Low-order Moments Research Articles

Fractional Low-Order Moments Based DOA Estimation With Co-Prime Array in Presence of Impulsive Noise

Direction of arrival (DOA) estimation with co-prime array is a hot issue in array signal processing. By using co-prime array structure, the degrees of freedom has been greatly improved, which can estimate much more sources than that of the conventional array structure. In the scenario of impulsive noise, the high-order (2-order or 4-order) moments of received signal do not exist, fractional low-order moments (FLOM) can be applied. In this paper, the concept of the co-prime array is extended to be applied on FLOM of received signal in presence of impulsive noise. And then MUSIC algorithm is used for DOA estimation. The proposed method is tested on the numerical data. The simulation results prove the effectiveness of the proposed method in impulsive noise scheme.

Cover-Lossless Robust Image Watermarking Against Geometric Deformations.

Cover-lossless robust watermarking is a new research issue in the information hiding community, which can restore the cover image completely in case of no attacks. Most countermeasures proposed in the literature usually focus on additive noise-like manipulations such as JPEG compression, low-pass filtering and Gaussian additive noise, but few are resistant to challenging geometric deformations such as rotation and scaling. The main reason is that in the existing cover-lossless robust watermarking algorithms, those exploited robust features are related to the pixel position. In this article, we present a new cover-lossless robust image watermarking method by efficiently embedding a watermark into low-order Zernike moments and reversibly hiding the distortion due to the robust watermark as the compensation information for restoration of the cover image. The amplitude of the exploited low-order Zernike moments are: 1) mathematically invariant to scaling the size of an image and rotation with any angle; and 2) robust to interpolation errors during geometric transformations, and those common image processing operations. To reduce the compensation information, the robust watermarking process is elaborately and luminously designed by using the quantized error, the watermarked error and the rounded error to represent the difference between the original and the robust watermarked image. As a result, a cover-lossless robust watermarking system against geometric deformations is achieved with good performance. Experimental results show that the proposed robust watermarking method can effectively reduce the compensation information, and the new cover-lossless robust watermarking system provides strong robustness to those content-preserving manipulations including scaling, rotation, JPEG compression and other noise-like manipulations. In case of no attacks, the cover image can be recovered without any loss.

Generalized covariance for non-Gaussian signal processing and GC-MUSIC under Alpha-stable distributed noise

Direction of arrival (DOA) estimation is one of the most important techniques applied in many practical engineering applications. Multiple signal classification (MUSIC) has gained increasing attention due to its high resolution in space. Many methods have been studied to handle the noise of Alpha-stable distribution within the framework of non-Gaussian signal processing. Inspired by a state-of-the-art concept, bounded nonlinear covariance (BNC), a more generalized concept, named generalized covariance (GC), is proposed. A series of existing concepts based on the fractional lower-order moment (FLOM), the correntropy, and the BNC are unified in the name of GC. Then, the convergence of GC under Alpha-stable distributed random variables is addressed. Furthermore, four different types of nonlinear functions are introduced for GC and BNC to handle impulsive noise, including sigmoid functions, score functions, FLOM mapping, Gaussian-like functions. These curves with different parameters are also exhibited in detail to illustrate their capabilities to suppress outliers. Also, GC-MUSIC is proposed, and its performances are compared with other 6 MUSIC-like algorithms in the presence of heavy-tailed impulsive noise. Besides, Cramer-Rao bound (CRB) of root mean squared error is also deduced and exhibited. Through Monte-Carlo simulations, the superiority of GC-MUSIC and BNC-MUSIC under Alpha-stable distributed noise is demonstrated.

Identification of ultra‐high‐frequency PD signals in gas‐insulated switchgear based on moment features considering electromagnetic mode

The feature extraction and pattern recognition techniques are of great importance to assess the insulation condition of gas-insulated switchgear. In this work, the ultra-high-frequency partial discharge (PD) signals generated from four types of typical insulation defects are analysed using S-transform, and the greyscale image in time-frequency representation is divided into five regions according to the cutoff frequencies of TE m 1 modes. Then, the three low-order moments of every subregion are extracted and the feature selection is performed based on the J criterion. To confirm the effectiveness of selected moment features after considering the electromagnetic modes, the support vector machine, k-nearest neighbour and particle swarm-optimised extreme learning machine (ELM) are utilised to classify the type of PD, and they achieve the recognition accuracies of 92, 88.5 and 95%, respectively. In addition, the results show that the ELM offers good generalisation performance at the fastest learning and testing speeds, thus more suitable for a real-time PD detection.

