Use Of Higher Order Moments Research Articles

Local order metrics for many-particle systems across length scales

Formulating order metrics that sensitively quantify the degree of order/disorder in many-particle systems in d-dimensional Euclidean space Rd across length scales is an outstanding challenge in physics, chemistry, and materials science. Since an infinite set of n-particle correlation functions is required to fully characterize a system, one must settle for a reduced set of structural information, in practice. We initiate a program to use the local number variance σN2(R) associated with a spherical sampling window of radius R (which encodes pair correlations) and an integral measure derived from it ΣN(Ri,Rj) that depends on two specified radial distances Ri and Rj. Across the first three space dimensions (d=1,2,3), we find these metrics can sensitively describe and categorize the degree of order/disorder of 41 different models of antihyperuniform, nonhyperuniform, disordered hyperuniform, and ordered hyperuniform many-particle systems at a specified length scale R. Using our local variance metrics, we demonstrate the importance of assessing order/disorder with respect to a specific value of R. These local order metrics could also aid in the inverse design of structures with prescribed length-scale-specific degrees of order/disorder that yield desired physical properties. In future work, it would be fruitful to explore the use of higher-order moments of the number of points within a spherical window of radius R [S. Torquato , ] to devise even more sensitive order metrics. Published by the American Physical Society 2024

Perceptual insensitivity to higher-order statistical moments of coherent random dot motion.

When the visual system analyzes distributed patterns of sensory inputs, what features of those distributions does it use? It has been previously demonstrated that higher-order statistical moments of luminance distributions influence perception of static surfaces and textures. Here, we tested whether the brain also represents higher-order moments of dynamic stimuli. We constructed random dot kinematograms, where dots moved according to probability distributions that selectively differed in terms of their mean, variance, skewness, or kurtosis. When viewing these stimuli, human observers were sensitive to the mean direction of coherent motion and to the variance of dot displacement angles, but they were insensitive to skewness and kurtosis. Observer behavior accorded with a model of directional motion energy, suggesting that information about higher-order moments is discarded early in the visual processing hierarchy. These results demonstrate that use of higher-order moments is not a general property of visual perception.

A Double Recursion for Calculating Moments of the Truncated Normal Distribution and its Connection to Change Detection

The integral ${\int }_{0}^{\infty }x^{m} e^{-\frac {1}{2}(x-a)^{2}}dx$ appears in likelihood ratios used to detect a change in the parameters of a normal distribution. As part of the mth moment of a truncated normal distribution, this integral is known to satisfy a recursion relation, which has been used to calculate the first four moments of a truncated normal. Use of higher order moments was rare. In more recent times, this integral has found important applications in methods of changepoint detection, with m going up to the thousands. The standard recursion formula entails numbers whose values grow quickly with m, rendering a low cap on computational feasibility. We present various aspects of dealing with the computational issues: asymptotics, recursion and approximation. We provide an example in a changepoint detection setting.

Increasing the lensing figure of merit through higher order convergence moments

The unprecedented quality, the increased dataset, and the wide area of ongoing and near future weak lensing surveys allows to move beyond the standard two points statistics thus making worthwhile to investigate higher order probes. As an interesting step towards this direction, we expolore the use of higher order moments (HOM) of the convergence field as a way to increase the lensing Figure of Merit (FoM). To this end, we rely on simulated convergence to first show that HOM can be measured and calibrated so that it is indeed possible to predict them for a given cosmological model provided suitable nuisance parameters are introduced and then marginalized over. We then forecast the accuracy on cosmological parameters from the use of HOM alone and in combination with standard shear power spectra tomography. It turns out that HOM allow to break some common degeneracies thus significantly boosting the overall FoM. We also qualitatively discuss possible systematics and how they can be dealt with.

Semi-analytic model to evaluate non-regularized stresses causing unfolding failure in composites

A novel semi-analytic model is developed to determine interlaminar stresses in laminates including highly curved parts, for obtaining a rapid, accurate and reliable design tool for evaluating the reserve factor of a certain aeronautic composite component prone to unfolding failure. The model assumes plane conditions and is based, firstly, in an approximation of the 2D displacement components by using a series of functional products of separate variables and, secondly, in the use of higher-order moments of the stress components, which allow an enriched beam model to be constructed. The solution is obtained as the sum of products of a set of known functions, written in terms of the eigenvalues and eigenvectors of the matrix governing the model, and a set of integration constants. These constants can be determined with the boundary conditions, in isolated parts, and the continuity conditions, in laminates containing parts with different curvatures. In comparison with an existing analytical method, the present model has the advantage of offering an extremely accurate solution at the zones in which the analytical method may yield errors of even a 100%. In comparison with a detailed finite elements model, the present model obtains a similar accuracy in much lower computational times.

