In a secondary result, we implement the Garnett-Killip-Schul construction of a doubling measure in R n {{\mathbb{R}}}^{n} that charges a rectifiable curve in an arbitrary complete, doubling, locally quasiconvex metric space.

Spectral Profile Partial Least-Squares (SP-PLS): Local multivariate pansharpening on spectral profiles

The compatibility of multispectral (MS) pansharpening algorithms with hyperspectral (HS) data is limited. With the recent development in HS satellites, there is a need for methods that can provide high spatial and spectral fidelity in both HS and MS scenarios.The present article presents a fast pansharpening method, based on the division of similar hyperspectral data in spectral subgroups using k-means clustering and Spectral Angle Mapper (SAM) profiling. Local Partial Least-Square (PLS) models are calibrated for each spectral subgroup against the respective pixels of the panchromatic image. The models are inverted to retrieve high-resolution pansharpened images. The method is tested against different methods that are able to handle both MS and HS pansharpening and assessed using reduced- and full-resolution evaluation methodologies. Based on a statistical multivariate approach, the proposed method is able to render uncertainty maps for spectral or spatial fidelity - functionality not reported in any other pansharpening study.

Level Set Learning with Pseudoreversible Neural Networks for Nonlinear Dimension Reduction in Function Approximation

Due to the curse of dimensionality and the limitation on training data, approximating high-dimensional functions is a very challenging task even for powerful deep neural networks. Inspired by the Nonlinear Level-set Learning (NLL) method, which uses the reversible residual network (RevNet), in this paper, we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudoreversible neural network (PRNN) module, which effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and active subspace methods, especially when the target function possesses critical points in the interior of its input domain.

A machine learning‐based multiple imputation method for the health and aging brain among latino elders study

AbstractBackgroundThe Health and Aging Brain among Latino Elders Study (HABLE) project seeks to understand the biological, social and environmental factors that impact brain aging among diverse communities. HABLE, like many other NIH funded data‐sharing projects, has important data assets for various uses, including social, environmental and behavioral data, and multiple data flow pathways. A common methodological issue for data collection in such a large survey‐based, behavior, behavioral, or environmental epidemiology study is missing data. It is impossible to achieve machine learning (ML) if data contains missing values. Therefore, developing new imputation methodology has become a very urgent task for HABLE.MethodIn this study, we developed a ML‐Based Multiple Imputation method. First, we built four ML‐Based Imputation models (Support Vector Machine (SVM), Random Forest (RF), eXtreme Gradient Boosting (XGB) and lasso and elastic‐net regularized Generalized Linear Model (GLMNET)). Then we adapted the above four ML‐Based methods to multiple imputation. Lastly, we evaluated our ML‐Based Multiple Imputation method in comparison with lowest of detection (LOD), random drawing from a lef‐censored normal distribution (ND), k‐nearest neighbors (kNN), local least squares (LLS), and random forest (RF), Singular value decomposition (SVD), Bayesian principal component analysis (BPCA).ResultOur results show that ML‐Based Multiple Imputation method can have lower Normalized Root Mean Square Error (NRMSE) in missing value imputation and higer prediciton performance in machine learning.ConclusionThe choice of missing handling methodology has a significant impact on the accompanying statistical analyses of alzheimer’s disease. The Machine Learning‐Based Multiple Imputation Method performs well for our alzheimer’s disease models. It can also be applied to other disease data analysis.

Theoretical predictions and experimental verifications of SERS detection in colorants.

Synthetic colorants added during food processing not only fail to provide nutrients, but also can be harmful to human health when used in excess. To establish a simple, convenient, rapid and low-cost surface-enhanced Raman spectroscopy (SERS) detection method for colorants, an active surface-enhanced substrate of colloidal gold nanoparticles (AuNPs) was prepared in this study. The density functional theory (DFT) method of B3LYP with 6-31G(d) was applied to determine the theoretical Raman spectra of erythrosine, basic orange 2, 21 and 22, and to attribute their characteristic spectral peaks. The SERS spectra of the four colorants were pre-processed using local least squares (LLS) and morphological weighted penalized least squares (MWPLS), and multiple linear regression (MLR) models were established to quantify the four colorants in beverages. The results showed that the prepared AuNPs with a particle size of about 50 nm were reproducible and stable, with a good enhancement of the SERS spectrum of rhodamine 6G at 10-8 mol L-1. The theoretical Raman frequencies were in good agreement with the experimental Raman frequencies, and the peak position differences of the main characteristic peaks of the four colorants were within 20 cm-1. The MLR calibration models for the concentrations of the four colorants showed relative errors of prediction (REP) of 2.97-8.96%, root mean square errors of prediction (RMSEP) of 0.03-0.94, R2 of 0.973-0.999, and limits of detection of 0.06 μg mL-1. The present method could be used to quantify erythrosine, basic orange 2, 21, and 22, revealing its wide range of applications in food safety.

