Hankel Matrix Research Articles

Recursive Delta-Operator-Based Subspace Identification with Fixed Data Size

This paper proposes a recursive delta-operator-based subspace identification method with fixed data size. The majority of existing subspace identification methods are constrained to discrete-time systems because of the disparity in Hankel matrices. Additionally, due to the storage cost, LQ-factorization and singular value decomposition in identification methods are best suited for batch processing rather than online identification. The continuous-time systems are transformed into state space models based on the delta-operator to address these issues. These models approach the original systems when the sampling interval approaches zero. The size of the data matrices is fixed to reduce the computing load. By fading the impact of past data on future data, the amount of data storage can be decreased. The effectiveness of the proposed method is illustrated by the continuous stirred tank reactor system.

A Combined Noise Reduction Method for Floodgate Vibration Signals Based on Adaptive Singular Value Decomposition and Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

To address the issue of the vibration characteristic signals of floodgates being affected by background white noise and low-frequency water flow noise, a noise reduction method combining the improved adaptive singular value decomposition algorithm (ASVD) and the improved complete ensemble EMD with adaptive noise (ICEEMDAN) is proposed. Firstly, a Hankel matrix is constructed based on the collected discrete time signals. After performing SVD on the Hankel matrix, the ASVD algorithm is used to automatically select the effective singular values to filter out most of the background white noise and retain the useful frequency components with similar energy in the signal. Then, ICEEMDAN combined with the Spearman correlation coefficient method is used to further filter out residual white noise and low-frequency water flows. The noise reduction performance of this combined method is verified through simulation experiments. Filtered by the ASVD-ICEEMDAN method, the signal-to-noise ratio of the simulation signal (50% noise level) is increased from 4.417 to 16.237, and the root mean square error is reduced from 2.286 to 0.586. Based on the practically measured vibration signals of a floodgate at a large hydropower station, the result shows that the ASVD-ICEEMDAN method exhibits good noise reduction performance and feature information extraction abilities for floodgate vibration signals, and can provide support for operational mode analysis and damage identification of practical structures under complex interference conditions.

Decomposition of kth Order Slant Toeplitz Operators

In this paper, we establish the block matrix decomposition of kth order slant Toeplitz operators. We also establish some relations between the compressions of kth order slant Toeplitz and kth order slant Hankel operators on H2. In the last section, we introduce the notion of kth order slant Toeplitz graphs.

Modal analysis of earthquake records for dams using stochastic subspace based on error analysis

AbstractSystem identification based on seismic data is crucial for updating numerical analyses and implementing health monitoring strategies for critical structures such as dams. However, most system identification methods have been developed with the assumption of white noise and stationarity of excitations, which is inconsistent with the nature of seismic data. As a result, the systems estimated from such methods are often associated with high uncertainties, making obtaining accurate and reliable results challenging. In this study, a new figure of merit (FOM) is proposed for tracking the ill‐conditioning of systems by analyzing the errors in stochastic subspace identification (SSI) through the inversion process. Unlike the condition number, which measures the ratio between the largest and smallest singular values, this FOM uses all singular values to assess the system's ill‐conditioning comprehensively. A semi‐automatic identification method based on SSI has been developed using the proposed FOM. The optimal dimensions of the Hankel matrix are obtained so that none of the system error components dominate. Frequency/damping clustering is used to overcome the identified system's over‐determination and remove outliers. Finally, the complexity of the modal shapes resulting from clustering is examined to validate the structural modal characteristics. The procedure was validated using seismic data from the Finite Element Model of Koyna Dam. It identified the first four modal frequencies of the structure with less than 2% error rate. The average error in estimating damping ratios was below 20%. The algorithm was then used to analyze the seismic monitoring data of Pacoima Dam for the Chino Hill 2008 and Granada Hills 2020 earthquakes. Despite the closely spaced Pacoima Dam modes, the study demonstrated excellent accuracy and robust methodology performance. By distinguishing the structural modes related to the dam body from other modes of the “dam‐foundation‐reservoir” system, the proposed method helps to determine the existing model's nonlinearity.

Sea Clutter Suppression Using Smoothed Pseudo-Wigner–Ville Distribution–Singular Value Decomposition during Sea Spikes

The detection of small targets within the background of sea clutter is a significant challenge faced in radar signal processing. Small target echoes are weak in energy, and can be submerged by sea clutter and sea spikes, which are caused by overturning waves and breaking waves. This severely affects the radar target detection performance. This paper proposes a smoothed pseudo-Wigner–Ville distribution–singular value decomposition (SPWVD-SVD) method for sea clutter suppression. This method determines the instantaneous frequency range of the target by contrasting the time–frequency characteristics of the sea spike and the target. Subsequently, it employs a singular value difference spectrum to reduce the rank of the Hankel matrix, thereby reducing the computational burden of the instantaneous frequency estimation step in the experiment. Based on the instantaneous frequency range of the target in the time–frequency domain, the singular values of the target signal are retained, while the singular values of clutter are set to zero. This process accomplishes the reconstruction of radar echo signals and effectively achieves the suppression of sea clutter. The suppression effect is verified using simulation data alongside ten sets of Intelligent Pixel processing X-band (IPIX) radar data against the background of sea spikes. By contrasting the clutter amplitudes before and after suppression, the SPWVD-SVD algorithm demonstrated an average clutter suppression of 15.06 dB, which proves the effectiveness of the proposed algorithm in suppressing sea clutter.

