Fourth Moment Research Articles

Noninvasive polarization-based technique for hematocrit monitoring

Subject of study. Polarization images corresponding to the 2D distributions of polarization state parameters for light scattered by biological specimens with different hematocrit levels were obtained by shining laser light in a given polarization state of blood-containing tissue through the nail bed of the left fifth finger. Aim of work. The aim of this work is to determine whether active video polarimetry is suitable for noninvasive real-time monitoring of blood hematocrit and to develop data-based criteria to identify hematocrit levels corresponding to various body states of human test subjects. Method. Physical models of changes in the parameters of polarized radiation during interaction with a biological specimen are used. A prototype optical design was developed for a Stokes video polarimeter capable of recording the irradiance distribution within a single analysis field on the sensitive area of a photodetector array. The digital images were processed using the Stokes–Mueller matrix formalism. A comparative statistical analysis of the 3D distributions for various parameters describing the polarization state (degree of polarization and the azimuth and ellipticity of the polarization ellipse) was performed to determine whether it is possible to detect differences in the hematocrit levels of human blood. Main results. A new approach to noninvasive hematocrit monitoring based on optical polarization is proposed. We obtained 3D distributions and histograms of light polarization state parameters for two human test subjects known to have different hematocrits. Statistical analysis revealed that significant differences in the second–fourth order statistical moments of the light polarization parameters are the most relevant diagnostic indicators for comparative analysis of biological tissues having different blood hematocrit levels. Practical significance. With further development, the proposed method can be applied to both medical diagnosis and emergency treatment. Timely detection of significant deviations from the normal hematocrit based on automated calculation and analysis of the data parameters identified will enable continuous hematocrit monitoring during surgical operations and effective real-time adjustment of blood viscosity.

Pilot estimators for a kind of sparse covariance matrices with incomplete heavy-tailed data

<abstract><p>This paper investigates generalized pilot estimators of covariance matrix in the presence of missing data. When the random samples have only bounded fourth moment, two kinds of generalized pilot estimators are provided, the generalized Huber estimator and the generalized truncated mean estimator. In addition, we construct thresholding generalized pilot estimator for a kind of sparse covariance matrices and establish the convergence rates in terms of probability under spectral and Frobenius norms respectively. Moreover, the convergence rates in sense of expectation are also given under an extra condition. Finally, simulation studies are conducted to demonstrate the superiority of our method.</p></abstract>

Dynamic event-triggered adaptive control for uncertain stochastic nonlinear systems

In this paper, the dynamic event-triggered adaptive tracking control problem is investigated via backstepping technology for uncertain stochastic nonlinear systems. First, the stochastic nonlinear system with unknown parameter is considered. By introducing an additional dynamic variable, a dynamic event-triggered adaptive controller is designed such that the closed-loop signals are uniformly ultimately bounded in the sense of the fourth moment. Then, a more general partially unknown stochastic nonlinear system is further considered, and the designed adaptive neural network control scheme ensures that the closed-loop signals are fourth moment semi-globally uniformly ultimately bounded (SGUUB). The proposed dynamic event-triggering mechanism (DETM) guarantees that the lengths of time intervals between each two consecutive events are lower-bounded by a positive constant. It is necessary to point out that the DETM is better at saving resources than the static event-triggering mechanism (SETM). Finally, two simulations are conducted to show the validity of the control strategies for these two systems, respectively.

