Original Data Matrix Research Articles

Centroidous Method for Determining Objective Weights

When using multi-criteria decision-making methods in applied problems, an important aspect is the determination of the criteria weights. These weights represent the degree of each criterion’s importance in a certain group. The process of determining weight coefficients from a dataset is described as an objective weighting method. The dataset considered here contains quantitative data representing measurements of the alternatives being compared, according to a previously determined system of criteria. The purpose of this study is to suggest a new method for determining objective criteria weights and estimating the proximity of the studied criteria to the centres of their groups. It is assumed that the closer a criterion is to the centre of the group, the more accurately it describes the entire group. The accuracy of the description of the entire group’s priorities is interpreted as the importance, and the higher the value, the more significant the weight of the criterion. The Centroidous method suggested here evaluates the importance of each criterion in relation to the centre of the entire group of criteria. The stability of the Centroidous method is examined in relation to the measures of Euclidean, Manhattan, and Chebyshev distances. By slightly modifying the data in the original normalised data matrix by 5% and 10% 100 and 10,000 times, stability is examined. A comparative analysis of the proposed Centroidous method obtained from the entropy, CRITIC, standard deviation, mean, and MEREC methods was performed. Three sets of data were generated for the comparative study of the methods, as follows: the mean value for alternatives with weak and strong differences and criteria with linear dependence. Additionally, an actual dataset from mobile phones was also used for the comparison.

Evaluating different CFD surrogate modelling approaches for fast and accurate indoor environment simulation

Indoor Environment Quality (IEQ) holds significant importance in building design and operation, and the simulation of indoor environments playing a crucial role in enhancing IEQ. Although Computational Fluid Dynamics (CFD) has been widely employed for simulating building environments, it is computationally demanding, particularly for large spaces. To tackle this challenge, we conducted a systematic evaluation of three surrogate models for accelerating CFD: Proper Orthogonal Decomposition (POD), Artificial Neural Networks (ANN), and a combined POD-ANN approach. Our evaluation criteria focused on assessing the model accuracy, the model size, computational time and extrapolation ability. A validated CFD case model and a real campus building are employed for model evaluation. The findings demonstrate that the top five modes can reconstruct the original data matrix accurately, and the POD-ANN significantly reduces model complexity and computation time by reducing the number of parameters in the neural network, the POD-ANN parameters is only 0.14 % of the ANN, and computation time is reduced by 63 %. In addition, the combination with ANN helps increase the extrapolation ability of POD significantly. In conclusion, this research proves that the POD-ANN can enhance the efficiency of CFD calculations with the advantages of both ANN and POD. By applying the POD-ANN to predict indoor temperature, we achieve faster predictions without compromising model accuracy, and an excellent extrapolation ability is achieved. This approach also reduces model complexity, highlighting its practical value for indoor environment prediction, particularly for large and complicated spaces.

A DEIM-CUR factorization with iterative SVDs

A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties like nonnegativity and sparsity, a CUR decomposition can better preserve them compared to the SVD. An essential aspect of this approach is the methodology used for selecting a subset of columns and rows from the original matrix. This study investigates the effectiveness of one-round sampling and iterative subselection techniques and introduces new iterative subselection strategies based on iterative SVDs. One provably appropriate technique for index selection in constructing a CUR factorization is the discrete empirical interpolation method (DEIM). Our contribution aims to improve the approximation quality of the DEIM scheme by iteratively invoking it in several rounds, in the sense that we select subsequent columns and rows based on the previously selected ones. Thus, we modify A after each iteration by removing the information that has been captured by the previously selected columns and rows. We also discuss how iterative procedures for computing a few singular vectors of large data matrices can be integrated with the new iterative subselection strategies. We present the results of numerical experiments, providing a comparison of one-round sampling and iterative subselection techniques, and demonstrating the improved approximation quality associated with using the latter.

