Robust Singular Value Decomposition Research Articles

A robust attitude estimation algorithm for seabed inverted ultra-short baseline

The ultra-short baseline(USBL) is fixed upside down on the seafloor to position an autonomous underwater vehicle, which is the inverted ultra-short baseline (i-USBL). Three attitude angles between the i-USBL coordinate system and the navigation coordinate system will be produced when the USBL is installed on the seabed. Significant positioning error will occur if the USBL attitude is undefined. Besides, it also affects the accuracy of attitude angles estimation that the outliers produced when using USBL. In this paper, we propose a robust singular value decomposition (RSVD) method for estimating the attitude angles. Specifically, we develop an attitude calibration model based on the i-USBL positioning mode. The USBL attitude angles is solved using singular value decomposition (SVD) according to the least squares method. The new attitude angles are used to calculate the residuals of the coordinates at each measurement point. The residuals are used to calculate the weights of each point by the IGG III weighting function. The recalculation of attitude angles is performed by using up-to-date weights and SVD. The performance of SVD, the conventional Gaussian Newton iterative method (GN), and RSVD are compared in the simulations. The simulations show that the RSVD has high calibration accuracy and robustness.

APPLICATION OF BIPLOT ANALYSIS WITH ROBUST SINGULAR VALUE DECOMPOSITION TO POVERTY DATA IN SULAWESI ISLAND

Poverty is defined as an inability of the individual to meet basic needs for a decent life. According to BPS data in2020, Sulawesi Island ranks fifth as the poorest island in Indonesia. This study aims to find out the mapping of areas and indicators of poverty in Sulawesi Island using Biplot Analysis with Robust Singular Value Decomposition approach for outlier research data. Based on the results of the study, there are five objects that are outlier and the information provided by the biplot amounted 98.45%. District/city that have similar characteristics are divided into 4 groups. The indicator of poverty that has the most diversity is the School Old Expectations Numbers (Var 4) and the one with the least diversity is Poor Households Using Clean Water (Var 8). Indicators of poverty that are positively correlated are Literacy Numbers (Var 1) and Non-Working Poor Population (Var 5), while the negative correlated are The Non-Working Poor Population (Var 5) and Poor Households Using Clean Water (Var 8). There are 19 districts/cities that have literacy values above the average of all districts/cities and 11 districts/cities that have a per capita expenditure value below the average of all districts/cities.

Blockwise Robust Singular Value Decomposition for MRI Brain Image Denoising

MRI is one of the high-dimensional high-throughput technologies that are playing a major role for diagnosing brain tumor. With the help of MRI images, brain tumor is diagnosed at advanced stages. Correct identification of brain tumor or abnormality, image processing in MRI of brain is highly essential that can reduce the chance of fatal stage. Occasionally, these MR images are introduced with noise during acquisition which reduces the image quality and limits the accuracy in diagnosis. Therefore, preliminary diagnosis of MRI brain images from the hospital may not be always reliable for further analysis because of the presence of noise. Reduction/Elimination of noise in medical images is an important task in preprocessing which is one of the previous crucial parts for further image analysis. Although, several denoising techniques are available in the literature including median filter, box filter, low pass filter, and high pass filter. In this paper, we have measured the influence of the above methods for denoising. We have also proposed a new blockwise robust singular value decomposition technique for denoising image. The results of our analysis showed that blockwise robust singular value decomposition technique gives the better performance compared to the other methods. Therefore, our recommendation is to use the Blockwise robust singular value decomposition technique for brain image denoising.

Kernel robust singular value decomposition

Singular value decomposition (SVD) is one of the most widely used algorithms for dimensionality reduction and performing principal component analysis, which represents an important tool used in many pattern recognition problems. However, in the case of data contamination with outlying observations, the classical SVD is not appropriate. To overcome this limitation, several robust SVD algorithms have been proposed, usually based on different types of norms or projection strategies. In this paper, we propose a kernel robust SVD algorithm based on the exponential-type Gaussian kernel, where four estimators are considered for the width hyper-parameters. Differently from the existing approaches that deal with kernel in principal component analysis and SVD, our proposal operates in the original space, instead of the feature space, being the kernel applied in a robust linear regression framework to obtain the robust estimates for the singular values and left and right singular vectors. Simulations show that the proposed algorithm outperforms the classical and robust SVD algorithms under consideration. We also illustrate the merits of the proposed algorithm in an application to image recovery due to the presence of noise.

