Kurtosis Estimator Research Articles

Developing Freak Wave Prediction Model with Directional Effects and its Validation

Quasi-resonant four-wave interactions may influence the statistical properties of deep-water surface gravity waves such as a freak wave. The freak wave prediction method was developed by Mori and Janssen (2006) based on the quasi-resonant wave theory assuming unidirectional wave condition. In this study, the directional effect is considered for kurtosis estimation from directional wave spectra assuming a semi-empirical relation. The validation of the kurtosisestimation including directional information is performed by operational wave forecasts and observed data.

Accuracy of $q$-Space Related Parameters in MRI: Simulations and Phantom Measurements

The accuracy of q-space measurements was evaluated at a 3.0-T clinical magnetic resonance imaging (MRI) scanner, as compared with a 4.7-T nuclear magnetic resonance (NMR) spectrometer. Measurements were performed using a stimulated-echo pulse-sequence on n-decane as well as on polyethylene glycol (PEG) mixed with different concentrations of water, in order to obtain bi-exponential signal decay curves. The diffusion coefficients as well as the modelled diffusional kurtosis K(fit) were obtained from the signal decay curve, while the full-width at half-maximum (FWHM) and the diffusional kurtosis K were obtained from the displacement distribution. Simulations of restricted diffusion, under conditions similar to those obtainable with a clinical MRI scanner, were carried out assuming various degrees of violation of the short gradient pulse (SGP) condition and of the long diffusion time limit. The results indicated that an MRI system can not be used for quantification of structural sizes less than about 10 microm by means of FWHM since the parameter underestimates the confinements due to violation of the SGP condition. However, FWHM can still be used as an important contrast parameter. The obtained kurtosis values were lower than expected from theory and the results showed that care must be taken when interpreting a kurtosis estimate deviating from zero.

Improving the performance of kurtosis estimator

In this communication, sample measures of kurtosis adapted by various software packages are compared for data from normal and non-normal populations. Further, two improved estimators of population kurtosis are proposed and their performance is compared with the currently used measures. The suggested estimators have considerably lower mean squared error ( MSE) for various sampling designs in our simulation study. Two empirical examples are given to illustrate the usefulness of suggested estimators in practice.

Avoidance of speckle noise in laser vibrometry by the use of kurtosis ratio: Application to mechanical fault diagnostics

This paper presents a statistical technique to enhance vibration signals measured by laser Doppler vibrometry (LDV). The method has been optimised for LDV signals measured on bearings of universal electric motors and applied to quality control of washing machines. Inherent problems of LDV are addressed, particularly the speckle noise occurring when rough surfaces are measured. The presence of speckle noise is detected using a new scalar indicator kurtosis ratio (KR), specifically designed to quantify the amount of random impulses generated by this noise. The KR is a ratio of the standard kurtosis and a robust estimate of kurtosis, thus indicating the outliers in the data. Since it is inefficient to reject the signals affected by the speckle noise, an algorithm for selecting an undistorted portion of a signal is proposed. The algorithm operates in the time domain and is thus fast and simple. The algorithm includes band-pass filtering and segmentation of the signal, as well as thresholding of the KR computed for each filtered signal segment. Algorithm parameters are discussed in detail and instructions for optimisation are provided. Experimental results demonstrate that speckle noise is effectively avoided in severely distorted signals, thus improving the signal-to-noise ratio (SNR) significantly. Typical faults are finally detected using squared envelope analysis. It is also shown that the KR of the band-pass filtered signal is related to the spectral kurtosis (SK).