A breakdown-free block conjugate gradient method for large-scale discriminant analysis

<abstract><p>Rayleigh-Ritz discriminant analysis (RRDA) is an effective algorithm for linear discriminant analysis (LDA), but there are some drawbacks in its implementation. In this paper, we first improved Rayleigh-Ritz discriminant analysis (IRRDA) to make its framework more concise, and established the equivalence theory of the solution space between our discriminant analysis and RRDA. Second, we proposed a new model based on positive definite systems of linear equations for linear discriminant analysis, and certificated the rationality of the new model. Compared with the traditional linear regression model for linear discriminant analysis, the coefficient matrix of our model avoided forming a centralized matrix or appending the original data matrix, but the original matrix itself, which greatly reduced the computational complexity. According to the size of data matrix, we designed two solution schemes for the new model based on the block conjugate gradient method. Experiments in real-world datasets demonstrated the effectiveness and efficiency of our algorithm and it showed that our method was more efficient and faster than RRDA.</p></abstract>

Application of traditional symbols in interior decoration based on VR technology

Abstract Traditional symbols, as visual symbols, are not only extremely widely used in traditional fields but are also constantly used in interior design. This paper first investigates the use of traditional symbols in interior decoration and the trend of integration of traditional symbols with modern design. Then, an index system is constructed to evaluate the level of development of interior design incorporating traditional symbols by establishing the original data matrix and determining the index weights. Finally, a fusion coordination model was established, and the data were analyzed for reliability, validity, and consistency tests to analyze the richness, taste, and emotional expression of interior design incorporating traditional symbols. The overall integration of traditional symbols with interior design increased from 0.4621 to 0.7346. The effect of traditional symbols in interior design based on VR technology on emotional performance in office scenes increased by 20% on average. The average of those who rated very conforming in different scenes was 20%, those who rated more conforming was 30%, and those who rated average performance was 10-20%. The research in this paper has important references and reference values for the use of traditional symbols in interior design.

Clustering of Asphalt Pavement Maintenance Sections Based on 3D Ground-Penetrating Radar and Principal Component Techniques

Asphalt pavement maintenance section classification is an important prerequisite for accurately determining asphalt pavement maintenance needs and formulating accurate maintenance plans. This paper introduces the three-dimensional (3D) ground-penetrating radar (GPR) pavement internal crack rate index on the basis of an original road surface performance data matrix, and the dimensionality of the road section classification data matrix was reduced through the principal component technique. An analysis of variance was used to compare the significance of the differences in the results for road section classification using different clustering methods and different clustering data and to investigate the influence of the clustering method, principal component technique and crack rate index on the maintenance road section classification results. The results showed that the principal component technique could reduce the dimensionality of the data matrix by 33% and retain more than 84% of the information. There was a genetic relationship between the clustering data and the technical characteristics of the classified sub-sections, and the internal crack rate was important for the characterisation of internal defects in asphalt pavement sub-sections and the determination of maintenance needs. The results of section classification varied considerably between clustering methods, and the choice of clustering method had a relationship to the pavement maintenance objectives. The dynamic clustering method combined with principal component analysis could significantly improve the significance of the differences in the clustering results, effectively improving the division of maintenance sections.

Kernelizing: A way to increase accuracy in trilinear decomposition analysis of multiexponential signals