Cross-Validation for Lower Rank Matrices Containing Outliers

Several statistical techniques for analyzing data matrices use lower rank approximations to these matrices, for which, in general, the appropriate rank must first be estimated depending on the objective of the study. The estimation can be conducted by cross-validation (CV), but most methods are not designed to cope with the presence of outliers, a very common problem in data matrices. The literature suggests one option to circumvent the problem, namely, the elimination of the outliers, but such information removal should only be performed when it is possible to verify that an outlier effectively corresponds to a collection or typing error. This paper proposes a methodology that combines the robust singular value decomposition (rSVD) with a CV scheme, and this allows outliers to be taken into account without eliminating them. For this, three possible rSVD’s are considered and six resistant criteria are proposed for the choice of the rank, based on three classic statistics used in multivariate statistics. To test the performance of the various methods, a simulation study and an analysis of real data are described, using an exclusively numerical evaluation through Procrustes statistics and critical angles between subspaces of principal components. We conclude that, when data matrices are contaminated with outliers, the best estimation of rank is the one that uses a CV scheme over a robust lower rank approximation (RLRA) containing as many components as possible. In our experiments, the best results were obtained when this RLRA was calculated using an rSVD that minimizes the L2 norm.

Missing value imputation using least squares techniques in contaminated matrices

This paper describes strategies to reduce the possible effect of outliers on the quality of imputations produced by a method that uses a mixture of two least squares techniques: regression and lower rank approximation of a matrix. To avoid the influence of discrepant data and maintain the computational speed of the original scheme, pre-processing options were explored before applying the imputation method. The first proposal is to previously use a robust singular value decomposition, the second is to detect outliers and then treat the potential outliers as missing. To evaluate the proposed methods, a cross-validation study was carried out on ten complete matrices of real data from multi-environment trials. The imputations were compared with the original data using three statistics: a measure of goodness of fit, the squared cosine between matrices and the prediction error. The results show that the original method should be replaced by one of the options presented here because outliers can cause low quality imputations or convergence problems.•The imputation algorithm based on Gabriel's cross-validation method uses two least squares techniques that can be affected by the presence of outliers. The inclusion of a robust singular value decomposition allows both to robustify the procedure and to detect outliers and consider them later as missing. These forms of pre-processing ensure that the algorithm performs well on any dataset that has a matrix form with suspected contamination.

Techniques for Robust Imputation in Incomplete Two-Way Tables

We describe imputation strategies resistant to outliers, through modifications of the simple imputation method proposed by Krzanowski and assess their performance. The strategies use a robust singular value decomposition, do not depend on distributional or structural assumptions and have no restrictions as to the pattern or missing data mechanisms. They are tested through the simulation of contamination and unbalance, both in artificially generated matrices and in a matrix of real data from an experiment with genotype-by-environment interaction. Their performance is assessed by means of prediction errors, the squared cosine between matrices, and a quality coefficient of fit between imputations and true values. For small matrices, the best results are obtained by applying robust decomposition directly, while for larger matrices the highest quality is obtained by eliminating the singular values of the imputation equation.

Missing‐value imputation using the robust singular‐value decomposition: Proposals and numerical evaluation

AbstractA common problem in the analysis of data from multi‐environment trials is imbalance caused by missing observations. To get around this problem, Yan proposed a method for imputing the missing values based on the singular‐value decomposition (SVD) of a matrix. However, this SVD can be affected by outliers and produce low quality imputations. In this article, we propose four extensions of the Yan method that are resistant to outliers, replacing the standard SVD method with four robust SVD extensions. We evaluate these methods, using exclusively numerical criteria in a simulation study and in a cross‐validation study based on real data. We conclude that in the presence of outliers, the standard SVD method should not be used; instead, the best alternatives are the robust SVD methods based on sub‐sampling when the percentage of contamination is less than 2% following a completely random missing data mechanism. In any other case, methods that either minimize the L2 norm or that involve L1 regressions are preferable.

