Singular-Value-Based Cluster Number Detection Method

This paper concerns accurate computation of the singular value decomposition (SVD) of an $m\times n$ matrix $A$ . As is well known, cross-product matrix based SVD algorithms compute large singular values accurately but generally deliver poor small singular values. A new novel cross-product matrix based SVD method is proposed: (a) Use a backward stable algorithm to compute the eigenpairs of $A^{\rm T}A$ and take the square roots of the large eigenvalues of it as the large singular values of $A$ ; (b) form the Rayleigh quotient of $A^{\rm T}A$ with respect to the matrix consisting of the computed eigenvectors associated with the computed small eigenvalues of $A^{\rm T}A$ ; (c) compute the eigenvalues of the Rayleigh quotient and take the square roots of them as the small singular values of $A$ . A detailed quantitative error analysis is conducted on the method. It is proved that if small singular values are well separated from the large ones then the method can compute the small ones accurately up to the order of the unit roundoff $\epsilon$ . An algorithm is developed that is not only cheaper than the standard Golub–Reinsch and Chan SVD algorithms but also can update or downdate a new SVD by adding or deleting a row and compute certain refined Ritz vectors for large matrix eigenproblems at very low cost. Several variants of the algorithm are proposed that compute some or all parts of the SVD. Typical numerical examples confirm the high accuracy of our algorithm.

We monitored the occupancy of a functionally important non-coordinated water molecule in the distal heme pocket of sperm whale myoglobin over the pH range 4.3-9.4. Water occupancy was assessed by using time-resolved spectroscopy to detect the perturbation of the heme visible band absorption spectrum caused by water entry after CO photodissociation ( Goldbeck, R. A., Bhaskaran, S., Ortega, C., Mendoza, J. L., Olson, J. S., Soman, J., Kliger, D. S., and Esquerra, R. M. (2006) Proc. Natl. Acad. Sci. U. S. A. 103, 1254-1259 ). We found that the water occupancy observed during the time interval between ligand photolysis and diffusive recombination decreased by nearly 20% as the pH was lowered below 6. This decrease accounted for most of the concomitant increase in the observed CO bimolecular recombination rate constant, as the lower water occupancy presented a smaller kinetic barrier to CO entry into the pocket at lower pH. These results were consistent with a model in which the distal histidine, which stabilizes the water molecule within the distal pocket by accepting a hydrogen bond, tends to swing out of the pocket upon protonation and destabilize the water occupancy at low pH. Extrapolation of this model to lower pH suggests that the additional increase in ligand association rate constant observed previously in stopped-flow studies at pH 3 may also be due in part to reduced distal water occupancy concomitant with further His64 protonation and coupled protein conformational change.

In order to solve the problems of low prediction accuracy and long training time that are common in the existing analog circuit fault diagnosis models, this paper proposes a new combination of wavelet packet feature extraction, singular value decomposition(SVD) and dimensionality reduction and support vector machine(SVM) classification method. This method selects wavelet packet analysis with higher accuracy than traditional wavelet analysis, extracts features of analog circuit fault data, and normalizes the extracted feature data; then uses singular value decomposition method to perform fault data matrix decompose to achieve the purpose of dimensionality reduction. The size of the singular value obtained by decomposition reflects the characteristics of the fault information. Selecting the matrix with the largest singular value as a sample can express the fault characteristics more accurately and efficiently; finally, use the support vector machine to decompose the fault after the singular value. The matrix is trained and classified, so as to realize the fault diagnosis of the analog circuit. The simulation experiment results show that, compared with the current diagnosis models such as BAGRNN, the SVD model proposed in this paper improves the fault diagnosis rate of analog circuits, effectively reduces the amount of matrix calculation, and speeds up the diagnosis.

We present four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture. We emphasize Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left- and right-singular vectors) for sparse matrices arising from two practical applications: information retrieval and seismic reflection tomography. The target architectures for our implementations are the CRAY-2S/4–128 and Alliant FX/80. The sparse SVD problem is well motivated by recent information-retrieval techniques in which dominant singular values and their corresponding singular vectors of large sparse term-document matrices are desired, and by nonlinear inverse problems from seismic tomography applications which require approximate pseudo-inverses of large sparse Jacobian matrices. This research may help advance the development of future out-of-core sparse SVD methods, which can be used, for example, to handle extremely large sparse matrices 0 × (106) rows or columns associated with extremely large databases in query-based information-retrieval applications.

