Higher Order Statistical Moments Research Articles

Confounder-adjusted covariances of system outputs and applications to structural health monitoring

Automated damage detection is an integral component of each structural health monitoring (SHM) system. Typically, measurements from various sensors are collected and reduced to damage-sensitive features, and diagnostic values are generated by statistically evaluating the features. Since changes in data do not only result from damage, it is necessary to determine the confounding factors (environmental or operational variables) and to remove their effects from the measurements or features. Many existing methods for correcting confounding effects are based on different types of mean regression. This neglects potential changes in higher-order statistical moments, but in particular, the output covariances are essential for generating reliable diagnostics for damage detection. This article presents an approach to explicitly quantify the changes in the covariance, using conditional covariance matrices based on a non-parametric, kernel-based estimator. The method is applied to the Munich Test Bridge and the KW51 Railway Bridge in Leuven, covering both raw sensor measurements (acceleration, strain, inclination) and extracted damage-sensitive features (natural frequencies). The results show that covariances between different vibration or inclination sensors can significantly change due to temperature changes, and the same is true for natural frequencies. To highlight the advantages, it is explained how conditional covariances can be combined with standard approaches for damage detection, such as the Mahalanobis distance and principal component analysis. As a result, more reliable diagnostic values can be generated with fewer false alarms.

Advancing objective functions in hydrological modelling: Integrating knowable moments for improved simulation accuracy

Runoff prediction is a crucial aspect of water resource management and risk mitigation. Despite hydrological modelling plays a vital role in accurately representing catchment behaviour, calibration still poses a significant challenge. This study explores the use of knowable moments (KMoments), a category of high-order statistical moments, as part of the core of objective functions in hydrological model calibration. Traditional objective functions, such as Nash-Sutcliffe Efficiency (NSE) and Kling-Gupta Efficiency (KGE), often make assumptions about data distribution and are sensitive to outliers. KMoments offer a promising alternative by enabling reliable estimation and effective description of high-order statistics from typical hydrological samples and therefore, reducing uncertainty in their estimation and computation of the objective functions in question. Three daily lumped hydrological models (GR4J, VIC, and HYMOD) were employed to test the performance of different calibration strategies using KMoments-based objective functions and compare them with conventional approaches. The hydrological consistency of the simulations was also assessed through 27 hydrological signatures. Our findings highlight the advantages of using KMoments, including improved performance metrics and enhanced hydrological signature reproduction. The findings contribute to advancing hydrological modelling techniques and provide valuable insights for researchers and practitioners seeking to enhance simulation accuracy and reliability.

An active multiphase probabilistic power flow based on a clustering approach

Probabilistic methods for analyzing power systems have become more widespread, mainly due to the increase in load and generation uncertainties and the current need for planning problems that are sufficiently accurate to take appropriate actions. However, some characteristics need to be better investigated in the Distribution Systems (DS) analysis, such as the input random variables’ modeling using real databases and, mainly, the probabilistic treatment of unbalance and scalability studies. This paper proposes an active Multiphase Probabilistic Power Flow (MPPF) evaluation, considering real measurement databases to represent customer demand uncertainties. It is called Multiphase Probabilistic Power Flow Based on K-Means (MPPFKM), and it encompasses a mixture of a clustering technique and an analytical method. The K-Means algorithm divides the input database into a set of clusters that models the scenarios to be evaluated in the multiphase power flow problem, which considers the distribution network with its inherent unbalance. Subsequently, the output variables’ higher order statistical moments are calculated using the obtained responses for each cluster. Finally, their PDFs and CDFs are recovered using the Gram-Charlier Expansion analytical method. While the clustering algorithm allows working with real data, giving the proposed work greater proximity to real-world results, the analytical tool provides an accurate recovery of the probability functions, because it detects their distortions caused by the nature of the multiphase problem. Estimating the correct number of clusters is a key factor in obtaining the precision required by power systems engineers and achieving significant computational time reduction, especially for large-scale unbalanced systems. Simulations are carried out using several DS feeders to test the proposed method, which specifically addresses DS issues not yet well explored in the literature, such as probabilistic simulations under different unbalance situations, applications with single-phase DGs, and scalability studies are conducted using the IEEE 8500 test feeder. Comparisons are made with usual MPPF algorithms, including Monte Carlo Simulation (MCS) and Point Estimation Methods (PEM). Simulations using large-scale unbalanced systems proved that the output variables' probability functions might present distortions, becoming more challenging for other methods. Under this perspective, MPPFKM presents better exactness in the calculation of higher order statistical moments and, consequently, can estimate the PDFs of the MPPF output variables with great precision, which is crucial for statistical models that use actual input data.

