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Multiscale Gaussian graphical models and algorithms for large-scale inference

Choi, Myung Jin, S.M. Massachusetts Institute of Technology
Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 123 p.
ENG
Relevância na Pesquisa
36.1%
Graphical models provide a powerful framework for stochastic processes by representing dependencies among random variables compactly with graphs. In particular, multiscale tree-structured graphs have attracted much attention for their computational efficiency as well as their ability to capture long-range correlations. However, tree models have limited modeling power that may lead to blocky artifacts. Previous works on extending trees to pyramidal structures resorted to computationally expensive methods to get solutions due to the resulting model complexity. In this thesis, we propose a pyramidal graphical model with rich modeling power for Gaussian processes, and develop efficient inference algorithms to solve large-scale estimation problems. The pyramidal graph has statistical links between pairs of neighboring nodes within each scale as well as between adjacent scales. Although the graph has many cycles, its hierarchical structure enables us to develop a class of fast algorithms in the spirit of multipole methods. The algorithms operate by guiding far-apart nodes to communicate through coarser scales and considering only local interactions at finer scales. The consistent stochastic structure of the pyramidal graph provides great flexibilities in designing and analyzing inference algorithms. Based on emerging techniques for inference on Gaussian graphical models...

Projective integration of expensive multiscale stochastic simulation; Projective integration of expensive stochastic processes

Chen, X.; Roberts, A.; Kevrekidis, I.
Fonte: Cambridge University Press; United Kingdom Publicador: Cambridge University Press; United Kingdom
Tipo: Conference paper
Publicado em //2011 EN
Relevância na Pesquisa
46.27%
Consider the case when a microscale simulator is too expensive for long time simulations necessary to determine macroscale dynamics. Projective integration uses bursts of the microscale simulator, using microscale time steps, and computes an approximation to the system over a macroscale time step by extrapolation. Projective integration has the potential to be an effective method to compute the long time dynamic behaviour of multiscale systems. However, many multiscale systems are influenced by noise. Thus it is important to consider the projective integration of such systems. By the maximum likelihood estimation, we estimate a linear stochastic differential equation from short bursts of data. The analytic solution of the linear stochastic differential equation then estimates the solution over a macroscale time step. We explore how the noise affects the projective integration in different methods. Monte Carlo simulation suggests design parameters offering stability and accuracy for the algorithms. The algorithms developed here may be applied to compute the long time dynamic behaviour of multiscale systems with noise.; http://conferences.science.unsw.edu.au/CTAC2010/index.php; Xiaopeng Chen, Anthony J. Roberts and Ioannis Kevrekidis; The 15th Biennial Computational Techniques and Applications Conference...

Optimal Sampling Strategies for Multiscale Stochastic Processes

Ribeiro, Vinay Joseph; Riedi, Rudolf H.; Baraniuk, Richard G.; Ribeiro, Vinay Joseph; Riedi, Rudolf H.; Baraniuk, Richard G.
Fonte: Universidade Rice Publicador: Universidade Rice
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
96.42%
Journal Paper; This paper studies multiscale stochastic processes which are random processes organized on the nodes of a tree. The random variables at different levels on the tree represent time series of samples of a stochastic process at different temporal or spatial cales. We focus on classes of multiscale processes with additional statistical structure connecting scales and seek an optimal linear estimator of coarse scale nodes using an incomplete set of nodes at a finer time scale. We prove that the optimal solution for any tree with so-called independent innovations is readily given by a polynomial-time algorithm which we term the water-filling algorithm. The optimal solutions vary dramatically with the correlation structure of the multiscale process. For so-called scale-invariant trees and processes with positive correlation progression through scales, uniformly spaced leaves are optimal and clustered leaves are the worst possible. For processes with negative correlation progression, uniformly spaced leaves are the worst possible. Our results have implications for network traffic estimation, sensor network design, and environmental monitoring.