The unmixing of multiexponential decay signals into monoexponential components using soft modelling approaches is a challenging task due to the strong correlation and complete window overlap of the profiles. To solve this problem, slicing methodologies, such as PowerSlicing, tensorize the original data matrix into a three-way data array that can be decomposed based on trilinear models providing unique solutions. Satisfactory results have been reported for different types of data, e.g., nuclear magnetic resonance or time-resolved fluorescence spectra. However, when decay signals are described by only a few sampling (time) points, a significant degradation of the results can be observed in terms of accuracy and precision of the recovered profiles.In this work, we propose a methodology called Kernelizing that provides a more efficient way to tensorize data matrices of multiexponential decays. Kernelizing relies on the invariance of exponential decays, i.e., when convolving a monoexponential decaying function with any positive function of finite width (hereafter called “kernel”), the shape of the decay (determined by the characteristic decay constant) remains unchanged and only the preexponential factor varies. The way preexponential factors are affected across the sample and time modes is linear, and it only depends on the kernel used. Thus, using kernels of different shapes, a set of convolved curves can be obtained for every sample, and a three-way data array generated, for which the modes are sample, time and kernelizing effect. This three-way array can be afterwards analyzed by a trilinear decomposition method, such as PARAFAC-ALS, to resolve the underlying monoexponential profiles. To validate this new approach and assess its performance, we applied Kernelizing to simulated datasets, real time-resolved fluorescence spectra collected on mixtures of fluorophores and fluorescence-lifetime imaging microscopy data. When the measured multiexponential decays feature few sampling points (down to fifteen), more accurate trilinear model estimates are obtained than when using slicing methodologies.

土壤有机质含量的高光谱成像预测研究

Soil nutrient content is an important index to evaluate the growing environment of crops. Rapid access to soil nutrient information is an important requirement for the development of modern precision agriculture, while the detection of soil organic matter content is a necessary condition for understanding the basic soil fertility and implementing crop precision cultivation. In this paper, the soil of rural fields in the southeast of Shanxi Province before sowing was taken as the research object. 111 soil samples to be tested were collected. After the process of drying, impurity removal and grinding, the hyperspectral data of the Region of interest (ROI) of the samples were collected, and then the chemical determination of soil organic matter content was conducted. The original spectral data matrix was pretreated by numerical transformation operations, such as arithmetic mean, average deviation, 1st derivation, natural logarithm and mixed multiplication, and a Partial least square regression (PLSR) quantitative analysis model was established. In these models, the obtained prediction set RP value under the pretreatment of F(A)*ln(AD) was the highest, reaching 0.8859. For spectral data preprocessed by F(A)* Ln (AD), the Competitive adaptive reweighted sampling (CARS) algorithm and Random frog (RF) algorithm were used to select key variables. The PLSR model was established by using F(A)* Ln (AD)&CARS data processing method, and the RP value was increased to 0.9545. The prediction results can accurately reflect the real content of soil organic matter. The results of this study can provide theoretical support for the application of hyperspectral imaging technology in the determination of soil organic matter content, and provide a reference for the rapid detection of other soil components.

Femtosecond laser marking depth of field 3D code

In order to reduce the market in a large number of counterfeit imitation products, product traceability anti-counterfeiting by more and more attention. Therefore, it is imperative to study product traceability and anti-counterfeiting marking technology. However, for products within a limited size, the data capacity of traditional Data Matrix (DM) code is insufficient to meet the storage requirements, and it is easy to clone and copy, which limits the development of laser marking technology. In order to improve the replication difficulty and information storage capacity of the Data Matrix code, the depth-of-field (DOF) 3D code is devised by adding Z-axis depth information to the original Data Matrix code, which can accommodate a larger information storage capacity and improve the reproduction difficulty of DM code, thus meeting the current commercial demand. In this paper, cemented carbide is taken as the research object, and DOF 3D code is marked on its surface by femtosecond laser, and the optimal process parameters are determined. At the same time, the recognition and information reading of DOF 3D code is studied, and the correctness and reliability of its information acquisition are tested.

Design of Computer-Aided Assessment and Recognition Platform for College Students' Innovative Training Plan with Data Matrix Mining

In this paper, an improved association rule algorithm based on vector product is used. This algorithm only needs to scan the transaction database once and does not generate a large number of frequent item candidate sets. The algorithm first maps the set of data transactions into a matrix of vectors, each column vector representing an attribute. Matrix factorization algorithms usually decompose the original data matrix into two or more low-dimensional matrices. This paper analyzes the common problems existing in society, colleges, and families in labor education, and innovatively proposes a four-dimensional comprehensive evaluation index system for labor education in colleges and universities, including “subject-dominated colleges—family education support—social environment support—college students’ individual characteristics”.

