Spectral Matrices Research Articles

Petrov-Galerkin Lucas Polynomials Procedure for the Time-Fractional Diffusion Equation

Herein, we build and implement a combination of Lucas polynomials basis that fulfills the spatial homogenous boundary conditions. This basis is then used to solve the time-fractional diffusion equation spectrally. The elements of all spectral matrices are explicitly obtained in terms of the Gauss hypergeometric function. The convergence and error analysis of the proposed Lucas expansion is studied. Numerical results indicate the high accuracy and applicability of the suggested algorithm.

On the vector-valued generalized autoregressive models

The classical autoregressive type models are widely used in time series modelling. Recently, a class of models known as generalized autoregressive, recognized by an additional parameter, has been proposed in order to reveal some hidden features which cannot be characterized by the standard autoregressive models. In this paper, the generalized autoregressive models are extended to the vector-valued autoregressive models which provide a flexible framework for modelling the dependent data. The properties of the new model such as stationary conditions, some explicit form of the auto-covariance function and the spectral density matrices are investigated. Unknown parameters are then estimated and compared with other kinds of traditional methods. The numerical results obtained by means of simulation studies are then reported. Finally, the traditional autoregressive model and generalized autoregressive model are fitted to a well-known bivariate time series, respectively, and the performance of the proposed models and the estimation methods are discussed.

√2-estimation for smooth eigenvectors of matrix-valued functions

Summary Modern statistical methods for multivariate time series rely on the eigendecomposition of matrix-valued functions such as time-varying covariance and spectral density matrices. The curse of indeterminacy or misidentification of smooth eigenvector functions has not received much attention. We resolve this important problem and recover smooth trajectories by examining the distance between the eigenvectors of the same matrix-valued function evaluated at two consecutive points. We change the sign of the next eigenvector if its distance with the current one is larger than the square root of 2. In the case of distinct eigenvalues, this simple method delivers smooth eigenvectors. For coalescing eigenvalues, we match the corresponding eigenvectors and apply an additional signing around the coalescing points. We establish consistency and rates of convergence for the proposed smooth eigenvector estimators. Simulation results and applications to real data confirm that our approach is needed to obtain smooth eigenvectors.

Dipole–monopole alternative as the precursor of pseudo-excitonic chargeless half-mode in an integrable nonlinear exciton–phonon system on a regular one-dimensional lattice

The new form of an integrable exciton–phonon nonlinear dynamical system distinguished by the two physical parameters is suggested. The system is settled along an infinite one-dimensional regular lattice and it admits the semi-discrete Lax representation in terms of 3 × 3 auxiliary spectral and evolution matrices. The explicit analytical four-component solution to the system’s dynamical equations found by means of Darboux–Bäcklund dressing technique turns out to be of broken PT-symmetry giving rise to the crossover between the monopole and dipole regimes of system’s dynamics. The crossover effect is proved to be inseparable from the mutual influence between the interacting subsystems in the form of specific nonlinear superposition of two qualitatively distinct types of traveling waves characterized by the two spatial scales and by the two velocities of two physically distinct origins. The threshold value of localization parameter separating the monopole and dipole dynamical regimes is strictly established in terms of the basic physical parameters. The phenomenon of dipole–monopole alternative for the spatial distribution of pseudo-excitons is shown to initiate the partial splitting between the pseudo-excitonic and vibrational subsystems in the so-called threshold dynamical regime specified by the threshold value of localization parameter. This partial splitting is manifested by the complete elimination of one pseudo-excitonic component accompanied by the actual conversion of another pseudo-excitonic component into the pseudo-excitonic chargeless half-mode. Relying upon the inherent PT-symmetry of system’s dynamical equations the four-component symmetry broken PT-conjugated counterpart solution is also analytically recovered. The integrable nonlinear exciton–phonon system under study is expected to be useful in modeling the nonlinear dynamical properties of PT-symmetric metamaterials. The critical overviews of several relevant and several irrelevant model systems are given to rebuff certain misleading but dogmatically accepted notions and statements circulating in scientific literature.

