Inference for dynamics of continuous variables: the Extended Plefka Expansion with hidden nodes

We consider the problem of a subnetwork of observed nodes embedded into a larger bulk of unknown (i.e. hidden) nodes, where the aim is to infer these hidden states given information about the subnetwork dynamics. The biochemical networks underlying many cellular and metabolic processes are important realizations of such a scenario as typically one is interested in reconstructing the time evolution of unobserved chemical concentrations starting from the experimentally more accessible ones. We present an application to this problem of a novel dynamical mean field approximation, the Extended Plefka Expansion, which is based on a path integral description of the stochastic dynamics. As a paradigmatic model we study the stochastic linear dynamics of continuous degrees of freedom interacting via random Gaussian couplings. The resulting joint distribution is known to be Gaussian and this allows us to fully characterize the posterior statistics of the hidden nodes. In particular the equal-time hidden-to-hidden variance -- conditioned on observations -- gives the expected error at each node when the hidden time courses are predicted based on the observations. We assess the accuracy of the Extended Plefka Expansion in predicting these single node variances as well as error correlations over time, focussing on the role of the system size and the number of observed nodes.Comment: 30 pages, 6 figures, 1 Appendi

Efficient white noise sampling and coupling for multilevel Monte Carlo with non-nested meshes

When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in a MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In a MLMC setting, a good coupling is enforced and the telescoping sum is respected.Comment: 28 pages, 10 figure

Kinetics of the chiral phase transition in a linear σ\sigma model

We study the dynamics of the chiral phase transition in a linear quark-meson $\sigma$ model using a novel approach based on semiclassical wave-particle duality. The quarks are treated as test particles in a Monte-Carlo simulation of elastic collisions and the coupling to the $\sigma$ meson, which is treated as a classical field. The exchange of energy and momentum between particles and fields is described in terms of appropriate Gaussian wave packets. It has been demonstrated that energy-momentum conservation and the principle of detailed balance are fulfilled, and that the dynamics leads to the correct equilibrium limit. First schematic studies of the dynamics of matter produced in heavy-ion collisions are presented.Comment: 15 pages, 12 figures, accepted by EPJA, dedicated to memory of Walter Greiner; v2: corrected typos, added references and an acknowledgmen

A variational approach to the QCD wave functional:Dynamical mass generation and confinement

We perform a variational calculation in the SU(N) Yang Mills theory in 3+1 dimensions. Our trial variational states are explicitly gauge invariant, and reduce to simple Gaussian states in the zero coupling limit. Our main result is that the energy is minimized for the value of the variational parameter away form the perturbative value. The best variational state is therefore characterized by a dynamically generated mass scale $M$. This scale is related to the perturbative scale $\Lambda_{QCD}$ by the following relation: $\alpha_{QCD}(M)={\pi\over 4}{1\over N}$. Taking the one loop QCD $\beta$- function and $\Lambda_{QCD}=150 Mev$ we find (for N=3) the vacuum condensate ${\alpha\over \pi}= 0.008 Gev^4$.Comment: 37 pages, (1 Figure available upon request), preprint LA-UR-94-2727, PUPT-149

U.S. stock market interaction network as learned by the Boltzmann Machine

We study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented analysis shows that binarization preserves market correlation structure. Properties of distributions of external fields and couplings as well as industry sector clustering structure are studied for different historical dates and moving window sizes. We found that a heavy positive tail in the distribution of couplings is responsible for the sparse market clustering structure. We also show that discrepancies between the model parameters might be used as a precursor of financial instabilities.Comment: 15 pages, 17 figures, 1 tabl