Compressive Ghost Imaging of the Moving Object Using the Low-Order Moments

Ghost imaging reconstructs the image based on the second-order correlation of the repeatedly measured light fields. When the observed object is moving, the consecutive sampling procedure leads to a motion blur in the reconstructed images. To overcome this defect, we propose a novel method of ghost imaging to obtain the motion information of moving object with a small number of measurements, in which the object could be regarded as relatively static. Our method exploits the idea of compressive sensing for a superior image reconstruction, combining with the low-order moments of the images to directly extract the motion information, which has the advantage of saving time and computation. With the gradual motion estimation and compensation during the imaging process, the experimental results show the proposed method could effectively overcome the motion blur, also possessing the advantage of reducing the necessary measurement number for each motion estimation and improving the reconstructed image quality.

Observations of upper ocean sound-speed structures in the North Pacific and their effects on long-range acoustic propagation at low and mid-frequencies.

Three 1000-km long, high resolution conductivity, temperature, depth sections in the North Pacific Ocean obtained by the ship towed vehicle SeaSoar are analyzed to quantify 2005 March/April upper-ocean sound-speed structure and determine the effects on low to mid-frequency transmission loss (TL) through numerical simulation. The observations reveal a variable mixed layer acoustic duct (MLAD) with a mean sonic layer depth of 91-m, and an even higher variability, 80-m-average-thickness transition layer connecting the mixed layer (ML) with the main thermocline. The sound-speed structure is hypothesized to be associated with thermohaline processes such as air-sea fluxes, eddies, submesoscale, fronts, internal waves, turbulence, and spice, but the analysis does not isolate these factors. Upper-ocean variability is quantified using observables of layer depth, ML gradient, and sound speed to compute low order moments, probability density functions, horizontal wavenumber spectra, and empirical orthogonal function decomposition. Coupled mode acoustic propagation simulations at 400 and 1000 Hz were carried out using the sound-speed observations from the upper 400-m appended to climatology, which reveal propagation physics associated with diffraction, random media effects, and deterministic feature scattering. Statistics of TL reveal important energy transfers between the MLAD and the deep sound channel.

Retrieval of lower-order moments of the drop size distribution using CSU-CHILL X-band polarimetric radar: a case study

Abstract. The lower-order moments of the drop size distribution (DSD) have generally been considered difficult to retrieve accurately from polarimetric radar data because these data are related to higher-order moments. For example, the 4.6th moment is associated with a specific differential phase and the 6th moment with reflectivity and ratio of high-order moments with differential reflectivity. Thus, conventionally, the emphasis has been to estimate rain rate (3.67th moment) or parameters of the exponential or gamma distribution for the DSD. Many double-moment “bulk” microphysical schemes predict the total number concentration (the 0th moment of the DSD, or M0) and the mixing ratio (or equivalently, the 3rd moment M3). Thus, it is difficult to compare the model outputs directly with polarimetric radar observations or, given the model outputs, forward model the radar observables. This article describes the use of double-moment normalization of DSDs and the resulting stable intrinsic shape that can be fitted by the generalized gamma (G-G) distribution. The two reference moments are M3 and M6, which are shown to be retrievable using the X-band radar reflectivity, differential reflectivity, and specific attenuation (from the iterative correction of measured reflectivity Zh using the total Φdp constraint, i.e., the iterative ZPHI method). Along with the climatological shape parameters of the G-G fit to the scaled/normalized DSDs, the lower-order moments are then retrieved more accurately than possible hitherto. The importance of measuring the complete DSD from 0.1 mm onwards is emphasized using, in our case, an optical array probe with 50 µm resolution collocated with a two-dimensional video disdrometer with about 170 µm resolution. This avoids small drop truncation and hence the accurate calculation of lower-order moments. A case study of a complex multi-cell storm which traversed an instrumented site near the CSU-CHILL radar is described for which the moments were retrieved from radar and compared with directly computed moments from the complete spectrum measurements using the aforementioned two disdrometers. Our detailed validation analysis of the radar-retrieved moments showed relative bias of the moments M0 through M2 was <15 % in magnitude, with Pearson’s correlation coefficient >0.9. Both radar measurement and parameterization errors were estimated rigorously. We show that the temporal variation of the radar-retrieved mass-weighted mean diameter with M0 resulted in coherent “time tracks” that can potentially lead to studies of precipitation evolution that have not been possible so far.