On the Averaging and Closure of Fiber Orientation Tensors in Virtual Process Chains

Fiber orientation tensors (FOT) are widely used to approximate statistical orientation distributions of fibers within fiber-reinforced polymers. The design process of components made of such fiber-reinforced composites is usually accompanied by a virtual process chain. In this virtual process chain, process-induced FOT are computed in a flow simulation and transferred to the structural simulation. Within the structural simulation, effective macroscopic properties are identified based on the averaged information contained in the FOT. Solving the field equations in flow simulations as well as homogenization of effective stiffnesses necessitates the application of a closure scheme, computing higher-order statistical moments based on assumptions. Additionally, non-congruent spatial discretizations require an intermediate mapping operation. This mapping operation is required, if the discretization, i.e., mesh, of the flow simulation differs from the discretization of the structural simulation. The main objective of this work is to give an answer to the question: Does the sequence of closure and mapping influence the achieved results? It will turn out, that the order influences the result, raising the consecutive question: Which order is beneficial? Both questions are addressed by deriving a quantification of the closure-related uncertainty. The two possible sequences, mapping followed by closure and closure followed by mapping, yield strongly different results, with the magnitude of the deviation even exceeding the magnitude of a reference result. Graphical consideration reveals that for both transversely isotropic and planar FOT-input, invalid results occur if the mapping takes place prior to closure. This issue is retrieved by orientation averaging stiffness tensors. As a by-product, we explicitly define for the first time the admissible parameter space of orthotropic fourth-order fiber orientation tensors and define a distance measure in this parameter space.

Brain Age Prediction Using 2D Projections Based on Higher-Order Statistical Moments and Eigenslices from 3D Magnetic Resonance Imaging Volumes.

Brain age prediction from 3D MRI volumes using deep learning has recently become a popular research topic, as brain age has been shown to be an important biomarker. Training deep networks can be very computationally demanding for large datasets like the U.K. Biobank (currently 29,035 subjects). In our previous work, it was demonstrated that using a few 2D projections (mean and standard deviation along three axes) instead of each full 3D volume leads to much faster training at the cost of a reduction in prediction accuracy. Here, we investigated if another set of 2D projections, based on higher-order statistical central moments and eigenslices, leads to a higher accuracy. Our results show that higher-order moments do not lead to a higher accuracy, but that eigenslices provide a small improvement. We also show that an ensemble of such models provides further improvement.

Integral Turbulent Length and Time Scales of Higher Order Moments

Turbulent length and time scales represent a fundamental quantity for analysing and modelling turbulent flows. Although higher order statistical moments have been conveniently used for decades to describe the mean behaviour of turbulent fluid flow, the definition of the integral turbulent scales seems to be limited to the velocity or its fluctuation itself (i.e. the first moment). Higher order moments are characterized by smaller integral scales and a framework is proposed for estimating autocorrelation functions and integral turbulent length or time scales of higher order moments under the assumption that the probability distribution of the velocity field is Gaussian. The new relations are tested for synthetic turbulence as well as for DNS data of a turbulent plane jet at Reynolds number 10000. The present results in particular suggest that the length or time scales of higher order moments can be markedly smaller than those of the turbulent variable itself, which has implications for statistical uncertainty estimates of higher order moments.

