Third-order Cumulants Research Articles

Prediction of extended x-ray-absorption fine-structure spectra from molecular interaction models:Na+(H2O)n−MgO(100) interface

A theoretical approach is presented that can be used to model extended x-ray-absorption fine-structure (EXAFS) spectra for complex systems at finite temperatures. This method has the ability to compute, directly, the effects of thermal motion on EXAFS spectra and to deconvolve the EXAFS signal into contributions from individual scattering paths. Classical molecular-dynamics simulations of the interface are used to generate configurations corresponding to a given temperature and are used to compute the Debye-Waller factors for all (single and multiple) scattering paths. Interface geometries and first- ${(r}_{\mathrm{eff}}),$ second- $({\ensuremath{\sigma}}^{2}),$ and third-order cumulants are computed directly from the configurations and input into the multiple-scattering x-ray-absorption fine-structure code, FEFF6, which calculates the EXAFS spectra. This method is applied to predict EXAFS spectra of sodium ions adsorbed at the MgO (100) interface at three types of surface sites---flat terrace, step edge, and step corner (kink). The calculations indicate that the experiments should only be able to detect signals from sodium ions adsorbed onto defect sites---step edges and corners. In situ spectra are computed from models of ${\mathrm{Na}}^{+}$ at the aqueous-MgO interfaces using sodium ions imbedded in finite water clusters. The strong overlap of features attributable to the water oxygens and the nearest-neighbor surface oxygens may complicate the structural analysis from the experimental spectra; however, the scattering from the magnesium atoms and focusing multiple-scattering paths at longer distances are clear signatures of the interface.

2-D signal modelling and reconstruction using third-order cumulants

An algorithm has been designed for AR modelling any arbitrary 2-D data string. Slices of the third-order cumulant matrix of the data have been used for this purpose. A Toeplitz structure of the cumulant matrix has been generated for reducing computational load. Input signal is assumed to be non-Gaussian white noise. Knowledge of input statistics is desired but not necessary. The algorithm is seen to possess a robustness against additive Gaussian noise. A reconstructed bispectrum of the data is available as a by-product. The use of the algorithm as a data encoder/compressor has been demonstrated. Satisfactory results have been obtained after application of the algorithm on synthetic image data of different complexities.

A Navier-Stokes-like traffic model

In a previous paper (C. Wagner et al., Phys. Rev. E 54 (1996) 5073), we derived a macroscopic traffic model from a Boltzmann-like equation by neglecting third- and higher-order cumulants. To find higher-order approximations, we now employ a Chapman-Enskog expansion to calculate the third-order cumulants. These skewness terms are then expressed in terms of the lower-order cumulants and their derivatives, thus leading to a Navier-Stokes-like model similar to classical hydrodynamics.

Third-order cumulant RLS algorithm for nonminimum ARMA systems identification

As higher-order cumulants preserve both the magnitude and the phase information of received signals, higher-order statistics have been considered as powerful signal processing tools for identifying nonminimum phase systems. A third-order cumulant-based recursive least-squares (CRLS) algorithm for the identification of time-invariant as well as time-variant nonminimum phase ARMA systems has been successfully developed in this paper. A cost function based on the third-order cumulant and/or the third-order cross cumulant is defined for the development of the CRLS system identification algorithm. The CRLS algorithm are then applied to different MA and ARMA models. In the case of identifying the parameters of an MA model, a direct application of the CRLS algorithm is adequate. When dealing with an ARMA model, the poles and the zeroes are estimated separately. In estimating the zeroes of the ARMA model, the construction of a residual time-series sequence for the MA part is required. In the present investigation, the strong convergency and consistency of the CRLS algorithm are analyzed and simulations on the time-invariant and time-variant nonminimum phase systems show that the results of applying the proposed CRLS approach are quite satisfactory. In addition, because of the third-order cumulant properties, the CRLS algorithm can also suppress the Gaussian noise and is capable of providing an unbiased estimate under a noisy environment.

