A unified statistical treatment for the multi-variate joint probability expression of general random processes in the form of finite expansion terms

In this paper, for a multi-variate random process of arbitrary distribution type which can be considered to be a sum of two different random processes as a result of the natural internal mechanism of the fluctuation, or of an artificial analytical classification of the fluctuation, a unified statistical treatment for the multi-variate joint probability distribution and the multi-variate joint moment with arbitrary order of the resultant random process is introduced exactly in the form of finite expansion terms. The validity of the present theory has been experimentally confirmed not only by means of digital simulation but also by Hiroshima City street noise data. The experimental results clearly show the usefulness of the theory, and also the importance of the exact correction to the truncation error of the expansion expression.     

A simplified method for detecting information on linear correlation latent in random noise or vibration waves

It is well-known that linear correlation information is very useful in establishing the statistical properties of random noise or vibration waves. As the amount of data increases, however, calculating the linear correlation function by the usual multiplicative operations for large amounts of data can be very troublesome, and so some simplified practical methods of detecting the correlation function are desirable. In this paper, a practical method of detection of a linear correlation function latent in random noise or vibration waves is proposed, based on use of the conditional probability distribution. First, a general expression for the bivariate joint probability distribution for discrete level sampling is introduced in the form of a series expansion in orthonormal functions. The expansion coefficients give principally the information on linear and non-linear correlations, so that a general expression for the correlation function can be derived by conditional averaging. A new simplified version of this detection method is then presented, which does not require higher order statistical information. Finally, the effectiveness of the method is confirmed experimentally by applying it to some actual street noise in Hiroshima City.     

On stochastic complex beam-beam interaction models with Gaussian colored noise

This paper is to continue our study on complex beam-beam interaction models in particle accelerators with random excitations Y. Xu, W. Xu, G.M. Mahmoud, On a complex beam-beam interaction model with random forcing [Physica A 336 (2004) 347-360]. The random noise is taken as the form of exponentially correlated Gaussian colored noise, and the transition probability density function is obtained in terms of a perturbation expansion of the parameter. Then the method of stochastic averaging based on perturbation technique is used to derive a Fokker-Planck equation for the transition probability density function. The solvability condition and the general transforms using the method of characteristics are proposed to obtain the approximate expressions of probability density function to order ε.Also the exact stationary probability density and the first and second moments of the amplitude are obtained, and one can find when the correlation time equals to zero, the result is identical to that derived from the Stratonovich-Khasminskii theorem for the same model under a broad-band excitation in our previous work.     

Elastic Half-Plane under Random Displacement Excitations on the Boundary

We study in this paper a respond of an elastic half-plane to random boundary excitations. We treat both the white noise excitations and more generally, homogeneous random fluctuations of displacements prescribed on the boundary. Solutions to these problems are inhomogeneous random fields which are however homogeneous with respect to the longitudinal coordinate. This is used to represent the displacements as series expansions involving a complete set of deterministic functions with corresponding random coefficients. We construct the Karhunen-Loève (K-L) series expansion which is based on the eigen-decomposition of the correlation operator. The K-L expansion can be used to calculate the statistical characteristics of other functionals of interest, in particular, the strain and stress tensors and the elastic energy tensor. This work is supported partly by the RFBR Grant N 06-01-00498. I. Shalimova acknowledges the host institute WIAS, Berlin, and the support of DFG, under Grant SA 861/6-1 of 2008.     

An improved technique on the stochastic functional approach for randomly rough surface scattering –Numerical-analytical Wiener analysis

This paper proposes an improved technique on the stochastic functional approach for randomly rough surface scattering. Its first application is made on a TE plane wave scattering from a Gaussian random surface having perfect conductivity with infinite extent. The random wavefield becomes a ‘stochastic Floquet form’ represented by a Wiener–Hermite expansion with unknown expansion coefficients called Wiener kernels. From the effective boundary condition as a model of the random surface, a series of integral equations determining the Wiener kernels are obtained. By applying a quadrature method to the first three order hierarchical equations, a matrix equation is derived. By solving that matrix equation, the exact Wiener kernels up to second order are numerically obtained. Then the incoherent scattering cross-section and the optical theorem are calculated. A prediction is that the optical theorem always holds, which is derived from previous work is confirmed in a numerical sense. It is then concluded that the improved technique is useful.     

Approximation of the Duffing oscillator frequency response function using the FPK equation

Although a great deal of work has been carried out on nonlinear structural dynamic systems under random excitation, there has been a comparatively small amount of this work concentrating on the calculation of the quantities commonly measured in structural dynamic tests. Perhaps the most fundamental of these quantities is the frequency response function (FRF). A number of years ago, Yar and Hammond took an interesting approach to estimating the FRF of a Duffing oscillator system which was based on an approximate solution of the Fokker-Planck-Kolmogorov equation. Despite reproducing the general features of the statistical linearisation estimate, the approximation failed to show the presence of the poles at odd multiples of the primary resonance which are known to occur experimentally. The current paper simply extends the work of Yar and Hammond to a higher-order of approximation and is thus able to show the existence of a third ‘harmonic’ in the FRF. A comparison is made with previous work where an approximation to the FRF was computed using the Volterra series.     