Well-conditioned ultraspherical and spectral integration methods for resolvent analysis of channel flows of Newtonian and viscoelastic fluids

Modal and nonmodal analyses of fluid flows provide fundamental insight into the early stages of transition to turbulence. Eigenvalues of the dynamical generator govern temporal growth or decay of individual modes, while singular values of the frequency response operator quantify the amplification of disturbances for linearly stable flows. In this paper, we develop well-conditioned ultraspherical and spectral integration methods for frequency response analysis of channel flows of Newtonian and viscoelastic fluids. Even if a discretization method is well-conditioned, we demonstrate that calculations can be erroneous if singular values are computed as the eigenvalues of a cascade connection of the frequency response operator and its adjoint. To address this issue, we utilize a feedback interconnection of the frequency response operator with its adjoint to avoid computation of inverses and facilitate robust singular value decomposition. Specifically, in contrast to conventional spectral collocation methods, the proposed method (i) produces reliable results in channel flows of viscoelastic fluids at high Weissenberg numbers (∼500); and (ii) does not require a staggered grid for the equations in primitive variables.

ASRA: Automatic singular value decomposition-based robust fingerprint image alignment

Fingerprint-based user identification and authentication are now used in many applications, but achieving absolute accuracy (eliminating false matches) still remains an issue. One of the reasons behind this issue is inappropriate image alignment prior to the feature extraction. In this paper, a robust Singular Value Decomposition (SVD) based fingerprint alignment method is proposed which automatically aligns the segmented and rotated image within the angular range − 900 to 900. Further, it overcomes the limitations of the existing fingerprint alignment methods as it neither depends on the quality of the image nor requires any reference image. The effectiveness of the approach has been tested with the standard fingerprint image databases FVC2002 (DB1, DB2, DB3, and DB4), FVC2004 (DB1, DB2, DB3, and DB4) and captured sensor images in an uncontrolled environment. The proposed approach was found to be efficient both in terms of accuracy and computational time. Also, it worked well for both database images and captured sensor images.

Frequency-based damage detection method for excitation signals above work frequencies: theoretical aspects and experimental results

This work presents a cost-effective methodology to determine small damages using frequency response functions (FRFs) in the high-frequency range. The main objective is to determine a failure in its initial stage, allowing it to be analyzed and tracked. A theoretical analysis is performed, relating changes in the structural frequency response to small damages. In the analytic study, the strain modal energy change is estimated when the modal wavelength is similar to the damage rift dimension. The frequency change magnitude and its observability are analyzed based on practical FRF parameters. To demonstrate that a small damage can be noticeable in a high-frequency spectrum, an experimental procedure is performed to confirm it. Using a robust singular value decomposition algorithm a failure identification at approximately 0.16% and 0.34% of structural mass change is achieved. Employing measurement datasets composed of reference and damage sets obtained from a single measurement point, these results show the effectiveness and possibilities of the presented method.

Robust Singular Value Decomposition Method on Minor Outlier Data

In multivariate statistics, Singular Value Decomposition (SVD) for a data matrix containing outliers does not provide data that can be analyzed optimally. This study aims to overcome outlier data using the Robust Singular Value Decomposition (RSVD) method and compare it with the SVD method. The analysis using the RSVD method includes several steps, namely determining the initial predictive value of the vector u and regressing it then normalizing the estimator vector β and carrying out the iteration process until convergent results are obtained. The results of this study indicate that the RSVD for dealing with minor outliers data is not influenced by initial estimates. The RSVD method is strongly influenced by the large amount of outliers data, the more extreme outliers data, the more iterations are.