EXAFS Cumulants and Interatomic Potentials

Temperature dependent EXAFS spectra of simple linear, octahedral and tetrahedral molecules have systematically been analyzed for the first and higher nearest neighbor (NN) shells including multiple-scattering contributions. The interatomic distances and the second-order cumulants obtained are in excellent agreement with the values given by the crystallographic and vibrational data, and the third-order cumulants have been estimated successfully. The stretching and bending anharmonic force constants have consequently been determined. The contribution of the stretching and bending force constants to the cumulants of higher NN shells was found to depend noticeably on molecular geometry. The dynamical effect of the scattering amplitude was also discussed.

Development of an adaptive generalised trimmed mean estimator to compute third-order cumulants

An adaptive generalised trimmed mean (AGTM) estimator for both symmetric and asymmetric unimodal distributions is developed in this paper. For asymmetric probability density functions (pdf 's) the problem is to decide where to truncate the pdf on the left and the right side. To solve this problem, an extended generalised Gaussian distribution that covers a wide range of length of tails and asymmetry by varying two parameters is employed to approximate the distribution of the available data. From this approximated distribution, truncation points on both sides of the measured distributions are obtained. The AGTMs of the discrete time series are employed in the estimation of the third-order cumulants of many zero-mean symmetric as well as asymmetric data, including the output from many autoregressive (AR), moving average (MA) and autoregressive moving average (ARMA) systems with simulated input signals from various pdf's.

Coverage dependence of surface structures and vibrations of [formula omitted] studied by Cl K-edge SEXAFS

The coverage dependence of surface structures and vibrations of Cl Ni(100) have been investigated by means of Cl K-edge SEXAFS (surface extended X-ray absorption fine structure) spectroscopy. In the 0.25 ML phase, Cl was found to locate on the fourfold hollow site with little or no surface reconstructions, this being similar to the c(2 × 2) phase with 0.50 ML Cl. The ClNi bond distance was determined to be 2.31 ± 0.02 A ̊ for 0.25 ML, which is slightly shorter than that of 2.35 ± 0.02 A ̊ for 0.50 ML. The temperature dependence of the SEXAFS spectra was analyzed to yield the second- and third-order cumulants which provide harmonic and anharmonic force constants. The ClNi chemical bond was consequently found to be stiffer and less anharmonic in the 0.25 ML state than in the 0.50 ML state, this corresponding to a slightly shorter bond distance at a lower coverage. Such a trend is discussed in terms of charge transfer from substrate to adsorbate and the repulsive interaction between adsorbates.

(Almost) periodic FIR system identification usingthird-order cyclic-statistics

Two new normal equations relating the coefficients of an (almost) periodic FIR system and the time-varying third-order cumulants of the system output are derived, from which two new linear algebraic algorithms are presented for parameter estimation. Simulation results show that the new algorithms perform better than the recently published linear closed-form algorithms.

Higher order cumulants-based least squares for nonminimum-phase systems identification

A third-order cumulants based adaptive recursive least-squares (CRLS) algorithm for the identification of time-invariant nonminimum phase systems, as well as time-variant nonminimum phase systems, has been successfully developed. As higher order cumulants preserve both the magnitude and the phase information of received signals, they have been considered as powerful signal processing tools for nonminimum phase systems. In this paper, the development of the CRLS algorithm is described and examined. A cost function based on the third-order cumulant and the third-order cross cumulant is defined for the development of the CRLS system identification algorithm. The CRLS algorithm is then applied to different moving average (MA) and autoregressive moving average (ARMA) models. In the case of identifying the parameters of an MA model, a direct application of the CRLS algorithm is adequate. When dealing with an ARMA model, the poles and the zeros are estimated separately. In estimating the zeros of the ARMA model, the construction of a residual time-series sequence for the MA part is required. Simulation results indicate that the CRLS algorithm is capable of identifying nonminimum phase and time-varying systems. In addition, because of the third-order cumulant properties, the CRLS algorithm can suppress Gaussian noise and is capable of providing an unbiased estimate in a noisy environment.