Continuum-extrapolated NNLO valence PDF of the pion at the physical point

We present lattice QCD calculations of valence parton distribution function (PDF) of pion employing next-to-next-leading-order (NNLO) perturbative QCD matching. Our calculations are based on three gauge ensembles of 2+1 flavor highly improved staggered quarks and Wilson--Clover valance quarks, corresponding to pion mass $m_\pi=140$~MeV at a lattice spacing $a=0.076$~fm and $m_\pi=300$~MeV at $a=0.04, 0.06$~fm. This enables us to present, for the first time, continuum-extrapolated lattice QCD results for NNLO valence PDF of the pion at the physical point. Applying leading-twist expansion for renormalization group invariant (RGI) ratios of bi-local pion matrix elements with NNLO Wilson coefficients we extract $2^{\mathrm{nd}}$, $4^{\mathrm{th}}$ and $6^{\mathrm{th}}$ Mellin moments of the PDF. We reconstruct the Bjorken-$x$ dependence of the NNLO PDF from real-space RGI ratios using a deep neural network (DNN) as well as from momentum-space matrix elements renormalized using a hybrid-scheme. All our results are in broad agreement with the results of global fits to the experimental data carried out by the xFitter and JAM collaborations.

Parametric testing for normality against bimodal and unimodal alternatives using higher moments

This study examines population and small sample properties of the standardized fifth and sixth moments – the “higher moments” – for assessing univariate normality against bimodal and selected unimodal alternatives. Population parameters and distributions for selected bimodal mixtures are calculated and contrasted with those for the normal distribution. Using Gram-Charlier series expansion methods, an omnibus goodness of fit test incorporating the higher moments is specified and Monte Carlo simulation used to compare test power with parametric tests based on the standardized third and fourth sample moments: the asymptotic and size corrected versions of the Jarque-Bera score test and the omnibus D’Agostino K2 test. The studentized range and directional tests using the third through sixth moments are also considered. The results demonstrate that incorporating the fifth and sixth moments can provide enhanced parametric normality test power for bimodal normal mixture alternatives but not for various unimodal alternatives.

Properties of eigenvalues and eigenvectors of large-dimensional sample correlation matrices

This paper is to study the properties of eigenvalues and eigenvectors of high-dimensional sample correlation matrices. We first improve the result of Jiang (Sankhyā 66 (2004) 35–48), Xiao and Zhou (J. Theoret. Probab. 23 (2010) 1–20) and the Theorem 1 of El Karoui (Ann. Appl. Probab. 19 (2009) 2362–2405), both concerning the limiting spectral distribution and the extreme eigenvalues of sample correlation matrices, by allowing a more general fourth moment condition. Then, we establish a central limit theorem (CLT) for the linear statistics of the eigenvectors of large sample correlation matrices. We discover that the difference between the functional CLT of the sample covariance matrix and the sample correlation matrix is fundamentally influenced by the direction of a nonrandom projection vector. In the special case where the square root of the correlation matrix is identity, the difference will be determined by the sum of the fourth powers of the entries of the projection vector. These results also indicate that the eigenmatrix of sample correlation matrices is not asymptotically Haar if the underlying distribution is Gaussian. In other words, the normalization based on the sample variances affects the asymptotic properties of the eigenmatrix of the Wishart matrix. Furthermore, we establish a theorem concerning CLT for the linear statistics of the eigenvectors of large sample covariance matrices. This theorem improves the main results in Bai, Miao and Pan (Ann. Probab. 35 (2007) 1532–1572), which requires the assumption that the fourth moment of the underlying variable matches the one of Gaussian distribution, as well as Theorem 1.3 in Pan and Zhou (Ann. Appl. Probab. 18 (2008) 1232–1270), which relaxed the Gaussian like fourth moment requirement but assumes the maximum entries of the projection vector converge to 0 uniformly. We illustrate the usefulness of the theoretical results through an application in communications.