Mining Gene Expression Profile with Missing Values: An Integration of Kernel PCA and Robust Singular Values Decomposition

Background: Gene expression profiling and transcriptomics provide valuable information about the role of genes that are differentially expressed between two or more samples. It is always important and challenging to analyse High-throughput DNA microarray data with a number of missing values under various experimental conditions. </P><P> Objectives: Graphical data visualizations of the expression of all genes in a particular cell provide holistic views of gene expression patterns, which improve our understanding of cellular systems under normal and pathological conditions. However, current visualization methods are sensitive to missing values, which are frequently observed in microarray-based gene expression profiling, potentially affecting the subsequent statistical analyses. Methods: We addressed in this study the problem of missing values with respect to different imputation methods using gene expression biplot (GE biplot), one of the most popular gene visualization techniques. The effects of missing values for mining differentially expressed genes in gene expression data were evaluated using four well-known imputation methods: Robust Singular Value Decomposition (Robust SVD), Column Average (CA), Column Median (CM), and K-nearest Neighbors (KNN). Frobenius norm and absolute distances were used to measure the accuracy of the methods. Results: Three numerical experiments were performed using simulated data (i) and publicly available colon cancer (ii) and leukemia data (iii) to analyze the performance of each method. The results showed that CM and KNN performed better than Robust SVD and CA for identifying the index gene profile in the biplot visualization in both the simulation study and the colon cancer and leukemia microarray datasets. Conclusion: The impact of missing values on the GE biplot was smaller when the data matrix was imputed by KNN than by CM. This study concluded that KNN performed satisfactorily in generating a GE biplot in the presence of missing values in microarray data.

A robust approach to singular spectrum analysis

AbstractSingular spectrum analysis (SSA) is a nonparametric method for time series analysis and forecasting that incorporates elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems, and signal processing. Although this technique has shown to be advantageous over traditional model‐based methods, in particular, one of the steps of the SSA algorithm, which refers to the singular value decomposition (SVD) of the trajectory matrix, is highly sensitive to data contamination. Specifically, and because SVD decomposition is least squares based, the presence of a single outlier, if extreme, may be enough to draw the leading principal component towards itself resulting in possible misinterpretations, which may subsequently, in particular and in the case of time series analysis, lead to reduced quality of model fit and forecast accuracy. In order to alleviate this problem, a robust SSA algorithm is proposed, where a robust SVD procedure replaces the least squares based one in the original SSA procedure. The SSA and the robust SSA approaches are compared in terms of model fit quality via Monte Carlo simulations that contemplate both clean and noisy/contaminated time series. The adequacy and value of the proposed approach is then compared with the standard SSA using real data about the industrial production index.

SVD and ASCII Character Encoding-Based Compression of Multiple Biosignals for Remote Healthcare Systems.

Advancements in electronics and miniaturized device fabrication technologies have enabled simultaneous acquisition of multiple biosignals (MBioSigs), but the area of compression of MBioSigs remains unexplored to date. This paper presents a robust singular value decomposition (SVD) and American standard code for information interchange (ASCII) character encoding-based algorithm for compression of MBioSigs for the first time to the best of our knowledge. At the preprocessing stage, MBioSigs are denoised, down sampled and then transformed to a two-dimensional (2-D) data array. SVD of the 2-D array is carried out and the dimensionality of the singular values is reduced. The resulting matrix is then compressed by a lossless ASCII character encoding-based technique. The proposed compression algorithm can be used in a variety of modes such as lossless, with or without using the down sampling operation. The compressed file is then uploaded to a hypertextpreprocessor (PHP)-based website for remote monitoring application. Evaluation results show that the proposed algorithm provides a good compression performance; in particular, the mean opinion score of the reconstructed signal falls under the category "very good" as per the gold standard subjective measure.

Adjustable Robust Singular Value Decomposition: Design, Analysis and Application to Finance

The Singular Value Decomposition (SVD) is a fundamental algorithm used to understand the structure of data by providing insight into the relationship between the row and column factors. SVD aims to approximate a rectangular data matrix, given some rank restriction, especially lower rank approximation. In practical data analysis, however, outliers and missing values maybe exist that restrict the performance of SVD, because SVD is a least squares method that is sensitive to errors in the data matrix. This paper proposes a robust SVD algorithm by applying an adjustable robust estimator. Through adjusting the tuning parameter in the algorithm, the method can be both robust and efficient. Moreover, a sequential robust SVD algorithm is proposed in order to decrease the computation volume in sequential and streaming data. The advantages of the proposed algorithms are proved with a financial application.