Assessment of label-free quantification and missing value imputation for proteomics in non-human primates

BackgroundReliable and effective label-free quantification (LFQ) analyses are dependent not only on the method of data acquisition in the mass spectrometer, but also on the downstream data processing, including software tools, query database, data normalization and imputation. In non-human primates (NHP), LFQ is challenging because the query databases for NHP are limited since the genomes of these species are not comprehensively annotated. This invariably results in limited discovery of proteins and associated Post Translational Modifications (PTMs) and a higher fraction of missing data points. While identification of fewer proteins and PTMs due to database limitations can negatively impact uncovering important and meaningful biological information, missing data also limits downstream analyses (e.g., multivariate analyses), decreases statistical power, biases statistical inference, and makes biological interpretation of the data more challenging. In this study we attempted to address both issues: first, we used the MetaMorphues proteomics search engine to counter the limits of NHP query databases and maximize the discovery of proteins and associated PTMs, and second, we evaluated different imputation methods for accurate data inference. We used a generic approach for missing data imputation analysis without distinguising the potential source of missing data (either non-assigned m/z or missing values across runs).ResultsUsing the MetaMorpheus proteomics search engine we obtained quantitative data for 1622 proteins and 10,634 peptides including 58 different PTMs (biological, metal and artifacts) across a diverse age range of NHP brain frontal cortex. However, among the 1622 proteins identified, only 293 proteins were quantified across all samples with no missing values, emphasizing the importance of implementing an accurate and statiscaly valid imputation method to fill in missing data. In our imputation analysis we demonstrate that Single Imputation methods that borrow information from correlated proteins such as Generalized Ridge Regression (GRR), Random Forest (RF), local least squares (LLS), and a Bayesian Principal Component Analysis methods (BPCA), are able to estimate missing protein abundance values with great accuracy.ConclusionsOverall, this study offers a detailed comparative analysis of LFQ data generated in NHP and proposes strategies for improved LFQ in NHP proteomics data.

Iterative bicluster-based Bayesian principal component analysis and least squares for missing-value imputation in microarray and RNA-sequencing data.

Microarray and RNA-sequencing (RNA-seq) techniques each produce gene expression data that can be expressed as a matrix that often contains missing values. Thus, a process of missing-value imputation that uses coherence information of the dataset is necessary. Existing imputation methods, such as iterative bicluster-based least squares (bi-iLS), use biclustering to estimate the missing values because genes are only similar under correlative experimental conditions. Also, they use the row average to obtain a temporary complete matrix, but the use of the row average is considered to be a flaw. The row average cannot reflect the real structure of the dataset because the row average only uses the information of an individual row. Therefore, we propose the use of Bayesian principal component analysis (BPCA) to obtain the temporary complete matrix instead of using the row average in bi-iLS. This alteration produces new missing values imputation method called iterative bicluster-based Bayesian principal component analysis and least squares (bi-BPCA-iLS). Several experiments have been conducted on two-dimension independent gene expression datasets, which are microarray (e.g., cell-cycle expression dataset of yeast saccharomyces cerevisiae) and RNA-seq (gene expression data from schizosaccharomyces pombe) datasets. In the case of the microarray dataset, our proposed bi-BPCA-iLS method showed a significant overall improvement in the normalized root mean square error (NRMSE) values of 10.6% from the local least squares (LLS) and 0.6% from the bi-iLS. In the case of the RNA-seq dataset, our proposed bi-BPCA-iLS method showed an overall improvement in the NRMSE values of 8.2% from the LLS and 3.1% from the bi-iLS. The additional computational time of bi-BPCA-iLS is not significant compared to bi-iLS.

A Simple Optimization Workflow to Enable Precise and Accurate Imputation of Missing Values in Proteomic Data Sets.