A vectorial Darboux transformation for the Fokas–Lenells system

We first reveal that there is a Riccati-type Miura transformation from the Fokas–Lenells (FL) system to the first member of the negative part of the AKNS hierarchy. We derive a vectorial Darboux transformation for the FL system by using the bidifferential graded algebra approach. Two types of reductions lead to the FL equation and nonlocal FL (nFL) equation, respectively. From vanishing seed solutions, N-soliton solutions for the FL equation and nFL equation are provided, with spectral matrices of the form of diagonal. Finally, as an application, we perform an asymptotic analysis for the bright–bright solitons of the nonlocal case.

Detecting the changes of wood properties with respect to elapsed years since wood formation by the eigenvalue distribution of near infrared spectral matrices

The effects of aging on wood properties are usually evaluated based on cambial age, with the assumption that the properties do not change once the wood has formed. This study examined changes in wood properties with the number of years elapsed since wood formation. Near-infrared (NIR) spectra were acquired from various positions of 20 sample trees, and changes in the spectra were examined with respect to elapsed years. We considered changes in multiple traits inclusively, rather than individually, using the distribution of eigenvalues calculated from NIR spectral matrices. The diffusion of eigenvalues with an increase in elapsed years followed Dyson’s Brownian motion. The gradual increase in the first eigenvalue, which is equivalent to the Helmholtz-free energy, indicates that the xylem in the heartwood changes to a more ordered physical state over time. The variations in Shannon entropy and density matrix with elapsed years revealed the irreversibility of the aging process. The proposed method is independent of a specific coordinate system and can, therefore, be applied using a wide variety of information other than that obtained from NIR spectra.

Automatic Emotion Recognition From Multi-Band EEG Data Based on a Deep Learning Scheme With Effective Channel Attention

Automatic emotion recognition using electroencephalogram (EEG) has obtained a wide range of attention in the domain of human-computer interaction (HCI) owing to the notable differences in brain activities in the event of different types of emotions. In this paper, a novel emotion recognition approach is proposed based on a deep learning scheme utilizing the temporal, spatial, and frequency effects of the EEG signal. As neural firing provides a pathway to elicit emotions, temporal, spatial, and frequency sub-band analysis of EEG signals uncovers salient information to categorize different classes of emotions. In this regard, temporal data from each channel are divided into major spectral bands and 2D signal matrices are constructed by combining the temporal information of different frequency band signals. After concatenating all signal matrices obtained from the available channels, a 3D feature space is obtained, which can better characterize emotion variations and, thus, better classification performance is obtained. The feature space is applied to a 2D deep neural network where the band information is passed to the depth dimension of the neural network. In order to highlight the important channels, a channel attention mechanism is proposed with the neural network to distribute the weights among the channels according to the contribution. Hence, the modified feature space effectively captures distinctive information about specific channels in the context of emotion recognition. In this study, detailed and extensive experimentations are carried out on a publicly available DEAP dataset and a very satisfactory performance is obtained for the valence and the arousal domain in 2-class scenario for the subject-dependent case. The average accuracies obtained for valence and arousal domain in binary class problem are 97.06% and 96.93%, respectively.

Efficient Variational Bayes Learning of Graphical Models With Smooth Structural Changes.

Estimating a sequence of dynamic undirected graphical models, in which adjacent graphs share similar structures, is of paramount importance in various social, financial, biological, and engineering systems, since the evolution of such networks can be utilized for example to spot trends, detect anomalies, predict vulnerability, and evaluate the impact of interventions. Existing methods for learning dynamic graphical models require the tuning parameters that control the graph sparsity and the temporal smoothness to be selected via brute-force grid search. Furthermore, these methods are computationally burdensome with time complexity O(NP3) for P variables and N time points. As a remedy, we propose a low-complexity tuning-free Bayesian approach, named BASS. Specifically, we impose temporally dependent spike and slab priors on the graphs such that they are sparse and varying smoothly across time. An efficient variational inference algorithm based on natural gradients is then derived to learn the graph structures from the data in an automatic manner. Owing to the pseudo-likelihood and the mean-field approximation, the time complexity of BASS is only O(NP2). To cope with the local maxima problem of variational inference, we resort to simulated annealing and propose a method based on bootstrapping of the observations to generate the annealing noise. We provide numerical evidence that BASS outperforms existing methods on synthetic data in terms of structure estimation, while being more efficient especially when the dimension P becomes high. We further apply the approach to the stock return data of 78 banks from 2005 to 2013 and find that the number of edges in the financial network as a function of time contains three peaks, in coincidence with the 2008 global financial crisis and the two subsequent European debt crisis. On the other hand, by identifying the frequency-domain resemblance to the time-varying graphical models, we show that BASS can be extended to learning frequency-varying inverse spectral density matrices, and further yields graphical models for multivariate stationary time series. As an illustration, we analyze scalp EEG signals of patients at the early stages of Alzheimer's disease (AD) and show that the brain networks extracted by BASS can better distinguish between the patients and the healthy controls.