Optimal Sampling Strategies for Multiscale Stochastic Processes

Ribeiro, Vinay Joseph; Riedi, Rudolf H.; Baraniuk, Richard G.; Ribeiro, Vinay Joseph; Riedi, Rudolf H.; Baraniuk, Richard G.
Fonte: Universidade Rice Publicador: Universidade Rice
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
76.22%
Journal Paper; In this paper, we determine which non-random sampling of fixed size gives the best linear predictor of the sum of a finite spatial population. We employ different multiscale superpopulation models and use the minimum mean-squared error as our optimality criterion. In a multiscale superpopulation tree models, the leaves represent the units of the population, interior nodes represent partial sums of the population, and the root node represents the total sum of the population. We prove that the optimal sampling pattern varies dramatically with the correlation structure of the tree nodes. While uniform sampling is optimal for trees with "positive correlation progression," it provides the worst possible sampling with "negative correlation progression." As an analysis tool, we introduce and study a class of independent innovations trees that are of interest in their own right. We derive a fast water-filling algorithm to determine the optimal sampling of the leaves to estimate the root of an independent innovations tree.

Multiscale queuing analysis, sampling theory, and network probing

Ribeiro, Vinay
Fonte: Universidade Rice Publicador: Universidade Rice
ENG
Relevância na Pesquisa
46.09%
Multiscale techniques model and analyze phenomena at multiple scales in space or time. This thesis develops novel multiscale solutions for problems in queuing theory, sampling theory, and network inference. First, we study the tail probability of an infinite-buffer queue fed with an arbitrary traffic source. The tail probability is a critical quantity for the design of computer networks. We propose a multiscale framework that uses traffic statistics at only a fixed finite set of time scales and derive three approximations for the tail probability. Theory and simulations strongly support the use of our approximations in different networking applications. Second, we design strategies to optimally sample a process in order to estimate its global average. Our results have implications for Internet measurement, sensor network design, environmental monitoring, etc. We restrict our analysis to linear estimation of certain multiscale stochastic processes---independent innovations trees and covariance trees. Our results demonstrate that the optimal solution depends strongly on the correlation structure of the tree. We also present an efficient "water-filling" solution for arbitrary independent innovations trees. Third, we present two probing tools that estimate the available bandwidth of network paths and locate links with scarce bandwidth. These tools aid network operations and network-aware applications such as grid computing. We use novel packet trains called "chirps" that simultaneously probe the network at multiple bit-rates which improves the efficiency of the tools. We validate the tools through simulations and Internet experiments.

All-Atom Multiscale Computational Modeling Of Viral Dynamics

Miao, Yinglong
Fonte: [Bloomington, Ind.] : Indiana University Publicador: [Bloomington, Ind.] : Indiana University
Tipo: Doctoral Dissertation
EN
Relevância na Pesquisa
36.12%
Thesis (Ph.D.) - Indiana University, Chemistry, 2009; Viruses are composed of millions of atoms functioning on supra-nanometer length scales over timescales of milliseconds or greater. In contrast, individual atoms interact on scales of angstroms and femtoseconds. Thus they display dual microscopic/macroscopic characteristics involving processes that span across widely-separated time and length scales. To address this challenge, we introduced automatically generated collective modes and order parameters to capture viral large-scale low-frequency coherent motions. With an all-atom multiscale analysis (AMA) of the Liouville equation, a stochastic (Fokker-Planck or Smoluchowski) equation and equivalent Langevin equations are derived for the order parameters. They are shown to evolve on timescales much larger than the 10^(-14)-second timescale of fast atomistic vibrations and collisions. This justifies a novel multiscale Molecular Dynamics/Order Parameter eXtrapolation (MD/OPX) approach, which propagates viral atomistic and nanoscale dynamics simultaneously by solving the Langevin equations of order parameters implicitly without the need to construct thermal-average forces and friction/diffusion coefficients. In MD/OPX, a set of short replica MD runs with random atomic velocity initializations estimate the ensemble average rate of change in order parameters...

