Columns Of Matrix Research Articles

Strong consistency of an estimator by the truncated singular value decomposition for an errors-in-variables regression model with collinearity

In this paper, we prove strong consistency of an estimator by the truncated singular value decomposition for a multivariate errors-in-variables linear regression model with collinearity. This result is an extension of Gleser's proof of the strong consistency of total least squares solutions to the case with modern rank constraints. While the usual discussion of consistency in the absence of solution uniqueness deals with the minimal norm solution, the contribution of this study is to develop a theory that shows the strong consistency of a set of solutions. The proof is based on properties of orthogonal projections, specifically properties of the Rayleigh-Ritz procedure for computing eigenvalues. This makes it suitable for targeting problems where some row vectors of the matrices do not contain noise. Therefore, this paper gives a proof for the regression model with the above condition on the row vectors, resulting in a natural generalization of the strong consistency for the standard TLS estimator.

Structural design and surface networking analysis of a truss antenna based on hexagonal frustum deployable mechanism units

With the development of space technology, deployable mechanism has been widely applied in large aperture space antennas. In this paper, a hexagonal frustum deployable antenna unit and an innovative modular assembling concept are proposed for large parabolic truss antenna. Screw topology diagram and screw constraint matrix are established to solve degree of freedom (DOF) of the antenna mechanism. By separating column vectors of screw constraint matrix, whole actuations of antenna mechanism are obtained according to unique solution condition. A new hexagon antenna network method based on hexagon frustum units and rotation transformation matrix is proposed, and correctness and efficiency of the modeling principle are proved by simulation. Finally, a prototype of the proposed hexagonal frustum truss deployable antenna mechanism is constructed, and a deployment experiment is conducted, which demonstrate the mobility and deployment performance of the mechanism. The mechanism proposed in this paper provides a theoretical reference for the parabolic truss deployable antenna mechanism, which is potentially valuable in the application of large aperture deployable antennas.

Clustering Longitudinal Data for Growth Curve Modelling by Gibbs Sampler and Information Criterion

Clustering longitudinal data for growth curve modelling is considered in this paper, where we aim to optimally estimate the underpinning unknown group partition matrix. Instead of following the conventional soft clustering approach, which assumes the columns of the partition matrix to have i.i.d. multinomial or categorical prior distributions and uses a regression model with the response following a finite mixture distribution to estimate the posterior distribution of the partition matrix, we propose an iterative partition and regression procedure to find the best partition matrix and the associated best growth curve regression model for each identified cluster. We show that the best partition matrix is the one minimizing a recently developed empirical Bayes information criterion (eBIC), which, due to the involved combinatorial explosion, is difficult to compute via enumerating all candidate partition matrices. Thus, we develop a Gibbs sampling method to generate a Markov chain of candidate partition matrices that has its equilibrium probability distribution equal the one induced from eBIC. We further show that the best partition matrix, given a priori the number of latent clusters, can be consistently estimated and is computationally scalable based on this Markov chain. The number of latent clusters is also best estimated by minimizing eBIC. The proposed iterative clustering and regression method is assessed by a comprehensive simulation study before being applied to two real-world growth curve modelling examples involving longitudinal data clustering.

Discovering the signal subgraph: An iterative screening approach on graphs

Supervised learning on graphs is a challenging task due to the high dimensionality and inherent structural dependencies in the data, where each edge depends on a pair of vertices. Existing conventional methods are designed for standard Euclidean data and do not account for the structural information inherent in graphs. In this paper, we propose an iterative vertex screening method to achieve dimension reduction across multiple graph datasets with matched vertex sets and associated graph attributes. Our method aims to identify a signal subgraph to provide a more concise representation of the full graphs, potentially benefiting subsequent vertex classification tasks. The method screens the rows and columns of the adjacency matrix concurrently and stops when the resulting distance correlation is maximized. We establish the theoretical foundation of our method by proving that it estimates the true signal subgraph with high probability. Additionally, we establish the convergence rate of classification error under the Erdos-Renyi random graph model and prove that the subsequent classification can be asymptotically optimal, outperforming the entire graph under high-dimensional conditions. Our method is evaluated on various simulated datasets and real-world human and murine graphs derived from functional and structural magnetic resonance images. The results demonstrate its excellent performance in estimating the ground-truth signal subgraph and achieving superior classification accuracy.