Bivariate q-normal distribution for transition matrix elements in quantum many-body systems

Recently, it was established, via lower order moments, that the univariate q-normal distribution, which is the weighting function for q-Hermite polynomials, describes the ensemble-averaged eigenvalue density from many-particle random matrix ensembles generated by k-body interactions (Vyas and Kota 2019 J. Stat. Mech. 103103). These ensembles are generically called embedded ensembles of k-body interactions [EE(k)] and their Gaussian orthogonal ensemble (GOE) and Gaussian unitary ensemble (GUE) versions are called embedded Gaussian orthogonal ensemble (EGOE), EGOE(k), and embedded Gaussian unitary ensemble (EGUE), EGUE(k), respectively. Going beyond the earlier work, this paper examines the lower-order bivariate reduced moments of the distribution of the square of the transition matrix elements (usually called transition strengths) generated by the action of a transition operator (such as a dipole operator in atoms and molecules, a quadrupole operator in atomic nucleic, etc) on the eigenstates generated by a k-body Hamiltonian H(k). To this end, H is represented by EGOE(k) [or EGUE(t)] and , that is to say, a t-body, by an independent EGOE(t) [or EGUE(t)]. Given this, it is shown that the ensemble-averaged distribution of transition strengths follows a bivariate q-normal, giving a first application of the bivariate q-normal distribution in the statistical physics of quantum systems. Also presented are the formulas for the bivariate correlation coefficient ρ and the q value, that define a bivariate q-normal as a function of the particle number m, the number of single particle states N that the particles are occupying and the body ranks k and t of H and , respectively. Finally, using the bivariate q-normal form, a formula is presented for the chaos measure, number of principal components, in the transition strengths from a state with energy E.

Calculation of mode lifetimes in weakly anharmonic solids using self-consistent ensemble eigenstates of the Liouvillian

Recent schemes for the efficient calculation of vibrational mode frequencies and lifetimes in weakly anharmonic systems such as solid crystal structures have shown promise, with simple relationships being observed between low order moments of the Liouvillian and the observed lifetime. While only shown empirically, two parameters seemed sufficient to determine the lifetime to high accuracy over a large range of temperatures for a few simple low degree of freedom models and a Lennard–Jones solid. In this paper we analytically reproduce the numerically observed relationship using a few generally applicable assumptions about the density of states. This advance is made by studying the autocorrelation based on an “ensemble eigenstate” of the Liouvillian. We propose the use of this ensemble eigenstate as the basis for autocorrelations to simplify calculations in richer systems. An analysis is also performed of the relationship between canonical and microcanonical autocorrelation functions and a long standing error in the long time behavior of such functions originally presented by Sen, Sinkovits, and Chakravarti is corrected.