Convergence of stochastic gene networks to hybrid piecewise deterministic processes

Crudu, Alina; Debussche, Arnaud; Muller, Aurélie; Radulescu, Ovidiu
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/01/2011
Relevância na Pesquisa
36.05%
We study the asymptotic behavior of multiscale stochastic gene networks using weak limits of Markov jump processes. Depending on the time and concentration scales of the system we distinguish four types of limits: continuous piecewise deterministic processes (PDP) with switching, PDP with jumps in the continuous variables, averaged PDP, and PDP with singular switching. We justify rigorously the convergence for the four types of limits. The convergence results can be used to simplify the stochastic dynamics of gene network models arising in molecular biology.

Diffusion in multiscale spacetimes

Calcagni, Gianluca
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.01%
We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples and the most general spectral-dimension profile of multifractional spaces is constructed.; Comment: 23 pages, 5 figures. v2: discussion improved, typos corrected, references added

Stochastic Dynamics of Bionanosystems: Multiscale Analysis and Specialized Ensembles

Pankavich, Stephen; Miao, Yinglong; Ortoleva, Jamil; Shreif, Zeina; Ortoleva, Peter
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 08/01/2010
Relevância na Pesquisa
36.23%
An approach for simulating bionanosystems, such as viruses and ribosomes, is presented. This calibration-free approach is based on an all-atom description for bionanosystems, a universal interatomic force field, and a multiscale perspective. The supramillion-atom nature of these bionanosystems prohibits the use of a direct molecular dynamics approach for phenomena like viral structural transitions or self-assembly that develop over milliseconds or longer. A key element of these multiscale systems is the cross-talk between, and consequent strong coupling of, processes over many scales in space and time. We elucidate the role of interscale cross-talk and overcome bionanosystem simulation difficulties with automated construction of order parameters (OPs) describing supra-nanometer scale structural features, construction of OP dependent ensembles describing the statistical properties of atomistic variables that ultimately contribute to the entropies driving the dynamics of the OPs, and the derivation of a rigorous equation for the stochastic dynamics of the OPs. Since the atomic scale features of the system are treated statistically, several ensembles are constructed that reflect various experimental conditions. The theory provides a basis for a practical...

Statistical Inference for Perturbations of Multiscale Dynamical Systems

Gailus, Siragan; Spiliopoulos, Konstantinos
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 28/04/2015
Relevância na Pesquisa
36.01%
In this paper, we study statistical inference for small-noise perturbations of multiscale dynamical systems. We prove the asymptotic consistency and asymptotic normality of an appropriately constructed maximum likelihood estimator (MLE) for a parameter of interest, identifying precisely its limiting variance. We allow unbounded coefficients in the equation for the slow process and assume neither periodicity nor that the fast process is compact. Ergodicity of the fast process is guaranteed by imposing a recurrence condition. Moreover, we allow full dependence of the coefficients on the slow and fast components and allow correlation between the driving noises of the slow and fast components. The results provide a theoretical basis for calibration of small-noise perturbed multiscale stochastic dynamical systems and related diffusion processes. In the course of the proof we also derive exponential bounds and ergodic theorems that may be of independent interest. Data from numerical simulations are provided to supplement and illustrate the theory.

Elimination of Intermediate Species in Multiscale Stochastic Reaction Networks

Cappelletti, Daniele; Wiuf, Carsten
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.14%
We study networks of biochemical reactions modelled by continuous-time Markov processes. Such networks typically contain many molecular species and reactions and are hard to study analytically as well as by simulation. Particularly, we are interested in reaction networks with intermediate species such as the substrate-enzyme complex in the Michaelis-Menten mechanism. These species are virtually in all real-world networks, they are typically short-lived, degraded at a fast rate and hard to observe experimentally. We provide conditions under which the Markov process of a multiscale reaction network with intermediate species is approximated in finite dimensional distribution by the Markov process of a simpler reduced reaction network without intermediate species. We do so by embedding the Markov processes into a one-parameter family of processes, where reaction rates and species abundances are scaled in the parameter. Further, we show that there are close links between these stochastic models and deterministic ODE models of the same networks.