Missing values in proteomic data sets have real consequences on downstream data analysis and reproducibility. Although several imputation methods exist to handle missing values, no single imputation method is best suited for a diverse range of data sets, and no clear strategy exists for evaluating imputation methods for clinical DIA-MS data sets, especially at different levels of protein quantification. To navigate through the different imputation strategies available in the literature, we have established a strategy to assess imputation methods on clinical label-free DIA-MS data sets. We used three DIA-MS data sets with real missing values to evaluate eight imputation methods with multiple parameters at different levels of protein quantification: a dilution series data set, a small pilot data set, and a clinical proteomic data set comparing paired tumor and stroma tissue. We found that imputation methods based on local structures within the data, like local least-squares (LLS) and random forest (RF), worked well in our dilution series data set, whereas imputation methods based on global structures within the data, like BPCA, performed well in the other two data sets. We also found that imputation at the most basic protein quantification level-fragment level-improved accuracy and the number of proteins quantified. With this analytical framework, we quickly and cost-effectively evaluated different imputation methods using two smaller complementary data sets to narrow down to the larger proteomic data set's most accurate methods. This acquisition strategy allowed us to provide reproducible evidence of the accuracy of the imputation method, even in the absence of a ground truth. Overall, this study indicates that the most suitable imputation method relies on the overall structure of the data set and provides an example of an analytic framework that may assist in identifying the most appropriate imputation strategies for the differential analysis of proteins.

Improved filtering approach for identification of protein-coding regions in eukaryotes by background noise reduction using S–G filter

Unlike prokaryotic genomes, the arrangement of protein-coding regions in eukaryotic genomes is not continuous. It is interrupted by the non-coding DNA sequences called introns. Therefore, the identification of protein-coding regions in a eukaryotic DNA sequence is one of the challenging research issues in bioinformatics. Signal processing-based computational tools such as short-time discreet Fourier transforms (STDFT) and narrow bandpass digital filters has been successfully used to resolve this problem. Filtering techniques are popularly used because of its faster response than the transform techniques. However, the prediction accuracy of the filtering approach is still limited due to background noise present in its spectrum. Background noise masks the discriminative three base periodicity (TBP) features and increases the chances of false prediction. Several de-noising techniques have been proposed so far. Recently, second-order moving average filter has been used to diminish the effect of background noise. However, the problem with moving average filter is that it operates similarly on coding and non-coding regions, and therefore, along with noise reduction, it also affects the spectral features of protein-coding regions which lead to inaccurate prediction results. In this work, we used the Savitzky–Golay (SG) filter for background noise reduction and compared the performance with other existing de-noising techniques. S–G filter works on the local least-squares polynomial approximation principal and act as a weighted moving average filter. This investigation shows that along with noise reduction, S–G filter can preserve the spectral values of coding regions with a greater extent, and therefore, provide more accurate prediction results than other de-noising techniques.

Airfoil profile reconstruction from unorganized noisy point cloud data

Abstract Airfoil blades are typically inspected in sections to verify their conformance to the geometric tolerances specified on their nominal design. To maintain the accuracy of geometric error evaluation, in particular, for the position and orientation errors of the airfoil sections, sectional airfoil profiles should be reconstructed from the inspection data points. This paper presents a new method to automatically reconstruct the airfoil profile from unorganized noisy sectional data points of 3D scanned blades. A three-step airfoil profile reconstruction approach is presented. First, the algorithm thins the scattered set of sectional data points by projecting them onto the local curves fitted to them. For this purpose, a recursive weighted local least-squares scheme is proposed to fit local curves within the measurement uncertainty constraint of inspection data. Then, to order the thinned set of data points, the profile polygon is generated and imperfect nodes are modified by evaluation of the angular deviation of edges. Finally, a closed nonperiodic B-spline curve is fitted to the thinned and ordered set of data points to construct the smooth airfoil profile. A series of case studies have been carried out to demonstrate the effectiveness of the proposed airfoil profile reconstruction method. Implementation results have demonstrated that the proposed method is accurate and robust to noise. In addition to blade inspection, other applications such as repair and adaptive machining of aero-engine blades can equally benefit from the proposed method for automatic airfoil profile reconstruction.

A true PML approach for steady-state vibration analysis of an elastically supported beam under moving load by a DLSFEM formulation

This paper concerns a computational implementation for solving a Moving Load (ML) problem on an infinite Euler–Bernoulli elastic beam on a Pasternak visco-elastic support. A steady-state dynamic response in convected coordinate is sought, by a numerical approach with discretization over a finite domain, implying spurious boundary reflections of non-evanescent waves. This is effectively solved by: (a) analytically formulating a new, true Perfectly Matched Layer (PML) approach, toward handling the underlying fourth-order differential problem and the corresponding far-field conditions, without adopting special boundary conditions; (b) outlining a local Discontinuous Least-Squares Finite Element Method (DLSFEM) formulation, apt to provide a robust approach for the present non self-adjoint problem and to conveniently handle the jump condition in the shear force at the concentrated ML position. Consistent numerical results are illustrated and compared to an available analytical solution, showing a perfect match, with a complete removal of spurious boundary effects and a proof of theoretical a priori error estimates. Further results are produced for a case with multiple MLs. The paper shows that the present innovative DLSFEM-PML formulation is effectively suitable to numerically solve a steady-state ML problem on an infinite beam, setting up a new computational tool in such a challenging mechanical context.