Kernel methods and flexible inference for complex stochastic dynamics

Approximation theory suggests that series expansions and projections represent standard tools for random process applications from both numerical and statistical standpoints. Such instruments emphasize the role of both sparsity and smoothness for compression purposes, the decorrelation power achieved in the expansion coefficients space compared to the signal space, and the reproducing kernel property when some special conditions are met. We consider these three aspects central to the discussion in this paper, and attempt to analyze the characteristics of some known approximation instruments employed in a complex application domain such as financial market time series. Volatility models are often built ad hoc, parametrically and through very sophisticated methodologies. But they can hardly deal with stochastic processes with regard to non-Gaussianity, covariance non-stationarity or complex dependence without paying a big price in terms of either model mis-specification or computational efficiency. It is thus a good idea to look at other more flexible inference tools; hence the strategy of combining greedy approximation and space dimensionality reduction techniques, which are less dependent on distributional assumptions and more targeted to achieve computationally efficient performances. Advantages and limitations of their use will be evaluated by looking at algorithmic and model building strategies, and by reporting statistical diagnostics.     

Generalized unscented Kalman filtering based radial basis function neural network for the prediction of ground radioactivity time series with missing data

On the assumption that random interruptions in the observation process are modeled by a sequence of independent Bernoulli random variables, we firstly generalize two kinds of nonlinear filtering methods with random interruption failures in the observation based on the extended Kalman filtering (EKF) and the unscented Kalman filtering (UKF), which were shortened as GEKF and GUKF in this paper, respectively. Then the nonlinear filtering model is established by using the radial basis function neural network (RBFNN) prototypes and the network weights as state equation and the output of RBFNN to present the observation equation. Finally, we take the filtering problem under missing observed data as a special case of nonlinear filtering with random intermittent failures by setting each missing data to be zero without needing to pre-estimate the missing data, and use the GEKF-based RBFNN and the GUKF-based RBFNN to predict the ground radioactivity time series with missing data. Experimental results demonstrate that the prediction results of GUKF-based RBFNN accord well with the real ground radioactivity time series while the prediction results of GEKF-based RBFNN are divergent.     

干涉型光纤传感器随机相位漂移统计特性研究

提出描述干涉型光纤传感器随机相位漂移Δφ（ｔ）物理模型，给出了Δφ（ｔ）的自相关函数和功率谱密度函数，还给出了随机相位漂移周期的统计分布．     

Calculation of Partition Function of QCD at Finite Chemical Potential

In equilibrium statistical field theory, the partition function has fundamental importance. In this paper we propose a direct and general method for calculating the partition function and equation of state of QCD at finite chemical potential. It is found that the partition function is totally determined by the dressed quark propagator at finite chemical potential up to a multiplicative constant. From this a criterion for the phase transition between the Nambu and the Wigner phases is obtained. This general method is applied to two specific cases： the free quark theory and QCD with a model dressed quark propagator having confinement features. In the first case, the standard Fermi distribution at T = 0 is reproduced. In the second case, we apply the conclusion in previous works to obtain the dressed quark propagator at finite chemical potential and find the unphysical result that the baryon number density vanishes for all values of chemical potential. The reason for this result is discussed.     

Adaptive sparse polynomial chaos expansion based on least angle regression

Polynomial chaos (PC) expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable (so-called polynomial chaos) basis. The number of terms to be computed grows dramatically with the size of the input random vector, which makes the computational cost of classical solution schemes (may it be intrusive (i.e. of Galerkin type) or non intrusive) unaffordable when the deterministic finite element model is expensive to evaluate.To address such problems, the paper describes a non intrusive method that builds a sparse PC expansion. First, an original strategy for truncating the PC expansions, based on hyperbolic index sets, is proposed. Then an adaptive algorithm based on least angle regression (LAR) is devised for automatically detecting the significant coefficients of the PC expansion. Beside the sparsity of the basis, the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon. The accuracy of the PC metamodel is checked using an estimate inspired by statistical learning theory, namely the corrected leave-one-out error. As a consequence, a rather small number of PC terms are eventually retained (sparse representation), which may be obtained at a reduced computational cost compared to the classical “full” PC approximation. The convergence of the algorithm is shown on an analytical function. Then the method is illustrated on three stochastic finite element problems. The first model features 10 input random variables, whereas the two others involve an input random field, which is discretized into 38 and 30 ? 500 random variables, respectively.     