Quenched Large Deviations for Multiscale Diffusion Processes in Random Environments

Spiliopoulos, Konstantinos
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.01%
We consider multiple time scales systems of stochastic differential equations with small noise in random environments. We prove a quenched large deviations principle with explicit characterization of the action functional. The random medium is assumed to be stationary and ergodic. In the course of the proof we also prove related quenched ergodic theorems for controlled diffusion processes in random environments that are of independent interest. The proof relies entirely on probabilistic arguments, allowing to obtain detailed information on how the rare event occurs. We derive a control, equivalently a change of measure, that leads to the large deviations lower bound. This information on the change of measure can motivate the design of asymptotically efficient Monte Carlo importance sampling schemes for multiscale systems in random environments.

Error Analysis of Diffusion Approximation Methods for Multiscale Systems in Reaction Kinetics

Cotter, Simon; Erban, Radek
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 18/12/2014
Relevância na Pesquisa
36.21%
Several different methods exist for efficient approximation of paths in multiscale stochastic chemical systems. Another approach is to use bursts of stochastic simulation to estimate the parameters of a stochastic differential equation approximation of the paths. In this paper, multiscale methods for approximating paths are used to formulate different strategies for estimating the dynamics by diffusion processes. We then analyse how efficient and accurate these methods are in a range of different scenarios, and compare their respective advantages and disadvantages to other methods proposed to analyse multiscale chemical networks.; Comment: 17 pages, 8 figures

Normal form transforms separate slow and fast modes in stochastic dynamical systems

Roberts, A. J.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 22/01/2007
Relevância na Pesquisa
36.09%
Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from detailed microscale dynamics. We explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems. Thus we not only must reduce the dimensionality of the dynamics, but also endeavour to separate all slow processes from all fast time processes, both deterministic and stochastic. Quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes. The results will help us accurately model, interpret and simulate multiscale stochastic systems.

Exponential L\'evy-type models with stochastic volatility and stochastic jump-intensity

Lorig, Matthew; Lozano-Carbassé, Oriol
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.02%
We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential L\'evy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors -- one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of \citet*{fpss} to models of the exponential L\'evy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility. To illustrate the flexibility of our modeling framework we extend five exponential L\'evy processes to include stochastic volatility and jump-intensity. For each of the extended models, using a single fast-varying factor of volatility and jump-intensity, we perform a calibration to the S&P500 implied volatility surface. Our results show decisively that the extended framework provides a significantly better fit to implied volatility than both the traditional exponential L\'evy models and the fast mean-reverting stochastic volatility models of \citet{fpss}.; Comment: 24 pages...

Optimal sampling strategies for multiscale stochastic processes

Ribeiro, Vinay J.; Riedi, Rudolf H.; Baraniuk, Richard G.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/11/2006
Relevância na Pesquisa
46.01%
In this paper, we determine which non-random sampling of fixed size gives the best linear predictor of the sum of a finite spatial population. We employ different multiscale superpopulation models and use the minimum mean-squared error as our optimality criterion. In multiscale superpopulation tree models, the leaves represent the units of the population, interior nodes represent partial sums of the population, and the root node represents the total sum of the population. We prove that the optimal sampling pattern varies dramatically with the correlation structure of the tree nodes. While uniform sampling is optimal for trees with ``positive correlation progression'', it provides the worst possible sampling with ``negative correlation progression.'' As an analysis tool, we introduce and study a class of independent innovations trees that are of interest in their own right. We derive a fast water-filling algorithm to determine the optimal sampling of the leaves to estimate the root of an independent innovations tree.; Comment: Published at http://dx.doi.org/10.1214/074921706000000509 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

Irreversible Thermodynamics in Multiscale Stochastic Dynamical Systems

Santillan, Moises; Qian, Hong
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.04%
This work extends the results of the recently developed theory of a rather complete thermodynamic formalism for discrete-state, continuous-time Markov processes with and without detailed balance. We aim at investigating the question that whether and how the thermodynamic structure is invariant in a multiscale stochastic system. That is, whether the relations between thermodynamic functions of state and process variables remain unchanged when the system is viewed at different time scales and resolutions. Our results show that the dynamics on a fast time scale contribute an entropic term to the "internal energy function", $u_S(x)$, for the slow dynamics. Based on the conditional free energy $u_S(x)$, one can then treat the slow dynamics as if the fast dynamics is nonexistent. Furthermore, we show that the free energy, which characterizes the spontaneous organization in a system without detailed balance, is invariant with or without the fast dynamics: The fast dynamics is assumed to reach stationarity instantaneously on the slow time scale; they have no effect on the system's free energy. The same can not be said for the entropy and the internal energy, both of which contain the same contribution from the fast dynamics. We also investigate the consequences of time-scale separation in connection to the concepts of quasi-stationaryty and steady-adiabaticity introduced in the phenomenological steady-state thermodynamics.