Local (T)HB-spline projectors via restricted hierarchical spline fitting

This paper is devoted to techniques for adaptive spline projection via quasi-interpolation, enabling the efficient approximation of given functions. We employ local least-squares fitting in restricted hierarchical spline spaces to establish novel projection operators for hierarchical splines of degree p. This leads to efficient spline projectors that require O(pd) floating point operations and O(1) evaluations of the given function per degree of freedom, while providing essentially the same accuracy as global approximation. Our spline projectors are based on a unifying framework for quasi-interpolation in hierarchical spline spaces. We present a detailed comparison with the scheme of Speleers and Manni (2016).

Local least–squares element differential method for solving heat conduction problems in composite structures

In this article, a completely new numerical method called the Local Least-Squares Element Differential Method (LSEDM), is proposed for solving general engineering problems governed by second order partial differential equations. The method is a type of strong-form finite element method. In this method, a set of differential formulations of the isoparametric elements with respect to global coordinates are employed to collocate the governing differential equations and Neumann boundary conditions of the considered problem to generate the system of equations for internal nodes and boundary nodes of the collocation element. For each outer boundary or element interface, one equation is generated using the Neumann boundary condition and thus a number of equations can be generated for each node associated with a number of element interfaces. The least-squares technique is used to cast these interface equations into one equation by optimizing the local physical variable at the least-squares formulation. Thus, the solution system has as many equations as the total number of nodes of the present heat conduction problem. The proposed LSEDM can ultimately guarantee the conservativeness of the heat flux across element surfaces and can effectively improve the solution stability of the element differential method in solving problems with hugely different material properties, which is a challenging issue in meshfree methods. Numerical examples on two- and three-dimensional heat conduction problems are given to demonstrate the stability and efficiency of the proposed method.

Investigation of the time elapsed effect on residual stress measurement in a composite plate by DIC method

In composite plates, there are generally amounts of residual stresses that can have adverse effects in final parts. So, it is very important to know about the residual stress's type, location and amounts in composite plates. The measurement methods of the residual stress are commonly indirect and the displacement and strain is calculated at first. Two-dimensional digital image correlation (2D DIC) is a practical and effective tool for quantitative in-plane deformation measurement of a planar object surface and now widely accepted and commonly used in the field of experimental mechanics. In the hole drilling method, the residual stress is measured immediately just after drilling and the other researchers have paid less attention to the amount and pattern of changes in released stress around the hole after drilling at different times. It can be stated that the novelty of this research may identify the optimal measurement time after the drilling and the holes with different diameters is considered and the displacement and strain fields are measured at different times. So in this research, the residual stress measurement of a composite plate in the time elapsed is discussed. Also, the effects of hole's diameter and elapsed time after drilling in amount of released residual stresses are investigated by 2D DIC method. The point-wise local least-squares fitting technique (PLS) is used for strain estimation from displacement fields. The residual stress is calculated by the estimated strain, determination of calibration constants in orthotropic materials and considering the properties of composite plate that has already been calculated. The finite element method (FEM) is used for determination of calibration constants and the correction factor in FEM is obtained by comparing of the FEM and experimental tensile test results.

An Efficient Online-Offline Method for Elliptic Homogenization Problems

We present a new numerical method for solving the elliptic homogenization problem. The main idea is that the missing effective matrix is reconstructed by solving the local least-squares in an offline stage, which shall be served as the input data for the online computation. The accuracy of the proposed method are analyzed with the aid of the refined estimates of the reconstruction operator. Two dimensional and three dimensional numerical tests confirm the efficiency of the proposed method, and illustrate that this online-offline strategy may significantly reduce the cost without loss of accuracy.

DLSFEM–PML formulation for the steady-state response of a taut string on visco-elastic support under moving load

The numerical solution of the steady-state response of a uniform taut string on visco-elastic support under a concentrated transverse moving load is addressed. By recasting the governing second-order differential equation as a first-order system in convected coordinate, a local Discontinuous Least-Squares Finite Element Method (DLSFEM) formulation is developed within a complex-valued function space, to overcome numerical instabilities linked to high-velocity loads and handle far-field conditions through an effective Perfectly Matched Layer (PML) implementation. As an original advancement of the present DLSFEM–PML formulation, a coercivity theorem is proven for any first-order ordinary differential system and uniform error estimates are established for the finite element approximation for both$$L^2$$- and$$H^1$$-norms. Thus, the formulation newly joins a DLSFEM approach and a PML implementation, for solving the above-mentioned moving load problem. Numerical examples illustrate feasibility and accuracy of the method in reproducing the expected trends of solution and a priori error estimates.

