Third-order Moments Research Articles

On the generation of sequences simulating higher order white noise for system identification

The generation of binary sequences that exhibit the characteristics of higher order white noise signals is considered in this paper. Maximal length sequences fail to approximate higher order white noise signals due to the existence of peaks in their higher order statistics. Three classes of binary sequences are examined in detail: dual-BCH sequences, Gold sequences, and sequences generated by the term-by-term modulo 2 addition of two maximal length (and/or Gold) sequences whose least periods are relatively prime. The second and third-order moments of each construction are mainly investigated and useful results are obtained. It is shown that in all of the cases the spectra of the moments of order two (autocorrelation) and three are determined by the crosscorrelation function of the component sequences used in each construction. The number of peaks appearing in third-order moments is significantly reduced or vanished in all classes. These results can be readily generalized to higher order (greater than three) moments. Finally, the quality and efficiency of sequences from each class in simulating higher order white noise signals is demonstrated by providing simulation results in the context of bilinear input–output system identification.

Turbulent Energy Scale-Budget Equations for nearly Homogeneous Sheared Turbulence

For moderate Reynolds numbers, the isotropic relation between second-order and third-order moments for velocity increments (Kolmogorov's equation) is not respected, reflecting a non-negligible correlation between the scales responsible for the injection, transfer and dissipation of the turbulent energy. For (shearless) grid turbulence, there is only one dominant large-scale phenomenon, which is the non-stationarity of statistical moments resulting from the decay of energy downstream of the grid. In this case, the extension of Kolmogorov's analysis, as carried out by Danaila, Anselmet, Zhou and Antonia, J. Fluid Mech. 391, 1999 359-369) is quite straightforward. For shear flows, several large-scale phenomena generally coexist with similar amplitudes. This is particularly the case for wall-bounded flows, where turbulent diffusion and shear effects can present comparable amplitudes. The objective of this work is to quantify, in a fully developed turbulent channel flow and far from the wall, the influence of these two effects on the scale-by-scale energy budget equation. A generalized Kolmogorov equation is derived. Relatively good agreement between the new equation and hot-wire measurements is obtained in the outer region (40 < x + 3 < 150) of the channel flow, for which the turbulent Reynolds number is R λ≈ 36.

A method of simulating multivariate nonnormal distributions by the Pearson distribution system and estimation

A method of simulating multivariate nonnormal distributions by using the Pearson distribution system which can represent wide class of distributions with various skewness and kurtosis is proposed. Firstly, a procedure of generating random numbers from the Pearson distribution system including type IV distribution which was difficult to implement is derived. Secondly, the Pearson distribution system is applied to generate random numbers from the multivariate nonnormal distributions with specified marginal skewness and kurtosis. Thirdly, the third-order moment matrix of them is obtained. This matrix makes it possible to construct the approximated multivariate nonnormal distributions with the various third- and fourth-order moment matrix. Fourthly, the fitting procedure for empirical data by both method of moments and maximum likelihood method is proposed. Fifthly, the fact that using Mersenne Twister is more precise and more practical than using typical congruential generators is found. In conclusion these methods can be widely applied to multivariate analysis and statistical modeling. Moreover, the Pearson distribution system will play a further important role in statistics area.

Effect of collisions on the dispersed phase fluctuation in a dilute tube flow: Experimental and theoretical analysis

Measurements of particle fluctuation in a fully developed pipe flow at moderate Reynolds number is performed in this study. The present data are obtained by using a two-component phase Doppler anemometer. The radial profiles are measured at a distance of 0.2 diameter downstream the exit of the tube. At this location, the core flow still carries all the properties of the tube turbulence. A low mass loading of partly responsive particles is considered. The Stokes number of these partly responsive particles is of order 3 when the integral turbulent time scale on the axis of the tube flow is used. The velocity statistics are analyzed up to the third-order moments and we show that the radial turbulent transport of fluctuating kinetic energy is much higher for the particles than for the fluid. Radial balances of longitudinal and radial kinetic stresses of the particles are examined. Particle–particle collisions have a negligible direct effect on the evolution of the longitudinal fluctuating velocity. However, even at this low mass loading, we prove that particle–particle collisions and redistribution from the very large streamwise velocity variance to the radial velocity variance in the near wall region strongly influence the radial fluctuation of the particles. In the core region, a quadrant analysis enables the detection of low streamwise velocities focusing toward the axis and the corresponding quadrants are strongly dominant for the glass beads. We expect that the partly responsive particles, because of their inertia, keep some memory of the lower streamwise velocity existing in the near wall region while they fly across the tube. The collisions in the near wall region are, therefore, expected to have a strong indirect influence on the whole kinetic-energy balance in the tube by partly driving the radial transport of the fluctuating kinetic energy of the particles. This effect should be particularly strong in this circular geometry because events from any azimuthal directions converge in the central region.