Sequentially extending space-filling experimental designs by optimally permuting and stacking columns of the design matrix

Researchers make available computationally expensive designs for computer experiments for widespread use by cataloging them and providing online links. This paper presents an algorithm that augments space-filling designs (SFDs) by optimally permuting and stacking columns of the design matrix to minimize the maximum absolute pairwise correlation among columns in the new extended design. The algorithm enables researchers to augment an SFD sequentially with batches of additional design points, which improves column orthogonality and adds more degrees of freedom for fitting metamodels. We show this method improves the correlation and space-filling properties of the resulting designs and allows us to extend some classes of designs to higher dimensions that are not easily obtainable. Moreover, the resulting extended designs compare well with many leading software-generated SFDs created from scratch in the extended design space.

A study on partial pivot ACA boundary element method for elasticity problems

The boundary element method (BEM) employing the partial pivot adaptive cross approximation (PACA) algorithm has been observed to experience convergence failures and reduced solution accuracy when solving elasticity problems, especially at large scales. To address this issue, this paper proposes an improved algorithm for both 2D and 3D elasticity problems.Our investigations revealed that when the normal of an element is parallel to the coordinate axes and the element is in the same plane as the source point, zero entries with regular distributions will appear in the coefficient matrix. This results in the premature satisfaction of the convergence criterion of the PACA algorithm, leading to the omission of some rows and columns of the coefficient matrix. To address this issue, this paper proposes improvements to the PACA algorithm through the Coordinate Rotation Method (CRM) and the Component Traversal Method (CTM), and validates the effectiveness of these two improved methods.Furthermore, we investigate reasons behind the decrease in accuracy when employing the CTM to solve large-scale problems. We attribute this to the oscillatory nature of the estimated relative error and propose the modified PACA (M-PACA) algorithm. M-PACA not only maintains the advantages of the PACA algorithm in reducing storage space and accelerating coefficient matrix generation but also addresses its limitations, ensuring more accurate and reliable solutions for elasticity problems.

Explainable graph clustering via expanders in the massively parallel computation model

Explainable clustering provides human-understandable reasons for decisions in black-box learning models. In a previous work, a decision tree built on the set of dimensions was used to define ranges of values for k-means clusters. For explainable graph clustering, we use expander graphs instead of dense subgraphs since powering an expander graph is guaranteed to result in a clique after at most a logarithmic number of steps.Consider a set of multi-dimensional points labeled with k labels. We introduce the heat map sorting problem as reordering the rows and columns of an input matrix (each point is a column and each row is a dimension) such that the labels of the entries of the matrix form connected components (clusters). A cluster is preserved if it remains connected, i.e., if it is not split into several clusters and no two clusters are merged. In the massively parallel computation model (MPC), each machine has a sublinear memory and the total memory of the machines is linear.We prove the problem is NP-hard. We give a fixed-parameter algorithm in MPC and an approximation algorithm based on expander decomposition. We empirically compare our algorithm with explainable k-means on several graphs of email and computer networks.

Secure Stego Method to Protect SMS Transmission

A novel method of message steganography will be proposed. The method will use a digital speech file as a covering media. The 32 binary bits of each speech sample will be used for data hiding, using these bits will increase the hiding capacity, and changing them will not much affect the sample value, thus the quality of the stego file will be always high. The proposed method will use a private key to protect the hidden message, the protection will be applied by shuffling the rows of the speech binary matrix using the contents of the generated indices key, The induces key will be generated by running a chaotic logistic map model using the values of the chaotic parameters included in the private key. The proposed method will be tested and implemented using various messages, the quality, sensitivity and speed of the proposed method will be analyzed to prove the enhancements provided by the proposed method.