POLYNOMIAL ESTIMATION OF DATA MODEL PARAMETERS WITH NEGATIVE KURTOSIS

Context. The paper focuses on the problem of estimating the center of distribution of the random component of experimental data for density models with a negative kurtosis.
 Objective. The goal of this research is to develop methods to improve the efficiency of polynomial estimation of parameters of experimental data with a negative kurtosis coefficient.
 Method. The study applies a relatively new approach to obtaining estimates for the center of the probability distribution from the results of experimental data with a stochastic component. This approach is based on polynomial estimation methods that rely on the mathematical apparatus of Kunchenko's stochastic polynomials and the description of random variables by higher-order statistics (moments or cumulants). A number of probability density distributions with a negative kurtosis coefficient are used as models of the random component.
 As a measure of efficiency, the ratio of variance of the estimates for the center of the distribution found using polynomial and classical methods based on the parameter of amount of information obtained is used.
 The relative accuracy of polynomial estimates in comparison with the estimates of the mean, median and quantile estimates (center of curvature) is researched using the Monte Carlo method for multiple tests.
 Results. Polynomial methods for estimating the distribution center parameter for data models of probability distribution density with a negative kurtosis coefficient have been constructed.
 Conclusions. The research carried out in this paper confirms the potentially high efficiency of polynomial estimates of the coordinates of the center of the experimental data, which are adequately described by model distributions with a negative kurtosis. Statistical modeling has confirmed the effectiveness of the obtained estimates in comparison with the known non-parametric estimates based on the statistics of the mean, median, and quantile, even with small sample sizes.

Higher-order statistical moments of predicted sonic boom waveforms through turbulence

Sonic booms generated by supersonic aircraft are affected by turbulence in the atmospheric boundary layer through which they propagate. Turbulence effects lead to random variability of the sonic boom waveforms measured on the ground, complicating the prediction of such waveforms. As an initial effort to predict the waveform variability, the solution of the coherent or mean sonic boom waveform has previously been formulated by the author. The current paper extends the formulation to derive an expression of the second-order statistical moment necessary to calculate the variance of the spectral amplitudes. Since the derivation uses a full wave equation, results using the derived expression will be compared with those obtained using a parabolic approximation. In order to fully quantify the uncertainty of our predictions, the higher order statistical moments are also formulated and are shown to be approximately zero. Consequently, the probability density function (pdf) of the spectral amplitudes is predicted to be Gaussian. The formulation is further extended to determine the pdf of the loudness of sonic booms and quantify the uncertainties associated with the loudness prediction. Results from the formulation are compared with available flight test data.

Strategies for Computational Fluid Dynamics Validation Experiments

Abstract The Benchmark Validation Experiment for Reynolds-averaged Navier–Stokes (RANS)/large eddy simulation (LES) Investigations (BeVERLI) aims to produce an experimental dataset of three-dimensional non-equilibrium turbulent boundary layers with various levels of separation that, for the first time, meets the most exacting requirements of computational fluid dynamics validation. The application of simulations and modeling in high-consequence engineering environments has become increasingly prominent in the past two decades, considerably raising the standards and demands of model validation and forcing a significant paradigm shift in the design of corresponding validation experiments. In this paper, based on the experiences of project BeVERLI, we present strategies for designing and executing validation experiments, hoping to ease the transition into this new era of fluid dynamics experimentation and help upcoming validation experiments succeed. We discuss the selection of a flow for validation, the synergistic use of simulations and experiments, cross-institutional collaborations, and tools, such as model scans, time-dependent measurements, and repeated and redundant measurements. The proposed strategies are shown to successfully mitigate risks and enable the methodical identification, measurement, uncertainty quantification, and characterization of critical flow features, boundary conditions, and corresponding sensitivities, promoting the highest levels of model validation experiment completeness per Oberkampf and Smith [1]. Furthermore, the applicability of these strategies to estimating critical and difficult-to-obtain bias error uncertainties of different measurement systems, e.g., the underprediction of high-order statistical moments from particle image velocimetry velocity field data due to spatial filtering effects, and to systematically assessing the quality of uncertainty estimates is shown.

