Sequence Of Singular Values Research Articles

Damage quantification in composite laminates based on sparsity-difference estimation of the normal time–frequency representation

Damage index (DI) is of vital importance for the health monitoring of composite structures. However, most of the existing DI construction methods cannot reveal the rich information hidden in Lamb waves, such as the time-varying characteristics of the signal amplitude and phase, and thus cannot well characterize the evolutionary trend of damage. To address this limitation, this study proposes a novel DI construction method for damage identification in composite structures. The normal time–frequency transform is first applied to obtain the unbiased time–frequency distribution of the received Lamb wave signal, and then the singular value decomposition method is used to obtain the singular value sequence of the two-dimensional time–frequency matrix to characterize its matrix sparsity. Next, the dynamic time warping method is used to measure the distance between singular value sequences of the current and baseline signals. A quantitative DI is finally defined using these distances. Two experimental studies are utilized to validate the effectiveness of the proposed method. Results show that the proposed method can reveal the rich information hidden in the Lamb wave signals and has better trend-matching performance in fitting the actual damage propagation trend than traditional methods.

Diffraction Separation and Imaging Using an Improved Singular Value Decomposition Method

Faults, fractures, karst caves, and other small-scale geological targets are critical for carbonate oil and gas exploration. However, seismic responses of these small-scale geological targets are low-energy diffractions and it is difficult for traditional method to image them with high resolution. To identify these targets, the diffraction separation is a key technology. Based on the difference of diffractions and reflections in both kinematic and dynamic properties, the singular value decomposition (SVD) method can separate diffractions and reflections effectively. However, how to select the appropriate singular value sequence for diffractions and reflections is a key issue in the application. Based on the analysis of the singular value spectrum characteristics, we propose a second-order difference spectrum strategy to improve the SVD method. The improved SVD method can separate the diffractions and reflections with minimal error. It is stable when seismic data contain Gaussian noise. Reverse time migration (RTM) method is used in the diffraction imaging because it can keep the true shape of subsurface scatterers when diffractions information is complete. Synthetic examples demonstrate the effectiveness of this method.

Random Discretization of the Finite Fourier Transform and Related Kernel Random Matrices

Abstract— The finite Fourier transform operator, and in particular its singular values, have been extensively studied in relation with band-limited functions. We study here the sequence of singular values of a random discretization of the finite Fourier transform in relation with applications to wireless communication. We prove that, with high probability, this sequence is close to the sequence of singular values of the finite Fourier transform itself. This also leads us to develop 2 estimates for the spectrum of kernel random matrices. This seems to be new to our knowledge. As applications, we give fairly good approximations of the number of degrees of freedom and the capacity of an approximate model of a MIMO wireless communication network. We provide the reader with some numerical examples that illustrate the theoretical results of this paper.

Fast Reliability Assessment Algorithm of Distribution Network based on Topological Similarity Analysis

In this paper, the concept of topological similarity is introduced to realize a fast and accurate reliability assessment of distribution network. Based on the singular value sequence as the similarity index, the mathematical relationship between similarity of network topology and power supply reliability of distribution network is developed. Firstly, the matrix representation, similarity index and its determination method of distribution network topology are presented, while the compression and reconstruction of matrix are analyzed to ensure the topological characteristics of adjacent matrix. Then, a fast reliability assessment algorithm of distribution network based on topological similarity is proposed, which uses root mean square method to determine similarity according to topology database and updates reliability indexes, together with the calculation flow. Lastly, case study of IEEE 33-node distribution network is undertaken, which shows that the calculation efficiency of the proposed algorithm is seven times higher than that of other conventional algorithms under the same calculation accuracy.

An Improved Denoising Method for Partial Discharge Signals Contaminated by White Noise Based on Adaptive Short-Time Singular Value Decomposition

To extract partial discharge (PD) signals from white noise efficiently, this paper proposes a denoising method for PD signals, named adaptive short-time singular value decomposition (ASTSVD). First, a sliding window was moved along the time axis of a PD signal to cut a whole signal into segments with overlaps. The singular value decomposition (SVD) method was then applied to each segment to obtain its singular value sequence. The minimum description length (MDL) criterion was used to determine the number of effective singular values automatically. Then, the selected singular values of each signal segment were used to reconstruct the noise-free signal segment, from which the denoised PD signal was obtained. To evaluate ASTSVD, we applied ASTSVD and two other methods on simulated, laboratory-measured, and field-detected noisy PD signals, respectively. Compared to the other two methods, the denoised PD signals of ASTSVD contain less residual noise and exhibit smaller waveform distortion.