Moments of superellipsoids and their application to range image registration

Cartesian moments are frequently used global geometrical features in computer vision for object pose estimation and recognition. We derive a closed form expression for 3-D Cartesian moment of order p+q+r of a superellipsoid in its canonical coordinate system. We also show how 3-D Cartesian moment of a globally deformed superellipsoid in general position and orientation can be computed as a linear combination of 3-D Cartesian moments of the corresponding nondeformed superellipsoid in canonical coordinate system. Additionally, moments of objects that are compositions of superellipsoids can be computed as simple sums of moments of individual parts. To demonstrate practical application of the derived results we register pairs of range images based on moments of recovered compositions of superellipsoids. We use a standard technique to find centers of gravity and principal axes in pairs of range images while third-order moments are used to resolve the four-way ambiguity. Experimental results show expected improvement of recovered rigid transformation based on moments of recovered superellipsoids as compared to the registration based on moments of raw range image data. Besides object pose estimation the presented results can be directly used for object recognition with moments and/or moment invariants as object features.

Third-order moment spectrum and weighted fuzzy classifier for robust 2-D object recognition

In this paper, a robust position, scale, and rotation invariant system for the recognition of closed 2-D noise corrupted images using the bispectral features of a contour sequence and the weighted fuzzy classifier are derived. The higher-order spectrum based on third-order moment, called a bispectrum, is applied to the contour sequences of an image to extract a 15-dimensional feature vector for each of the 2-D images. This bispectral feature vector, which is invariant to shape translation, scale, and rotation transformation, can be used to represent a 2-D planar image and is fed into a weighted fuzzy classifier for the recognition process. The experiments with eight different shapes of aircraft images are presented to illustrate the high performance of the proposed system even when the image is significantly corrupted by noise.

Blind deconvolution of spotted image based on slice selection of third-order moment

The 1D blind deconvolution algorithm using maximum time delay slice of the third-order moment ((MTDS-TOM)[Lu, W.]) is extended to 2D blind deconvolution for spotted image deblurring. A scaled and shifted version of the image is obtained using a special slice selected from its third-order moment, which is estimated using a 4D blind deconvolution. An application of the proposed method for removing the optical blur of a microarray image is given.

The kurtosis of the cosmic shear field

We study the fourth-order moment of the cosmic shear field using the dark matter halo approach to describe the nonlinear gravitational evolution of structure in the universe. Since the third-order moment of the shear field vanishes because of symmetry, non-Gaussian signatures in its one-point statistics emerge at the fourth-order level. We argue that the shear kurtosis parameter S_4 = <gamma^4>/<gamma^2>^3 may be more directly applicable to realistic data than the well-studied higher-order statistics of the convergence field, since obtaining the convergence requires a non-local reconstruction from the measured shear field. We compare our halo model predictions for the variance, skewness and kurtosis of lensing fields with ray-tracing simulations of cold dark matter models and find good agreement. The shear kurtosis calculation is made tractable by developing approximations for fast and accurate evaluations of the 8-dimensional integrals needed to obtain the kurtosis. We show that on small scales it is dominated by correlations within halos more massive than 10^14 solar masses. The shear kurtosis is sensitive to the mass density parameter of the universe, Omega, and has relatively weak dependences on other parameters. The approximations we develop for the third- and fourth-order moments allow for accurate halo model predictions for the 3-dimensional mass distribution as well. We demonstrate their accuracy in the small scale regime, below 2 Mpc, where analytical approaches used in the literature so far cease to be accurate.

Multiresolution-based watersheds for efficient image segmentation

This paper presents an efficient method for image segmentation based on a multiresolution application of a wavelet transform and watershed segmentation algorithm. The procedure toward complete segmentation consists of four steps: pyramid representation, image segmentation, region merging and region projection. First, pyramid representation creates multiresolution images using a wavelet transform. Second, image segmentation segments the lowest-resolution image of the pyramid using a watershed segmentation algorithm. Third, region merging merges the segmented regions using the third-order moment values of the wavelet coefficients. Finally, the segmented low-resolution image with label is projected into a full-resolution image (original image) by inverse wavelet transform. Experimental results of the presented method can be applied to the segmentation of noise or degraded images as well as reduce over-segmentation. In addition, we applied our method to human face detection with accurate and closed boundaries.