Robust and fuzzy ensemble framework via spectral learning for random projection-based fuzzy-c-means clustering

The ensembles of random projection-based fuzzy-c-means (RP-FCM) can handle high-dimensional data efficiently. However, the performance of these ensemble frameworks is still hindered by some issues, such as misaligned membership matrices, information loss of co-similar matrices, large storage space, unstable ensemble results due to the additional re-clustering, etc. To address these issues, we propose a robust and fuzzy ensemble framework via spectral learning for RP-FCM clustering. After using random projection to generate different dimensional datasets and obtaining the membership matrices via fuzzy-c-means, we first convert these membership matrices into regularized graphs and approximates the affinity matrices of these graphs by spectral matrices. This step not only avoids the alignment problems of membership matrices but also excludes the storage of large-scale graphs. The spectral matrices of the same size are used as the features of membership matrices for the ensemble, avoiding the possible information loss by applying co-similar matrix transformations. More importantly, an optimization model is designed in our framework to learn the fusion of spectral features. In this model, the proportion of each base clustering is adjusted adaptively through a fuzzification exponent, and the effect of outliers is also suppressed by a robust norm. Finally, the Laplacian rank constraint in the model guarantees the ensemble can achieve the exact final partition. An efficient algorithm for this model is derived, and its time complexity and convergence are also analyzed. Competitive experimental results on benchmark data demonstrate the effectiveness of the proposed ensemble framework in comparison to state-of-the-art methods.

Moment functional based spectral density functional theory

We describe a density-functional method which aims at computing the ground state electron density and the spectral function at the same time. One basic ingredient of our method is the construction of the spectral function from the first four spectral moment matrices. The second basic ingredient is the construction of the spectral moment matrices from density functionals. We call our method moment-functional based spectral density-functional theory (MFbSDFT), because it is based on density-functionals for the spectral moments and because it allows us to compute the spectral function. If it is implemented in second variation our method consumes only a fraction more computer time than a standard DFT calculation with the PBE functional. We show that MFbSDFT captures correlation effects such as the valence-band satellite in Ni and the formation of lower and upper Hubbard bands in SrVO$_3$. For the purpose of constructing the spectral function from the first four $N\times N$ spectral moment matrices we describe an efficient algorithm based on the diagonalization of one hermitean $2N\times 2N$ matrix.

Multiple orthogonal polynomials: Pearson equations and Christoffel formulas

Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss–Borel factorization of the moment matrix is given. It is shown a hidden freedom exhibited by the spectral system related to the multiple orthogonal polynomials. Pearson equations are discussed, a Laguerre–Freud matrix is considered, and differential equations for type I and II multiple orthogonal polynomials, as well as for the corresponding linear forms are given. The Jacobi–Piñeiro multiple orthogonal polynomials of type I and type II are used as an illustrating case and the corresponding differential relations are presented. A permuting Christoffel transformation is discussed, finding the connection between the different families of multiple orthogonal polynomials. The Jacobi–Piñeiro case provides a convenient illustration of these symmetries, giving linear relations between different polynomials with shifted and permuted parameters. We also present the general theory for the perturbation of each weight by a different polynomial or rational function aka called Christoffel and Geronimus transformations. The connections formulas between the type II multiple orthogonal polynomials, the type I linear forms, as well as the vector Stieltjes–Markov vector functions is also presented. We illustrate these findings by analyzing the special case of modification by an even polynomial.