Reliability assessment of distribution networks through graph theory, topology similarity and statistical analysis

This study aims to investigate the topology characteristic indices which have a close relationship with reliability of distribution networks via mathematical tools. First, a practical network is extracted into a topology and presented by weighted adjacency matrix associated with a failure degree matrix, which contains more significant information than that of original adjacency matrix. Second, singular value sequence and vertex/edge sharing are proposed as two similarity indices. Finally, similarity functions are applied on two cases to establish the relationship with reliability by curve fitting and clustering. optimisation strategy based on modified contingency-load-loss index is elaborated to improve the reliability and guide to distribution network planning.

Bearing fault diagnosis method based on the generalized S transform time–frequency spectrum de-noised by singular value decomposition

In view of the fact that the random noise interferes with the characteristic extraction of a rolling bearing fault signal, a new method of fault feature extraction is proposed based on the combination of the generalized S transform and singular value decomposition (SVD). Firstly, the 2D time–frequency spectrum bearing fault signal is obtained by applying the generalized S transform, and the time–frequency spectrum matrix is used as the objective matrix of SVD to solve the singular values. Then the K-means clustering algorithm is used to classify the singular value sequence, and the singular values for reconstruction are determined. Finally, the de-noised matrix is carried out the generalized S inversion transform to get the de-noised fault signal, and the power spectrum is calculated to finish the fault diagnosis. By analyzing the simulated signal and the actual bearing fault data, results show that the proposed method can effectively identify typical faults of rolling bearings and improve the diagnosis effect of rolling bearing faults. And it provides a new way to realize the fault diagnosis of rolling bearings under noise.

Separation of Single Frequency Component Using Singular Value Decomposition

It is proved that for an arbitrary frequency component, no matter how much the frequency and its amplitude are, this frequency component always generates only two nonzero singular values. Based on this relationship, an approach based on singular value decomposition (SVD) is proposed to separate the single frequency, and the condition for SVD to separate the single frequency is that the amplitude of each frequency is not equal to each other. To separate the frequencies with the same amplitude, it is proposed to add the white noise to the original signal, thus the amplitudes of frequencies become different, and the influence of noise on the singular values is studied. Three principles for selecting singular values are proposed. The quantitative relation among the singular values and frequency parameters is obtained, so the place of the nonzero singular values of each frequency in the singular value sequence can be located. Under these conditions, as long as a frequency can be distinguished in the amplitude spectrum of original signal, it can always be separated from the original signal by SVD. Simulation and practical signal separation examples verify the effectiveness of this approach, and compared with the existing methods, SVD has higher frequency separation accuracy.

A novel scheme on multi-channel mechanical fault signal diagnosis based on augmented quaternion singular spectrum analysis

In this paper, a novel multi-channel mechanical failure signal classification method based on augmented quaternion singular spectrum analysis (AQSSA) is proposed. Quaternion is used to couple four channels signal, and the quaternion trajectory matrixes can be developed as augmented quaternion matrix by using the feature of the quaternion. The singular value sequence including characteristic information can be extracted by quaternion singular value decomposition (QSVD) of the augmented trajectory matrix using its covariance matrix. The method of traditional singular spectrum analysis (SSA) can only analyze the single channel signal, however, AQSSA can fully use the correlation of multi-channel and reduce the loss of the effective information. Additionally, the main singular values are defined by some methods such as difference spectrum aimed, which has the limitation that major singular values can’t be obtained under the high background noise. Thus, a concept of partial mean of singular value sequence is proposed, and it can be set as the standard of evaluating the trend of singular value sequence. In order to testify the performance of the proposed method, the numerical simulation signal and the fault vibration signal of bearing are simultaneously adopted to verify its effectiveness. The results indicate that the effectiveness of mechanical fault classification by the proposed method is superior to the traditional SSA method and the method of permutation entropy.