On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed Errors

In this paper, we study the estimation of a rank-one spiked tensor in the presence of heavy tailed noise. Our results highlight some of the fundamental similarities and differences in the tradeoff between statistical and computational efficiencies under heavy tailed and Gaussian noise. In particular, we show that, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> th order tensors, the tradeoff manifests in an identical fashion as the Gaussian case when the noise has finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4(p-1)$ </tex-math></inline-formula> th moment. The difference in signal strength requirements, with or without computational constraints, for us to estimate the singular vectors at the optimal rate, interestingly, narrows for noise with heavier tails and vanishes when the noise only has finite fourth moment. Moreover, if the noise has less than fourth moment, tensor SVD, perhaps the most natural approach, is suboptimal even though it is computationally intractable. Our analysis exploits a close connection between estimating the rank-one spikes and the spectral norm of a random tensor with iid entries. In particular, we show that the order of the spectral norm of a random tensor can be precisely characterized by the moment of its entries, generalizing classical results for random matrices. In addition to the theoretical guarantees, we propose estimation procedures for the heavy tailed regime, which are easy to implement and efficient to run. Numerical experiments are presented to demonstrate their practical merits.

An Efficient High-Order-Moment-Based Reliability Method Combining the Maximum Likelihood Point and Cubic Normal Transformation

Moment-based methods can measure the safety degrees of mechanical systems affected by unavoidable uncertainties, utilizing only the statistical moments of random variables for reliability analysis. For the conventional derivation of the first four statistical moments based on the second-order Taylor expansion series evaluated at the most likelihood point (MLP), skewness and kurtosis involve the higher fourth raw moments of random variables and thus are unfavorable for engineering applications. This paper develops new computing formulae for the first four statistical moments which require only the first four central moments of random variables, and the probability distribution of the performance function is approximated using cubic normal transformation. Several numerical examples are given to demonstrate the accuracy of the proposed methods. Comparisons of the two proposed approaches and the maximum entropy method (ME) are also made regarding reliability assessment.

Navigation With Time Limits in Transportation Networks: A Fourth Moment Approach

This paper investigates the stochastic on-time arrival (SOTA) problem in transportation networks. We propose a fourth moment approach (FMA), which calculates the tight lower bound of a given routing policy’s on-time-arrival probability, through estimating the first four moments of the policy’s travel time. Then, we employ the generalized policy iteration (GPI) scheme to gradually improve the policy towards the optimal one. Different from state-of-the-art algorithms for the SOTA problem, which require the full travel time distribution and usually incur high computational cost due to the convolution integration operation, FMA only requires the moments of travel-time statistics, which are easily estimated from the statistics perspective. Moreover, the algorithm’s computational complexity analysis indicates the relatively light computational load requirement of FMA. Experimental results in a range of transportation networks show FMA’s superior performance over state of the arts.

Bitcoin and S&P500: Co-movements of high-order moments in the time-frequency domain.

Interactions between stock and cryptocurrency markets have experienced shifts and changes in their dynamics. In this paper, we study the connection between S&P500 and Bitcoin in higher-order moments, specifically up to the fourth conditional moment, utilizing the time-scale perspective of the wavelet coherence analysis. Using data from 19 August 2011 to 14 January 2022, the results show that the co-movement between Bitcoin and S&P500 is moment-dependent and varies across time and frequency. There is very weak or even non-existent connection between the two markets before 2018. Starting 2018, but mostly 2019 onwards, the interconnections emerge. The co-movements between the volatility of Bitcoin and S&P500 intensified around the COVID-19 outbreak, especially at mid-term scales. For skewness and kurtosis, the co-movement is stronger and more significant at mid- and long-term scales. A partial-wavelet coherence analysis underlines the intermediating role of economic policy uncertainty (EPU) in provoking the Bitcoin-S&P500 nexus. These results reflect the co-movement between US stock and Bitcoin markets beyond the second moment of return distribution and across time scales, suggesting the relevance and importance of considering fat tails and return asymmetry when jointly considering US equity-Bitcoin trading or investments and the policy formulation for the sake of US market stability.

Interaction of two axisymmetric turbulent wakes

The interaction between turbulent axisymmetric wakes plays an important role in many industrial applications, notably in the modeling of wind farms. This work proposes an experimental study on the interaction between two bluff bodies. We find that a simple mathematical expression for the wake interaction length based on nonequilibrium turbulence scalings can be used to collapse the streamwise developments of the second, third, and fourth moments of the streamwise fluctuating velocity.

