Integer Moments Research Articles

On Statistics of Bi-Orthogonal Eigenvectors in Real and Complex Ginibre Ensembles: Combining Partial Schur Decomposition with Supersymmetry

We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated ‘non-orthogonality overlap factor’ (also known as the ‘eigenvalue condition number’) of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size {Ntimes N}. First we derive the general finite N expression for the JPD of a real eigenvalue {lambda} and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its ‘bulk’ and ‘edge’ scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue z and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its ‘bulk’ scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig (Phys Rev Lett 81(16):3367–3370, 1998), and we provide the ‘edge’ scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach (The distribution of overlaps between eigenvectors of Ginibre matrices, 2018. arXiv:1801.01219).

Conditional hyperbolic quadrature method of moments for kinetic equations

The conditional quadrature method of moments (CQMOM) was introduced by Yuan and Fox (2011) [4] to reconstruct a velocity distribution function (VDF) from a finite set of its integer moments. The reconstructed VDF takes the form of a sum of weighted Dirac delta functions in velocity phase space, and provides a closure for the spatial flux term in the corresponding kinetic equation. The CQMOM closure for the flux leads to a weakly hyperbolic system of moment equations. In subsequent work by Chalons et al. (2010) [8], the Dirac delta functions were replaced by Gaussian distributions, which make the moment system hyperbolic but at the added cost of dealing with continuous distributions. Here, a hyperbolic version of CQMOM is proposed that uses weighted Dirac delta functions. While the moment set employed for multi-Gaussian and conditional HyQMOM (CHyQMOM) are equivalent, the latter is able to access all of moment space whereas the former cannot (e.g. arbitrary values of the fourth-order velocity moment in 1-D phase space with two nodes). By making use of the properties of CHyQMOM in 2-D phase space, it is possible to control a symmetrical subset of the optimal moments from Fox (2009) [24]. Furthermore, the moment sets for 2-D problems are smaller for CHyQMOM than in the original CQMOM thanks to a judicious choice of the velocity abscissas in phase space.

A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures

A theory of intermittency differentiation is developed for a general class of 1D Infinitely Divisible Multiplicative Chaos measures. The intermittency invariance of the underlying infinitely divisible field is established and utilized to derive a Feynman-Kac equation for the distribution of the total mass of the limit measure by considering a stochastic flow in intermittency. The resulting equation prescribes the rule of intermittency differentiation for a general functional of the total mass and determines the distribution of the total mass and its dependence structure to the first order in intermittency. A class of non-local functionals of the limit measure extending the total mass is introduced and shown to be invariant under intermittency differentiation making the computation of the full high temperature expansion of the total mass distribution possible in principle. For application, positive integer moments and covariance structure of the total mass are considered in detail.

Ultrasound parametric imaging of hepatic steatosis using the homodyned-K distribution: An animal study