A method for estimating intensity and impulse response functions of filtered Poisson processes

This paper presents a method for estimating the intensity and impulse response (IR) function of filtered Poisson processes. The case is considered where the filtered Poisson process is modeled as an output of the linear constant-coefficient ordinary differential equations having poles and zeros driven by Poisson impulse processes. It is shown that an explicit formula for estimating the intensity is derived by combining second-and third-order cumulants of the residual time series generated from the discretized filtered Poisson process. It is also shown that the IR function can be estimated from the parameters of the discretized filtered Poisson process. Then, Monte Carlo simulations demonstrate the validity of the proposed method in some specific examples. It is concluded that the proposed method can be extensively applied to actual phenomena appearing in engineering and science.

A high-speed systolic array for computing third-order cumulants

A high-speed systolic array for computing third-order cumulants

Anharmonicity of the Bending and Stretching Vibrations Observed in Extended X-Ray Absorption Fine Structure of Tetrahedral Molecules

We have measured and analyzed the temperature dependence of extended X-ray absorption fine structure (EXAFS) spectra of tetrahedral systems MBr 4 ( M=C, Si, Ge). The EXAFS analysis by means of the cumulant expansion technique enables one to obtain information about force constants including the third-order anharmonicity. The second-order cumulants obtained experimentally are in excellent agreement with the values expected by the vibrational data and the third-order cumulants have been determined successfully. For the first nearest neighbor (NN) Br–M shells the stretching motions are apparently dominant to describe EXAFS, while for the second NN Br–Br shell the bending modes are found to contribute significantly to the cumulants especially for the third-order anharmonicity. The obtained force constants are compared to each other and the origin of observed bending anharmonicity is discussed.

MA parameter estimation and cumulant enhancement

This paper addresses the problem of estimating the parameters of a moving average (MA) model from either only third- or fourth-order cumulants of the noisy observations of the system output. The system is driven by an independent and identically distributed non-Gaussian sequence that is not observed. The unknown model parameters are obtained using a batch least squares method. Recursive methods are also developed and used to claim the uniqueness of the batch least squares solutions. A novel technique for the enhancement of third-order cumulants of MA processes is introduced. This new technique is based on the concept of composite property mappings and helps reduce the variance of the estimates of third- (or fourth)-order cumulants of MA processes. Simulation results are presented that demonstrate the performance of the new methods and compare them with a range of existing techniques.

Unified approach to trimmed mean estimation and its application to bispectrum estimation of EEG signals

The trimmed mean estimator truncates both sides of the probability distribution by the same amount. For asymmetric distributions one needs to truncate the distribution on the left and right tails by different amounts which depend on the detailed properties—like the length of tails and asymmetry—of probability distributions. An adaptive generalised trimmed mean estimator for unimodal distributions is developed here to provide a unified approach to trimmed mean estimation. As realisations of different random processes vary very much in length of tails and asymmetry, an asymmetric generalised Gaussian distribution that covers a wide range of tails and asymmetry is employed to approximate the distributions of discrete data. Using this distribution, truncation points of the discrete distribution on both sides have been obtained. The adaptive generalised trimmed mean estimator is employed to estimate the autocorrelations and the third-order cumulants of simulated signals from various distributions as well as output signals from MA and ARMA systems. Finally the algorithm is applied to the bispectral analysis of event-related EEG-signals.

Lattice algorithms for recursive instrumental variable methods

We develop recursive lattice algorithms for the estimation of the AR parameters of an ARMA process using an instrumental variable (IV) such as z(n)=y2(n), which leads to AR estimates based on the third-order cumulants, or z(n)=y(n-q), which leads to AR estimates based on the higher-order Yule-Walker equations. In the non-recursive mode the resulting set of linear equations involves the cross-correlation matrix of the observed process y(n), and the instrumental process z(n) and hence this matrix is generally not Hermitian. Our time- and order-recursive algorithm leads to a pair of lattices, one excited by y(n) and the other by z(n), the lattices being coupled through order and time update equations. We show that IV-based AR model fitting is equivalent to linear prediction with non-conventional orthogonality conditions. Extensions to ARMA processes are discussed. The joint-process estimation problem is also treated. Some statistical analysis is carried out and convergence results are given when y(n) and z(n) can be modelled as stationary processes. The only assumption that we make about z(n), the associated process, is that the cross-correlation of y(n) and z(n) suffices for the estimation of the AR parameters, i.e. we assume that z(n) is an instrumental process. Particular choices of z(n) legal to estimates based on the higher-order cumulants, which are useful when the non-Gaussian signal is corrupted by additive coloured Gaussian noise.