Spectral fluctuations in the Sachdev-Ye-Kitaev model

We present a detailed quantitative analysis of spectral correlations in the Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Matrix Theory (RMT) behavior are due to a small number of long-wavelength fluctuations (of the order of the number of Majorana fermions N) from one realization of the ensemble to the next one. These modes can be parameterized effectively in terms of Q-Hermite orthogonal polynomials, the main contribution being due to scale fluctuations for which we give a simple analytical estimate. Our numerical results for N = 32 show that only the lowest eight polynomials are needed to eliminate the nonuniversal part of the spectral fluctuations. The covariance matrix of the coefficients of this expansion can be obtained analytically from low-order double-trace moments. We evaluate the covariance matrix of the first six moments and find that it agrees with the numerics. We also analyze the spectral correlations in terms of a nonlinear σ-model, which is derived through a Fierz transformation, and evaluate the one and two-point spectral correlation functions to two-loop order. The wide correlator is given by the sum of the universal RMT result and corrections whose lowest-order term corresponds to scale fluctuations. However, the loop expansion of the σ-model results in an ill-behaved expansion of the resolvent, and it gives universal RMT fluctuations not only for q = 4 or higher even q-body interactions, but also for the q = 2 SYK model albeit with a much smaller Thouless energy while the correct result in this case should have been Poisson statistics. In our numerical studies we analyze the number variance and spectral form factor for N = 32 and q = 4. We show that the quadratic deviation of the number variance for large energies appears as a peak for small times in the spectral form factor. After eliminating the long-wavelength fluctuations, we find quantitative agreement with RMT up to an exponentially large number of level spacings for the number variance or exponentially short times in the case of the spectral form factor.

Higher-order analysis of probabilistic long-term loss under nonstationary hazards

Quantification of hazard-induced losses plays a significant role in risk assessment and management of civil infrastructure subjected to hazards in a life-cycle context. A rational approach to assess long-term loss is of vital importance. The loss assessment associated with stationary hazard models and low-order moments (i.e., expectation and variance) has been widely investigated in previous studies. This paper proposes a novel approach for the higher-order analysis of long-term loss under both stationary and nonstationary hazards. An analytical approach based on the moment generating function is developed to assess the first four statistical moments of long-term loss under different stochastic models (e.g., homogeneous Poisson process, non-homogeneous Poisson process, renewal process). Based on the law of total expectation, the developed approach expands the application scope of the moment generating function to nonstationary models and higher-order moments (i.e., skewness and kurtosis). Furthermore, by employing the convolution technique, the proposed approach effectively addresses the difficulty of assessing higher-order moments in a renewal process. Besides the loss analysis, the mixed Poisson process, a relatively new stochastic model, is introduced to consider uncertainty springing from the stochastic occurrence rate. Two illustrative examples are presented to demonstrate practical implementations of the developed approach. Ultimately, the proposed framework could aid the decision-maker to select the optimal option by incorporating higher-order moments of long-term loss within the decision-making process.

A new framework for rainfall downscaling based on EEMD and an improved fractal interpolation algorithm

In this research, a novel framework is proposed for temporal downscaling rainfall records to address the problem of insufficient temporal resolution of rainfall during the refined hydrological simulation. This framework consists of three main steps: (1) The existence of the fractal features of the original rainfall series with different temporal scales (month, week, day, h) can be judged by the dominant frequency of Hilbert marginal spectrum, and geometric shape similarity of different time scales on the basis of the ensemble empirical mode decomposition (EEMD). (2) Based on the previous fractal interpolation theory, an improved algorithm that couples with a threshold to control the number of interpolation points for the specific interpolation intervals and a parameter to captures the high intermittentness of rainfall series is developed, which can achieve the conservation of the total rainfall quantity. (3) EEMD method is applied again to compare and evaluate the fit of the downscaled records to observed records from the perspective of IMF components, and various statistical indicators (e.g., root mean square error, correlation coefficient, Nash–Sutcliffe efficiency coefficient, fractal dimension) are applied to quantify the performance of improved fractal interpolation algorithm. Three sets of observed hourly rainfall series (from 2016 to 2018) of Chaohu city, Anhui, China, were investigated using the new framework, EF (EEMD + improved fractal interpolation algorithm). First, in order to validate the applicability of the EF framework, the observed hourly rainfall sets were aggregated into daily, weekly, and monthly sets and subsequently, the EF framework was employed to downscale above sets back into daily and hourly scale. Then, a long-duration rainfall event with hourly scale in early July 2016 was downscaled to 5-min time series using EF framework and equipartition method, respectively. Finally, simulation results based on MIKE URBAN software were compared for EF framework and equipartition method. The results of our work reveal that: (a) EEMD is a reasonable identification tool for fractal characteristics of rainfall records. (b) the improved fractal interpolation algorithm not only reasonably preserves key statistical indices (e.g., low-order moments, autocorrelation, power spectra), but also captures the overall trends and inherent details of the rainfall. (c) Based on the digital simulation and model analysis, the storm sewer peak flows obtained by the EF framework are larger than that of equipartition method.