Imputation using higher order moments of an auxiliary variable

ABSTRACTIn this article, we propose a new method of imputation that makes use of higher order moments of an auxiliary variable while imputing missing values. The mean, ratio, and regression methods of imputation are shown to be special cases and less efficient than the newly developed method of imputation, which makes use of higher order moments. Analytical comparisons show that the first-order mean squared error approximation for the proposed new method of imputation is always smaller than that for the regression method of imputation. At the end, the proposed higher order moments-based imputation method has been applied to a real dataset.

Correlations and Fluctuations of Pions at the LHC

The intriguing possibility of Bose-Einstein condensation of pions at the LHC is examined with the use of higher order moments of the multiplicity distribution. The scaled variance, skewness and kurtosis are calculated for the pion system. The obtained results show that the normalized kurtosis has a significant increase for the case of the pion condensation.

Note on Friedrichs' inequality in N-star-shaped domains

We establish an explicit constant in the Friedrichs inequality‖u−1|S|∫Su(x)dx‖Lp(Ω)≤C(Ω,S,B)‖∇u‖Lp(Ω) for u∈W1,p(Ω), 1≤p<∞, a domain Ω⊂Rn star-shaped with respect to a convex set B, and a measurable subset S⊂Ω with |S|>0, n≥2. This result will be generalized to so-called N-star-shaped domains which can be written as union of N domains star-shaped with respect to convex sets. A second generalization concerns the use of higher order moments when replacing the integral mean of u over S by the mean of higher order Taylor polynomials. The proofs are direct and elementary.

Advanced Combustor Exit Plane Temperature Diagnostics Based on Large Eddy Simulations

The accurate description and flow calibration at the exit of a gas turbine combustor is a key point of turbine design. Indeed, the temperature field at the turbine inlet directly impacts both the work extraction by the turbine stages as well as drives the wall heat transfer occurring on the vanes and rotor surfaces. Properly simulating or measuring this critical planar interface has been an extensive topic of research. Despite such efforts, industrial exploitation of the various measures at that interface usually resumes to a quantification of the temperature heterogeneities through the Local Radial Temperature Distortion Factor (LRTDF), thereby loosing major spatial and unsteady information. As lean burn configurations become a standard for aero-engine combustors, the absence of dilution holes reduces the intensity of mixing and the use of more advanced statistical tools should propose an exhaustive representation of the temperature map and mixing quality. Current state-of-the-art numerical design of gas turbine combustor relies on Large Eddy Simulations (LES) giving access to a fully 3-D temporally dependent flow field throughout the combustion chamber. Going beyond LRTDF is therefore possible and advanced analyses of the combustor-turbine interface are accessible to better qualify proposed design changes. In this work, a simple statistical representation of the temperature distribution is introduced and applied for the analysis of a lean burn combustor simulator interface. It is shown that at the exit of this chamber, the temperature samples are distributed following a non-Gaussian shape, highlighting a most probable local temperature largely different from the conventional mean value. The use of high order moments explains such deviations and indicates the presence of strongly segregated zones governed by the technological implementation of effusion cooling systems. Based on this diagnostic, a new and complementary representation of the LRTDF is proposed, accounting for the statistical dimension of the classical (radius, temperature) formulation.

Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology

Abstract. The need of understanding and modelling the space–time variability of natural processes in hydrological sciences produced a large body of literature over the last thirty years. In this context, a multifractal framework provides parsimonious models which can be applied to a wide-scale range of hydrological processes, and are based on the empirical detection of some patterns in observational data, i.e. a scale invariant mechanism repeating scale after scale. Hence, multifractal analyses heavily rely on available data series and their statistical processing. In such analyses, high order moments are often estimated and used in model identification and fitting as if they were reliable. This paper warns practitioners against the blind use in geophysical time series analyses of classical statistics, which is based upon independent samples typically following distributions of exponential type. Indeed, the study of natural processes reveals scaling behaviours in state (departure from exponential distribution tails) and in time (departure from independence), thus implying dramatic increase of bias and uncertainty in statistical estimation. Surprisingly, all these differences are commonly unaccounted for in most multifractal analyses of hydrological processes, which may result in inappropriate modelling, wrong inferences and false claims about the properties of the processes studied. Using theoretical reasoning and Monte Carlo simulations, we find that the reliability of multifractal methods that use high order moments (&gt;3) is questionable. In particular, we suggest that, because of estimation problems, the use of moments of order higher than two should be avoided, either in justifying or fitting models. Nonetheless, in most problems the first two moments provide enough information for the most important characteristics of the distribution.