Fourth moments of the code-weights

In this paper we give the fourth joint moments of the weights of random linear codes. We calculate the fourth moment using mutual position of 4 lines. The solution of the 4-subspace problem is known and we obtain the information on line configurations from the set of indecomposables of the 4-subspace problem.

Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions

Let X be an M×N random matrix consisting of independent M-variate elliptically distributed column vectors x1,…,xN with general population covariance matrix Σ. In the literature, the quantity XX∗ is referred to as the sample covariance matrix after scaling, where X∗ is the transpose of X. In this article, we prove that the limiting behavior of the scaled largest eigenvalue of XX∗ is universal for a wide class of elliptical distributions, namely, the scaled largest eigenvalue converges weakly to the same limit regardless of the distributions that x1,…,xN follow as M,N→∞ with M∕N→ϕ0>0 if the weak fourth moment of the radius of x1 exists. In particular, via comparing the Green function with that of the sample covariance matrix of multivariate normally distributed data, we conclude that the limiting distribution of the scaled largest eigenvalue is the celebrated Tracy-Widom law.

Asymptotics of AIC, BIC and Cp model selection rules in high-dimensional regression

Variable selection in multivariate linear regression is essential for the interpretation, subsequent statistical inferences and predictions of the statistical problem at hand. It has a long history of being studied, and many regressor selection criteria have been proposed. Most commonly used criteria include the Akaike information criterion (AIC), Bayesian information criterion (BIC), and Mallow’s Cp and their modifications. It is well-known that if the true model is among the candidate models, then BIC is strongly consistent while AIC is not when only the sample size tends to infinity and the numbers of response variables and regressors remain fixed; a setting often described as large-sample. Increasingly, more and more datasets are viewed as high-dimensional in the sense that the number of response variables (p), the number of regressors (k) and the sample size (n) tend to infinity such that p∕n→c∈(0,1) and k∕n→α∈[0,1) with α+c<1. A few recent works reported that, under high dimension, the asymptotic properties of AIC, BIC and Cp selection rules in the large-sample setting do not necessarily carry over in the high-dimensional setting. In this paper, we clarify their asymptotic properties and provide sufficient conditions for which a selection rule is strongly consistent, almost surely under specify and over specify a true model. We do not assume normality in the errors, and we only require finite fourth moment. The main tool employed is random matrix theory techniques. A consequence of this work states that, under certain mild high-dimensional conditions, if the BIC selection rule is strongly consistent then the AIC selection rule is also strongly consistent, but not vice versa. This result is in stark contrast to the large-sample result.

Empirical spectral processes for stationary state space models

In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. Furthermore, we show the weak convergence in both the space of continuous functions and in the dual space to a Gaussian process and give an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander–Rosenblatt statistic and the Cramér–von Mises statistic. We present the exact limit distributions of both statistics and show their performance through a simulation study.

The fourth moment of Dirichlet L-functions at the central value

The asymptotic formula of the fourth moment of Dirichlet L-functions at the central value was predicted in a conjecture by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, and the prime moduli case was proved by M. P. Young in 2011. This work establishes this asymptotic formula for general moduli. The work relies on the study of a special divisor sum function, called \({\mathcal {D}}_q\)-function, which plays a key role in deducing the main terms. Another key ingredient is an application of new bounds for double sums in Kloosterman sums, applied in the estimate of the error terms.