Multiple-order moments of the transient electromagnetic response of a one-dimensional earth with finite conductance – theory

The concept of moments of the electromagnetic response is useful in electromagnetic interpretation. Analytic formulae exist for low-order moments of a few conductivity models, enabling source parameters such as time constant, depth, conductance and conductivity to be estimated from the measured moments of the electromagnetic response. However, most models for which analytic formulae exist have conductivity varying abruptly as a function of depth or position. In this paper, we have derived a procedure that allows moments of any order to be calculated for a conductivity which has finite conductance but can otherwise vary arbitrarily with depth. The horizontal loop transient electromagnetic step response is computed as a sum of residues. Integration of the step response over time yields a mathematical expression for a moment of any order. We illustrate the procedure for a Gaussian conductivity function which varies smoothly with depth. The Gaussian model produces results that agree in specific limits with the thin sheet, thick sheet and half-space cases.

Demonstration of a quantized acoustic octupole topological insulator

Recently introduced quantized multipole topological insulators (QMTIs) reveal new types of gapped boundary states, which themselves represent lower-dimensional topological phases and host symmetry protected zero-dimensional corner states. Inspired by these predictions, tremendous efforts have been devoted to the experimental observation of quantized quadrupole topological phase. However, due to stringent requirements of anti-commuting reflection symmetries, it is challenging to achieve higher-order quantized multipole moments, such as octupole moments, in a three-dimensional structure. Here, we overcome this challenge, and experimentally realize the acoustic analogue of a quantized octupole topological insulator using negatively coupled resonators. We confirm by first-principle studies that our design possesses a quantized octupole topological phase, and experimentally demonstrate spectroscopic evidence of a hierarchy of boundary modes, observing 3rd order topological corner states. Furthermore, we reveal topological phase transitions from higher- to lower-order multipole moments. Our work offers a pathway to explore higher-order topological states in 3D classical platforms.

A Gaussian moment method and its augmentation via LSTM recurrent neural networks for the statistics of cavitating bubble populations

Phase-averaged dilute bubbly flow models require high-order statistical moments of the bubble population. The method of classes, which directly evolve bins of bubbles in the probability space, are accurate but computationally expensive. Moment-based methods based upon a Gaussian closure present an opportunity to accelerate this approach, particularly when the bubble size distributions are broad (polydisperse). For linear bubble dynamics a Gaussian closure is exact, but for bubbles undergoing large and nonlinear oscillations, it results in a large error from misrepresented higher-order moments. Long short-term memory recurrent neural networks, trained on Monte Carlo truth data, are proposed to improve these model predictions. The networks are used to correct the low-order moment evolution equations and improve prediction of higher-order moments based upon the low-order ones. Results show that the networks can reduce model errors to less than 1% of their unaugmented values.

A Novel Extended Higher-Order Moment Multi-Factor Framework for Forecasting the Carbon Price: Testing on the Multilayer Long Short-Term Memory Network

Predicting the carbon price accurately can not only promote the sustainability of the carbon market and the price driving mechanism of carbon emissions, but can also help investors avoid market risks and increase returns. However, previous research has only focused on the low-order moment perspective of the returns for predicting the carbon price, while ignoring the shock of extreme events and market asymmetry originating from its pricing factor markets. In this paper, a novel extended higher-order moment multi-factor framework (EHM-APT) was formed to improve the prediction and to capture the driving mechanism of the carbon price. Furthermore, a multi-layer and multi-variable Long Short-Term Memory Network (Multi-LSTM) model was constructed so that the parameters and structure could be determined experimentally for testing the performance of the proposed framework. The results show that the pricing framework considers the shock of extreme events and market asymmetry and can improve the prediction compared with a framework that does not consider the shock of higher-order moment terms. Additionally, the Multi-LSTM model is more competitive for prediction than other benchmark models. This conclusion proves the rationality and accuracy of the proposed framework. The application of the pricing framework encourages investors and financial institutions to pay more attention to the pricing factor of extreme events and market asymmetry for accurate price prediction and investment analysis.