Vapor concentration and bimodal distributions of turbulent fluctuations in cavitating flow around a hydrofoil

The article presents results of a statistical analysis of the original PIV data for the cavitating flow around the 2D symmetric hydrofoil mimicking a guide vane of a Francis turbine which were previously reported in Timoshevskiy et al. (2020). We employ the procedure of statistical vector filtration (Heinz et al., 2004) to eliminate vector outliers from the measured velocity fields, which allows us to retrieve genuine spatial distributions of higher-order statistical moments of velocity fluctuations, and the method of vapor phase detection (Pervunin et al., 2021) to extract time-averaged fields of vapor in the cavitating flow. The spatial distributions of the probability of vapor phase occurrence allowed us to unambiguously distinguish the extension of the flow area occupied by the dispersed phase (cavitation) together with the time-averaged concentration of the vapor phase in the three characteristic regimes of the cavitating flow as well as different features of cavitation evolution, including the location and length of an attached cavity and the place of detachments of cloud cavities in the case of unsteady cloud cavitation. The development of cavitation is also reflected in the fields of higher-order statistical moments of turbulent fluctuations by the example of coefficients of asymmetry and excess. With a transition to unsteady cloud cavitation, the distribution of velocity fluctuations transforms, taking a nonequilibrium shape, with an increase in magnitudes of the asymmetry and excess coefficients over a significant flow area. In a region of the attached cavity, around the place of formation of vapor clouds and downstream, where the cloud cavities are carried away by the flow, the distribution of velocity fluctuations becomes bimodal. Through visual inspection of the raw PIV images, realizations related to two characteristic phases of the cavity oscillation cycle were extracted, providing quasi-phase-averaged data. As a result, the bimodal shape of the distributions in certain areas of the unsteady cavitating flow was demonstrated to be most probably linked with periodic detachments and downstream advection of the vapor clouds as the predominant fundamental process determining the flow structure and dynamics.

Investigating the Number of Monte Carlo Simulations for Statistically Stationary Model Outputs

The number of random fields required to capture the spatial variability of soil properties and their impact on the performance of geotechnical systems is often varied. However, the number of random fields required to obtain higher-order statistical moments of model outputs has not yet been studied. This research aims to investigate the number of Monte Carlo simulations needed to achieve stationary higher-order statistics of a pore pressure head in an unsaturated soil slope under steady-state infiltration. The study recommends using at least 500 Monte Carlo samples for the probabilistic analysis of geotechnical engineering models. A more conservative choice for up to second-moment analysis is 1000 samples. The analysis reveals significant variations in skewness, which become stationary for all mesh grids when the number of samples exceeds 15,000. Kurtosis stabilizes only when the number of samples reaches 25,000. The pore pressure head in the unsaturated zone is less uncertain. Additionally, the probability density function of the pore pressure head follows a leptokurtic distribution.

An Experimental Study of Non-Gaussian Properties of Tornado-like Loads on a Low-Rise Building Model

This paper focuses on the non-Gaussian properties of tornado pressure loading on a Texas Tech University (TTU) building model. External pressures of the model were measured at multiple radial positions in a two-celled tornado-like vortex generated by a tornado-like vortex simulator. High-order statistical moments of pressure coefficients were studied. The spatial distributions of non-Gaussian zones were presented and compared with the figure for boundary layer wind. Four exponential models were used to fit the probability density of tornado pressures. The peak factors obtained by five methods were also investigated. Results indicate that non-Gaussian regions of a low-rise building in a tornado-like vortex significantly differ from that in boundary layer wind. The peak pressure coefficients exhibit a maximum value near the end of windward eaves when the model is located at a tornadic core radius. The probability density of tornado pressure time series cannot be fitted by a single exponential distribution. The gamma distribution and generalized extreme value (GEV) distribution can describe the probabilistic behavior of pressure coefficients at the most unfavorable load position. Compared with other methods, the skewness-dependent peak factor method exhibits the advantages of reliable results, easy calculation, and wide applicability.

Optical anisotropy composition of benign and malignant prostate tissues revealed by Mueller-matrix imaging.

