Mixed Moments Research Articles

New classes of density estimates of low bias

A desirable property of a kernel used for a density estimate is that it has high order as this gives low bias. Given a kernel K, a basis pj(z),j≥0, and q≥1, we show how to choose a=a0,…,aq−1′ such that K̂q(z)=K(z)∑j=0q−1ajpj(z) is a kernel of order at least q. a is given in terms of the mixed moments, EZjpk(Z). Some of the kernels involve the complex unit. A simulation study is performed to compare density estimates based on some of the proposed kernels to competitive ones including those in Withers and Nadarajah [1]. The proposed estimates are shown to have smaller biases and smaller mean squared errors.

Homogeneity constraints on the mixed moments of velocity gradient and pressure Hessian in incompressible turbulence

We derive a relation for the correlation between pressure Hessian and velocity gradient in homogeneous flows. This relation provides restrictions to the parameters in the closure models of pressure hessian in velocity gradient dynamics, and together with the Poisson equation, we could obtain an integral expression for the fourth-order moment of velocity gradient in isotropic flows. Furthermore, this relation could be generalized to shear flows, and it is approximately satisfied even in the presence of a shear and of a wall, as occurs in turbulent channel flows.

Robust image hashing based on quaternion polar complex exponential transform and image energy

To apply the quaternion moments to the image feature extraction process and improve the performance of the image hashing algorithm, we propose a hashing algorithm based on quaternion polar complex exponential transform and image energy. First, the input color image is preprocessed to form a secondary image. The secondary image is subjected to fast quaternion generic polar complex exponential transform to obtain the mixed low-order moment coefficients as the global features of the image. Second, the algorithm extracts energy change features on the edges of the secondary image and energy point features in the center of the image as energy features of the image under the YCbCr color space. Finally, the image’s mixed low-order moment and energy features are combined and scrambled to form the final hash sequence. The experimental results show that the proposed algorithm balances robustness and differentiation well. Moreover, the hash sequence is the shortest, and the algorithm has better check-all and check-accuracy rates compared with other algorithms in the copy detection experiments.

Advanced Modeling and Signal Processing Methods in Brain-Computer Interfaces Based on a Vector of Cyclic Rhythmically Connected Random Processes.

In this study is substantiated the new mathematical model of vector of electroencephalographic signals, registered under the conditions of multiple repetitions of the mental control influences of brain-computer interface operator, in the form of a vector of cyclic rhythmically connected random processes, which, due to taking into account the stochasticity and cyclicity, the variability and commonality of the rhythm of the investigated signals have a number of advantages over the known models. This new model opens the way for the study of multidimensional distribution functions; initial, central, and mixed moment functions of higher order such as for each electroencephalographic signal separately; as well as for their respective compatible probabilistic characteristics, among which the most informative characteristics can be selected. This provides an increase in accuracy in the detection (classification) of mental control influences of the brain-computer interface operators. Based on the developed mathematical model, the statistical processing methods of vector of electroencephalographic signals are substantiated, which consist of statistical evaluation of its probabilistic characteristics and make it possible to conduct an effective joint statistical estimation of the probability characteristics of electroencephalographic signals. This provides the basis for coordinated integration of information from different sensors. The use of moment functions of higher order and their spectral images in the frequency domain, as informative characteristics in brain-computer interface systems, are substantiated. Their significant sensitivity to the mental controlling influence of the brain-computer interface operator is experimentally established. The application of Bessel's inequality to the problems of reducing the dimensions (from 500 to 20 numbers) of the vectors of informative features makes it possible to significantly reduce the computational complexity of the algorithms for the functioning of brain-computer interface systems. Namely, we experimentally established that only the first 20 values of the Fourier transform of the estimation of moment functions of higher-order electroencephalographic signals are sufficient to form the vector of informative features in brain-computer interface systems, because these spectral components make up at least 95% of the total energy of the corresponding statistical estimate of the moment functions of higher-order electroencephalographic signals.