Comparative study of moment based parameterization for morphological texture description

The two principal morphological texture descriptors, granulometry and morphological covariance, rely on the common principle of successive filtering of an image using a variety of structuring elements, from which feature vectors are subsequently computed. A crucial stage of their computation is the numerical characterization or parameterization of each of the filtered images. In this regard, the zero-th statistical moment is the traditional measure, while the use of higher order moments has also been reported. In this paper, we present the results of a comparative study, concentrating on the potential of various statistical moments for the task of parameterization, while additionally investigating the contribution of Fourier transform moments. The experiments are conducted with focus on texture description effectiveness and on noise robustness, using publicly available texture collections: Outex, CUReT and KTH-TIPS2b, where it is shown that the combination of moments leads to superior classification performance even at high noise levels.

Estimating an endpoint with high-order moments

We present a new method for estimating the endpoint of a unidimensional sample when the distribution function decreases at a polynomial rate to zero in the neighborhood of the endpoint. The estimator is based on the use of high-order moments of the variable of interest. It is assumed that the order of the moments goes to infinity, and we give conditions on its rate of divergence to get the asymptotic normality of the estimator. The good performance of the estimator is illustrated on some finite sample situations.

Utilization of higher order moments of scrambling variables in randomized response sampling

In this paper, a new estimator for estimating the proportion of a potentially sensitive attribute in survey sampling has been introduced. The proposed estimator makes use of higher order moments of the scrambling variable at the estimation stage. The proposed estimator has been found to be more efficient than the estimator due to Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] and Franklin [1989. A comparison of estimators for randomized response sampling with continuous distributions from a dichotomous population. Comm. Statist. Theory Methods 18, 489–505] type estimators in randomized response sampling. Recently, Guerriero and Sandri [2007. A note on the comparison of some randomized response procedures. J. Statist. Plann. Inference 137, 2184–2190] have shown that the family of randomized response models proposed by Kuk [1990. Asking sensitive questions indirectly. Biomerika 77(2), 436–438] is better than the Simmons’ family in terms of efficiency and protection.

A Generic Pulsar Radio Emission Mechanism

A generic interpretation of pulsar radio emission that does not rely on the identification of a specific emission mechanism is explored. Coherence is quantified in terms of a coherence factor, which implies a maximum brightness temperature; the possible significance of Poincare invariants is pointed out, and the potential use of higher order moments of the intensity to measure the coherence is discussed. The effect of the Lorentz boost between the plasma rest frame and the pulsar frame, the suppression of emission at low frequencies due to curvature of the field lines, and a natural frequency of the pulsar plasma are incorporated into a generic model for pulsar emission, and three illustrative examples of its possible use are given.

Statistical moments of autoregressive model residuals for damage localisation

Monitoring structural health is a problem with significant importance in the world today. Aging civil infrastructure and aircraft fleets have made non-destructive evaluation an important research topic. Non-destructive techniques based on dynamic signatures have struggled to gain widespread acceptance due to the perceived difficulty in applying these methods, as well as the mixed results they can produce. A simple and reliable method that is useful without in-depth knowledge of the structure is necessary to transition dynamic response-based health monitoring into the industrial mainstream.Modal parameters, including shifting frequencies, damping ratios, and mode shapes have received considerable attention as damage indicators. The results have been mixed and require an expert to carry out the testing and interpretation. Detailed knowledge of the structure before it becomes damaged is required, either in the form of experimental data or an analytical model.A method based on vector autoregressive (ARV) models is proposed. These models accurately capture the predictable dynamics present in the response. They leave the unpredictable portion, including the component resulting from unmeasured input shocks, in the residual.An estimate of the autoregressive model residual series standard deviation provides an accurate diagnosis of damage conditions. Additionally, a repeatable threshold level that separates damaged from undamaged is identified, indicating the possibility of damage identification and localisation without explicit knowledge of the undamaged structure. Similar statistical analysis applied to the raw data necessitates the use of higher-order moments that are more sensitive to disguised outliers, but are also prone to false indications resulting from overemphasising rarely occurring extreme values.Results are included from data collected using an eight-degree of freedom damage simulation test-bed, built and tested at Los Alamos National Laboratory (LANL). Confidence bounds on each moment are computed for the available data sets and are included to illustrate “significant” differences.

