Terms Of Cumulants Research Articles

Correcting event-by-event fluctuations in heavy-ion collisions for exact global conservation laws with the generalized subensemble acceptance method

We introduce the subensemble acceptance method 2.0 (SAM-2.0) -- a procedure to correct cumulants of a random number distribution inside a subsystem for the effect of exact global conservation of a conserved quantity to which this number is correlated, with applications to measurements of event-by-event fluctuations in heavy-ion collisions. The method expresses the corrected cumulants in terms of the cumulants inside and outside the subsystem that are not subject to the exact conservation. The derivation assumes that all probability distributions associated with the cumulants are peaked at the mean values but are otherwise of arbitrary shape. The formalism reduces to the original SAM [V. Vovchenko et al., Phys.Lett.B 811 (2020) 135868 [arXiv:2003.13905]] when applied to a coordinate space subvolume of a uniform thermal system. As the new method is restricted neither to the uniform systems nor to the coordinate space, it is applicable to fluctuations measured in heavy-ion collisions at various collision energies in different momentum space acceptances. The SAM-2.0 thus brings the experimental measurements and theoretical calculations of event-by-event fluctuations closer together, as the latter are typically performed without the account of exact global conservation.

Cumulant–Cumulant Relations in Free Probability Theory from Magnus’ Expansion

Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work, we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process, we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker–Campbell–Hausdorff formula due to A. Murua.

Skewness, kurtosis, and the fifth and sixth order cumulants of net baryon-number distributions from lattice QCD confront high-statistics STAR data

We present new results on up to $6^{th}$ order cumulants of net baryon-number fluctuations at small values of the baryon chemical potential, $\mu_B$, obtained in lattice QCD calculations with physical values of light and strange quark masses. Representation of the Taylor expansions of higher order cumulants in terms of the ratio of the two lowest order cumulants, $M_B/\sigma_B^2=\chi_1^B(T,\mu_B)/\chi_2^B(T,\mu_B)$, allows for a parameter free comparison with data on net proton-number cumulants obtained by the STAR Collaboration in the Beam Energy Scan at RHIC. We show that recent high statistics data on skewness and kurtosis ratios of net proton-number distributions, obtained at beam energy $\sqrt{s_{_{NN}}}=54.4$ GeV, agree well with lattice QCD results on cumulants of net baryon-number fluctuations close to the pseudo-critical temperature, $T_{pc}(\mu_B)$, for the chiral transition in QCD. We also present first results from a next-to-leading order expansion of $5^{th}$ and $6^{th}$ order cumulants on the line of pseudo-critical temperatures.

About the foundation of the Kubo generalized cumulants theory: a revisited and corrected approach

More than fifty years ago, in a couple of seminal works (Kubo 1962 J. Phys. Soc. Japan 17 1100–20; Kubo 1963 J. Math. Phys. 4 174–83) Kubo introduced the important idea of generalized cumulants, extending to stochastic operators this concept implicitly introduced by Laplace in 1810. Kubo’s idea has been applied in several branches of physics, where the result of the average process is a Lioville operator or an effective time evolution operator for the density matrix of spin systems or the reduced density matrix for boson-fermions etc. Despite this success, the theoretical developments in these Kubo works pose problems that were highlighted many years ago by Fox and van Kampen and never solved. These weaknesses and errors, in particular concerning the factorization property of exponentials of cumulants and the explicit expressions that give generalized cumulants in terms of generalized moments and vice-versa, caused some perplexity (and confusion) about the possible application of this procedure, limiting its use, in practice. In the present paper, we give a sound ground to the approach to cumulant operators, working in a general framework that shows the potentiality of the old Kubo idea, today not yet fully exploited. It results that, for the same moment operators, different definitions of generalized cumulants can be adopted. A general Kubo–Meeron closed-form formula giving cumulant operators in terms of moment operators cannot be obtained, but the reverse one, cumulants in terms of operators, is given and, noticeably, formally it does not depend on the specific nature of the moments, but just on the definition of the generalized cumulants.

