Inverse problem for multivariate time series using dynamical latent variables

Factor analysis is a well known statistical method to describe the variability among observed variables in terms of a smaller number of unobserved latent variables called factors. While dealing with multivariate time series, the temporal correlation structure of data may be modeled by including correlations in latent factors, but a crucial choice is the covariance function to be implemented. We show that analyzing multivariate time series in terms of latent Gaussian processes, which are mutually independent but with each of them being characterized by exponentially decaying temporal correlations, leads to an efficient implementation of the expectation–maximization algorithm for the maximum likelihood estimation of parameters, due to the properties of block-tridiagonal matrices. The proposed approach solves an ambiguity known as the identifiability problem, which renders the solution of factor analysis determined only up to an orthogonal transformation. Samples with just two temporal points are sufficient for the parameter estimation: hence the proposed approach may be applied even in the absence of prior information about the correlation structure of latent variables by fitting the model to pairs of points with varying time delay. Our modeling allows one to make predictions of the future values of time series and we illustrate our method by applying it to an analysis of published gene expression data from cell culture HeLa.     

New Correlation Duality Relations for the Planar Potts Model

We introduce a new method to generate duality relations for correlation functions of the Potts model on a planar graph. The method extends previously known results, by allowing the consideration of the correlation function for arbitrarily placed vertices on the graph. We show that generally it is linear combinations of correlation functions, not the individual correlations, that are related by dualities. The method is illustrated in several non-trivial cases, and the relation to earlier results is explained. A graph-theoretical formulation of our results in terms of rooted dichromatic, or Tutte, polynomials is also given.     

Variational pair theory of the attractive and extended Hubbard model ind-dimensions

We present a variational approach for treating the Hubbard Hamiltonian in one, two and three dimensions. It is based on 2M-fermion wavefunctions which are allowed to form correlated spin-singlet pairs. Expressions for the ground state energy and correlation functions are derived in terms of general pair coefficient functions. The presented approach offers a convenient starting point for improved variational treatments that allow to include different specific types of pair correlations. We present first applications to the attractive and to the extended Hubbard model using a very simple ansatz for the pair coefficient functions. The ground state energy, chemical potential, order parameter, momentum distribution as well as spin-spin and density-density correlation functions follow from a system of coupled nonlinear equations that has to be solved selfconsistently. All quantities are given for arbitrary band-filling in one, two and three dimensions. Our results are compared with those of other approximations and for the one-dimensional case with the exact results of Krivnov and Ovchinnikov.     

The large amplitude oscillatory strain response of aqueous foam: Strain localization and full stress Fourier spectrum

The stochastic motion of a two-dimensional vesicle in linear shear flow is studied at finite temperature. In the limit of small deformations from a circle, Langevin-type equations of motion are derived, which are highly nonlinear due to the constraint of constant perimeter length. These equations are solved in the low-temperature limit and using a mean-field approach, in which the length constraint is satisfied only on average. The constraint imposes non-trivial correlations between the lowest deformation modes at low temperature. We also simulate a vesicle in a hydrodynamic solvent by using the multi-particle collision dynamics technique, both in the quasi-circular regime and for larger deformations, and compare the stationary deformation correlation functions and the time autocorrelation functions with theoretical predictions. Good agreement between theory and simulations is obtained.     

Modeling subfilter soot-turbulence interactions in Large Eddy Simulation: An a priori study