Intelligent Diagnosis Method for Rotating Machinery Using Dictionary Learning and Singular Value Decomposition.

Rotating machinery is widely used in industrial applications. With the trend towards more precise and more critical operating conditions, mechanical failures may easily occur. Condition monitoring and fault diagnosis (CMFD) technology is an effective tool to enhance the reliability and security of rotating machinery. In this paper, an intelligent fault diagnosis method based on dictionary learning and singular value decomposition (SVD) is proposed. First, the dictionary learning scheme is capable of generating an adaptive dictionary whose atoms reveal the underlying structure of raw signals. Essentially, dictionary learning is employed as an adaptive feature extraction method regardless of any prior knowledge. Second, the singular value sequence of learned dictionary matrix is served to extract feature vector. Generally, since the vector is of high dimensionality, a simple and practical principal component analysis (PCA) is applied to reduce dimensionality. Finally, the K-nearest neighbor (KNN) algorithm is adopted for identification and classification of fault patterns automatically. Two experimental case studies are investigated to corroborate the effectiveness of the proposed method in intelligent diagnosis of rotating machinery faults. The comparison analysis validates that the dictionary learning-based matrix construction approach outperforms the mode decomposition-based methods in terms of capacity and adaptability for feature extraction.

Banach limits and traces on [formula omitted

We introduce a new approach to traces on the principal ideal L1,∞ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J. Dixmier, the new approach provides the explicit construction of every trace of every operator in L1,∞ in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on L1,∞ and the set of all translation invariant functionals on l∞. This bijection allows us to identify all known and commonly used subsets of traces (Dixmier traces, Connes–Dixmier traces, etc.) in terms of invariance properties of linear functionals on l∞, and definitively classify the measurability of operators in L1,∞ in terms of qualified convergence of sums of eigenvalues. This classification has led us to a resolution of several open problems (for the class L1,∞) from [7]. As an application we extend Connes' classical trace theorem to positive normalised traces.

Decomposition of color images into singular components

У цій статті запропоновано сингулярне розкладання кольорових зображень на адитивні компоненти. Дослідження отриманих адитивних компонент на множині різних зображень показали, що кількість крупномасштабних сегментів на зображенні можна встановити, проаналізувавши компоненти власних векторів, які отримані у результаті сингулярного розкладання

Video Watermarking Algorithm Based on DCT and SVD

This paper presents an original video based on DCT (Discrete Cosine Transform) and SVD (singular value decomposition) algorithm combined adaptive watermark embedding algorithm. Algorithm will first split the video stream into different scenes, each frame of the discrete cosine transform, and its singular value decomposition of the DC coefficient; the original watermark image and after scrambling algorithm using singular value decomposition singular value sequence, and finally watermarked image. Experiments show that the algorithm meet the invisibility requirements also meet the robustness requirements, efficiently against attacks as frame dropping, Gaussian Noise and Compression attack.

Selection of effective singular values using difference spectrum and its application to fault diagnosis of headstock

The noise reduction effect of singular value decomposition (SVD) relies on the selection of effective singular values. The characteristic of singular values of normal signal and noise being studied, it is pointed out that there is a sudden change in the singular values of normal signal, but not in the ones of noise. The concept of difference spectrum of singular value is put forward, which consists of the forward differences of singular value sequence and can describe the sudden change status of singular values of a complicated signal. The automatic selection of effective singular values can be realized by the peak of the difference spectrum. If the maximum peak of difference spectrum is located in the first coordinates, it means that a strong direct current (DC) component is contained in original signal and the number of effective singular values will be determined by the second maximum peak coordinates, while what the first singular value corresponds to is the DC component, or else the number of effective singular values is determined by the maximum peak coordinates. The relationship between column number of matrix and noise removing quantity of SVD is also studied using difference spectrum and the result shows that this relationship is like a symmetrical parabola. By dint of the difference spectrum, the hidden modulation feature caused by gear vibration in headstock is isolated from a turning force signal and the fault gear is accurately located by this modulation feature.