Application of MMI-SVP theory in spindle fault feature extraction

Singular value decomposition (SVD) is an effective method for estimating and separating noise in a signal. However, its performance depends on several factors, including the construction of the Hankel matrix, the number of retained reconstructed components, the length of data analysis, and even the fault feature frequency of the monitored object. This paper aims to discuss these issues and proposes an adaptive selection method of singular value pairs (SVP) based on minimum mutual information (MMI). Firstly, SVD is used to obtain singular values (SV), and the anti-angle averaging method is employed to generate a set of sub-signals for reconstruction. SV is utilized to characterize the energy of the sub-signals and determine the SVP. Subsequently, mutual information is introduced to quantitatively evaluate mutation signals and adaptively select SVP to avoid excessive or insufficient noise reduction. Additionally, the optimal dimension for the Hankel matrix decomposition is automatically determined by considering the singular value ratio and MMI index. Finally, based on prior knowledge of bearing size and fault characteristics, the minimum sample length of the Hankel matrix is determined. The effectiveness of MMI-SVP is verified through simulation analysis and real bearing fault cases.

Reconstruction of GPS Coordinate Time Series Based on Low-Rank Hankel Matrix Recovery

Reconstruction of GPS Coordinate Time Series Based on Low-Rank Hankel Matrix Recovery

IDA and Hankel operators on Fock spaces

IDA and Hankel operators on Fock spaces

One-Shot Generative Prior in Hankel-k-Space for Parallel Imaging Reconstruction.

Magnetic resonance imaging serves as an essential tool for clinical diagnosis. However, it suffers from a long acquisition time. The utilization of deep learning, especially the deep generative models, offers aggressive acceleration and better reconstruction in magnetic resonance imaging. Nevertheless, learning the data distribution as prior knowledge and reconstructing the image from limited data remains challenging. In this work, we propose a novel Hankel-k-space generative model (HKGM), which can generate samples from a training set of as little as one k-space. At the prior learning stage, we first construct a large Hankel matrix from k-space data, then extract multiple structured k-space patches from the Hankel matrix to capture the internal distribution among different patches. Extracting patches from a Hankel matrix enables the generative model to be learned from the redundant and low-rank data space. At the iterative reconstruction stage, the desired solution obeys the learned prior knowledge. The intermediate reconstruction solution is updated by taking it as the input of the generative model. The updated result is then alternatively operated by imposing low-rank penalty on its Hankel matrix and data consistency constraint on the measurement data. Experimental results confirmed that the internal statistics of patches within single k-space data carry enough information for learning a powerful generative model and providing state-of-the-art reconstruction.

Robust estimation of synchronized spontaneous otoacoustic emission via singular value decomposition and optimal shrinkage.

We investigate matrix signal processing techniques for estimating synchronized spontaneous otoacoustic emission (OAE) in noise. Responses to repeated clicks are first stored in a matrix, and singular value decomposition is either applied in the time domain or the frequency domain after constructing a Hankel matrix at every frequency. The singular values are subject to optimal shrinkage (OS) which maximizes the signal-to-noise ratio. Human OAE data were analyzed, and the Hankel matrix method outperforms the time-domain OS method in synchronized spontaneous otoacoustic emission estimation, but not in the estimation of transient-evoked otoacoustic emission. Reasons for the performance discrepancy are discussed.

On a Particular Solution of the σ-Commutation Problem () for Toeplitz and Hankel Matrices

A unified approach is proposed to the construction of matrix pairs (T,H) that solve the ‑commutation problem for Toeplitz and Hankel matrices. For a certain particular case, a family of solutions is derived.

On Some Algebras of Truncated Hankel and Asymmetric Truncated Hankel Operators

On Some Algebras of Truncated Hankel and Asymmetric Truncated Hankel Operators

Prediction of landslide step-like displacement using factor preprocessing-based hybrid optimized SVR model in the Three Gorges Reservoir, China

Artificial intelligence (AI) is gaining popularity in landslide displacement modeling due to its strong generalization capacity and ability to accurately represent complex, nonlinear laws. However, few studies have examined the validity of factor selection and the impact of data noise on the model. To address these issues, we employ various analyses, including Pearson correlation coefficient and grey correlation degree analyses to diagnose redundant and highly correlated factors, respectively. Singular spectrum analysis (SSA) is employed for denoising the data, leveraging the benefits of the hankel matrix and minimal parameters. This technique facilitates the decomposition of displacement into trend and periodic terms. The trend term displacement is fitted with a polynomial, and the periodic term displacement is predicted using a support vector regression (SVR) model. The hybrid-optimized SVR by particle swarm optimization-gravitational search algorithm (PSO-GSA) is used to improve the model's performance and prediction ability. We apply the proposed method to the Bazimen and Baishuihe landslides in Zigui County for model validation. The results show that denoising the input factor data reduces the root mean square error (RMSE) and mean absolute error (MAE) and increases the correlation coefficient (R). Furthermore, the PSO-GSA algorithm exhibits excellent generalization ability and fast optimization speed in optimizing parameters. The correlation coefficient increase to 0.99 when using the hybrid-optimized SVR model for displacement prediction.