An Algebraic Estimator for Large Spectral Density Matrices

We propose a new estimator of high-dimensional spectral density matrices, called ALgebraic Spectral Estimator (ALSE), under the assumption of an underlying low rank plus sparse structure, as typically assumed in dynamic factor models. The ALSE is computed by minimizing a quadratic loss under a nuclear norm plus l 1 norm constraint to control the latent rank and the residual sparsity pattern. The loss function requires as input the classical smoothed periodogram estimator and two threshold parameters, the choice of which is thoroughly discussed. We prove consistency of ALSE as both the dimension p and the sample size T diverge to infinity, as well as the recovery of latent rank and residual sparsity pattern with probability one. We then propose the UNshrunk ALgebraic Spectral Estimator (UNALSE), which is designed to minimize the Frobenius loss with respect to the pre-estimator while retaining the optimality of the ALSE. When applying UNALSE to a standard U.S. quarterly macroeconomic dataset, we find evidence of two main sources of comovements: a real factor driving the economy at business cycle frequencies, and a nominal factor driving the higher frequency dynamics. The article is also complemented by an extensive simulation exercise. Supplementary materials for this article are available online.

Spectral representation-based efficient simulation method for fully non-stationary spatially varying ground motions

As one of the most commonly-used simulation methods, the spectral representation method (SRM) has been utilized to generate fully non-stationary spatially varying ground motions. However, this method usually fails to simulate these ground motions at a large number of interesting locations because of the heavy computational cost of the decomposition of spectral matrices and the superposition of harmonic functions, especially for simulations with time-frequency coupled modulation functions. To address this issue, this study develops an efficient SRM-based method based on a simple interpolation strategy and a quadratic regularization projected Barzilai–Borwein assisted non-negative matrix factorization (QRPBB-NMF). Concretely, the distribution of interpolation nodes is predetermined via spectral feature analysis and only the spectral matrix decomposition at these nodes is required. Then, QRPBB-NMF is introduced to decouple the decomposed spectra into the sum of the products of various time and frequency expressions. Finally, a simple spline interpolation is carried out for these expressions, thereby approximately achieving the SRM-required matrix decomposition in a decoupled manner. The developed method can decrease the operand of the matrix decomposition and invoke the Fast Fourier Transform to speed up the harmonic superposition. A case study in which the fully non-stationary spatially varying ground motions of a tunnel are simulated proves the effectiveness of the developed method in terms of accuracy and efficiency.

Seismic analysis of a I-sectioned steel frame based on an improved time-domain spectral element method

An improved time-domain spectral element method (ITSEM) is developed for seismic analysis of I-sectioned steel structures in this paper. The developed method considers the effect of section type on spectral stiffness using shear correction coefficient. In the calculation process of the time-domain spectral stiffness and mass matrices, the invariant integral operations of the element matrices are extracted and saved in advance. Thus, the global mass and stiffness matrices of the structures can be obtained only by simple addition, subtraction, multiplication and division operations, which greatly improves computational efficiency. Next, the natural frequencies of a I-sectioned beam are computed and compared with the finite element method (FEM) and analytical results to validate the ITSEM. Then, the seismic responses of a ten-story steel frame using ITSEM are obtained and compared with those by FEM, which proves the effectiveness, accuracy and high efficiency of ITSEM in elastic seismic analysis.

Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems

The Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for tough problems, we have to loosen the default Chebfun tolerance in Newton’s solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic, the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how the Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).

Time series graphical lasso and sparse VAR estimation

A two-stage sparse vector autoregression method is proposed. It relies on the more recent and powerful technique of time series graphical lasso to estimate sparse inverse spectral density matrices in the first stage, and its second stage refines non-zero entries of the AR coefficient matrices using a false discovery rate (FDR) procedure. Compared to a recent approach, the method has the advantage of avoiding the inversion of the spectral density matrix, but has to deal with optimization over Hermitian matrices with complex-valued entries. Such modifications significantly improve the computational time with a little loss in forecasting performance. The algorithmic and computational properties of the method have been studied and the performance of the two methods is compared using simulated and a real macro-economic dataset. The simulation results show that the proposed modification is preferred over the existing method when the goal is to learn the structure of the AR coefficient matrices while the latter outperforms the former when forecasting is the ultimate task.