Hepatic steatosis is an abnormal state where excess lipid mass is accumulated in hepatocyte vesicles. Backscattered ultrasound signals received from the liver contain useful information regarding the degree of steatosis in the liver. The homodyned-K (HK) distribution has been demonstrated as a general model for ultrasound backscattering. The estimator based on the first three integer moments (denoted as “FTM”) of the intensity has potential for practical applications because of its simplicity and low computational complexity. This study explored the diagnostic performance of HK parametric imaging based on the FTM method in the assessment of hepatic steatosis. Phantom experiments were initially conducted using the sliding window technique to determine an appropriate window size length (WSL) for HK parametric imaging. Subsequently, hepatic steatosis was induced in male Wistar rats fed a methionine- and choline-deficient (MCD) diet for 0 (i.e., normal control), 1, 2, 4, 6, and 8 weeks (n = 36; six rats in each group). After completing the scheduled MCD diet, ultrasound B-mode and HK imaging of the rat livers were performed in vivo and histopathological examinations were conducted to score the degree of hepatic steatosis. HK parameters μ (related to scatterer number density) and k (related to scatterer periodicity) were expressed as functions of the steatosis stage in terms of the median and interquartile range (IQR). Receiver operating characteristic (ROC) curve analysis was conducted to assess the diagnostic performance levels of the μ and k parameters. The results showed that an appropriate WSL for HK parametric imaging is seven times the pulse length of the transducer. The median value of the μ parameter increased monotonically from 0.194 (IQR: 0.18–0.23) to 0.893 (IQR: 0.64–1.04) as the steatosis stage increased. Concurrently, the median value of the k parameter increased from 0.279 (IQR: 0.26–0.31) to 0.5 (IQR: 0.41–0.54) in the early stages (normal to mild) and decreased to 0.39 (IQR: 0.29–0.45) in the advanced stages (moderate to severe). The areas under the ROC curves obtained using (μ, k) were (0.947, 0.804), (0.914, 0.575), and (0.813, 0.604) for the steatosis stages of ≥mild, ≥moderate, and ≥severe, respectively. The current findings suggest that ultrasound HK parametric imaging based on FTM estimation has great potential for future clinical diagnoses of hepatic steatosis.

Asymptotic Normality Distribution of Simulated Minimum Hellinger Distance Estimators for Continuous Models

Certain distributions do not have a closed-form density, but it is simple to draw samples from them. For such distributions, simulated minimum Hellinger distance (SMHD) estimation appears to be useful. Since the method is distance-based, it happens to be naturally robust. This paper is a follow-up to a previous paper where the SMHD estimators were only shown to be consistent; this paper establishes their asymptotic normality. For any parametric family of distributions for which all positive integer moments exist, asymptotic properties for the SMHD method indicate that the variance of the SMHD estimators attains the lower bound for simulation-based estimators, which is based on the inverse of the Fisher information matrix, adjusted by a constant that reflects the loss of efficiency due to simulations. All these features suggest that the SMHD method is applicable in many fields such as finance or actuarial science where we often encounter distributions without closed-form density.

An effective approach for probabilistic lifetime modelling based on the principle of maximum entropy with fractional moments

The paper presents a fractional moment method for probabilistic lifetime modelling of uncertain engineering systems. A novel feature of the method is the use of fractional moments, as opposed to integer moments commonly used so far in the structural reliability literature. The fractional moments are calculated from a small, simulated sample of remaining useful life of the system. And the fractional exponents that are used to model the system lifetime distribution are determined through the entropy maximization process, rather than assigned by an analyst in prior. Together with the theory of copula, the efficiency and accuracy of the proposed method are illustrated by the probabilistic lifetime modelling of several dynamical and discontinuous stochastic systems.

New scheme for implementing the method of moments with interpolative closure

ABSTRACTThis article presents a new scheme for implementing the method of moments with interpolative closure (MOMIC), which was proposed by Frenklach and Harris in 1987. The new scheme can be implemented with arbitrary moment sequences. A new interpolation scheme, which is implemented among the square root of logarithms of newly defined normalized moments, was verified to be suitable for interpolation with only three known data points. The new scheme was applied to studies on Brownian coagulation in both free-molecular and continuum-slip regimes. The new MOMIC with integer moment sequence was verified to exhibit nearly identical accuracy to its original version as well as other state-of-the-art numerical methods, including QMOM, TEMOM, and log MM, and higher accuracy than these referenced numerical methods as the fractional moment sequence was employed. The study enhances the capability of the MOMIC for solving population balance equations.© 2017 American Association for Aerosol Research

Application of the Discrete Geometrical Invariants to the Quantitative Monitoring of the Electrochemical Background