Modeling (almost) periodic moving average processes using cyclic statistics

Estimating parameters of almost cyclostationary non-Gaussian moving average (MA) processes using noisy output-only data is considered. It is shown that second-order cyclic correlations of the output are generally insufficient in uniquely characterizing almost periodically time-varying MA(q) models, while third-order and higher order cumulants can be used to estimate their model parameters within a scale factor. Both linear and nonlinear identification algorithms for fixed and time-varying order q(t) are presented. Statistical model order determination procedures are also derived. Implementation issues are discussed and resistance to noise is claimed when the signal of interest has cycles distinct from the additive noise. Simulations are performed to verify the theoretical results.

Adaptive detection of noisy speech using third-order statistics

Detection of speech in noisy recordings is challenging, especially when the noise does not follow the usual whiteness, stationarity and high signal-to-noise ratio assumptions. A robust speech detector can affect significantly the performance of several speech-processing tasks, such as endpoint detection, segmentation and finally recognition, if we deal with real life data as opposed to laboratory or controlled environment recordings. The detector proposed in this paper is based on a Gaussianity test that employs third-order cumulants of the data to decide on the binary hypotheses of noise only versus speech plus noise. Speech intervals are detected by exploiting the third-order information present in the speech signal. The detector can tolerate a large family of additive noises thanks to its third-order statistics basis. The sample-adaptive and decision feedback variations proposed here provide the detector with tracking ability with respect to both the time variations of speech and the possible non-stationarity of noise. Experiments carried out using real data recorded in a moving car interior show satisfactory performance of the proposed algorithms down to -6 dB signal-to-noise ratio.

Blind separation of impulsive noise form corruptedaudio signal

The authors present a new method for separating the impulsive noise from a corrupted audio signal. It is shown that if the audio signal is assumed to be Gaussian distributed, the third-order cumulants of the impulsive noise can be separated, based upon the corrupted audio signal. The impulsive noise can then be reconstructed via the relationship between the bispectrum and the Fourier spectrum. Finally, the audio signal is restored by simply subtracting the reconstructed impulsive noise from the signal.

On the identification of non-minimum phase non-gaussian MA and ARM A models using a third-order cumulant

As higher-order cumulants preserve both the magnitude and the phase information of received signals, a higher-order cumulant has been considered as a powerful signal processing tool for a non-minimum phase system. This paper describes the development of a third-order cumulant-based adaptive recursive least-square algorithm for the identification of a time-invariant non-minimum phase system and a time-variant non-minimum phase with non-gaussian input. The third-order cumulant-based algorithm has its basis in a cost function defined in the third-order cumulant and the third-order cross cumulant. The algorithm is applied to non-minimum phase ARM A and MA models system identification. The experimental results indicate that the third-order cumulant-based adaptive algorithm is capable of identifying the non-minimum phase and time-varying system. In addition, because of the third-order cumulant properties, the algorithm can suppress the gaussian noise and is capable of providing an unbiased estimate under a nois...

Cumulant-based autocorrelation estimates of non-Gaussian linear processes

Autocorrelation of linear random processes can be expressed in terms of their cumulants. Theoretical insensitivity of the latter to additive Gaussian noise of unknown covariance, is exploited in this paper to develop (within a scale) autocorrelation estimators of linear non-Gaussian time series using cumulants of order higher than two. Windowed projections of third-order cumulants are shown to yield strongly consistent estimators of the autocorrelation sequence. Both batch and recursive algorithms are derived. Asymptotic variance expressions of the proposed estimators are also presented. Simulations are provided to illustrate the performance of the proposed algorithms and compare them with conventional approaches.