Radio Frequency Interference Excision Using Spectral‐Domain Statistics

ABSTRACTA radio frequency interference (RFI) excision algorithm based on spectral kurtosis, a spectral variant of time‐domain kurtosis, is proposed and implemented in software. The algorithm works by providing a robust estimator for Gaussian noise that, when violated, indicates the presence of non‐Gaussian RFI. A theoretical formalism is used that unifies the well‐known time‐domain kurtosis estimator with past work related to spectral kurtosis, and leads naturally to a single expression encompassing both. The algorithm accumulates the first two powers of M power spectral density (PSD) estimates, obtained via Fourier transform, to form a spectral kurtosis (SK) estimator whose expected statistical variance is used to define an RFI detection threshold. The performance of the algorithm is theoretically evaluated for different time‐domain RFI characteristics and signal‐to‐noise ratios η. The theoretical performance of the algorithm for intermittent RFI (RFI present in R out of M PSD estimates) is evaluated and shown to depend greatly on the duty cycle, d = R/M. The algorithm is most effective for d = 1/(4 + η), but cannot distinguish RFI from Gaussian noise at any η when d = 0.5. The expected efficiency and robustness of the algorithm are tested using data from the newly designed FASR Subsystem Testbed radio interferometer operating at the Owens Valley Solar Array. The ability of the algorithm to discriminate RFI against the temporally and spectrally complex radio emission produced during solar radio bursts is demonstrated.

Denoising using local projective subspace methods

In this paper we present denoising algorithms for enhancing noisy signals based on Local ICA (LICA), Delayed AMUSE (dAMUSE) and Kernel PCA (KPCA). The algorithm LICA relies on applying ICA locally to clusters of signals embedded in a high-dimensional feature space of delayed coordinates. The components resembling the signals can be detected by various criteria like estimators of kurtosis or the variance of autocorrelations depending on the statistical nature of the signal. The algorithm proposed can be applied favorably to the problem of denoising multi-dimensional data. Another projective subspace denoising method using delayed coordinates has been proposed recently with the algorithm dAMUSE. It combines the solution of blind source separation problems with denoising efforts in an elegant way and proofs to be very efficient and fast. Finally, KPCA represents a non-linear projective subspace method that is well suited for denoising also. Besides illustrative applications to toy examples and images, we provide an application of all algorithms considered to the analysis of protein NMR spectra.

Erratum: Second-order gravitational effects of local inhomogeneities on CMB anisotropies and non-Gaussian signatures [Phys. Rev. D72, 103506 (2005)

Based on the second-order nonlinear theory of perturbations in non-zero \Lambda flat cosmological models, we study the gravitational effects of local inhomogeneities on cosmic microwave background (CMB) anisotropies. As the local inhomogeneities we consider firstly large-scale dipole and quadruple distributions of galaxies around us and next an isolated cluster-scale matter distribution. It is found that, due to the second-order integral Sachs-Wolfe effect, the north-south asymmetry of CMB anisotropies and non-Gaussian signatures (in terms of scale-dependent estimators of kurtosis) in a spot-like object are caused from these matter distributions along light paths. Our theoretical results seem to be consistent with recent various observational results which have been shown by Hansen et al., Eriksen et al., Vielva et al. and Cruz et al.

Second-order gravitational effects of local inhomogeneities on CMB anisotropies and non-Gaussian signatures

Based on the second-order nonlinear theory of perturbations in nonzero $\ensuremath{\Lambda}$ flat cosmological models, we study the gravitational effects of local inhomogeneities on cosmic microwave background (CMB) anisotropies. As the local inhomogeneities we consider first large-scale dipole and quadrupole distributions of galaxies around us and next an isolated cluster-scale matter distribution. It is found that, due to the second-order integral Sachs-Wolfe effect, the north-south asymmetry of CMB anisotropies and non-Gaussian signatures (in terms of scale-dependent estimators of kurtosis) in a spotlike object are caused from these matter distributions along light paths. Our theoretical results seem to be consistent with recent various observational results which have been shown by Hansen et al., Eriksen et al., Vielva et al. and Cruz et al.

Crack detection in beams using kurtosis

A new technique for crack detection in beam structures based on kurtosis is presented. The fundamental vibration mode of the cracked cantilever beam is analyzed and both the location and size of the crack are estimated. The location of the crack is determined by the abrupt changes in the spatial variation of the analyzed response, while the size of the crack is related to the estimate of kurtosis. The proposed prediction method was validated by experiments on cracked Plexiglas beams. The influence of added noise on the estimation accuracy of the method has been investigated. Compared to existing methods for crack detection, the proposed kurtosis-based prediction scheme is attractive due to low computational complexity and inherent robustness against noise.

