Order Central Moments Research Articles

The Kohonen network incorporating explicit statistics and its application to the travelling salesman problem

In this paper we introduce a new self-organizing neural network, the Kohonen Network Incorporating Explicit Statistics (KNIES) that is based on Kohonen's Self-Organizing Map (SOM). The primary difference between the SOM and the KNIES is the fact that every iteration in the training phase includes two distinct modules—the attracting module and the dispersing module. As a result of the newly introduced dispersing module the neurons maintain the overall statistical properties of the data points. Thus, although in SOM the neurons individually find their places both statistically and topologically, in KNIES they collectively maintain their mean to be the mean of the data points, which they represent. Although the scheme as it is currently implemented maintains the mean as its invariant, the scheme can easily be generalized to maintain higher order central moments as invariants. The new scheme has been used to solve the Euclidean Travelling Salesman Problem (TSP). Experimental results for problems taken from TSPLIB [Reinelt, G. (1991). TSPLIB—A travelling salesman problem library. ORSA Journal on Computing, 3, pp. 376–384] indicate that it is a very accurate NN strategy for the TSP—probably the most accurate neural solutions available in the literature.

A moment-based approach for deskewing rotationally symmetric shapes

The covariance matrix of a pattern is composed by its second order central moments. For a rotationally symmetric shape, its covariance matrix is a scalar identity matrix. In this work, we apply this property to restore the skewed shape of rotational symmetry. The relations between the skew transformation matrix and the covariance matrices of original and skewed shapes are derived. By computing the covariance matrix of the skewed shape and letting the covariance matrix of the original shape be a scalar identity matrix, the skew transformation matrix can be solved. Then, the rotationally symmetric shape can be recovered by multiplying the inverse transformation matrix with the skewed shape. The method does not rely on continuous contours and is robust to noise, because only the second-order moments of the input shape are required. Experimental results are also presented.

MULTIRESOLUTIONAL TEXTURE ANALYSIS FOR ULTRASOUND TISSUE CHARACTERIZATION

Multiresolutional texture analysis techniques using wavelet transforms have been used for predicting percentage intramuscular fat (IMFAT) of beef ribeye muscle. Ultrasound B-mode images have been collected from 207 live beef animals and digitized through frame grabber connected to a personal computer. Since fat has acoustic properties that are appreciably different from that of muscle, transmitted ultrasound is reflected from the interface between fat and muscle. As fat deposits increase, the speckle content of ultrasound B-mode images also increases. Since the speckle alters the texture of the image, IMFAT can be estimated using texture analysis methods. The Haar wavelet was used as a basis function for generating the fast wavelet transform and several features were calculated from the 2-D wavelet decomposed ultrasound images. Significant features included energy ratios, second and fourth order central moments. Correlation coefficients of the features with the actual IMFAT were calculated to select features for predicting IMFAT. Linear regression (LR) models were used as a tool for predicting IMFAT from the selected features.

Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks.

Dynamical stochastic models of single neurons and neural networks often take the form of a system of n\ensuremath{\ge}2 coupled stochastic differential equations. We consider such systems under the assumption that third and higher order central moments are relatively small. In the general case, a system of 1/2n(n+3) (generally) nonlinear coupled ordinary differential equations holds for the approximate means, variances, and covariances. For the general linear system the solutions of these equations give exact results---this is illustrated in a simple case. Generally, the moment equations can be solved numerically. Results are given for a spiking Fitzhugh-Nagumo model neuron driven by a current with additive white noise. Differential equations are obtained for the means, variances, and covariances of the dynamical variables in a network of n connected spiking neurons in the presence of noise. \textcopyright{} 1996 The American Physical Society.

Energy loss straggling of low energy alpha particles in a medium composed of polyatomic and diatomic molecules

The results of the measurement of the energy straggling of low energy alpha particles in the polyatomic and diatomic molecule media are reported. A general analytical formula for straggling widths of light ions, obtained by solving Symon's equation for the second and third order central moments of the straggling distributions in non-relativistic approximation, has shown remarkably good agreement with experimental data at energy losses less than 70% of the initial projectile energies.