A glimpse of hydrodynamics beyond the Navier–Stokes equations

A solution of Maxwell’s moment equations is obtained for the steady, one-dimensional flow through a normal shock wave in a Maxwellian gas. The solution method makes use of a new representation for the conservation equations, a closure relation for a single third-order moment, and a spatially integrated form of Maxwell’s equations of transfer. The integrated equations of transfer act as constraints in evaluating two sets of coefficients introduced in the conservation equations and the single closure relation. A total of only four coefficients are found to be fully sufficient when comparisons are made with results from the direct simulation Monte Carlo method, which show excellent agreement across all hydrodynamic variables including two fourth-order moments which the theory predicts.

Effect of vorticity on second- and third-order statistics of passive scalar gradients.

The influence of vorticity on second- and third-order moments of the spatial derivatives of a forced, passive scalar field has been studied in the framework of a simplified problem; the analysis is restricted to dominating rotation and molecular diffusion is represented by a linear model. The results reveal that, in the case of a passive scalar experiencing forcing in an isotropic medium, both vorticity and diffusion counteract anisotropy imposed on the scalar field. Anisotropy at the level of second-order moments appears to be destroyed essentially by the action of vorticity.

Subspace design of low-rank estimators for higher-order statistics

Higher-order statistics (HOS) are well known for their robustness to additive Gaussian noise and ability to preserve phase. HOS estimates, on the other hand, have been criticized for high complexity and the need for long data in order to maintain small variance. Since rank reduction offers a general principle for reduction of estimator variance and complexity, we consider the problem of designing low-rank estimators for HOS. We propose three methods for choosing the transformation matrix that reduces the mean-square error (MSE) associated with the low-rank HOS estimates. We also demonstrate the advantages of using low-rank third-order moment estimates for blind system estimation. Results indicate that the full rank MSE corresponding to some data length N can be attained by a low-rank estimator corresponding to a length significantly smaller than N.

Using the moments of a thick layer to map conductance and conductivity from airborne electromagnetic data

The moments of the airborne electromagnetic impulse response have been derived for a thick horizontal layer. These moments can be written in an analytic form for the first- and second-order moments, both for the horizontal component and the vertical component. In the limiting case of a thin sheet, the horizontal component has first-, second- and third-order moments, while the vertical component has first- and second-order moments. On the other hand, the limiting case of a half space only has first-order moments for both components. The analytic expressions for the moments can be inverted analytically, so that the half space conductivity, thin-sheet conductance, and conductivity and thickness can be estimated. The images derived from the higher-order moments better identify a deeper conductor. This is consistent with the higher-order moments being more responsive to deeper material.

Blind deconvolution using maximum time delay slice of third order moment

A new algorithm is presented for blind deconvolution of a non-minimum phase system. Instead of solving a one-dimensional blind deconvolution problem, the proposed method solves a two-dimensional blind deconvolution problem. It recovers a scaled and time-shifted version of the input sequence by the maximum time delay slice of the third order moment (TOM) of the input sequence. The only channel information required in this method is its TOM, which can be estimated from the output signal. The main advantages of the proposed method are that it does not require the procedures of optimising and signal preprocessing (such as spectral prewhitening).

The moments of a sphere in a uniform field, a versatile model

The moments of a sphere in a uniform field can be used to approximately model the moments of a sphere in a dipolar field. The numerical computations are trivial and the approximation is especially good for higher-order moments. The greatest discrepancy is seen on the zeroth-order moment at large radii, however, the zeroth-order moment is not commonly used for interpreting airborne electromagnetic data. The sphere in a uniform field can also be used to approximate the response of a body that has its currents constrained to flow in a specific dip plane. This means that plate-like bodies or anisotropic spheres can also be modelled. The third-order moment of MEGATEMO data acquired over the Reid-Mahaffy test site shows an anomalous response than can be modelled by a sphere at 170 m depth with a conductivity of 15 S/m and a radius of 40 m. The currents flowing in the sphere are constrained to flow in a vertical plane. This model is consistent with the geology of the area and a hole drilled to test the anomalous zone.

A Langevin equation for turbulent velocity increments

Recently, Friedrich and Peinke demonstrated empirically that a Fokker–Planck equation describes the scale dependence of probability distribution functions of longitudinal velocity increments vr in fully developed turbulent flows. Thanks to the analysis of an experimental velocity signal, the stochastic process vr is further investigated by examining the related Langevin equation. This process is found to be Markovian in scale because the turbulent velocity field is correlated over distances much larger than the correlation length ρ of its spatial derivative. A Gaussian approximation for the random force yields evolution equations for the structure functions 〈vrn〉. Analytic solutions are obtained, in agreement with experimental data for even-order moments when the scale r is larger than a few times ρ. The third-order moment 〈vr3〉 is found linear in r, as predicted by Kolmogorov’s four-fifths law.