Spectral properties of the Cauchy transform on [formula omitted

Let h m , p ( z ) , ( m , p ) ∈ Z + × Z + , be the Landau orthogonal basis of the Hilbert space on L 2 ( C , e − | z | 2 d λ ( z ) ) where λ ( z ) is the usual Lebesgue measure on the complex plane. In this paper we give some spectral properties of the Cauchy transform on the orthogonal complement of Bargmann space Λ 0 ( C ) in L 2 ( C , e − | z | 2 d λ ( z ) ) . In particular for m fixed, we consider the orthogonal projection operator on the Hilbert subspace spanned by h m , p ( z ) , p = 0 , 1 , 2 , … , and we give explicitly the sequence of singular values of its composition with the Cauchy transform in L 2 ( C , e − | z | 2 d λ ( z ) ) . As application of these of the Cauchy transform we get some identities for special functions which could be of independent interest.

Singular values of compressions, restrictions and dilations

For an operator A and a subspace E, denote by A:E the restriction of A to E and by AE the compression of A to E. A pair (S,T) of hermitian operators is said to be monotone if there exist two nondecreasing functions f, g and a hermitian Z such that S=f(Z) and T=g(Z). Using this notion: (1) We give a simple analytic characterisation of invertible operators X such that Sing(A:X(E))⩾Sing(A:E) for all subspaces E, where Sing(·) stands for the sequence of singular values. Equivalently, if A is hermitian we characterise invertible operators X satisfying Eig(AX(E))⩾Eig(AE) for all subspaces E, Eig(·) standing for the sequence of eigenvalues. (2) We prove many inequalities involving monotone pairs of positive operators. For instance we show that detAE·detBE⩽det(AB)E. In some other circumstances, the opposite inequality holds. We also show that pairs of hermitians can be dilated into monotone pairs.

Weighted Tensor Product Algorithms for Linear Multivariate Problems

We study the ε-approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables. A class of weighted tensor product (WTP) algorithms is defined which depends on a number of parameters. Two classes of permissible information are studied. Λall consists of all linear functionals while Λstd consists of evaluations of f or its derivatives. We show that these multivariate problems are sometimes tractable even with a worst-case assurance. We study problem tractability by investigating when a WTP algorithm is a polynomial-time algorithm, that is, when the minimal number of information evaluations is a polynomial in 1/ε and d. For Λall we construct an optimal WTP algorithm and provide a necessary and sufficient condition for tractability in terms of the sequence of weights and the sequence of singular values for d=1. ForΛstd we obtain a weaker result by constructing a WTP algorithm which is optimal only for some weight sequences.

Stability Rates for Linear Ill-Posed Problems with Compact and Non-Compact Operators

In this paper we deal with the ’strength’ of ill-posedness for ill-posed linear operator equations Ax = y in Hilbert spaces, where we distinguish according to M. Z. .Nashed the ill-posedness of type I if A is not compact, but we have R(A) \neq \bar{R(A)} for the range R(A) of A , and the ill-posedness of type II for compact operators A . From our considerations it seems to follow that-the problems with non-compact operators A are not ingeneral ’less’ ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators A as discussed by B. Hofmann and U. Tautenhahn are derived from the decay rate of the non-increasing sequence of singular values of A . Since singular values do not exist for non-compact operators A , we introduce stability rates in order to have a common measure for the compact and non-compact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in thenon-compact case. Moreover, using increasing rearrangements of multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for multiplication* operators. In this context, the character of sufficient conditions providing convergence rates of Tikhonov regularization are compared-for compact operators and multiplication. operators.

On log-subharmonicity of singular values of matrices

Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by $s_1,...,s_n$ the decreasing sequence of singular values of a matrix, we prove that the functions $log s_{1}(F(λ)) + ... + log s_{k}(F(λ

Superoptimal completions of triangular matrices

We construct the unique completion of a partial triangular matrix with compact operator entries that has the property that its sequence of singular values is minimal in lexicographical order among all completions. In addition some partial results regarding the singular values of this superoptimal completion are presented.