Comparing Variances and Other Measures of Dispersion

Testing hypotheses about variance parameters arises in contexts where uniformity is important and also in relation to checking assumptions as a preliminary to analysis of variance (ANOVA), dose-response modeling, discriminant analysis and so forth. In contrast to procedures for tests on means, tests for variances derived assuming normality of the parent populations are highly nonrobust to nonnormality. Procedures that aim to achieve robustness follow three types of strategies: (1) adjusting a normal-theory test procedure using an estimate of kurtosis, (2) carrying out an ANOVA on a spread variable computed for each observation and (3) using resampling of residuals to determine p values for a given statistic. We review these three approaches, comparing properties of procedures both in terms of the theoretical basis and by presenting examples. Equality of variances is first considered in the two-sample problem followed by the k-sample problem (one-way design).

On more robust estimation of skewness and kurtosis

For both the academic and the financial communities it is a familiar stylized fact that stock market returns have negative skewness and severe excess kurtosis. This stylized fact has been supported by a vast collection of empirical studies. Given that the conventional measures of skewness and kurtosis are computed as an average and that averages are not robust, we ask: “How useful are the measures of skewness and kurtosis used in previous empirical studies?” To answer this question, we provide a survey of robust measures of skewness and kurtosis from the statistics literature and carry out extensive Monte Carlo simulations that compare the conventional measures with the robust measures of our survey. An application of the robust measures to daily S&P500 index data indicates that the stylized facts might have been accepted too readily. We suggest that looking beyond the standard skewness and kurtosis measures can provide deeper and more accurate insight into market returns behavior.

Un metodo grafico per la trasformazione di Box-Cox: aspetti esplorativi ed inferenziali

The main iusse addressed in this paper is the location of a range of values for the parameter l of the Box-Cox trnsformation, by means of Normality and it exploits some results about asymmetry and kurtosis estimators. Thus, a graphical device is developed that yields an interval of values for lambda , so that the transformed observations are consistent with the hypothesis of Normality. Both the results from Monte Carlo studies and from applications to real data sets are illustrated to show the performance of the proposed techique with respect to the maximum likelihood method.

A novel kurtosis driven variable step-size adaptive algorithm

A new variable step-size LMS filter is introduced. The time-varying step-size sequence is adjusted, utilizing the kurtosis of the estimation error, therefore reducing performance degradation due to the existence of strong noise. The convergence properties of the algorithm are analyzed, and an adaptive kurtosis estimator that takes into account noise statistics and optimally adapts itself is also presented. Simulation results confirm the algorithm's improved performance and flexibility.

Approximate Goodness‐of‐Fit Tests of fitted generalized extreme value distributions using LH moments

Approximate goodness‐of‐fit tests of fitted generalized extreme value (GEV) distributions using LH moments are formulated on the basis of comparison of sample LH kurtosis estimates and theoretical LH kurtosis values of the fitted distributions. These tests are different from those that have been derived for testing the GEV distributions of which parameter values are known a priori. The tests are intended to answer the following questions: Does a fitted GEV distribution describe adequately a given data series? If not, can the GEV distribution function describe adequately the larger events in that data series for use for high quantile estimation? If so, what degree of emphasis on the larger events is needed in order that the GEV distribution becomes acceptable? The use of the GEV distribution in conjunction with the LH moment estimation method and the formulated tests should alleviate the need for finding the “correct” distribution. The tests are evaluated by Monte Carlo simulations using generated samples of both GEV and Wakeby distributions. Applications of the tests to observed annual maximum streamflow data are presented.

The Estimation of Skewness and Kurtosis of Random Effects in the Linear Model

Generalising the ANOVA method of estimating variance components in mixed linear models a simple procedure is presented to estimate skewness and kurtosis of the distributions of the random effects of the model. For the model II of a one-way classification this procedure is demonstrated explicitly.