Diagrammatic construction of an irreducible basis of algebraic invariants for the sixteen vertex model

According to a fundamental theorem by Hilbert the partition function and other physically relevant properties of the general sixteen vertex model must be expressible algebraically in terms of a small number of basic invariants with respect to the symmetry group of the model. The construction of such an irreducible set of algebraic invariants has been described in two earlier papers by Graff and Hijmans^1,2). In the present paper it is shown that this basic set can be obtained in a simpler and more elegant way by means of a diagrammatic technique, based on scalar and vectorial multiplication of three kinds of construction elements and the use of six reduction rules for diagrams.     

Scattering of a plane wave from a periodic random surface: a probabilistic approach

This paper deals with a probabilistic formulation of the wave scattering from a periodic random surface. When a plane wave is incident on a random surface described by a periodic stationary stochastic process, it is shown by a group-theoretic consideration that the scattered wave may have a stochastic Floquet form, i.e. a product of a periodic stationary random function and an exponential phase factor. Such a periodic stationary random function is then written by a harmonic series representation similar to a Fourier series, where Fourier coefficients are mutually correlated stationary processes instead of constants. The mutually correlated stationary processes are represented by Wiener - Hermite functional series with unknown coefficient functions called Wiener kernels. In case of a slightly rough surface and TE wave incidence, low-order Wiener kernels are determined from the boundary condition. Several statistical properties of the scattering are calculated and illustrated in figures.     

Comparing the structure of an emerging market with a mature one under global perturbation

In this paper we investigate the Tehran stock exchange (TSE) and Dow Jones Industrial Average (DJIA) in terms of perturbed correlation matrices. To perturb a stock market, there are two methods, namely local and global perturbation. In the local method, we replace a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series, whereas in the global method, we reconstruct the correlation matrix after replacing the original return series with Gaussian-distributed time series. The local perturbation is just a technical study. We analyze these markets through two statistical approaches, random matrix theory (RMT) and the correlation coefficient distribution. By using RMT, we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks. We find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with Gaussian-distributed time series, the mean values of correlation coefficients, the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the RMT prediction fall sharply in both markets. By comparing these two markets, we can see that the DJIA is more sensitive to global perturbations. These findings are crucial for risk management and portfolio selection.     

Stochastic Modeling of Indoor Air Temperature

Temperature is one of the main parameters describing thermal comfort and indoor air quality. In this paper we propose an approach, based on a modification of the continuous time random walk, to model the indoor air temperature. We perform a statistical analysis of the recorded time series, that allows us to point out the main statistical properties of the recorded variable. The obtained conclusions about the nature of the process lead to a continuous time random walk, that in contrast to the classical approach, models time dependence of the jumps distribution. Moreover, we show that the waiting times can be modeled by a tempered stable distribution, which yields a subdiffusive behavior in short times and diffusive behavior in longer times. Finally, by conducting a simulation study we illustrate possible applications of the presented approach in the thermal comfort monitoring and forecasting.     

Eigenvalue densities of real and complex Wishart correlation matrices

Wishart correlation matrices are the standard model for the statistical analysis of time series. The ensemble averaged eigenvalue density is of considerable practical and theoretical interest. For complex time series and correlation matrices, the eigenvalue density is known exactly. In the real case, a fundamental mathematical obstacle made it forbiddingly complicated to obtain exact results. We use the supersymmetry method to fully circumvent this problem. We present an exact formula for the eigenvalue density in the real case in terms of twofold integrals and finite sums.     

Exchange symmetry in a system of nonrelativistic spin-1/2 fermions in the Feynman quantum statistics representation

A compact representation is obtained for the quantum statistical sum of indistinguishable nonrelativistic spin-1/2 fermions in the form of Feynman path integrals which can be used as the basis to develop a fundamentally exact method of computer modeling for systems of strongly interacting electrons at nonzero temperature. A basis of symmetrized wave functions is constructed using Young symmetry operators. An exact permutation symmetrization procedure leads to an avalanche-like multiplication in the number of diagrams of linked Feynman integrals of the order of N!. The partition function can be simplified without introducing any approximations and this is performed numerically by computer by direct sorting of diagrams. The control tables obtained, containing combinatorial weights of diagrams, direct the Markov random walk process in virtual trajectory space which is achieved numerically by computer. The equilibrium characteristics of the quantum system are calculated by averaging. This approach is an expansion of the Monte Carlo-Metropolis method to systems of quantum indistinguishable particles with spin. Demonstration numerical calculations using this method were made for the simplest exchange systems, for a hydrogen molecule, a Be⁺ ion, and a Li atom. The ground state of the hydrogen molecule is reproduced with a statistical error of 0.2%. Exchange-correlation effects lead to nontrivial structural changes in the thermally excited electron shells of ions in a state of strong plasma compression.     

Solution of the Dirac Equation in Expanding Universes

Exact solutions of the Dirac equation in the Robertson–Walker space-time are obtained by an elementary separation method that represents a straightforward improvement of previous results. The radial equations are integrated by reporting them to hypergeometric equations. The separated time equations are solved exactly for three models of universe expansion and integrated by series in a case of the standard cosmological model. The integration of both radial and time equations represents an improvement of previous results.