This paper presents ATR results with High Range Resolution (HRR) profiles used for classification. It is shown that effective HRR-ATR performance can be achieved if the templates are formed via Singular Value Decomposition (SVD) of detected HRR profiles. It is demonstrated theoretically that in the mean-squared sense, the eigen-vectors represent the optimal feature set. SVD analysis of a large class of XPATCH and MSTAR HRR-data clearly indicates that significant proportion (> 90%) of target energy is accounted for by the eigen-vectors of range correlation matrix, corresponding to only the largest singular value. The SV Decomposition also decouples the range and angle basis spaces. Furthermore, it is shown that significant clutter reduction can be achieved if HRR data is reconstructed using only the significant eigenvectors. ATR results with eigen-templates are compared with those based on mean-templates. Results are included for both XPATCH and MSTAR data using linear least- squares and matched-filter based classifiers.

Day‐to‐day variation of travel demand has been rarely studied, due to the limitation of traditional transport data collection methods and the difficulty in high‐dimensional data processing. In this study, singular value decomposition (SVD) is used to study the day‐to‐day regularity of metro travel demand, based on four one‐month datasets from the metro networks of Shanghai and Shenzhen, China. The results show that SVD is a tool to understand the intrinsic structure of daily metro travel demand. It is found that daily metro travel demand can be decomposed into three constituents: periodic part, burst part and other part. The periodic part varies weekly and accounts for a majority of the travel demand of origin–destination matrix. The burst part exhibits short‐lived spikes, which are caused by special events or holidays. Also, other part varies randomly and only contributes a fraction of travel demand. Moreover, the periodic part corresponding to the two largest singular values is very stable in 2 months, and accounts for most of the travel demand. Finally, the burst part is used to analyse the impact of a collision accident. This work is helpful for short‐term travel demand prediction, metro operation schedule and emergency management.

Grouping large number of text documents is a challenging task due to high dimensional representation of the vector space model. Higher dimensionality of text data leads to computational burden and inefficient cluster results. In this work we improve the quality of the text document clustering using Singular Value Decomposition technique along with dimensionality reduction. In this paper we have also proposed Singular Value Decomposition which helps in finding the appropriate number of clusters for k-means clustering technique according to the singular values (Eigen values) calculated in the Singular Value Decomposition method.

When seismic waves encounter abrupt points of stratum or lithology in the process of propagation, such as fault edges, pinch-out points, or the protrusion of unconformity surfaces, the Snell law will break down and many diffractions will be generated. However, the diffraction information is typically masked by strong reflections; thus, separating diffractions is one key issue for diffraction imaging. The traditional singular value decomposition (SVD) method with the ability for wavefield decomposition and reconstruction is useful in removing strong reflections of large singular values. However, the continuously changing characteristics of the singular value of steep-slope reflections make it difficult to choose a suitable parameter for separating diffractions. To solve the problem of singular value selection in diffractions reconstruction by the SVD method, we have developed a Cook-distance SVD method by mathematically least-squares measuring the contribution of every singular value. The Cook-distance SVD can amplify the difference between continuously changing singular values and easily find the singular values representing diffractions. Synthetic examples and field data applications demonstrate that our Cook-distance SVD method can avoid parameter testing and has good performance in diffraction separation and can highlight reservoir-involved small-scale geologic edges and scatterers.

In this letter, we put forth a novel strategy to boost the reliability of orthogonal frequency division multiplexing differential chaos shift keying (OFDM-DCSK) systems over doubly selective fading channels. Provided the accessibility of channel state information, singular value decomposition is exploited to partition the channel into distinct sub-channels, which is realized by multiplying the right singular vectors by the symbol matrix at the transmitter. Subsequently, the sub-channel associated with the largest singular value is allocated to the reference chaotic sequence, leading to a higher signal-to-noise ratio (SNR) of demodulated symbols and lower bit error rate (BER) at the receiver. Inspired by the water-filling principle, the BER can be further reduced by permuting the symbols in a strategic manner. This approach of channel decomposition integrated with symbol reallocation, referred to as singular-vector pre-coded OFDM-DCSK (SVP-OFDM-DCSK), is able to escalate the SNR over doubly selective channels, ultimately enhancing the system reliability. The BER of the SVP-OFDM-DCSK system is also analyzed. Simulation results demonstrate that the proposed system with and without symbol permutation step achieves lower BERs than existing schemes over doubly selective fading channels.