A Mueller matrix imaging approach is employed to disclose the three-dimensional composition framework of optical anisotropy within cancerous biotissues. Visualized by the Mueller matrix technique spatial architecture of optical anisotropy of tissues is characterised by high-order statistical moments. Thus, quantitative analysis of the spatial distribution of optical anisotropy, such as linear and circular birefringence and dichroism, is revealed by using high-order statistical moments, enabling definitively discriminate prostate adenoma and carcinoma. The developed approach provides greater (>90%) accuracy of diagnostic achieved by using either the 3-rd or 4-th order statistical moments of the linear anisotropy parameters. Noticeable difference is observed between prostate adenoma and carcinoma tissue samples in terms of the extinction coefficient and the degree of depolarisation. Juxtaposition to other optical diagnostic modalities demonstrates the greater accuracy of the approach described herein, paving the way for its wider application in cancer diagnosis and tissue characterization.

Statistical properties of extreme soliton collisions.

Synchronous collisions between a large number of solitons are considered in the context of a statistical description. It is shown that, during the interaction of solitons of the same signs, the wave field is effectively smoothed out. When the number of solitons increases and the sequence of their amplitudes decay slower, the focused wave becomes even smoother and the statistical moments get frozen for a long time. This quasi-stationary state is characterized by greatly reduced statistical moments and by the density of solitons close to some critical value. This state may be treated as the small-dispersion limit, what makes it possible to analytically estimate all high-order statistical moments. While the focus of the study is made on the Korteweg-de Vries equation and its modified version, a much broader applicability of the results to equations that support soliton-type solutions is discussed.

Isolating the modulation of mean warming and higher-order temperature changes on ozone in a changing climate over the contiguous United States

Surface ozone is substantially affected by climate change through the modulation of key meteorological parameters such as temperature. While the changes in temperature under a warming climate manifest in changes of both the mean and higher-order statistical moments, their individual role in ozone concentration changes has not been broadly investigated. To address this gap, we use a novel approach to isolate the impacts of mean warming and changes in higher-order moments of temperature on ozone over the southeastern U.S. (SEUS) and western U.S. (WUS) by the mid-21st century based on simulations under Technology Driver Model A1B and B2 scenarios from a regional modeling framework (Weather Research and Forecasting model coupled with Chemistry (WRF/Chem)). Mean warming generally dominates the impacts of climate change on ozone, and higher-order moment temperature changes can also counteract 25% of the ozone exceedance of 70 ppbv over SEUS, and may offset 48% of the mean warming induced increase of ozone exceedance in heat waves during 2046–2055 under A1B. The opposite changes in the higher-order moments over SEUS and WUS lead to opposite impacts on ozone exceedance in the two regions. Our results suggest that improving prediction of both the mean and higher-order temperature changes may be crucial to constraining the future changes in ozone concentration to better inform air quality policy.

On the Kurtosis of Modulation Formats for Characterizing the Nonlinear Fiber Propagation

Knowing only two high-order statistical moments of modulation symbols, often represented by the fourth moment called “kurtosis”, the overestimation of nonlinear interference (NLI) in a Gaussian noise (GN) model due to Gaussian signaling assumption can be corrected through an enhanced GN (EGN) model. However, in some modern optical communication systems where the transmitted modulation symbols are statistically correlated, such as in systems that use probabilistic constellation shaping (PCS) with finite-length sphere shaping, the kurtosis-based EGN model produces significant inaccuracies in analytical prediction of NLI. In this paper, we show that for correlated modulation symbols, the NLI can be more accurately estimated by substituting a statistical measure called windowed kurtosis into the EGN model, instead of the conventional kurtosis. Remarkably, the optimal window length for windowed kurtosis is found to be consistent with the self-phase modulation (SPM) and cross-phase modulation (XPM) characteristic times in various system configurations. The findings can be used in practice to analytically evaluate and design NLI-tolerant modulation formats.