Optimal programmed control in energy minimisation problem

This paper is devoted to the construction of optimal programmed control of the motion of a material point in a medium with a random coefficient of resistance in a homogeneous field of gravity of the Earth. The control objective is to minimise the mathematical expectation of the total mechanical energy of the material point. The problem is solved by two methods proposed by the author in previous works. Both methods assume approximation of the initial stochastic problem by deterministic optimal control problems. In the first case, the transition to approximate deterministic problems is carried out by replacing the parameters representing continuous random variables by discrete-type random variables converging in distribution to the original continuous ones (the method of distribution discretisation). The meaning of such a replacement is that the resulting approximate problem can be considered no longer as a stochastic, but as a deterministic optimal control problem, and can be solved, accordingly, with the help of known standard methods. In the second case, the initial stochastic dynamic system as a result of averaging its equations is replaced first by an infinite system and then, after zeroing all moments of sufficiently high order, by a finite system of equations for mixed moments of the solution and a random parameter (method of moments). The control functions and the optimal values of the quality functional obtained by the two methods were found to be almost identical, and both methods showed good convergence.

Motzkin path decompositions of functionals in noncommutative probability

We study the decomposition of free random variables in terms of their orthogonal replicas from a new perspective. First, we show that the mixed moments of orthogonal replicas with respect to the normalized linear functional [Formula: see text] are naturally described in terms of Motzkin paths identified with reduced Motzkin words [Formula: see text]. Using this fact, we demonstrate that the mixed moments of order [Formula: see text] of free random variables with respect to the free product of normalized linear functionals are sums of the mixed moments of the orthogonal replicas of these variables with respect to [Formula: see text] with summation extending over [Formula: see text], the set of reduced Motzkin paths of length [Formula: see text]. One of the applications of this formula is a decomposition formula for mixed moments of free random variables in terms of their boolean cumulants which corresponds to the decomposition of the lattice [Formula: see text] into sublattices [Formula: see text] of partitions which are monotonically adapted to colors in [Formula: see text]. The linear functionals defined by the mixed moments of orthogonal replicas and indexed by reduced Motzkin words play the role of a generating set of the space of product functionals in which the boolean product corresponds to constant Motzkin paths and the free product corresponds to all Motzkin paths.

Random Walk on Nonnegative Integers in Beta Distributed Random Environment

We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent $\mathrm{Beta}(\mu,\mu)$ distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter $\eta$. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary parameter $\eta$ affects the return probabilities. The second is an explicit series of integrals, and we show that non-rigorous critical point asymptotics suggest that the large deviation behaviour of this half-space random walk in random environment is the same as for the analogous random walk on $\mathbb{Z}$.

Bivariate Copulas Based on Counter-Monotonic Shock Method

This paper explores the properties of a family of bivariate copulas based on a new approach using the counter-monotonic shock method. The resulting copula covers the full range of negative dependence induced by one parameter. Expressions for the copula and density are derived and many theoretical properties are examined thoroughly, including explicit expressions for prominent measures of dependence, namely Spearman’s rho, Kendall’s tau and Blomqvist’s beta. The convexity properties of this copula are presented, together with explicit expressions of the mixed moments. Estimation of the dependence parameter using the method of moments is considered, then a simulation study is carried out to evaluate the performance of the suggested estimator. Finally, an application of the proposed copula is illustrated by means of a real data set on air quality in New York City.

Magnetocaloric effect and magnetic properties of Dy2CoMnO6: Monte Carlo study

ABSTRACT We studied the magnetic properties and magnetocaloric effect of Dy2CoMnO6 (DCMO) by using the Monte Carlo study. The magnetisation vs. the temperature was found. The ferrimagnetic-paramagnetic transition was obtained around the transition temperature T C = 82 K. The value of T C was found. dM/dT and internal energy vs. the temperature have been given for several magnetic fields. The magnetic entropy changes vs. temperature have been established. The maximum of magnetic entropy changes increases with increasing the magnetic field from 2.53, 3.43, 4.19, 4.93 and 5.63 J/kg. K with h = 1, 2, 3, 4 and 5 T, respectively. Relative cooling power was found and increases with increasing the value of the magnetic field from 18 (at 1 T) to 50 J/kg (at 5 T). The thermal adiabatic temperature was also determined. The magnetisation vs. the magnetic field has been deduced for several temperatures. Our results offer comprehensive information for understanding the roles of mixed valent magnetic ions and rare earth magnetic moments on the magnetic and magnetocaloric properties.

Bootstrapping Dirac ensembles

We apply the bootstrap technique to find the moments of certain multi-trace and multi-matrix random matrix models suggested by noncommutative geometry. Using bootstrapping we are able to find the relationships between the coupling constant of these models and their second moments. Using the Schwinger–Dyson equations, all other moments can be expressed in terms of the coupling constant and the second moment. Explicit relations for higher mixed moments are also obtained.