Mechanical properties of wood materials using near-infrared spectroscopy based on correlation local embedding and partial least-squares

This study used near-infrared (NIR) spectroscopy to predict mechanical properties of wood. NIR spectra were collected in wavelengths 900–1700 nm, and spectra averaged by radial and tangential surface spectra were used to establish a partial least square (PLS) model based on correlation local embedding (CLE). Mongolian oak (Quercus mongolica Fisch. ex Ledeb.) was used to test the effectiveness of the model. The cross-validation method was used to verify the robustness of the CLE–PLS model. Ninety samples were tested as the calibration set and forty-five as the validation set. The results show that the prediction coefficient of determination (R_{p}^{2}) is 0.80 for MOR, and 0.78 for MOE. The ratio of performance to deviation is 2.23 for MOR and 2.15 for MOE.

Adaptive estimation of heteroskedastic functional-coefficient regressions with an application to fiscal policy evaluation on asset markets

This article studies the adaptive estimation of the heteroskedastic functional-coefficient regressions. The motivation for such a theoretical study originates from the empirical analysis of Jansen et al., where the role of fiscal policy on the U.S. asset markets (treasury bonds) is evaluated via the functional-coefficient model. It is found that this model is subject to time-varying heteroskedasticity. As a result, the local least square (LLS) estimator suffers from efficiency loss. To overcome this problem, we propose an adaptive LLS (ALLS) estimator, which can adapt to heteroskedasticity of unknown form asymptotically. Simulation studies confirm that the ALLS estimator can achieve significant efficiency gain in finite samples, compared to the LLS estimator. Real data analysis reveals that the heteroskedastic functional-coefficient model provides adequate fit and better out-of-sample forecasting.

A nifty collaborative analysis to predicting a novel tool (DRFLLS) for missing values estimation

One of the important trends in an intelligent data analysis will be the growing importance of data processing. But this point faces problems similar to those of data mining (i.e., high-dimensional data, missing value imputation and data integration); one of the challenges in estimation missing value methods is how to select the optimal number of nearest neighbors of those values. This paper, attempting to search the capability of building a novel tool to estimate missing values of various datasets called developed random forest and local least squares (DRFLLS). By developing random forest algorithm, seven categories of similarity measures were defined. These categories are person similarity coefficient, simple similarity, and fuzzy similarity (M1, M2, M3, M4 and M5). They are sufficient to estimate the optimal number of neighborhoods of missing values in this application. Hereafter, local least squares (LLS) has been used to estimate the missing values. Imputation accuracy can be measured in different ways: Pearson correlation (PC) and NRMSE. Then, the optimal number of neighborhoods is associated with the highest value of PC and a smaller value of NRMSE. The experimental results were carried out on six datasets obtained from different disciplines, and DRFLLS proves the dataset which has a small rate of missing values gave the best estimation to the number of nearest neighbors by DRFPC and in the second degree by DRFFSM1 when r = 4, while if the dataset has high rate of missing values, then it gave the best estimation to number of nearest neighbors by DRFFSM5 and in the second degree by DRFFSM3. After that, the missing value was estimated by LLS, and the results accuracy was measured by NRMSE and Pearson correlation. The smallest value of NRMSE for a given dataset is corresponding to DRF correlation function which is a better function for a given dataset. The highest value of PC for a given dataset is corresponding to DRF correlation function which is a better function for a given dataset.

Networked distributed fusion estimation under uncertain outputs with random transmission delays, packet losses and multi-packet processing

Abstract This paper investigates the distributed fusion estimation problem for networked systems whose multisensor measured outputs involve uncertainties modelled by random parameter matrices. Each sensor transmits its measured outputs to a local processor over different communication channels and random failures –one-step delays and packet dropouts– are assumed to occur during the transmission. White sequences of Bernoulli random variables with different probabilities are introduced to describe the observations that are used to update the estimators at each sampling time. Due to the transmission failures, each local processor may receive either one or two data packets, or even nothing and, when the current measurement does not arrive on time, its predictor is used in the design of the estimators to compensate the lack of updated information. By using an innovation approach, local least-squares linear estimators (filter and fixed-point smoother) are obtained at the individual local processors, without requiring the signal evolution model. From these local estimators, distributed fusion filtering and smoothing estimators weighted by matrices are obtained in a unified way, by applying the least-squares criterion. A simulation study is presented to examine the performance of the estimators and the influence that both sensor uncertainties and transmission failures have on the estimation accuracy.