Standardized cumulants of flow harmonic fluctuations

The distribution of flow harmonics in heavy ion experiment can be characterized by standardized cumulants. We first model the ellipticity and power parameters of the elliptic-power distribution by employing MC-Glauber model. Then we use the elliptic-power distribution together with the hydrodynamic linear response approximation to study the two dimensional standardized cumulants of elliptic and triangular flow ($v_2$ and $v_3$) distribution. For the second harmonic, it turns out that finding two dimensional cumulants in terms of $2q$-particle correlation functions $c_2\{2q\}$ is limited to the skewness. We also show that $c_3\{2\}$, $c_3\{4\}$, and $c_3\{6\}$, are related to the second, fourth, and sixth standardized cumulants of the $v_3$ distribution, respectively. The cumulant $c_{n}\{2q\}$ can be also written in terms of $v_n\{2q\}$. Specifically, $-(v_3\{4\}/v_3\{2\})^4$ turns out to be the kurtosis of the $v_3$ event-by-event fluctuation distribution. We introduce a new parametrization for the distribution $p(v_3)$ with $v_3\{2\}$, kurtosis and sixth-order standardized cumulant being its free parameters. Compared to the Gaussian distribution, it indicates a more accurate fit with experimental results. Finally, we compare the kurtosis obtained from simulation with that of extracted from experimental data for the $v_3$ distribution.

Monotone, free, and boolean cumulants: A shuffle algebra approach

The theory of cumulants is revisited in the “Rota way”, that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each other using the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.

Relations between cumulants in noncommutative probability

We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.

Cumulants of the q-semicircular Law, Tutte Polynomials, and Heaps

AbstractThe q-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of itsmoments in terms ofmatchings, where q follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case q = 0 of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier–Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.

Cumulants as iterated integrals

A formula expressing cumulants in terms of iterated integrals of the distribution function is derived. It generalizes results of Jones and Balakrishnan who computed expressions for cumulants up to order 4.

Approximations of impulse response curves based on the generalized moving Gaussian distribution function

We derive approximations (in time) for impulse response curves not based on the Gaussian distribution (like the well-known Edgeworth expansions), but based on other suitable distribution functions. In general, we derive the full expansion for such an approximation based on an approximative function, the derivatives and the cumulants of this approximative function and on the cumulants of the impulse response curve itself, see formula (39). Suitable distribution functions exhibit a skewed profile which offers better opportunities for approximations of typical impulse response curves than approximations based on the symmetrical Gaussian distribution. As an example of such a suitable approximative distribution function we study more in detail the Generalized Moving Gaussian distribution Z t = 1 M 0 t ν exp − α t − β / t 2 , 0 < t < ∞ , α > 0 , β > 0 , − ∞ < ν < ∞ , with the normalization M 0 = 2 β α ν + 1 / 2 exp 2 α β K ν + 1 2 α β . A common feature in convection–diffusion problems is that the Laplace transform of an impulse response curve can be described by a single exponential as a( s) exp ( xb( s)). We characterize the first four cumulants in terms of the Taylor coefficients of the functions a( s) and b( s). Moreover, it is shown that the temporal cumulants are linear in the spatial variable. This material is applied to several examples and it is shown that the proposed approximative Generalized Moving Gaussian distributions for impulse responses in the field of convection–diffusion type cases perform in general better than the expansions based on the pure Gaussian distribution. Based on this type of approximation other properties based on the original impulse response curve can be found along analytical ways.

Genus Expansion for Real Wishart Matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central-limit-type theorem for the asymptotic distribution around the expected value.

Moments from cumulants and vice versa

Moments and cumulants are expressed in terms of each other using Bell polynomials. Inbuilt routines for the latter make these expressions amenable to use by algebraic manipulation programs. One of the four formulas given is an explicit version of Kendall's use of Faa di Bruno's chain rule to express cumulants in terms of moments.

Modeling the slowing of neurofilament transport along the mouse sciatic nerve

Neurofilaments are transported along axons in the slow component of axonal transport. The average rate of movement is generally quoted as several millimeters or tenths of a millimeter per day, but this rate is known to decrease while the neurofilaments are in transit due to spatial and temporal factors that are not understood. We have previously presented a stochastic model for neurofilament movement in vivo based on the transport kinetics of single neurofilaments observed by time-lapse fluorescence imaging in cultured neurons. The model took into account multiple velocity states and was only accessible through computational simulations. In simulations of the movement of a pulse of radiolabeled neurofilaments, this model generated a Gaussian wave which closely matched the experimental data. Here we present a simpler model with only three velocity states which is more amenable to analytical approaches. We show that the transport wave can be fully described by the mean and variance and we present analytical solutions for these cumulants in terms of the kinetic parameters of the model. We use the resulting expressions to examine the slowing of neurofilament transport in the mouse sciatic nerve. We show that the slowing is accompanied by an increase in the spread of the wave and that these changes are most readily explained by a change in the rate at which neurofilaments reverse their direction of movement. This suggests that the directionality of neurofilament transport in axons may be under spatial and/or temporal control and that alterations in the directionality of neurofilament transport could provide a mechanism for regulating the transport and distribution of these cytoskeletal polymers along axons.