A methodology for acoustic seafloor classification

A seafloor classification methodology, based on a parameterization of the reverberation probability density function in conjunction with neural network classifiers, is evaluated through computer simulations. Different seafloor provides are represented by a number of scatterer distributions exhibiting various degrees of departure from the nominal Poisson distribution. Using a computer simulation program, these distributions were insonified at different spatial scales by varying the transmitted pulse length. The statistical signature obtained consists of reverberation kurtosis estimates as a function of pulse length. Two neural network classifiers are presented with the task of discriminating among the various scatterer distributions based on obtained acoustic signatures. The results indicate that this approach offers considerable promise for practical, realizable solutions to the problem of remote seafloor classification. >

Covariance structure analysis with heterogeneous kurtosis parameters

SUMMARY This paper discusses the analysis of covariance structures in a wide class of multivariate distributions whose marginal distributions may have heterogeneous kurtosis parameters. Elliptical distributions often used as a generalization of the normal theory are members of this class. It is shown that a simple adjustment of the weight matrix of normal theory, using kurtosis estimates, results in an asymptotically efficient estimator of structural parameters within the class of estimators that minimize a general discrepancy function. Results are obtained for arbitrary covariance structures as well as those that meet a scale invariance assumption. Two real data sets are analyzed for illustrative purposes.

PRINCIPLES AND APPLICATION OF MAXIMUM KURTOSIS PHASE ESTIMATION1

ABSTRACTMethods of minimum entropy deconvolution (MED) try to take advantage of the non‐Gaussian distribution of primary reflectivities in the design of deconvolution operators. Of these, Wiggins’(1978) original method performs as well as any in practice. However, we present examples to show that it does not provide a reliable means of deconvolving seismic data: its operators are not stable and, instead of whitening the data, they often band‐pass filter it severely. The method could be more appropriately called maximum kurtosis deconvolution since the varimax norm it employs is really an estimate of kurtosis. Its poor performance is explained in terms of the relation between the kurtosis of a noisy band‐limited seismic trace and the kurtosis of the underlying reflectivity sequence, and between the estimation errors in a maximum kurtosis operator and the data and design parameters.The scheme put forward by Fourmann in 1984, whereby the data are corrected by the phase rotation that maximizes their kurtosis, is a more practical method. This preserves the main attraction of MED, its potential for phase control, and leaves trace whitening and noise control to proven conventional methods. The correction can be determined without actually applying a whole series of phase shifts to the data. The application of the method is illustrated by means of practical and synthetic examples, and summarized by rules derived from theory. In particular, the signal‐dominated bandwidth must exceed a threshold for the method to work at all and estimation of the phase correction requires a considerable amount of data.Kurtosis can estimate phase better than other norms that are misleadingly declared to be more efficient by theory based on full‐band, noise‐free data.

Nonparametric estimation of variance, skewness and kurtosis of the distribution of a statistic by jackknife and bootstrap techniques

While jackknife and bootstrap estimates of the variance of a statistic are well–known, the author extends these nonparametric maximum likelihood techniques to the estimation of skewness and kurtosis. In addition to the usual negative jackknife also a positive jackknife as proposed by BERAN (1984) receives interest in this work. The performance of the methods is investigated by a Monte Carlo study for Kendall's tau in various situations likely to occur in practice. Possible applications of these developments are discussed.

Asymptotic detection performance of discrete power and higher-order spectra estimates

Several detection statistics are compared in the frequency domain based on the asymptotic probability of detection (APD) criterion. They include second-order, fourth-order, normalized fourth-order, and kurtosis estimates. The results show that for randomly occurring signals which can be characterized as non-Gaussian, the fourth-order, normalized fourth-order, and kurtosis estimates can have higher asymptotic probability of detection levels compared with second-order estimates. But only for the normalized fourth-order and kurtosis estimates do the results seem significant. Moreover, if a second-order estimate of the noise is available to normalize a fourth-order estimate of signal and noise, the resultant normalized fourth-order estimate has higher asymptotic probability of detection levels even for Gaussian signals. This result holds only when there is a significant positive covariance between the numerator and the normalizing noise sample in the denominator. On the other hand, if an independent noise sample is used to normalize a second-order or fourth-order estimate, the overall performance based on the asymptotic probability of detection will be degraded compared with the unnormalized second-order or fourth-order estimates, respectively.