Schatten class operators on exponential weighted Bergman spaces

In this paper, we study Toeplitz and Hankel operators on exponential weighted Bergman spaces. For 0< p<infty , we obtain sufficient and necessary conditions for Toeplitz and Hankel operators to belong to Schatten-p class by the averaging functions of symbols. For a continuous increasing convex function h, the Schatten-h class Toeplitz and Hankel operators are also characterized.

Singular component decomposition and its application in rolling bearing fault diagnosis

In the rolling bearing fault diagnosis based on vibration signal, the feature information always exists in the specific sideband of the signal. Singular value, as a mathematical quantity can represent the characteristic information of data, provides the theoretical support for extracting fault information from signals. Based on this, this paper proposes a new method called Singular Component Decomposition (SCD). The signal is divided into components by the signal spectrum trend. Through linear transformation of Hankel matrix constructed by each component, the corresponding singular values are calculated. Combined with the amplitude filtering characteristic and the time domain negative entropy index, the effective singular values are screened out. The reconstructed components retain complete fault information for fault diagnosis of rolling bearings. The simulation and experimental results show that the SCD can extract the effective information to the maximum extent to realize the fault diagnosis of the signal.

Fredholm and Schatten Class Operators on Bergman Spaces with Exponential Weights

In this paper, we give a characterization of Fredholmness of the Toeplitz operators on the Bergman spaces A φ p with exponential weights in D when 0 &lt; p &lt; ∞ . Also, we obtain the sufficient and necessary conditions which the Toeplitz and Hankel operators on A φ 2 belong to h -Schatten class, where h is a continuous increasing convex function.

LU decomposition and Toeplitz decomposition of a neural network

Any matrix A has an LU decomposition up to a row or column permutation. Less well-known is the fact that it has a ‘Toeplitz decomposition’ A=T1T2⋯Tr where Ti's are Toeplitz matrices. We will prove that any continuous function f:Rn→Rm has an approximation to arbitrary accuracy by a neural network that maps x∈Rn to L1σ1U1σ2L2σ3U2⋯Lrσ2r−1Urx∈Rm, i.e., where the weight matrices alternate between lower and upper triangular matrices, σi(x)≔σ(x−bi) for some bias vector bi, and the activation σ may be chosen to be essentially any uniformly continuous nonpolynomial function. The same result also holds with Toeplitz matrices, i.e., f≈T1σ1T2σ2⋯σr−1Tr to arbitrary accuracy, and likewise for Hankel matrices. A consequence of our Toeplitz result is a fixed-width universal approximation theorem for convolutional neural networks, which so far have only arbitrary width versions. Since our results apply in particular to the case when f is a general neural network, we may regard them as LU and Toeplitz decompositions of a neural network. The practical implication of our results is that one may vastly reduce the number of weight parameters in a neural network without sacrificing its power of universal approximation. We will present several experiments on real data sets to show that imposing such structures on the weight matrices dramatically reduces the number of training parameters with almost no noticeable effect on test accuracy.

Characterizations of matrix-valued asymmetric truncated Hankel operators

Characterizations of matrix-valued asymmetric truncated Hankel operators

The variance of a restricted sum-of-squares function over short intervals in [formula omitted

For B∈Fq[T] of degree 2n≥2, consider the number of ways of writing B=E2+γF2, where γ∈Fq⁎ is fixed, and E,F∈Fq[T] with degE=n and degF=m<n. We denote this restricted sum-of-squares function by Sγ;m(B). We obtain an exact formula for the variance of Sγ;m(B) over short intervals in Fq[T] where q is an odd prime power. Interestingly, when m+1≤n≤2m, the variance vanishes on some (not all) short intervals, which is in contrast to the standard function that counts representations as a sum of two squares. We use the method of additive characters and Hankel matrices that we previously used for the variance and correlations of the divisor function. In Section 2, we give a short overview of our approach. Section 3 gives new results relating to values of quadratic forms, over Fq, that are defined from Hankel matrices.

Hankel operators on doubling Fock spaces

Suppose ϕ is a real-valued subharmonic function on C with ΔϕdA is a doubling measure. The doubling Fock space Fϕp is the family of holomorphic functions on C such that f(⋅)e−ϕ(⋅)∈Lp. We introduce the function space IDArs,q,α and discuss the decomposition theorem for this space. We use it to characterize the boundedness and compactness of Hankel operators from a doubling Fock space Fϕp to a weighted Lesbegue space Lϕq for all possible 1≤p,q<∞, which extends the results of [9] from the special case ρ≍1. We also obtain the relationship between the solution operators to ∂‾-equation and Hankel operator. As some applications, we obtain the characterizations on f for which Hankel operators Hf and Hf‾ are both bounded (or compact) from Fϕp to Lϕq.