THE METHOD OF STATISTICAL MOMENTS IN A POLYNOMIAL REGRESSION OF CORRELATION DEPENDENCE

The methods of probability theory substantiate a brief procedure that allows us to express the parameters of the polynomial regression of conditional mathematical expectation through mixed statistical moments of a system of random variables. Examples of linear and quadratic regression are implemented. In the second case, consideration is limited to the situation when the probability density of a random argument is an even function. The result was obtained without cumbersome calculations, since it was obtained using not the initial statistical moments resulting from the least squares method, but mixed central moments reflecting the type of regression curve. It is shown that in the general case, taking into account the nonlinearity of the correlation dependence only strengthens the inequality, which is the criterion for the adequacy of the regression approximation. The convergence of such a procedure is confirmed if the conditional expectation is not essentially a polynomial.

Analytical Formulas for Conditional Mixed Moments of Generalized Stochastic Correlation Process

This paper proposes a simple and novel approach based on solving a partial differential equation (PDE) to establish the concise analytical formulas for a conditional moment and mixed moment of the Jacobi process with constant parameters, accomplished by including random fluctuations with an asymmetric Wiener process and without any knowledge of the transition probability density function. Our idea involves a system with a recurrence differential equation which leads to the PDE by involving an asymmetric matrix. Then, by using Itô’s lemma, all formulas for the Jacobi process with constant parameters as well as time-dependent parameters are extended to the generalized stochastic correlation processes. In addition, their statistical properties are provided in closed forms. Finally, to illustrate applications of the proposed formulas in practice, estimations of parametric methods based on the moments are mentioned, particularly in the method of moments estimators.

A superfast Brillouin frequency shift estimation method based on the mixed moment for the distributed optical fiber sensor

To estimate Brillouin frequency shift with high accuracy and less computational burden, the mixed moment is used to estimate it for the distributed optical fiber sensor. Its computational burden is much less than that of the typical algorithms. Meanwhile, it is free of curve fitting and initialization of fitting parameters. The optimal order in the proposed algorithm is determined. The accuracy and computation time of the proposed algorithm, the typical spectrum fitting algorithms based on the pseudo-Voigt model, the quadratic polynomials, and the cross-correlation algorithm are compared for the simulated and collected Brillouin spectra with different signal-to-noise ratios. The results reveal that the computation time of the proposed algorithm is only 1/6464–1/3986, 1/33–1/24, 1/111–1/72 of that of the other three algorithms. However, its accuracy is close to that of the spectrum fitting algorithm based on the pseudo-Voigt profile and is a little higher than that of the other two algorithms.

The mixed second moment of quadratic Dirichlet L-functions over function fields

We investigate the mixed second moment involving the second derivative at the central point of a family of quadratic Dirichlet L-functions over 𝔽q(t), associated to the hyperelliptic curves of genus g. We compute the full degree five polynomial in the asymptotic expansion of the mixed second moment when the cardinality q of the finite field is fixed and the genus g tends to infinity. This is a partial analogue of classical Ingham’s result about the Riemann zeta function.

Differential Equations in Banach Spaces Multiplicatively Perturbed by Random Noise

We consider the problem to find the moment functions of the solution of the Cauchy problem for a first-order linear inhomogeneous differential equation with random coefficients in a Banach space. The problem is reduced to the initial problem for a nonrandom differential equation with ordinary and variational derivatives. We obtain explicit expressions for the mathematical expectation and the second-order mixed moment functions for the solution of the equation.

Almost Sure Large Fluctuations of Random Multiplicative Functions

Abstract We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum _{n \leq x} f(n)| \geq \sqrt{x} (\log \log x)^{1/4+o(1)}$. This is the first such bound that grows faster than $\sqrt{x}$, answering a question of Halász and proving a conjecture of Erd̋s. It is plausible that the exponent $1/4$ is sharp in this problem. The proofs work by establishing a multivariate Gaussian approximation for the sums $\sum _{n \leq x} f(n)$ at a sequence of $x$, conditional on the behaviour of $f(p)$ for all except the largest primes $p$. The most difficult aspect is showing that the conditional covariances of the sums are usually small, so the corresponding Gaussians are usually roughly independent. These covariances are related to a Euler product (or multiplicative chaos) type integral twisted by additive characters, which we study using various tools including mean value estimates for Dirichlet polynomials, high mixed moment estimates for random Euler products, and barrier arguments with random walks.