Third-Moment Standardization for Structural Reliability Analysis

First- and second-order reliability methods are generally considered to be among the most useful for computing structural reliability. In these methods, the uncertainties included in resistances and loads are generally expressed as continuous random variables that have a known cumulative distribution function. The Rosenblatt transformation is a fundamental requirement for structural reliability analysis. However, in practical applications, the cumulative distribution functions of some random variables are unknown, and the probabilistic characteristics of these variables may be expressed using only statistical moments. In the present study, a structural reliability analysis method with inclusion of random variables with unknown cumulative distribution functions is suggested. Normal transformation methods that make use of high-order moments are investigated, and an accurate third-moment standardization function is proposed. Using the proposed method, the normal transformation for random variables with unknown cumulative distribution functions can be realized without using the Rosenblatt transformation. Through the numerical examples presented, the proposed method is found to be sufficiently accurate to include the random variables with unknown cumulative distribution functions in the first- and second-order reliability analyses with little extra computational effort.

New Point Estimates for Probability Moments

There are many areas of structural safety and structural dynamics in which it is often desirable to compute the first few statistical moments of a function of random variables. The usual approximation is by the Taylor expansion method. This approach requires the computation of derivatives. In order to avoid the computation of derivatives, point estimates for probability moments have been proposed. However, the accuracy is quite low, and sometimes, the estimating points may be outside the region in which the random variable is defined. In the present paper, new point estimates for probability moments are proposed, in which increasing the number of estimating points is easier because the estimating points are independent of the random variable in its original space and the use of high-order moments of the random variables is not required. By using this approximation, the practicability and accuracy of point estimates can be much improved.

Fourier demixing filter based on statistical independencebetween signals

A new filter based on the statistical independence between signals is presented. The original signal is estimated by the weighted sum of each harmonic of the observation corrupted by noise. The filter is synthesised by adjusting the weighting parameters by learning with the use of higher order moments of observation noise without any information about the dynamics of a target system.

The generalized method of moments as applied to problems of flood frequency analysis: Some practical results for the log-Pearson type 3 distribution

The log-Pearson type 3 distribution is widely used in North America and Australia for fitting annual flood series. Four different versions of the method of moments used in fitting this distribution are compared using Monte Carlo simulated samples which reflect some of the characteristics of annual flood series observed at some Canadian rivers. The bias, standard error, root mean square error, and skew, of the parameter estimates, and of estimates of events associated with different probabilities of occurrence are examined. Also examined are the correlation coefficients between probabilities of occurrence are examined. Also examined are the correlation coefficients between the parameter estimates and between the sample moments that are used in each of the four methods of estimation. It is observed that variances, covariances and correlation coefficients calculated using the usual first-order asymptotic approximation might have considerable error and therefore should be used with caution. On the basis of mean square error of events with return period above the range covered by the sample it is observed that a method proposed earlier ★ ★ Bobée (1975). which uses moments of order 1, 2 and 3 in real space performs better than the other three methods although certain of the other methods follow the recommendation put forward by some investigators that higher order moments (moments of order 3 or more) should be avoided in flood frequency estimation. It is argued in the present study that the use of higher order moments should not be avoided simply because they have high variability because it is not only the variability of the moments which determines the degree of variability of the estimated design flood events but also the correlation that exists between these moments. Some recommendations are given at the end of the study aimed at achieving better efficiency in flood frequency research at a period where more and more distributions and methods of estimation are being proposed.

Moments of error-frequency response due to coefficient inaccuracy for sampled data filters

McCrady and Spence recently developed a general coefficient fabrication error model for sampled data filters which makes use of higher order moments of the perturbed frequency response. By using previous results on cumulants and moments of a random variable, we derive an algorithm to compute recursively the coefficients of the Fourier series of these moments. The algorithm avoids the cumbersome combinatorial approach and permits a systematic determination of statistics of the perturbed frequency response.