Statistical nuclear spectroscopy with [formula omitted]-normal and bivariate [formula omitted]-normal distributions and [formula omitted]-hermite polynomials

Statistical nuclear spectroscopy (also called spectral distribution method), introduced by J.B. French in late 60’s and developed in detail in the later years by his group and many other groups, is based on the Gaussian forms for the state (eigenvalue) and transition strength densities in shell model spaces with their extension to partial densities defined over shell model subspaces. The Gaussian forms have their basis in embedded random matrix ensembles with nuclear Hamiltonians consisting of a mean-field one-body part and a residual two-body part. However, following the recent random matrix results for the so called Sachdev–Ye–Kitaev model due to Verbaarschot et al, embedded random matrix ensembles with k-body interactions are re-examined and it is shown that the density of states, transition strength densities and strength functions (partial densities) in fact follow more closely the q-normal distribution (the parameter q is related to the fourth moment of these distributions with q=1 giving Gaussian and q=0 giving semi-circle form). The q-normal has the important property that it is bounded for 0≤q<1. The q-normal (also its bivariate and general multi-variate extensions) and the associated q-Hermite polynomials are studied for their properties by Bryc, Szablowski and others. Following these, in the present article, developed is statistical nuclear spectroscopy based on q-normal (univariate and bivariate) distributions and the associated q-Hermite polynomials. In particular, formulation is presented for nuclear level densities, shell model orbit occupancies, transition strengths (for electromagnetic and β and double β-decay type operators) and strength sums.

Robust Covariance Matrix Estimation for High-Dimensional Compositional Data with Application to Sales Data Analysis

Compositional data arises in a wide variety of research areas when some form of standardization and composition is necessary. Estimating covariance matrices is of fundamental importance for high-dimensional compositional data analysis. However, existing methods require the restrictive Gaussian or sub-Gaussian assumption, which may not hold in practice. We propose a robust composition adjusted thresholding covariance procedure based on Huber-type M-estimation to estimate the sparse covariance structure of high-dimensional compositional data. We introduce a cross-validation procedure to choose the tuning parameters of the proposed method. Theoretically, by assuming a bounded fourth moment condition, we obtain the rates of convergence and signal recovery property for the proposed method and provide the theoretical guarantees for the cross-validation procedure under the high-dimensional setting. Numerically, we demonstrate the effectiveness of the proposed method in simulation studies and also a real application to sales data analysis.

Modeling Realized Variance with Realized Quarticity

This paper proposes a model for realized variance that exploits information in realized quarticity. The realized variance and quarticity measures are both highly persistent and highly correlated with each other. The proposed model incorporates information from the observed realized quarticity process via autoregressive conditional variance dynamics. It exploits conditional dependence in higher order (fourth) moments in analogy to the class of GARCH models exploit conditional dependence in second moments.

On the Moments of Probability Distribution Function of Amplitude Scintillation in the Polar Region

AbstractThe distortions seen in the Global Navigation Satellite Systems radio signal caused by ionospheric irregularities in the polar region are examined through the lens of probability distributions of fluctuations in the recorded scintillation signals. The first four moments of the probability distribution function (PDF) of the amplitude scintillation are computed and analyzed for 106 events collected at Pond Inlet station [Magnetic Coordinates = (80.0°N, 2.6°E)]. As a starting point, the background variation level is investigated using similar computations carried out with about 400 30‐s window segments. The resulting distribution for the skewness (third moment) and the kurtosis (fourth moment) strongly indicate that the background fluctuations are nearly Gaussian; given the computed variances, the Rayleigh distribution hypothesis is ruled out. In the case where scintillation is present, we found that the skewness distribution nearly overlaps with the background while the excess kurtosis strongly indicates a departure from normality. More precisely, the PDF appear mostly symmetric and leptokurtic (positive kurtosis K) and exhibit, in some cases, significant flatness (K ∼ 1). These results reveal that the analyzed scintillations are statistically consistent with stationarity and therefore vanishing skewness. Moreover, the empirical moments are compared with those derived from probability densities referred in the literature regarding scintillations. We have found that none of the log‐normal, Nakagami, Rician and α–μ distributions account for the analyzed data. The generalized Gaussian distribution with a shape parameter 1 &lt; β &lt; 2 seems the most probable and the most adequate to account for the results.