Spatial Dispersion of Elastic Waves in Transversely Isotropic Media Using Lagrange Spectral Element Method

Dispersion is studied for the two-dimensional propagation of elastic waves in transversely isotropic media using the Lagrange spectral element method. Spectral element matrices are derived as the tensor product of elementary second-order tensors. Gauss-Lobatto-Legendre points are used for the interpolation of Lagrange-type shape functions as well as for the numerical integration to obtain elementary matrices. The Rayleigh quotient approximation technique is employed to find the solution of the eigenvalue problem, which is obtained from the semidiscretized form of the elastic wave equation for propagation of plane harmonic waves. Variations of errors in the phase/group velocities of bulk waves are depicted graphically with the order of interpolation polynomial, angle with the symmetry axis, and the time discretization. Error analysis clearly demonstrated the effectiveness of the Lagrange spectral element method for wave simulation in a transversely isotropic medium.

Numerical Design of Non-Reciprocal Bandpass Filters With the Aid of 3D Coupling Matrix for 5G Bands

In this brief, numerical analysis on spatio-temporally modulated (STM) non-reciprocal bandpass filters is methodically demonstrated to corroborate the underlying phenomenon. The concept of a 3D coupling matrix is illustrated for the first time, which provides an analytical insight into the distribution of RF energy among the intermodulation (IM) products generated in time-modulated resonators. Flagship equations based on coupling matrix are proposed that bypass the tedious analysis of spectral admittance and ABCD matrices. These equations can be modified to design any ordered filters and with varying cross-coupling instances. The experimental validation confirms the accuracy of the numerical equations, and the model can be utilized to further the development in this burgeoning yet eccentric research domain.

Calculation of lower and upper band boundaries for the feasible solutions of rank-deficient multivariate curve resolution problems

The computation of lower and upper band boundaries for the feasible solutions of multivariate curve resolution problems is an important and well-understood methodology. These techniques assume rank-regular spectral data matrices, namely the rank of the matrices equals the number of chemical species involved. For rank-deficient problems, which include linear dependencies within the pure component factors, band boundary calculations are much more complex. This paper deals with rank-deficient problems for which the rank-deficient factor is known and describes how to calculate band boundaries for the dual factor. The key tools for these band boundary computations are polytope constructions and linear programming problems to be solved for each spectral channel. Numerical studies are presented for a model problem and for two experimental data sets.

Evaluation of the age dependent variation of wood properties based on the eigenvalue distribution of near infrared spectral matrices

The age-dependent variation in wood properties is a very complex phenomenon because its pattern is highly dependent on tree species and wood traits. In this study, we evaluated the variation of multiple traits inclusively based on the distribution of eigenvalues calculated from the near-infrared spectral matrix at each cambial age. The experiments were conducted on four tree species with characteristic xylem structures, aiming to clarify the intrinsic behaviour of tree aging independent of tree species. The eigenvalues diffused with age in any species, such as in Dyson's Brownian motion. The gradual increase in the first eigenvalue, which is equivalent to the Helmholtz free energy, indicates that trees form a more ordered wood with age. As all the variations induced by various wood properties during the process of tree growth were aggregated into the set of eigenvalues, the Fokker-Planck equation representing the variation of eigenvalue distribution might provide a conclusive answer for the determination of demarcation between juvenile and mature wood. The age dependency of Shannon entropy and density matrix calculated from the probability associated with each energy eigenstate provided us with knowledge from the perspective of randomness; namely, tree aging from the perspective of the variation of wood properties was clearly an irreversible process. This result offers an important clue for sustainable forest management and the use of wood resources. The proposed method does not depend on a specific coordinate, thus, it will work well using data other than the near infrared spectrum.