Conditions for The Saturation of a Matrix Inequality

Matrix inequalities play a vital role in the study of linear algebra and information theory. In this paper, the author investigates the condition when a proven matrix inequality is saturated, and Pi’s are linearly independent matrices of the same size, and Si’s are also linearly independent matrices of the same size. The author constructs the conditions when Pi is a column vector or a 2 2 matrix, and Si is an arbitrary square matrix. The author finds a very potential result when investigating the column situation. The author also gets massive result in the 2 2 situation. In particular, the inequality is not saturated when Pi is a 2 2 matrix and K=2 or 3 and it could be saturated under the situation when K=4 if only if two certain matrices can enable the product of and form two established matrices. At the end of the essay, the author tries some more complex cases such as 2 3 matrices Pi when K=2.

The Algorithm of Gu and Eisenstat and D-Optimal Design of Experiments

This paper addresses the following problem: given m potential observations to determine n parameters, m>n, what is the best choice of n observations. The problem can be formulated as finding the n×n submatrix of the complete m×n observation matrix that has maximum determinant. An algorithm by Gu and Eisenstat for a determining a strongly rank-revealing QR factorisation of a matrix can be adapted to address this latter formulation. The algorithm starts with an initial selection of n rows of the observation matrix and then performs a sequence of row interchanges, with the determinant of the current submatrix strictly increasing at each step until no further improvement can be made. The algorithm implements rank-one updating strategies, which leads to a compact and efficient algorithm. The algorithm does not necessarily determine the global optimum but provides a practical approach to designing an effective measurement strategy. In this paper, we describe how the Gu–Eisenstat algorithm can be adapted to address the problem of optimal experimental design and used with the QR algorithm with column pivoting to provide effective designs. We also describe implementations of sequential algorithms to add further measurements that optimise the information gain at each step. We illustrate performance on several metrology examples.

The discrete empirical interpolation method in class identification and data summarization

AbstractThe discrete empirical interpolation method (DEIM) is well established as a means of performing model order reduction in approximating solutions to differential equations, but it has also more recently demonstrated potential in performing data class detection through subset selection. Leveraging the singular value decomposition (SVD) for dimension reduction, DEIM uses interpolatory projection to identify the representative rows and/or columns of a data matrix. This approach has been adapted to develop additional algorithms, including a CUR matrix factorization for performing dimension reduction while preserving the interpretability of the data. DEIM‐oversampling techniques have also been developed expressly for the purpose of index selection in identifying more DEIM representatives than would typically be allowed by the matrix rank. Even with these developments, there is still a relatively large gap in the literature regarding the use of DEIM in performing unsupervised learning tasks to analyze large datasets. Known examples of DEIM's demonstrated applicability include contexts such as physics‐based modeling/monitoring, electrocardiogram data summarization and classification, and document term subset selection. This overview presents a description of DEIM and some DEIM‐related algorithms, discusses existing results from the literature with an emphasis on more statistical‐learning‐based tasks, and identifies areas for further exploration moving forward.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Dimension Reduction Statistical Learning and Exploratory Methods of the Data Sciences > Exploratory Data Analysis Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification

Digital model reconstruction and geometric error analysis of additive manufacturing blisk enabled by 3D scanner

Additive manufacturing has advantages in manufacturing blisk. However, additive manufacturing blisk can deviate in size and shape from their original digital models. In this paper, the shape of a complex shaped blisk manufactured by additive manufacturing is measured by using a 3D scanner, and the geometric errors compared to the digital model are calculated by using a column vector row index-based K-Neighborhood search algorithm. The computed range for geometric errors falls within (0.0075 mm, 6.9177 mm). Approximately 94% of the data points have errors within 3 mm. Larger geometric errors are primarily concentrated at the edges of the blades, aligning with the characteristic features of additive manufacturing processes.