A new LES model for subfilter soot-turbulence interactions is developed based on an a priori analysis using large-scale DNS data of temporally evolving non premixed n-heptane jet flames at a jet Reynolds number of 15,000. In this work, soot formation is modeled in LES by solving explicit transport equations for soot moments, and the unclosed filtered soot moment source terms are closed by a presumed PDF approach. Due to the strong intermittency of soot fields, a previous modeling approach assumes the presumed PDF to be bimodal accounting for sooting and non-sooting subfilter regions but neglects any sub-structure of the soot distribution. In this work, the modeling framework is improved by a new presumed PDF model that explicitly accounts for the sub-structure of the sooting mode, which is modeled by a log-normal distribution. The previous and new models are assessed by means of their prediction of the filtered source terms and the filtered intermittency, and the log-normal distribution is found to significantly reduce modeling errors, in particular, for the coagulation source term. Introducing a log-normal distribution for the PDF of the sooting mode involves a large amount of additional model parameters, such as the width of the distribution and correlation coefficients among different soot moments, so model assumptions to reduce the number of model parameters are discussed by means of the DNS data. The conclusions are found to be robust with respect to a variation in the global Damköhler number in the DNS datasets. The final model formulation only requires solving two additional transport equations in LES compared to previous models, while significantly improved model predictions are obtained for the coagulation source term which is import for predicting the number of soot particles.     

The pairing correlations and nuclear shapes at very high angular momenta

The usual Strutinsky shell corrections include the pairing correlations in the BCS approach. At high-spin states the cranked intrinsic wave functions are not symmetric or antisymmetric under time reversal symmetry for general triaxial shapes. On the basis of the Hartree-Fock-Bogoliubov (HFB) approach two generalizations of the Strutinsky procedure are given to describe pairing correlations also for high-spin states and triaxial shapes. The method is applied to the neutron-deficient rare earth nucleus ¹⁵⁰Gd. It is found that pairing has an important effect on the change of nuclear deformation with increasing angular momentum. The proton pairing persists at least up to I ≈ 40.     

Decay of correlations in vibrational relaxation

We investigate the decay of initial vibrational correlations in a dilute gas mixture of diatomic molecules and structureless particles. We use the techniques of Davis and Oppenheim to derive an equation for vibrational relaxation which is suitable for correlated systems. We then use the Landau-Teller transition probabilities and solve for the one-and two-molecule distribution functions and the two-molecule correlation functions. We find that the correlations decay faster than the distribution functions, which agrees with the results of Oppenheim, Shuler,et al. for other systems.     

Diffusion-limited annihilation in inhomogeneous environments

We study diffusion-limited (on-site) pair annihilation A + A → 0 and (on-site) fusion A + A → A which we show to be equivalent for arbitrary space-dependent diffusion and reaction rates. For one-dimensional lattices with nearest neighbour hopping we find that in the limit of infinite reaction rate the time-dependent n-point density correlations for many-particle initial states are determined by the correlation functions of a dual diffusion-limited annihilation process with at most 2n particles initially. Furthermore, by reformulating general properties of annihilating random walks in one dimension in terms of fermionic anticommutation relations we derive an exact representation for these correlation functions in terms of conditional probabilities for a single particle performing a random walk with dual hopping rates. This allows for the exact and explicit calculation of a wide range of universal and non-universal types of behaviour for the decay of the density and density correlations.     

Dynamics of stochastic and nearly stochastic two-party competitions

We introduce a stochastic model to study the problem of time evolution and outcome of simple two-party competitions or battles on a lattice where each party randomly deploys its constituents or elements to the lattice. The elements have assigned strength levels that determine how competitive or effective they are against the opponent. In our models, the elements neutralize one another when they are at the same site with a combination of strength levels and numbers determining which one gains control of the site. The competitions last until complete dominance has been established by one side by eliminating the opponents or a draw is achieved (unless the time evolution is terminated via some ad hoc condition). A Markov chain approach is used to describe the time-dependent dynamics of such competitions. The advantage of the approach is that it allows us to develop a theoretical framework for describing competitive systems where a combination of random and correlated events decide the outcome. We use the approach here for studies of highly contentious stochastic battles and for that of a battle with correlated events along with stochastic events. We present the method, a simple illustrative example on how the method works and close by considering two non-trivial cases including one with a combination of stochasticity and correlations.     

AC*-algebra of the two-dimensional Ising model

We consider the two-dimensional Ising model and show how correlation functions are determined by a state of aC*-Clifford algebra. We describe how the phase transition manifests itself in terms of a jump in the index of a Fredholm operator. A connection with the Pfaffian approach is made through the theory of unitary dilations of contraction semigroups.     