Improved Approximation and Visualization of the Correlation Matrix

The graphical representation of the correlation matrix by means of different multivariate statistical methods is reviewed, a comparison of the different procedures is presented with the use of an example dataset, and an improved representation with better fit is proposed. Principal component analysis is widely used for making pictures of correlation structure, though as shown a weighted alternating least squares approach that avoids the fitting of the diagonal of the correlation matrix outperforms both principal component analysis and principal factor analysis in approximating a correlation matrix. Weighted alternating least squares is a very strong competitor for principal component analysis, in particular if the correlation matrix is the focus of the study, because it improves the representation of the correlation matrix, often at the expense of only a minor percentage of explained variance for the original data matrix, if the latter is mapped onto the correlation biplot by regression. In this article, we propose to combine weighted alternating least squares with an additive adjustment of the correlation matrix, and this is seen to lead to further improved approximation of the correlation matrix.

Weighted Sparseness-Based Anomaly Detection for Hyperspectral Imagery.

Anomaly detection of hyperspectral remote sensing data has recently become more attractive in hyperspectral image processing. The low-rank and sparse matrix decomposition-based anomaly detection algorithm (LRaSMD) exhibits poor detection performance in complex scenes with multiple background edges and noise. Therefore, this study proposes a weighted sparse hyperspectral anomaly detection method. First, using the idea of matrix decomposition in mathematics, the original hyperspectral data matrix is reconstructed into three sub-matrices with low rank, small sparsity and representing noise, respectively. Second, to suppress the noise interference in the complex background, we employed the low-rank, background image as a reference, built a local spectral and spatial dictionary through the sliding window strategy, reconstructed the HSI pixels of the original data, and extracted the sparse coefficient. We proposed the sparse coefficient divergence evaluation index (SCDI) as a weighting factor to weight the sparse anomaly map to obtain a significant anomaly map to suppress the background edge, noise, and other residues caused by decomposition, and enhance the abnormal target. Finally, abnormal pixels are segmented based on the adaptive threshold. The experimental results demonstrate that, on a real-scene hyperspectral dataset with a complicated background, the proposed method outperforms the existing representative algorithms in terms of detection performance.

Learning an Optimal Bipartite Graph for Subspace Clustering via Constrained Laplacian Rank.

In this article, we focus on utilizing the idea of co-clustering algorithms to address the subspace clustering problem. In recent years, co-clustering methods have been developed greatly with many important applications, such as document clustering and gene expression analysis. Different from the traditional graph-based methods, co-clustering can utilize the bipartite graph to extract the duality relationship between samples and features. It means that the bipartite graph can obtain more information than other traditional graph methods. Therefore, we proposed a novel method to handle the subspace clustering problem by combining dictionary learning with a bipartite graph under the constraint of the (normalized) Laplacian rank. Besides, to avoid the effect of redundant information hiding in the data, the original data matrix is not used as the static dictionary in our model. By updating the dictionary matrix under the sparse constraint, we can obtain a better coefficient matrix to construct the bipartite graph. Based on Theorem 2 and Lemma 1, we further speed up our algorithm. Experimental results on both synthetic and benchmark datasets demonstrate the superior effectiveness and stability of our model.

Cluster quality analysis based on SVD, PCA-based k-means and NMF techniques: an online survey data

With the increase in computerisation in every field, a huge amount of data is collected from everywhere. Therefore, extracting useful information has become a necessary task in the present era. Data mining helps to extract the information and uncover the relationship among the data. Clustering is an unsupervised technique used for partitioning objects into several groups and discover the hidden relationship among the data. There are many techniques used for clustering. In this article, a comparative study and analysis of three famous clustering techniques are done: principal component analysis (PCA), singular value decomposition (SVD) and non-negative matrix factorisation (NMF) for the clustering of a database. The database collected through a set of questionnaire surveys related to day-to-day activities. Then a comparison of their natural clustering ability is being done. Also, the use of normalised mutual information (NMI) and purity as two-cluster quality evaluation measures are explored. Then an attempt is made to show the amount of information from the original data matrix that the approximated data matrix contains. Next, to verify the accuracy of the variance covered by the approximated data matrix, the Frobenius norm is used. At last, the results are compared with the variance covered by using singular values, and a detailed analysis of each data matrix is explained.