A new method for evaluating the decay ratios in a boiling water reactor (BWR) using the singular value decomposition (SVD) method had been proposed. In this method, a signal component closely related to the BWR stability can be extracted from independent components of the neutron noise signal decomposed by the SVD method. However, real-time stability monitoring by the SVD method requires an efficient procedure for screening such compoments. For efficient screening, an artificial neural network (ANN) with three layers was adopted. The trained ANN was actually applied to decomposed components of local power range monitor (LPRM) signals that were measured in stability experiments conducted in the Ringhals-1 BWR. In each LPRM signal, multiple candidates were screened from the decomposed components. However, decay ratios could be estimated by introducing appropriate criterions for selecting the most suitable component among the candidates. The estimated decay ratios are almost identical to those evaluated by visual screening in a previous study. The selected components commonly have the largest singular value, the largest decay ratio and the least squared fitting error among the candidates. By virtue of excellent screening performance of the trained ANN, the real-time stability monitoring by the SVD method can be applied in practice.

A method to extract latent semantic components from noisy categorical time-series data based on the Takens’ time-delay-embedding method and singular value decomposition is presented. A demonstration of this method to analyze a sleep stage time-series is demonstrated. The first component extracted by this method, i.e., the component with the largest singular value can be considered the circadian rhythm. The second component, which exhibits damping oscillation, can be interpreted as the ultradian rhythm, and matched with the moving-averaged $c_{2}$; which is an estimation of the cortico-thalamo-cortical loop strength calculated from the corresponding electroencephalogram data using the method previously reported. The sleep stage is treated as a nominal variable instead of an ordinal variable in this method; however, the quantitative variation of sleep state is extracted from the sleep stage time-series. We believe that this result suggests the validity and usefulness of both the methods, i.e., the method reported in the present study and the method reported in a previous study.

Modern deep neural networks (DNNs) often require high memory consumption and large computational loads. In order to deploy DNN algorithms efficiently on edge or mobile devices, a series of DNN compression algorithms have been explored, including factorization methods. Factorization methods approximate the weight matrix of a DNN layer with the multiplication of two or multiple low-rank matrices. However, it is hard to measure the ranks of DNN layers during the training process. Previous works mainly induce low-rank through implicit approximations or via costly singular value decomposition (SVD) process on every training step. The former approach usually induces a high accuracy loss while the latter has a low efficiency. In this work, we propose SVD training, the first method to explicitly achieve low-rank DNNs during training without applying SVD on every step. SVD training first decomposes each layer into the form of its full-rank SVD, then performs training directly on the decomposed weights. We add orthogonality regularization to the singular vectors, which ensure the valid form of SVD and avoid gradient vanishing/exploding. Low-rank is encouraged by applying sparsity-inducing regularizers on the singular values of each layer. Singular value pruning is applied at the end to explicitly reach a low-rank model. We empirically show that SVD training can significantly reduce the rank of DNN layers and achieve higher reduction on computation load under the same accuracy, comparing to not only previous factorization methods but also state-of-the-art filter pruning methods.

An improved ridge estimation (IRE) method for troposphere water vapor tomography

One key aspect of local fault diagnosis is how to effectively extract abrupt features from the vibration signals. This paper proposes a method to automatically extract abrupt information based on singular value decomposition and higher-order statistics. In order to observe the distribution law of singular values, a numerical analysis to simulate the noise, periodic signal, abrupt signal and singular value distribution is conducted. Based on higher-order statistics and spectrum analysis, a method to automatically choose the upper and lower borders of the singular value interval reflecting the abrupt information is built. And the selected singular values derived from this method are used to reconstruct abrupt signals. It is proven that the method is able to obtain accurate results by processing the rub-impact fault signal measured from the experiments. The analytical and experimental results indicate that the proposed method is feasible for automatically extracting abrupt information caused by faults like the rotor–stator rub-impact.

As one of the most important components in rotating machinery, gearbox is fragile due to its harsh working environment. A new feature extraction method for gearbox fault diagnosis integrating singular value decomposition (SVD) with multidimensional scaling (MDS) is proposed in this article. SVD is used to extract the singular components (SCs), which reflect the main energy of the vibration signal, and the kurtosis values of SCs are calculated to detect the fault sensitivity of each singular value. The kurtosis-weighted singular values are applied to classify different conditions of gearbox supplemented by the MDS and fuzzy C-means clustering (FCM) method. Five gearbox operating conditions including tooth breakage, tooth pitting, gear eccentricity and normal are simulated to test the performance of proposed feature extraction method in an experiment rig. Partition coefficient (PC) and partition entropy (PE) are used to evaluate the classification effect of weighted singular values. The result suggests that the proposed kurtosis-weighted singular values perform better in distinguishing the different conditions of gearbox than original singular values.