Exact irreducible moments of the Landau collision operator in the random-velocity moment expansion

Exact moments of the Landau collision operator are calculated for the irreducible Hermite polynomials written in terms of the random-velocity variable. We present closed, algebraic formulas which can be implemented in computer algebra systems. The formulas reproduce the results for the total-velocity moment expansion (J-Y Ji and E D Held, 2006 Phys. Plasmas 13 102 103) and for the random-velocity moment expansion with the small mass-ratio approximation (J-Y Ji and E D Held, 2008 Phys. Plasmas 15 102 101). For verification of the formulas, example calculations for several lowest order moments are presented. The collisional moments can be applied in the derivations of Braginskii and integral (nonlocal) closures for arbitrary relative flow velocity between electrons and ions.

Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter is analyzed, when the external force is periodic in space and given by , both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force , and the soliton width ls, which is a function of the nonlinearity as well as the initial frequency of the solitary wave, there is a transition (at fixed ) from trapped to unbound behavior of the soliton, which depends on the parameters r and K of the external force and the nonlinearity parameter . We previously studied this phenomena when (Quintero et al 2019 J. Phys. A: Math. Theor. 52 285201) where we showed that for the soliton oscillates in an effective potential, while for it moves uniformly as a free particle. In this paper we focus on the dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of r and K when at fixed value of the frequency . Since the solitary wave gets narrower for fixed as a function of , we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase .

On the accuracy of macroscopic equations for linearized rarefied gas flows

Many macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level, either from the mesoscopic Boltzmann equation or some physical arguments, including (i) Burnett, Woods, super-Burnett, augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation, (ii) Grad 13, regularized 13/26 moment equations, rational extended thermodynamics equations, and generalized hydrodynamic equations, where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials, and (iii) bi-velocity equations and “thermo-mechanically consistent" Burnett equations based on the argument of “volume diffusion”. This paper is dedicated to assess the accuracy of these macroscopic equations. We first consider the Rayleigh-Brillouin scattering, where light is scattered by the density fluctuation in gas. In this specific problem macroscopic equations can be linearized and solutions can always be obtained, no matter whether they are stable or not. Moreover, the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem. Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation. We find that (i) the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion, (ii) for the moment method, the more moments are included, the more accurate the results are, and (iii) macroscopic equations based on “volume diffusion" do not work well even when the Knudsen number is very small. Therefore, among about a dozen tested equations, the regularized 26 moment equations are the most accurate. However, for moderate and highly rarefied gas flows, huge number of moments should be included, as the convergence to true solutions is rather slow. The same conclusion is drawn from the problem of sound propagation between the transducer and receiver. This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function. This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.

A Novel Strategy for Identifying Oil Pollutants Based on Excitation-Emission Matrix Fluorescence Spectroscopy and Zernike Moments

Human health and ecological environment are badly affected due to oil pollution. A novel strategy for identifying oil pollutants has been proposed based on generic modular design. A total of three modules are included in the oil identification strategy and each module is consisted of a series of steps. The different steps of each module were experimented and evaluated by excitation-emission matrix fluorescence spectroscopy data set of oil. The experimental results show that the average accuracy with 13.6% was improved by using the histogram equalization than using thresholding in module 1. The average accuracy with 5% was improved by using the low-order Zernike moments than using high-order Zernike moments in module 2. The average accuracy with 28.9% was improved by using angle similarity measure in the nearest-neighbor classifier compared to the other six in module 3. The optimal accuracy with 95% was obtained by combining the margin features of excitation-emission matrix fluorescence spectroscopy extracted by low-order Zernike moments with the nearest-neighbor classifier applied to angle similarity measure. The combination also has a good specificity and sensitivity. The results provide references for identifying oil pollutants.