Metabolomic Biomarker Identification in Presence of Outliers and Missing Values.

Metabolomics is the sophisticated and high-throughput technology based on the entire set of metabolites which is known as the connector between genotypes and phenotypes. For any phenotypic changes, potential metabolite (biomarker) identification is very important because it provides diagnostic as well as prognostic markers and can help to develop new biomolecular therapy. Biomarker identification from metabolomics data analysis is hampered by the use of high-throughput technology that provides high dimensional data matrix which contains missing values as well as outliers. However, missing value imputation and outliers handling techniques play important role in identifying biomarker correctly. Although several missing value imputation techniques are available, outliers deteriorate the accuracy of imputation as well as the accuracy of biomarker identification. Therefore, in this paper we have proposed a new biomarker identification technique combining the groupwise robust singular value decomposition, t-test, and fold-change approach that can identify biomarkers more correctly from metabolomics dataset. We have also compared the performance of the proposed technique with those of other traditional techniques for biomarker identification using both simulated and real data analysis in absence and presence of outliers. Using our proposed method in hepatocellular carcinoma (HCC) dataset, we have also identified the four upregulated and two downregulated metabolites as potential metabolomic biomarkers for HCC disease.

Identification of Differentially Expressed Gene Using Robust Singular Value Decomposition

Identification of Differentially Expressed Gene Using Robust Singular Value Decomposition

A robust AMMI model for the analysis of genotype-by-environment data.

One of the most widely used models to analyse genotype-by-environment data is the additive main effects and multiplicative interaction (AMMI) model. Genotype-by-environment data resulting from multi-location trials are usually organized in two-way tables with genotypes in the rows and environments (location-year combinations) in the columns. The AMMI model applies singular value decomposition (SVD) to the residuals of a specific linear model, to decompose the genotype-by-environment interaction (GEI) into a sum of multiplicative terms. However, SVD, being a least squares method, is highly sensitive to contamination and the presence of even a single outlier, if extreme, may draw the leading principal component towards itself resulting in possible misinterpretations and in turn lead to bad practical decisions. Since, as in many other real-life studies the distribution of these data is usually not normal due to the presence of outlying observations, either resulting from measurement errors or sometimes from individual intrinsic characteristics, robust SVD methods have been suggested to help overcome this handicap. We propose a robust generalization of the AMMI model (the R-AMMI model) that overcomes the fragility of its classical version when the data are contaminated. Here, robust statistical methods replace the classic ones to model, structure and analyse GEI. The performance of the robust extensions of the AMMI model is assessed through a Monte Carlo simulation study where several contamination schemes are considered. Applications to two real plant datasets are also presented to illustrate the benefits of the proposed methodology, which can be broadened to both animal and human genetics studies. Source code implemented in R is available in the supplementary material under the function r-AMMI. paulocanas@gmail.com Supplementary data are available at Bioinformatics online.

Statistical inference based on robust low-rank data matrix approximation

The singular value decomposition is widely used to approximate data matrices with lower rank matrices. Feng and He [Ann. Appl. Stat. 3 (2009) 1634-1654] developed tests on dimensionality of the mean structure of a data matrix based on the singular value decomposition. However, the first singular values and vectors can be driven by a small number of outlying measurements. In this paper, we consider a robust alternative that moderates the effect of outliers in low-rank approximations. Under the assumption of random row effects, we provide the asymptotic representations of the robust low-rank approximation. These representations may be used in testing the adequacy of a low-rank approximation. We use oligonucleotide gene microarray data to demonstrate how robust singular value decomposition compares with the its traditional counterparts. Examples show that the robust methods often lead to a more meaningful assessment of the dimensionality of gene intensity data matrices.