Academic literature on the topic 'Bispectrum'

Bispectrum is a powerful tool for non-Gaussian signal processing and nonlinearity detection. However, it is difficult to use in practical applications due to that it is a 2-dimensional function. Bispectral slices are widely used reduction methods, and they can only represent a small part of the whole bispectral information. Integrated bispectrum contains more signal features than that of the bispectral slices, whereas the integration will lose the focus of some signal features. To overcome these problems, a new approach called maximal bispectrum is proposed to extract signal features. Maximal bispectrum is obtained by selecting the maximal values of every row of the magnitude bispectrum in the whole bispectral plane and it is a 1-dimensional function. Feature extraction based on maximal bispectrum is investigated and the maximal bispectrum is used to extract features of gear fault. Experimental results indicate that the maximal bispectrum is effective for diagnosing gear crack fault.

AbstractRelying on a separable modal expansion of the bispectrum, the implementation of a fast estimator for the full bispectrum of a 3d particle distribution is presented. The computational cost of accurate bispectrum estimation is negligible relative to simulation evolution, so the bispectrum can be used as a standard diagnostic whenever the power spectrum is evaluated. As an application, the time evolution of gravitational and primordial dark matter bispectra was measured in a large suite of N-body simulations. The bispectrum shape changes characteristically when the cosmic web becomes dominated by filaments and halos, therefore providing a quantitative probe of 3d structure formation. Our measured bispectra are determined by ∼ 50 coefficients, which can be used as fitting formulae in the nonlinear regime and for non-Gaussian initial conditions. We also compare the measured bispectra with predictions from the Effective Field Theory of Large Scale Structures (EFTofLSS).

The generated signals generally contain a large amount of background noise when the mechanical bearing fails, and the fault signals present nonlinear and non-Gaussian feature, which have heavy tail and belong to α -stable distribution ( 1 < α < 2 ); even the background noises are also α -stable distribution process. Then it is difficult to obtain reliable conclusion by using the traditional bispectral analysis method under α -stable distribution environment. Two improved bispectrum methods are proposed based on fractional lower-order covariation in this paper, including fractional low-order direct bispectrum (FLODB) method, fractional low-order indirect bispectrum (FLOIDB) method. In order to decrease the estimate variance and increase the bispectral flatness, the fractional lower-order autoregression (FLOAR) model bispectrum and fractional lower-order autoregressive moving average (FLOARMA) model bispectrum methods are presented, and their calculation steps are summarized. We compare the improved bispectrum methods with the conventional methods employing second-order statistics in Gaussian and S α S distribution environments; the simulation results show that the improved bispectrum methods have performance advantages compared to the traditional methods. Finally, we use the improved methods to estimate the bispectrum of the normal and outer race fault signal; the result indicates that they are feasible and effective for fault diagnosis.

As stated in the short note by R. O. Plaisted and H. G. Peña, they derived what they believed to be “a maximum entropy (MEM) representation for bispectra” and they claimed that their result “can be generalized to an Nth‐order MEM auto‐spectrum.” The purpose of this discussion is to point out the faults in the derivation and show that the claim of MEM is not correct. Furthermore, the parametric representation of the bispectrum that the authors actually attempted to derive is not new but can be found elsewhere (Huber et al., 1971). Plaisted and Peña stated that the random variables [Formula: see text] in their equation (1) can be assumed generally to be normally distributed with zero mean and variance [Formula: see text]. This is a serious mistake. With this assumption the third‐order moment sequence of the [Formula: see text] process given by their equation (3) vanishes, i.e., [Formula: see text] for all (k, l) and therefore the bispectrum of [Formula: see text] is identical to zero. The random variables [Formula: see text] in their equation (1) and hence [Formula: see text] have to be non‐Gaussian for the bispectrum of [Formula: see text] to exist. As a matter of fact, bispectral analysis has been used to obtain information regarding deviations of a process from Gaussianity (Godfrey, 1965).

ABSTRACT We present a study of the 21-cm signal bispectrum (which quantifies the non-Gaussianity in the signal) from the Cosmic Dawn (CD). For our analysis, we have simulated the 21-cm signal using radiative transfer code grizzly, while considering two types of sources (mini-QSOs and HMXBs) for Ly α coupling and the X-ray heating of the IGM. Using this simulated signal, we have, for the first time, estimated the CD 21-cm bispectra for all unique k-triangles and for a range of k modes. We observe that the redshift evolution of the bispectrum magnitude and sign follow a generic trend for both source models. However, the redshifts at which the bispectrum magnitude reaches their maximum and minimum values and show their sign reversal depends on the source model. When the Ly α coupling and the X-ray heating of the IGM occur simultaneously, we observe two consecutive sign reversals in the bispectra for small k-triangles (irrespective of the source models). One arising at the beginning of the IGM heating and the other at the end of Ly α-coupling saturation. This feature can be used in principle to constrain the CD history and/or to identify the specific CD scenarios. We also quantify the impact of the spin temperature (TS) fluctuations on the bispectra. We find that TS fluctuations have maximum impact on the bispectrum magnitude for small k-triangles and at the stage when Ly α coupling reaches saturation. Furthermore, we are also the first to quantify the impact of redshift space distortions (RSD), on the CD bispectra. We find that the impact of RSD on the CD 21-cm bispectra is significant ($\gt 20{{\ \rm per\ cent}}$) and the level depends on the stages of the CD and the k-triangles for which the bispectra are being estimated.

The bispectra of combination and blend yarns made on a spun silk spinning system are discussed, along with bispectra of mathematically idealized combination yarns. For comparison, the bispectra of a random combination yarn and two types of pseudorandom combination yarns generated by a computer simulation are calculated. Furthermore, the effect of periodic yarn irregularity on the bispectrum is estimated.