In this paper, we apply the statistics of the fractional moments (SFM) and discrete geometrical sets/invariants (DGI) for explain of the temporal evolution of the electrochemical background. For analysis of this phenomenon, we apply the internal correlation factor (ICF) and proved that integral curves expressed in the form of voltammograms (VAGs) are more sensitive in comparison with their derivatives. For analysis of the VAGs (integral curves), we propose the set of the quantitative parameters that form the invariant DGI curves of the second and the fourth orders, correspondingly. The method of their calculation based on the generalization of the well-known Pythagor’s theorem is described. The quantitative parameters that determine these DGI allow monitoring the background of the electrochemical solution covering the period of 1-1000 measurements for two types of electrode (Pt and C) and notice the specific peculiarities that characterize each electrode material. The total set of 1000 measurements was divided on 9 parts (1-100, 101-200, 201-300, …, 901-1000) and the duration of each hundred set was 1300 sec. The proposed algorithm is sensitive and has a “universal” character. It can be applied for a wide set of random curves (experimental measurements) that are needed to be compared in terms of a limited number of the integer moments. The qualitative peculiarities of the background behavior for two types of electrodes (Pt and C) based on the DGI can be explained quantitatively.

Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

We consider the Beta polymer, an exactly solvable model of directed polymer on the square lattice, introduced by Barraquand and Corwin (BC) (2016 Probab. Theory Relat. Fields 1–16). We study the statistical properties of its point to point partition sum. The problem is equivalent to a model of a random walk in a time-dependent (and in general biased) 1D random environment. In this formulation, we study the sample to sample fluctuations of the transition probability distribution function (PDF) of the random walk. Using the Bethe ansatz we obtain exact formulas for the integer moments, and Fredholm determinant formulas for the Laplace transform of the directed polymer partition sum/random walk transition probability. The asymptotic analysis of these formulas at large time t is performed both (i) in a diffusive vicinity, , of the optimal direction (in space-time) chosen by the random walk, where the fluctuations of the PDF are found to be Gamma distributed; (ii) in the large deviations regime, , of the random walk, where the fluctuations of the logarithm of the PDF are found to grow with time as t1/3 and to be distributed according to the Tracy–Widom GUE distribution. Our exact results complement those of BC for the cumulative distribution function of the random walk in regime (ii), and in regime (i) they unveil a novel fluctuation behavior. We also discuss the crossover regime between (i) and (ii), identified as . Our results are confronted to extensive numerical simulations of the model.

An approximate theoretical solution to particle coagulation and gelation using a method of moments

An approximate scheme that permits a much simplified application of method of moments to the study of aerosol coagulation and gelation processes has been proposed and verified in this work. The central idea underlying the scheme is to express a fractional moment in terms of its adjacent integer moments. In this manner the moment equations are then decoupled, and can be solved separately and analytically. The crucial rules for approximating fractional moments are proposed. A method of moments using the new scheme is applied to the study of aerosol coagulation characterized by various collision kernels, including Brownian kernels in the free-molecular and continuum regime and parametric gelling kernel. The computational accuracy of the proposed scheme is validated with a sectional model. The simulational results are extensively compared with previously published results.

The distributional zeta-function in disordered field theory

In this paper, we present a new mathematical rigorous technique for computing the average free energy of a disordered system with quenched randomness, using the replicas. The basic tool of this technique is a distributional zeta-function, a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which cannot be written as a series of the integer moments, can be made as small as desired. This result supports the use of integer moments of the partition function, computed via replicas, for expressing the average free energy of the system. One advantage of the proposed formalism is that it does not require the understanding of the properties of the permutation group when the number of replicas goes to zero. Moreover, the symmetry is broken using the saddle-point equations of the model. As an application for the distributional zeta-function technique, we obtain the average free energy of the disordered [Formula: see text] model defined in a [Formula: see text]-dimensional Euclidean space.

Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes

We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional Brownian motion with Hurst index $H \to 0$ (fBM0). In previous collaborations we obtained the probability distribution function (PDF) of the value of the global minimum (equivalently maximum) for the first two processes, using the {\it freezing-duality conjecture} (FDC). Here we study the PDF of the position of the maximum $x_m$ through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the $\beta$-Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary $\beta >0$ and positive integer $n$ in terms of sums over partitions. For positive moments, this expression agrees with a very recent independent derivation by Mezzadri and Reynolds. We check our results against a contour integral formula derived recently by Borodin and Gorin (presented in the Appendix A from these authors). The duality necessary for the FDC to work is proved, and on our expressions, found to correspond to exchange of partitions with their dual. Performing the limit $n \to 0$ and to negative Dyson index $\beta \to -2$, we obtain the moments of $x_m$ and give explicit expressions for the lowest ones. Numerical checks for the GUE polynomials, performed independently by N. Simm, indicate encouraging agreement. Some results are also obtained for moments in Laguerre, Hermite-Gaussian, as well as circular and related ensembles. The correlations of the position and the value of the field at the minimum are also analyzed.

Stochastic areas of diffusions and applications

In this paper we study the stochastic area swept by a regular time-homogeneous diffusion till a stopping time. This unifies some recent literature in this area. Through stochastic time-change we establish a link between the stochastic area and the stopping time of another associated time-homogeneous diffusion. Then we characterize the Laplace transform and integer moments of the stochastic area in terms of the eigenfunctions of the associated diffusion. We show applications of the results to a new structural model of default (Yildirim [28]) and the Omega risk model of bankruptcy in risk analysis (Gerber, Shiu and Yang [11]).

Generalized TEMOM Scheme for Solving the Population Balance Equation

This article proposes a novel generalized Taylor expansion method of moments (TEMOM) scheme for solving the population balance equation. The proposed scheme can completely overcome the shortcoming of the existing TEMOM and substantially improve the accuracy for both integer and fractional moments. In the generalized TEMOM, the optimal number of equations is 2+1, where is an integer greater than zero. The existing TEMOM is a special case of the generalized TEMOM when is 1. The generalized TEMOM was tested for aerosols undergoing Brownian coagulation in the continuum regime, and it was verified to achieve nearly the same accuracy as the quadrature method of moments (QMOM) with a fractional moment sequence, and higher accuracy than the QMOM with an integer moment sequence. Regarding accuracy and efficiency, the generalized TEMOM scheme was verified to be a competitive method for solving the population balance equation.Copyright 2015 American Association for Aerosol Research

Partial transpose of two disjoint blocks in XY spin chains

.We consider the partial transpose of the spin reduced density matrix of two disjoint blocks in spin chains admitting a representation in terms of free fermions, such as XY chains. We exploit the solution of the model in terms of Majorana fermions and show that such partial transpose in the spin variables is a linear combination of four Gaussian fermionic operators. This representation allows to explicitly construct and evaluate the integer moments of the partial transpose. We numerically study critical XX and Ising chains and we show that the asymptotic results for large blocks agree with conformal field theory predictions if corrections to the scaling are properly taken into account.

Closed-Form Representations of the Density Function and Integer Moments of the Sample Correlation Coefficient

This paper provides a simplified representation of the exact density function of R, the sample correlation coefficient. The odd and even moments of R are also obtained in closed forms. Being expressed in terms of generalized hypergeometric functions, the resulting representations are readily computable. Some numerical examples corroborate the validity of the results derived herein.

Moments of losses during busy-periods of regular and nonpreemptive oscillating $$M^X/G/1/n$$ M X / G / 1 / n systems

This work addresses loss characteristics associated to busy-periods of regular and nonpreemptive oscillating \(M^X/G/1/n\) systems. By taking advantage of the Markov regenerative structure of the number of customers in the system and resorting to results on moments of compound mixed Poisson distributions, it proposes a fast and easy to implement recursive procedure to compute integer moments of the number of customers lost in busy-periods initiated with multiple customers in the system.