Multiple scattering in random media. III. Coherent potential propagators and fluctuations

An earlier microscopic approach to the theory of the averaged resolvent operator for an electron interacting with impurities is formulated in terms of coherent propagators. We study the corrections to the coherent potential approximation arising from fluctuations. For uncorrelated positions of the impurities, the linear, restricted, and general two-body additive approximations to the treatments of fluctuations are studied. For general correlations, the linear and restricted two-body additive approximations are studied. For both coherent and bare propagators, corresponding treatments of fluctuations involve the same correlation functions for impurities.Work supported in part by the National Science Foundation under Contract No. NSF DMR 79-23213.     

Correlations without joint distributions in quantum mechanics

The use of joint distribution functions for noncommuting observables in quantum thermodynamics is investigated in the light of L. Cohen's proof that such distributions are not determined by the quantum state. Cohen's proof is irrelevant to uses of the functions that do not depend on interpreting them as distributions. An example of this, from quantum Onsager theory, is discussed. Other uses presuppose that correlations betweenp andq values depend at least on the state. But correlations may be fixed by the state even though the distribution varies from one ensemble to another represented by that state. Taking covariance as a measure of correlation, it is shown that the different commonly used joint distributions yield the same correlations for a given state. A general characterization is given for a family of distributions with this same covariance.     

Selfconsistent approximations in Mori's theory

The constitutive quantities in Mori's theory, the residual forces, are expanded in terms of time-dependent correlation functions and products of operators at t = 0, where it is assumed that the time derivatives of the observables are given by products of them. As a first consequence the Heisenberg dynamics of the observables are obtained as an expansion of the same type. The dynamic equations for correlation functions result to be selfconsistent nonlinear equations of the type known from mode-mode coupling approximations. The approach yields a necessary condition for the validity of the presented equations. As a third consequence the static correlations can be calculated from fluctuation-dissipation theorems, if the observables obey a Lie algebra. For a simple spin model the convergence of the expansion is studied. As a further test, dynamic and static correlations are calculated for a Heisenberg ferromagnet at low temperatures, where the results are compared to those of a Holstein-Primakoff treatment.     

Arbitrary truncated Levy flight: Asymmetrical truncation and high-order correlations

The generalized correlation approach, which has been successfully used in statistical radio physics to describe non-Gaussian random processes, is proposed to describe stochastic financial processes. The generalized correlation approach has been used to describe a non-Gaussian random walk with independent, identically distributed increments in the general case, and high-order correlations have been investigated. The cumulants of an asymmetrically truncated Levy distribution have been found. The behaviors of asymmetrically truncated Levy flight, as a particular case of a random walk, are considered. It is shown that, in the Levy regime, high-order correlations between values of asymmetrically truncated Levy flight exist. The source of high-order correlations is the non-Gaussianity of the increments: the increment skewness generates threefold correlation, and the increment kurtosis generates fourfold correlation.     

On the critical properties of the Edwards and the self-avoiding walk model of polymer chains

We study the two- and three-dimensional, superrenormalizable Edwards model and the self-avoiding walk model of polymers. Using a Schwinger-Dyson equation and upper and lower bounds on correlations in terms of “skeleton diagrams” [6] we establish the existence of a non-trivial continuum limit in the two- and three-dimensional, superrenormalizable Edwards model. We also prove that perturbation theory is asymptotic for the continuum correlations of these models.A fairly detailed analysis of the approach to the critical point in the self-avoiding walk model is presented. In particular, we show that η<1. In dimension d?4, we discuss rigorous consequences of the conjecture that η is non-negative: among other implications, we derive that the continuum limit is trivial and that γ=1, in d?5 dimensions, and that corrections to mean-field scaling laws are at most logarithmic in four dimensions.     

Correlations in the sand pile model: From the log-normal distribution to self-organized criticality

We have studied the approach of the Abelian sand pile model towards the stationary, self-organized criticality state. The uncorrelated limit is shown both numerically and by a simple analysis to follow the log-normal distribution. We introduce and evaluate several correlation fuctions to study the correlated region.