ABSTRACT The bispectrum can quantify the non-Gussianity present in the redshifted 21-cm signal produced by the neutral hydrogen ($\rm {H \small {I}}$) during the Epoch of Reionization (EoR). Motivated by this, we perform a comprehensive study of the EoR 21-cm bispectrum using simulated signals. Given a model of reionization, we demonstrate the behaviour of the bispectrum for all unique triangles in k space. For ease of identification of the unique triangles we parametrize the k-triangle space with two parameters, namely the ratio of the two arms of the triangle (n = k2/k1) and the cosine of the angle between them (cos θ). Furthermore, for the first time we quantify the impact of the redshift space distortions (RSD) on the spherically averaged EoR 21-cm bispectrum in the entire unique triangle space. We find that the real space signal bispectra for small and intermediate k1-triangles ($k_1 \le 0.6 \, \, {\rm Mpc}^{-1}$) is negative in most of the unique triangle space. It takes a positive sign for squeezed, stretched, and linear k1-triangles, specifically for large k1 values ($k_1 \ge 0.6 \, \, {\rm Mpc}^{-1}$). The RSD affects both the sign and magnitude of the bispectra significantly. It changes (increases/decreases) the magnitude of the bispectra by $50\!-\!100{{\ \rm per\ cent}}$ without changing its sign (mostly) during the entire period of the EoR for small and intermediate k1-triangles. For larger k1-triangles, RSD affects the magnitude by $100\!-\!200{{\ \rm per\ cent}}$ and also flips the sign from negative to positive. We conclude that it is important to take into account the impact of RSD for a correct interpretation of the EoR 21-cm bispectra.

We analyse the Planck full-mission cosmic microwave background (CMB) temperature and E-mode polarization maps to obtain constraints on primordial non-Gaussianity (NG). We compare estimates obtained from separable template-fitting, binned, and optimal modal bispectrum estimators, finding consistent values for the local, equilateral, and orthogonal bispectrum amplitudes. Our combined temperature and polarization analysis produces the following final results: fNLlocal = −0.9 ± 5.1; fNLequil = −26 ± 47; and fNLortho = −38 ± 24 (68% CL, statistical). These results include low-multipole (4 ≤ ℓ < 40) polarization data that are not included in our previous analysis. The results also pass an extensive battery of tests (with additional tests regarding foreground residuals compared to 2015), and they are stable with respect to our 2015 measurements (with small fluctuations, at the level of a fraction of a standard deviation, which is consistent with changes in data processing). Polarization-only bispectra display a significant improvement in robustness; they can now be used independently to set primordial NG constraints with a sensitivity comparable to WMAP temperature-based results and they give excellent agreement. In addition to the analysis of the standard local, equilateral, and orthogonal bispectrum shapes, we consider a large number of additional cases, such as scale-dependent feature and resonance bispectra, isocurvature primordial NG, and parity-breaking models, where we also place tight constraints but do not detect any signal. The non-primordial lensing bispectrum is, however, detected with an improved significance compared to 2015, excluding the null hypothesis at 3.5σ. Beyond estimates of individual shape amplitudes, we also present model-independent reconstructions and analyses of the Planck CMB bispectrum. Our final constraint on the local primordial trispectrum shape is gNLlocal = (−5.8 ± 6.5) × 104 (68% CL, statistical), while constraints for other trispectrum shapes are also determined. Exploiting the tight limits on various bispectrum and trispectrum shapes, we constrain the parameter space of different early-Universe scenarios that generate primordial NG, including general single-field models of inflation, multi-field models (e.g. curvaton models), models of inflation with axion fields producing parity-violation bispectra in the tensor sector, and inflationary models involving vector-like fields with directionally-dependent bispectra. Our results provide a high-precision test for structure-formation scenarios, showing complete agreement with the basic picture of the ΛCDM cosmology regarding the statistics of the initial conditions, with cosmic structures arising from adiabatic, passive, Gaussian, and primordial seed perturbations.

This paper aims to analyze the electrocardiography (ECG) signals for patient with atrial fibrillation (AF) by using bispectrum and extreme learning machine (ELM). AF is the most common irregular heart beat disease which may cause many cardiac diseases as well. Bispectral analysis was used to extract the nonlinear information in the ECG signals. The bispectral features of each ECG episode were determined and fed to the ELM classifier. The classification accuracy of ELM to distinguish nonterminating, terminating AF, and terminating immediately AF was 96.25%. In this study, the normal ECG signal was also compared with AF ECG signal due to the nonlinearity which was determined by bispectrum. The classification result of ELM was 99.15% to distinguish AF ECGs from normal ECGs.

The motor current signature analysis (MCSA) provides a nondestructive method for gear fault detection. The motor current in the faulty gear system not only involves the frequency information related to the fault but also the electric supply frequency and gear meshing-related frequency, which not only contaminates the fault characteristics but also increases the difficulty of fault extraction. To extract the fault characteristic frequency effectively, an innovative method based on the wavelet bispectrum (WB) is proposed. Bispectrum is an effective tool for identifying the fault-related quadratic phase coupling (QPC). However, it requires a large amount of data averaging, which is not suitable for short data analysis. In this paper, the wavelet bispectrum is introduced to motor current analysis and the problem of QPC extraction under variable speed conditions is preliminarily solved. Furthermore, a fault diagnostic approach for locomotive gears using the wavelet bispectrum and wavelet bispectral entropy is suggested. The presented method was effectively applied to the locomotive online running operations, and faults of the drive gear were successfully diagnosed.