Non-negative low-rank representation based on dictionary learning for single-cell RNA-sequencing data analysis

In the analysis of single-cell RNA-sequencing (scRNA-seq) data, how to effectively and accurately identify cell clusters from a large number of cell mixtures is still a challenge. Low-rank representation (LRR) method has achieved excellent results in subspace clustering. But in previous studies, most LRR-based methods usually choose the original data matrix as the dictionary. In addition, the methods based on LRR usually use spectral clustering algorithm to complete cell clustering. Therefore, there is a matching problem between the spectral clustering method and the affinity matrix, which is difficult to ensure the optimal effect of clustering. Considering the above two points, we propose the DLNLRR method to better identify the cell type. First, DLNLRR can update the dictionary during the optimization process instead of using the predefined fixed dictionary, so it can realize dictionary learning and LRR learning at the same time. Second, DLNLRR can realize subspace clustering without relying on spectral clustering algorithm, that is, we can perform clustering directly based on the low-rank matrix. Finally, we carry out a large number of experiments on real single-cell datasets and experimental results show that DLNLRR is superior to other scRNA-seq data analysis algorithms in cell type identification.

Fault Diagnosis Using Dynamic Principal Component Analysis and GA Feature Selection Modeling for Industrial Processes

With the continuous expansion of industrial production scale, most of the chemical process variables are nonlinear, multi-modal and dynamic. For some traditional multivariate statistical monitoring and fault diagnosis algorithms, such as principal component analysis (PCA), the premise of its application is that the process data is time-independent. To this end, a dynamic principal component analysis (DPCA) method is proposed. However, since the input matrix of DPCA fault diagnosis needs to add an augmented matrix to the original data matrix, the number of eigenvalues of the augmented matrix is too large and there are many redundant eigenvectors. Therefore, this paper proposes a fault diagnosis and monitoring algorithm combining feature selection and DPCA, which considers the dynamic characteristics of multivariate data and reduces the dimension of the input matrix. At present, the average modeling and diagnostic accuracy of PCA-based fault diagnosis on T2 statistic is 65.49%, and that on Q statistic is 76.78%. The average modeling and diagnostic accuracy of fault diagnosis based on DPCA on T2 statistic is 63.17%, and the average modeling and diagnostic accuracy on Q statistic is 83.65%. Finally, through a TE simulation process, this paper proves that the accuracy is greatly improved when using the method proposed in this paper compared with PCA and DPCA.

Exemplar-based large-scale low-rank matrix decomposition for collaborative prediction

Recommender system is one of the most effective tools to solve the problem of information overload. As a popular method for recommendation system, regularized singular value decomposition (RSVD) has the advantage of prediction accuracy. However, with the growing of the size of rating matrix A, RSVD suffers from both ‘out of memory’ and high computational cost. To alleviate these disadvantages, we utilize CUR decomposition approach to reduce the memory consumption before RSVD is applied. Additionally, the rating matrix A often is sparse, so we propose a novel column sampling algorithm and sparseness measure to solve the data sparse problem. The main computation cost in the original RSVD is to compute and respectively. While replacing N by CUR or C in RSVD, we can reduce the computational cost from to (where c is the size of data subspace, m and n are the size of the input matrix, t is the number of iterations, is the number of features). Because CUR is explicitly expressed in terms of a small number of actual columns and actual rows of the original data matrix, the result of matrix decomposition has better interpretability. The advantage of what we devised CUR+RSVD and C+RSVD collaborative prediction approaches is that, they not only can deal with the large scale matrix rapidly but also preserve the sparsity of the original matrix, more interestingly, our approaches have higher prediction accuracy. Experimental results on Movielens, Joke data set, etc. show that what we proposed methods can handle the sparsity issue on large-scale low-rank matrix effectively. Compared with RSVD, it can achieve far better prediction accuracy and recommendation result, meanwhile save about 70% training time at the same dataset.