COMBINATORIAL RELATION BETWEEN FREE CUMULANTS AND JACOBI PARAMETERS

We find a formula which expresses free and conditionally free cumulants in terms of Jacobi parameters. This leads to some necessary conditions for free and conditionally free infinite divisibility. We also express conditionally free cumulants of two measures in terms of their free cumulants.

Cumulants, lattice paths, and orthogonal polynomials

A formula expressing free cumulants in terms of Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and the Lagrange inversion formula. For the converse we discuss Gessel–Viennot theory to express Hankel determinants in terms of various cumulants.

Free Cumulants and Enumeration of Connected Partitions

A combinatorial formula is derived which expresses free cumulants in terms of classical cumulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected partitions, respectively. The proof relies on Möbius inversion on the partition lattice.

Free Cumulants and Enumeration of Connected Partitions

A combinatorial formula is derived which expresses free cumulants in terms of classical cumulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected partitions, respectively. The proof relies on Möbius inversion on the partition lattice.

Roughness distributions for 1/f alpha signals.

The probability density function (PDF) of the roughness, i.e., of the temporal variance, of 1/f(alpha) noise signals is studied. Our starting point is the generalization of the model of Gaussian, time periodic, 1/f noise, discussed in our recent Letter [Phys. Rev. Lett. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions (alpha < or = 1/2, 1/2 < alpha < or = 1, and 1< alpha), distinguished by the scaling of the cumulants in terms of the microscopic scale and the total length of the period. Various analytical representations of the PDF allow for a precise numerical evaluation of the scaling function of the PDF for any alpha. A simulation of the periodic process makes it possible to study also nonperiodic, thus experimentally more relevant, signals on relatively short intervals embedded in the full period. We find that for alpha < or = 1/2 the scaled PDFs in both the periodic and the nonperiodic cases are Gaussian, but for alpha > 1/2 they differ from the Gaussian and from each other. Both deviations increase with growing alpha. That conclusion, based on numerics, is reinforced by analytic results for alpha = 2 and alpha-->infinity, in the latter limit the scaling function of the PDF being finite for periodic signals, but developing a singularity for the aperiodic ones. Finally, an overview is given for the scaling of cumulants of the roughness and the various scaling regions in arbitrary dimensions. We suggest that our theoretical and numerical results open a different perspective on the data analysis of 1/f(alpha) processes.

Robust voice activity detection using higher-order statistics in the LPC residual domain

This paper presents a robust algorithm for voice activity detection (VAD) based on newly established properties of the higher order statistics (HOS) of speech. Analytical expressions for the third and fourth-order cumulants of the LPC residual of short-term speech are derived assuming a sinusoidal model. The flat spectral feature of this residual results in distinct characteristics for these cumulants in terms of phase, periodicity and harmonic content and yields closed-form expressions for the skewness and kurtosis. Important properties about these cumulants and their similarity with the autocorrelation function are revealed from this exploratory part. They show that the HOS of speech are sufficiently distinct from those of Gaussian noise and can be used as a basis for speech detection. Their immunity to Gaussian noise makes them particularly useful in algorithms designed for low SNR environments. The proposed VAD algorithm combines HOS metrics with second-order measures, such as SNR and LPC prediction error, to classify speech and noise frames. The variance of the HOS estimators is quantified and used to yield a likelihood measure for noise frames. Moreover, a voicing condition for speech frames is derived based on the relation between the skewness and kurtosis of voiced speech. The performance of the algorithm is compared to the ITU-T G.729B VAD in various noise conditions, and quantified using the probability of correct and false classifications. The results show that the proposed algorithm has an overall better performance than G.729B, with noticeable improvement in Gaussian-like noises, such as street and parking garage, and moderate to low SNR.

The Early History of the Cumulants and the Gram‐Charlier Series

SummaryThe early history of the Gram‐Charlier series is discussed from three points of view: (1) a generalization of Laplace's central limit theorem, (2) a least squares approximation to a continuous function by means of Chebyshev‐Hermite polynomials, (3) a generalization of Gauss's normal distribution to a system of skew distributions. Thiele defined the cumulants in terms of the moments, first by a recursion formula and later by an expansion of the logarithm of the moment generating function. He devised a differential operator which adjusts any cumulant to a desired value. His little known 1899 paper in Danish on the properties of the cumulants is translated into English in the Appendix.