On the factor ambiguity of MCR problems for blockwise incomplete data sets

Multivariate curve resolution (MCR) methods are sometimes faced with missing or erroneous data, e.g., due to sensor saturation. In some cases, an estimation of the missing data is possible, but often MCR works with the largest submatrix without missing entries. This ignores all rows and columns of the data matrix that contain missing values. A successful approach to deal with incomplete data multisets has been proposed by Alier and Tauler (2013), but it does not include a factor ambiguity analysis. Here, the missing data problem is addressed in combination with a factor ambiguity analysis. An approach is presented that minimizes the factor ambiguity by extracting a maximum of spectral information even from incomplete rows and columns of the spectral data matrix. The method requires a high signal-to-noise ratio. Applications are presented for UV/Vis and HSI data.

On the asymptotics of matrix coefficients of representations of [formula omitted]

We study matrix coefficients of the unitary (and also the completely bounded) representations of SL(2,R) and its universal covering group. We describe the asymptotic distribution of column vectors in terms of Whittaker functions, exhibiting also a relationship to the Fourier transform.

Statistical Inference For Noisy Matrix Completion Incorporating Auxiliary Information

This article investigates statistical inference for noisy matrix completion in a semi-supervised model when auxiliary covariates are available. The model consists of two parts. One part is a low-rank matrix induced by unobserved latent factors; the other part models the effects of the observed covariates through a coefficient matrix which is composed of high-dimensional column vectors. We model the observational pattern of the responses through a logistic regression of the covariates, and allow its probability to go to zero as the sample size increases. We apply an iterative least squares (LS) estimation approach in our considered context. The iterative LS methods in general enjoy a low computational cost, but deriving the statistical properties of the resulting estimators is a challenging task. We show that our method only needs a few iterations, and the resulting entry-wise estimators of the low-rank matrix and the coefficient matrix are guaranteed to have asymptotic normal distributions. As a result, individual inference can be conducted for each entry of the unknown matrices. We also propose a simultaneous testing procedure with multiplier bootstrap for the high-dimensional coefficient matrix. This simultaneous inferential tool can help us further investigate the effects of covariates for the prediction of missing entries. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Development and analysis of diethylaminoethyl silica-based adsorption column for removing antibiotic resistance genes from wastewater

The current study investigates the efficiency of diethyl aminoethyl silica gel (DEAE-Si) based adsorption column for removing antibiotic resistance genes from secondary treated wastewater. Various column parameters like packing shape, packing size, bed height, and flowrate were analyzed to study their effect and optimize the column performance. The adsorption capacity of the column varied in range of ∼4.5 to 7.8 μg/g of DNA adsorbed. For regular packing of spherical shape, 3 mm–5 mm beads size, 8 cm column height, and flow rate of 0.5 mL/min. Adsorption capacity of 6.23 μg/g with 20 h of break through time and 36 h of saturation time was attained. The column regeneration was performed using NaCl solution at pH 8.5 ± 0.5. High removal efficiency of around 90 % up to 3 regeneration cycles, 75 % for 4th regeneration cycle, and >50 % for 5th regeneration cycle was attained. For practical understanding and to check how the column would perform for increased volumes of wastewater 10 times larger sized column was analyzed in simulated and real secondary treated wastewater matrix. The adsorption capacity attained using large column was around 4.65 μg/g at higher flowrate of 5 mL/min. and constant empty bed contact time resulting in treating up to 17.1 L of wastewater. The overall performance of large column was satisfactory and comparable with the test column in simulated wastewater matrix. >80 % removal efficiency of ARGs of different size in real secondary treated wastewater matrix was achieved on maintaining low flowrate.