Quantum mean centering for block-encoding-based quantum algorithm

Mean Centering (MC) is an important data preprocessing technique, which has a wide range of applications in data mining, machine learning, and multivariate statistical analysis. When the data set is large, this process will be time-consuming. In this paper, we propose an efficient quantum MC algorithm based on the block-encoding technique, which enables the existing quantum algorithms can get rid of the assumption that the original data set has been classically mean-centered. Specifically, we first adopt the strategy that MC can be achieved by multiplying by the centering matrix C, i.e., removing the row means, column means and row-column means of the original data matrix X can be expressed as XC, CX and CXC, respectively. This allows many classical problems involving MC, such as Principal Component Analysis (PCA), to directly solve the matrix algebra problems related to XC, CX or CXC. Next, we can employ the block-encoding technique to realize MC. To achieve it, we first show how to construct the block-encoding of the centering matrix C, and then further obtain the block-encodings of XC, CX and CXC. Finally, we describe one by one how to apply our MC algorithm to PCA and other algorithms.

Matrix Factorization-Based Dimensionality Reduction Algorithms─A Comparative Study on Spectroscopic Profiling Data.

Spectroscopic profiling data used in analytical chemistry can be very high-dimensional. Dimensionality reduction (DR) is an effective way to handle the potential "curse of dimensionality" problem. Among the existing DR algorithms, many can be categorized as a matrix factorization (MF) problem, which decomposes the original data matrix X into the product of a low-dimensional matrix W and a dictionary matrix H. First, this paper provides a theoretical reformulation of relevant DR algorithms under a unified MF perspective, including PCA (principal component analysis), NMF (non-negative matrix factorization), LAE (linear autoencoder), RP (random projection), SRP (sparse random projection), VQ (vector quantization), AA (archetypical analysis), and ICA (independent component analysis). From this perspective, an open-sourced toolkit has been developed to integrate all of the above algorithms with a unified API. Second, we made a comparative study on MF-based DR algorithms. In a case study of TOF (time-of-flight) mass spectra, the eight algorithms extracted three components from the original 27,619 features. The results are compared by a set of DR quality metrics, e.g., reconstruction error, pairwise distance/ranking property, computational cost, local and global structure preservations, etc. Finally, based on the case study result, we summarized guidelines for DR algorithm selection. (1) For reconstruction quality, choose ICA. In the case study, ICA, PCA, and NMF have high reconstruction qualities (reconstruction error < 2%), ICA being the best. (2) To keep the pairwise topological structure, choose PCA. PCA best preserves the pairwise distance/ranking property. (3) For edge computing and IoT scenarios, choose RP or SRP if reconstruction is not required and the JL-lemma condition is met. The RP family has the best computational performance in the experiment, almost 10-100 times faster than its peers.

Deep alternating non-negative matrix factorisation

Non-negative matrix factorisation (NMF) is a promising data-mining technique for non-negative data. NMF achieves feature extraction by factorising the original data matrix into a basis matrix and coding matrix both with non-negative entries. Recently, multi-layer or deep NMF has been studied because of its ability to extract deep representative features which can help profoundly understand the original data. The existing deep NMF approaches are implemented by cascading the factorisation of the coding matrix at each layer. This paper proposes a novel scheme, in which the factorisations of the basis and coding matrices are alternated along layers. Based on this scheme, several deep NMF models to address various data scenarios are developed. Extensive experimental results on several machine learning tasks, including collaborative filtering, image inpainting, and community detection, show that the proposed alternating deep factorisation algorithms perform significantly better than existing state-of-the-art algorithms in most instances of these tasks.