On the Growth Rate of a Linear Stochastic Recursion with Markovian Dependence

We consider the linear stochastic recursion \(x_{i+1} = a_{i}x_{i}+b_{i}\) where the multipliers \(a_i\) are random and have Markovian dependence given by the exponential of a standard Brownian motion and \(b_{i}\) are i.i.d. positive random noise independent of \(a_{i}\). Using large deviations theory we study the growth rates (Lyapunov exponents) of the positive integer moments \(\lambda _q = \lim _{n\rightarrow \infty } \frac{1}{n} \log \mathbb {E}[(x_n)^q]\) with \(q\in \mathbb {Z}_+\). We show that the Lyapunov exponents \(\lambda _q\) exist, under appropriate scaling of the model parameters, and have non-analytic behavior manifested as a phase transition. We study the properties of the phase transition and the critical exponents using both analytic and numerical methods.

Multiple regression and inverse moments improve the characterization of the spatial scaling behavior of daily streamflows in the Southeast United States

AbstractUnderstanding the spatial structure of daily streamflow is essential for managing freshwater resources, especially in poorly gaged regions. Spatial scaling assumptions are common in flood frequency prediction (e.g., index‐flood method) and the prediction of continuous streamflow at ungaged sites (e.g. drainage‐area ratio), with simple scaling by drainage area being the most common assumption. In this study, scaling analyses of daily streamflow from 173 streamgages in the southeastern United States resulted in three important findings. First, the use of only positive integer moment orders, as has been done in most previous studies, captures only the probabilistic and spatial scaling behavior of flows above an exceedance probability near the median; negative moment orders (inverse moments) are needed for lower streamflows. Second, assessing scaling by using drainage area alone is shown to result in a high degree of omitted‐variable bias, masking the true spatial scaling behavior. Multiple regression is shown to mitigate this bias, controlling for regional heterogeneity of basin attributes, especially those correlated with drainage area. Previous univariate scaling analyses have neglected the scaling of low‐flow events and may have produced biased estimates of the spatial scaling exponent. Third, the multiple regression results show that mean flows scale with an exponent of one, low flows scale with spatial scaling exponents greater than one, and high flows scale with exponents less than one. The relationship between scaling exponents and exceedance probabilities may be a fundamental signature of regional streamflow. This signature may improve our understanding of the physical processes generating streamflow at different exceedance probabilities.

Log-gamma directed polymer with fixed endpoints via the replica Bethe Ansatz

We study the model of a discrete directed polymer (DP) on a square lattice with homogeneous inverse gamma distribution of site random Boltzmann weights, introduced by Seppalainen (2012 Ann. Probab. 40 19–73). The integer moments of the partition sum, , are studied using a transfer matrix formulation, which appears as a generalization of the Lieb–Liniger quantum mechanics of bosons to discrete time and space. In the present case of the inverse gamma distribution the model is integrable in terms of a coordinate Bethe Ansatz, as discovered by Brunet. Using the Brunet-Bethe eigenstates we obtain an exact expression for the integer moments of for polymers of arbitrary lengths and fixed endpoint positions. Although these moments do not exist for all integer n, we are nevertheless able to construct a generating function which reproduces all existing integer moments and which takes the form of a Fredholm determinant (FD). This suggests an analytic continuation via a Mellin–Barnes transform and we thereby propose a FD ansatz representation for the probability distribution function (PDF) of Z and its Laplace transform. In the limit of a very long DP, this ansatz yields that the distribution of the free energy converges to the Gaussian unitary ensemble (GUE) Tracy-Widom distribution up to a non-trivial average and variance that we calculate. Our asymptotic predictions coincide with a result by Borodin et al (2013 Commun. Math. Phys. 324 215–32) based on a formula obtained by Corwin et al (2011 arXiv:1110.3489) using the geometric Robinson–Schensted–Knuth (gRSK) correspondence. In addition we obtain the dependence on the endpoint position and the exact elastic coefficient at a large time. We argue the equivalence between our formula and that of Borodin et al. As we will discuss, this provides a connection between quantum integrability and tropical combinatorics.