Estimation of the third and fourth order central moments of measurement errors from sums of powers of least squares adjustment residuals

Estimation of the third and fourth order central moments of measurement errors from sums of powers of least squares adjustment residuals

Estimation of the third and fourth order central moments of measurement errors from sums of powers of least squares adjustment residuals

In the paper a method of determination of estimators of the central moments of the third and fourth order on the basis of the least squares adjustment is presented. These estimators concern the moments of the observation results as well as to readjustment (better) of observation sets with application of methods that take into consideration moments of higher order (not only the variance-covariance matrix). The obtained estimators are not biased, but their effectiveness was not studied. That is why the presented theory is not completed (in the sense of the estimators properties).

Estimation of shape characteristics of surface muscle signal spectra from time domain data.

Myoelectric manifestations of muscle fatigue have been described by monitoring the first-order moment (mean frequency) of the power spectral density function during voluntary or electrically elicited sustained contractions. Higher order central moments provide additional information about the width, skewness, and kurtosis of the spectrum and its shape changes, thereby providing a description of slow nonstationarities more accurate than that allowed by the mean frequency alone. In 1986, B. Saltzberg introduced a method of representing the moments of the power spectral density function of band limited signals, without computing the Fourier transform, as weighted sums of samples of the autocorrelation function. If we allow for oversampling of the signal (and therefore of its autocorrelation function), more efficient weighted sums can be found which give Saltzberg's formula as a limiting case. The faster rate of decay of the weights implies a faster convergence of the estimates and the need to compute fewer samples of the autocorrelation function. The algorithm is particularly suitable for: 1) analysis of evoked potentials (M-waves), because it does not need zero padding to increase resolution and operates on any number of samples, and 2) on-line implementation by dedicated microprocessors performing simultaneous spectral moment analysis on a number of parallel channels.

An energy loss straggling formula

The analytical formulae for straggling widths were obtained solving Symon's equations for the second and third order central moments of the straggling distributions in non-relativistic approximation, enabling prediction of peak widths and asymmetry parameters of charged particle energy loss distributions for thick targets. The respective calculations based on these formulae were found to be in good agreement with experimental values for particle energy losses of light ions up to 70% of the nominal projectile energy value.

Use of higher order signal moments and high speed digital sampling technique for neutron flux measurements

The second (conventional variance or Campbell signal), the third, and the modified fourth order central signal moments associated with the amplified and filtered currents from two electrodes of an ex-core neutron sensitive fission detector were measured versus the reactor power of the 1-MW TRIGA reactor in Mexico City. Two channels of a high-speed (400-MHz) multiplexing data sampler and an analog-to-digital converter with 12-b resolution and 1-Mword buffer memory were used. The data were further retrieved into a PC, and estimates for autocorrelation and cross-correlation moments up to the fifth order, coherence, skewness, excess, etc., quantities were calculated offline. Five-mode operation of the detector was achieved, including conventional counting rates and currents in agreement with theory and the authors' previous results with analog techniques. The signals are proportional to the neutron flux and reactor power in some flux ranges. The suppression of background noise is improved and the lower limit of the measurement range is extended as the order of moment is increased, in agreement with theory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Laws of iterated expectations for higher order central moments

Formulae for central moments in terms of conditional moments are derived. The formulae are applied to a model of a randomly stopped sum.

Variance and higher order signal moments in neutron flux measurements

Explicit expressions are derived for the second (also called variance or Campbell signal) and higher order central moments of signals from a neutron detector in a nuclear fission reactor. The signal is treated as a compound distribution, which is composed of detection rates of different types of events with fluctuating pulse amplitudes and with arbitrary pulse shapes. The technique with probability generating functions (pgf) is applied. The present results provide an extended theoretical basis for the utilization of different signal moments for flux monitoring in nuclear fission reactors.

Calculating geometric properties from fourier representation

The area, the location of the centroid and the second order moments of a region are determined and expressed in closed-form in terms of the Fourier-coefficients of its boundary function. The orientation of the central axes and the second order central moments can then be easily obtained. Bounds on the perimeter of the region are derived in terms of the Fourier coefficients as well.

On a moment theorem for complex Gaussian processes

A general theorem is provided for the moments of a complex Gaussian video process. This theorem is analogous to the well-known property of the multivariate normal distribution for real variables, which states that an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th order central product moment is zero if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is odd and is equal to a sum of products of covariances when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is even.