Gamma-ray blazar variability: new statistical methods of time-flux distributions

ABSTRACT Variable γ-ray emission from blazars, one of the most powerful classes of astronomical sources featuring relativistic jets, is a widely discussed topic. In this work, we present the results of a variability study of a sample of 20 blazars using γ-ray (0.1–300 GeV) observations from Fermi/LAT telescope. Using maximum likelihood estimation (MLE) methods, we find that the probability density functions that best describe the γ-ray blazar flux distributions use the stable distribution family, which generalizes the Gaussian distribution. The results suggest that the average behaviour of the γ-ray flux variability over this period can be characterized by log-stable distributions. For most of the sample sources, this estimate leads to standard lognormal distribution (α = 2). However, a few sources clearly display heavy tail distributions (MLE leads to α < 2), suggesting underlying multiplicative processes of infinite variance. Furthermore, the light curves were analysed by employing novel non-stationarity and autocorrelation analyses. The former analysis allowed us to quantitatively evaluate non-stationarity in each source – finding the forgetting rate (corresponding to decay time) maximizing the log-likelihood for the modelled evolution of the probability density functions. Additionally, evaluation of local variability allows us to detect local anomalies, suggesting a transient nature of some of the statistical properties of the light curves. With the autocorrelation analysis, we examined the lag dependence of the statistical behaviour of all the {(yt, yt + l)} points, described by various mixed moments, allowing us to quantitatively evaluate multiple characteristic time scales and implying possible hidden periodic processes.

“Ain’t the American Dream Grand”: Satirical Play in Rockstar’s Grand Theft Auto V

Rockstar Games is a video game publisher famous for its immersive and highly detailed action-adventure titles set in America, thanks to its two main franchises, Red Dead Redemption and Grand Theft Auto. This article explores how Rockstar employs the location and game play elements of Grand Theft Auto V (2013), a title that has now generated over $6 billion in worldwide sales, to critique elements of American culture, politics, and lifestyle. Presented by Rockstar co-founder Dan Houser as “the endpoint of the American dream,” Grand Theft Auto V provides a crime-laden journey through a simulated Southern California landscape (“San Andreas”). The game is awash with social and political commentary. Of interest here is how, through specific game mechanics such as “satirical play,” the title targets the concept of “the American Dream,” and exposes themes of excess consumption, fake-ness, and social decay in the Californian and broader American experience. Exposition of these themes is, however, compromised by Rockstar’s primary commitment to deliver a mainstream gameplay experience, resulting in mixed messages and moments of ludonarrative dissonance.

Multi-feature evaluation of financial contagion

Financial contagion refers to the spread of market turmoils, for example from one country or index to another country or another index. It is standardly assessed by modelling the evolution of the correlation matrix, for example of returns, usually after removing univariate dynamics with the GARCH model. However, significant events like crises visible in one financial market are usually reflected in other financial markets/countries simultaneously in several dimensions, i.e., in general, entire distributions of returns in other markets are affected. These distributions are determined/described by their expected value, variance, skewness, kurtosis and other statistics that determine the shape of the distribution function of returns, which can be based on higher (mixed) moments. These descriptive statistics are not constant over time, and, moreover, they can interreact within the given market and among the markets over time. In this article we propose, and use for the daily values of five indexes (CAC40, DAX30, DJIA, FTSE250 and WIG20) over the time period 2006–2017, a new, simple and computationally inexpensive methodology to automatically extend contagion evaluation from the evolution of the correlation matrix to the evolution of multiple higher mixed moments as well. Specifically, the joint distribution of normalized variables for each pair of indexes is modeled as a polynomial with time evolving coefficients estimated using an exponential moving average. As we can obtain any arbitrary number of evolving mixed moments this way, its dimensionality reduction using PCA (principal component analysis) is also discussed, obtaining a lower number of dominating and relatively independent features, which can each be interpreted through a polynomial that describes the corresponding perturbation of joint distribution. We obtain features that describe the interrelations among stock markets in several dimensions and that provide information about the current stage of crisis and the strength of the contagion process.

The fourth moment of derivatives of Dirichlet L-functions in function fields

We obtain the asymptotic main term of moments of arbitrary derivatives of L-functions in the function field setting. Specifically, we obtain the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus Q in {mathbb {F}}_q [T], and the asymptotic limit is as {{,mathrm{deg},}}Q longrightarrow infty . This extends the work of Tamam who obtained the asymptotic main term of low moments of L-functions, without derivatives, in the function field setting. It is also the function field q-analogue of the work of Conrey, who obtained the fourth moment of derivatives of the Riemann zeta-function.