A computational topology-based spatiotemporal analysis technique for honeybee aggregation

A primary challenge in understanding collective behavior is characterizing the spatiotemporal dynamics of the group. We employ topological data analysis to explore the structure of honeybee aggregations that form during trophallaxis, which is the direct exchange of food among nestmates. From the positions of individual bees, we build topological summaries called CROCKER matrices to track the morphology of the group as a function of scale and time. Each column of a CROCKER matrix records the number of topological features, such as the number of components or holes, that exist in the data for a range of analysis scales, at a given point in time. To detect important changes in the morphology of the group from this information, we first apply dimensionality reduction techniques to these matrices and then use classic clustering and change-point detection algorithms on the resulting scalar data. A test of this methodology on synthetic data from an agent-based model of honeybees and their trophallaxis behavior shows two distinct phases: a dispersed phase that occurs before food is introduced, followed by a food-exchange phase during which aggregations form. We then move to laboratory data, successfully detecting the same two phases across multiple experiments. Interestingly, our method reveals an additional phase change towards the end of the experiments, suggesting the possibility of another dispersed phase that follows the food-exchange phase.

Singular value thresholding two-stage matrix completion for drug sensitivity discovery

Incomplete data presents significant challenges in drug sensitivity analysis, especially in critical areas like oncology, where precision is paramount. Our study introduces an innovative imputation method designed specifically for low-rank matrices, addressing the crucial challenge of data completion in anticancer drug sensitivity testing. Our method unfolds in two main stages: Initially, the singular value thresholding algorithm is employed for preliminary matrix completion, establishing a solid foundation for subsequent steps. Then, the matrix rows are segmented into distinct blocks based on hierarchical clustering of correlation coefficients, applying singular value thresholding to the largest block, which has been proved to possess the largest entropy. This is followed by a refined data restoration process, where the reconstructed largest block is integrated into the initial matrix completion to achieve the final matrix completion. Compared to other methods, our approach not only improves the accuracy of data restoration but also ensures the integrity and reliability of the imputed values, establishing it as a robust tool for future drug sensitivity analysis.

Robust Tensor-Based DOA and Polarization Estimation in Conformal Polarization Sensitive Array with Bad Data.

Partially impaired sensor arrays pose a significant challenge in accurately estimating signal parameters. The occurrence of bad data is highly probable, resulting in random loss of source information and substantial performance degradation in parameter estimation. In this paper, a tensor variational sparse Bayesian learning (TVSBL) method is proposed for the estimate of direction of arrival (DOA) and polarization parameters jointly based on a conformal polarization sensitive array (CPSA), taking into account scenarios with the partially impaired sensor array. First, a sparse tensor-based received data model is developed for CPSAs that incorporates bad data. Then, a column vector detection method is proposed to diagnose the positions of the impaired sensors. In scenarios involving partially impaired sensor arrays, a low-rank matrix completion method is employed to recover the random loss of signal information. Finally, variational sparse Bayesian learning (VSBL) and minimum eigenvector methods are utilized sequentially to obtain the DOA and polarization parameters estimation, successively. Furthermore, the Cramér-Rao bound is given for the proposed method. Simulation results validated the effectiveness of the proposed method.

Intelligent design of sensor networks for data-driven sensor maintenance at railways

With rapid advances in digitization, many critical processes in transportation, industries, and our daily life rely on sensor measurements. With time, however, the measurements may get gradually biased and their precision deteriorates, leading to an enhanced risk of major disruptions caused by false sensor measurements. All single sensor measurements are uncertain and deviate from the true value. To detect malfunctioning sensors early on, a set of recent measurements of each sensor has to be constantly cross-checked against the measurements of a given number of other sensors, i.e., sensors should form a diagnosable network.In this article, we examine the intelligent positioning of safety-relevant sensors at railways such that the installed sensors can constantly cross-check each other and the number of the required sensors is minimized. The arising sensor positioning problem (SPP) belongs to the family of the coordinated set covering problems with two binary matrices: the choice of columns in one matrix implies the selection of specific columns and rows in the other matrix. We formulate an integer program, provide some formal analysis of the SPP and design a customized large neighborhood search metaheuristic RuM, which finds close-to-optimality solutions fast. In our computational experiments, we show that if we ignore the diagnosability requirement, the installed sensors cannot sufficiently cross-check each other in most cases. However, it costs only a few (or even no) additional sensors to ensure the diagnosability of the sensor network.