Technical note: Interpretation of field observations of point-source methane plume using observation-driven large-eddy simulations

Abstract. This study demonstrates the ability of large-eddy simulation (LES) forced by a large-scale model to reproduce plume dispersion in an actual field campaign. Our aim is to bring together field observations taken under non-ideal conditions and LES to show that this combination can help to derive point-source strengths from sparse observations. We analyze results from a single-day case study based on data collected near an oil well during the ROMEO campaign (ROmanian Methane Emissions from Oil and gas) that took place in October 2019. We set up our LES using boundary conditions derived from the meteorological reanalysis ERA5 and released a point source in line with the configuration in the field. The weather conditions produced by the LES show close agreement with field observations, although the observed wind field showed complex features due to the absence of synoptic forcing. In order to align the plume direction with field observations, we created a second simulation experiment with manipulated wind fields that better resemble the observations. Using these LESs, the estimated source strengths agree well with the emitted artificial tracer gas plume, indicating the suitability of LES to infer source strengths from observations under complex conditions. To further harvest the added value of LES, higher-order statistical moments of the simulated plume were analyzed. Here, we found good agreement with plumes from previous LES and laboratory experiments in channel flows. We derived a length scale of plume mixing from the boundary layer height, the mean wind speed and convective velocity scale. It was demonstrated that this length scale represents the distance from the source at which the predominant plume behavior transfers from meandering dispersion to relative dispersion.

Flood Risk Modeling under Uncertainties: The Case Study of Croatia

This study presents an approach for assessing the flood risk using the fully probabilistic description of the annual damage using the derived higher-order statistical moments of the annual damage random variable. The annual damage distribution is used to analyze the impact on the macro-scale agglomeration described with the unified damage function in the case study of the river Kupa, Croatia. The agglomeration damage function is derived through the unit damage function and estimated distribution of building thresholds within the studied agglomeration. The unit damage is described with the shape function, which can easily accommodate different forms of damage assessment depending on the available information. The estimated distribution of annual maximum flood levels is propagated through the agglomeration damage function into the annual damage distribution derived using higher-order statistical moments and presented with the exceedance probability, indicating the likelihood that a certain annual maximum damage can be exceeded. Besides the aleatory uncertainty of annual maximum flood levels, we analyzed the impact of the uncertainty in estimating the unit damage function upon the annual damage distribution. This uncertainty shows a significant impact on the shape of the annual damage distribution, particularly in the region of extreme flood events. Analyzing the annual damage distribution range, resulting from the uncertainty in damage functions, provides more information to policymakers in assessing the potential consequences on the future spatial planning programs, particularly from the strategic environmental assessment point of view.

A spatial averaging of sonic boom waveforms

Sonic booms generated by conventional supersonic aircraft are influenced by turbulence in the atmospheric boundary layer through which they propagate. The turbulence effects lead to random variability of the sonic boom waveforms measured on the ground, complicating the prediction of such waveforms. While some of the waveforms have been described as N-waves with spikes immediately after the shocks, others measured a few hundreds of feet away tend to be rounded. The current paper presents an analysis method that uses a spatial averaging of an ensemble of measured waveforms to obtain a mean waveform and higher-order statistical moments. The statistics obtained are anticipated to be useful in characterizing the waveforms generated by any supersonic vehicle. They will also be useful for validating theoretical and semi-empirical models used for predicting turbulence effects on sonic booms. This paper discusses the advantages and disadvantages of spatial averaging in the time domain and the frequency domain. Preliminary results indicate that the mean waveforms have rise times larger than those obtained using a sonic boom propagation model that does not account for the turbulence effects, suggesting that turbulence effects may not be ignored.

Solar panel scale inversion with higher-order statistical moments of echo laser pulse waveform

The space environment is becoming increasingly complicated; therefore, precise detection and identification of space targets are critical to preserving space security. Compared to optical and radar imaging, obtaining space target information using laser echo waveform is more efficient for detection. Based on the skew-normal distribution decomposition, the connection between sub-echoes and target scale following the decomposition of the space target pulse laser echo waveform is explored, and an inversion approach is suggested to identify the scale information of the solar panel utilizing the intersection of skewness and kurtosis contours in the coordinate system of the variables to be solved, based on the high-order moment characteristics of waveforms such as skewness and kurtosis. Using the proposed method, we realized the scale inversion of a 60-cm solar panel of a cube satellite inclined at 45 deg, with the turntable and the detection system placed 80 m apart. The results show that the skewness and kurtosis of the decomposed echo can represent the target size information, and the approach used here can successfully extract the solar panel scale information of a typical satellite, providing methods and data support for space target detection and classification.