Analysis of the Interacting Octad System of 12CH4

The octad system of methane 12CH4 has been analyzed with a fourth-order (459 parameters) Hamiltonian for energy levels and a third-order (28 parameters) dipole moment expansion for infrared intensities. Nearly 8000 assigned positions and 2500 measured line intensities (obtained with the Fourier transform spectrometer located at Kitt Peak National Observatory/National Solar Observatory) have been considered in the modeling. Infrared (Octad–Ground State) and hot bands (Octad–Dyad) data have been used. While the model does not reproduce the measurements at their inherent experimental precisions, it has been sufficient to interpret and substantially assign this spectral region of methane for the first time in over 30 years of investigation.

The Impulse-Response Moments of a Conductive Sphere in a Uniform Field, a Versatile and Efficient Electromagnetic Model

The impulse response moments of a conductive sphere in free space excited by a uniform magnetic field can be used to approximate the moments of a sphere in a dipolar field. The numerical computations are straightforward and the approximation is especially good for higher-order moments. The greatest discrepancy is seen on the zeroth-order moment when the radius of the sphere is large. It is possible to improve the accuracy for the zeroth-order moment by modelling the large-radius sphere (in a dipole field) as the combined response of multiple small-radius spheres (each in a locally uniform field). The small spheres are closely packed inside the larger sphere. The discrepancy can be reduced to less than 15 % in this manner.The sphere in a uniform field can also be used to approximate the response of a body that has its currents constrained to flow in a plane with a specific orientation. This means that plate-like bodies or anisotropic spheres can also be modelled.The third-order moment has been calculated from data acquired during a MEGATEM airborne electromagnetic survey of the Reid-Mahaffy test site. There is an anomalous response in the third-order moment that can be modelled by a sphere at 170 m depth with a conductivity of 15 S/m and a radius of 40 m. The currents flowing in the sphere are constrained to flow in a vertical plane. This model is consistent with the geology of the area and a hole drilled to test the anomalous zone.

Ocean Turbulence. Part I: One-Point Closure Model—Momentum and Heat Vertical Diffusivities

Since the early forties, one-point turbulence closure models have been the canonical tools used to describe turbulent flows in many fields. In geophysics, Mellor and Yamada applied such models using the 1980 state-of-the art. Since then, no improvements were introduced to alleviate two major difficulties: 1) closure of the pressure correlations, which affects the correct determination of the critical Richardson number Ri(sub cr) above which turbulent mixing is no longer possible and 2) the need to express the non-local third-order moments (TOM) in terms of lower order moments rather than via the down-gradient approximation as done thus far, since the latter seriously underestimates the TOMs. Since 1) and 2) are still being dealt with adjustable parameters which weaken the credibility of the models, alternative models, not based on turbulence modeling, have been suggested. The aim of this paper is to show that new information, partly derived from the newest 2-point closure model discussed, can be used to solve these shortcomings. The new one-point closure model, which in its simplest form is algebraic and thus simple to implement, is first shown to reproduce a variety of data. Then, it is used in a Ocean-General Circulation Model (O-GCM) where it reproduces well a large variety of ocean data. While phenomenological models are specifically tuned to ocean turbulence, the present model is not. It is first tested against laboratory data on stably stratified flows and then used in an O-GCM. It is more general, more predictive and more resilient, e.g., it can incorporate phenomena like wave-breaking at the surface, salinity diffusivity, non-locality, etc. One important feature that naturally comes out of the new model is that the predicted Richardson critical value Ri(sub cr) is Ri (sub cr approx. = 1) in agreement with both Large Eddy Simulations (LES) and empirical evidence while all previous models predicted Ri (sub cr approx. = 0.2) which led to a considerable underestimate of the extent of turbulent mixing and thus to an incorrect mixed layer depth. The predicted temperature and salinity profiles (vs. depth) are presented and compared with those of the Kolmogorov-Petruvsky-Piskunuv (KPP) model and Levitus data.

Influence of the surface roughness on the third-order moments of velocity fluctuations

Roughness wall effects in a zero pressure gradient turbulent boundary layers were investigated using hot-wire anemometry. The skewness and diffusion factors of u and v, the longitudinal and normal velocity fluctuations, were measured and represented using wall variables. The results indicate that the wall roughness removes the crossover point between sweep and ejection events to the outer region of the layer for a single Reynolds number Re θ > 3,000. This behaviour exhibits that the roughness surface favours the maintaining of sweep events obtained by a quadrant analysis. These results show that communication between the wall region and outer region of a turbulent boundary layer exists and the wall similarity hypothesis for a rough wall is questionable. The effect of the wall roughness on the position of the point crossover from sweep to ejection motions with respect to the wall seems to be the same as that obtained when the Reynolds number is higher.