Particle-based Multiscale Modeling of Calcium Puff Dynamics

Dobramysl, Ulrich; Rüdiger, Sten; Erban, Radek
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.11%
Intracellular calcium is regulated in part by the release of Ca$^{2+}$ ions from the endoplasmic reticulum via inositol-4,5-triphosphate receptor (IP$_3$R) channels (among other possibilities such as RyR and L-type calcium channels). The resulting dynamics are highly diverse, lead to local calcium ``puffs'' as well as global waves propagating through cells, as observed in {\it Xenopus} oocytes, neurons, and other cell types. Local fluctuations in the number of calcium ions play a crucial role in the onset of these features. Previous modeling studies of calcium puff dynamics stemming from IP$_3$R channels have predominantly focused on stochastic channel models coupled to deterministic diffusion of ions, thereby neglecting local fluctuations of the ion number. Tracking of individual ions is computationally difficult due to the scale separation in the Ca$^{2+}$ concentration when channels are in the open or closed states. In this paper, a spatial multiscale model for investigating of the dynamics of puffs is presented. It couples Brownian motion (diffusion) of ions with a stochastic channel gating model. The model is used to analyze calcium puff statistics. Concentration time traces as well as channel state information are studied. We identify the regime in which puffs can be found and develop a mean-field theory to extract the boundary of this regime. Puffs are only possible when the time scale of channel inhibition is sufficiently large. Implications for the understanding of puff generation and termination are discussed.; Comment: 20 pages...

Probabilistic methods for multiscale evolutionary dynamics

Luo, Shishi Zhige
Fonte: Universidade Duke Publicador: Universidade Duke
Tipo: Dissertação
Publicado em //2013
Relevância na Pesquisa
56.31%

Evolution by natural selection can occur at multiple biological scales. This is particularly the case for host-pathogen systems, where selection occurs both within each infected host as well as through transmission between hosts. Despite there being established mathematical models for understanding evolution at a single biological scale, fewer tractable models exist for multiscale evolutionary dynamics. Here I present mathematical approaches using tools from probability and stochastic processes as well as dynamical systems to handle multiscale evolutionary systems. The first problem I address concerns the antigenic evolution of influenza. Using a combination of ordinary differential equations and inhomogeneous Poisson processes, I study how immune selection pressures at the within-host level impact population-level evolutionary dynamics. The second problem involves the more general question of evolutionary dynamics when selection occurs antagonistically at two biological scales. In addition to host-pathogen systems, such situations arise naturally in the evolution of traits such as the production of a public good and the use of a common resource. I introduce a model for this general phenomenon that is intuitively visualized as a a stochastic ball-and-urn system and can be used to systematically obtain general properties of antagonistic multiscale evolution. Lastly...

Factorial kriging for multiscale modelling

Ma,Y.Z.; Royer,J.-J.; Wang,H.; Wang,Y.; Zhang,T.
Fonte: Journal of the Southern African Institute of Mining and Metallurgy Publicador: Journal of the Southern African Institute of Mining and Metallurgy
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/08/2014 EN
Relevância na Pesquisa
46.01%
SYNOPSIS This paper presents a matrix formulation of factorial kriging, and its relationships with simple and ordinary kriging. Similar to other kriging methods, factorial kriging can be applied to both stationary and intrinsic stochastic processes, and is often used as a local operator. Therefore, the concepts of intrinsic random function and local stationarity are first briefly reviewed. Kriging is presented in a block matrix form in which the kriging solution is useful not only for understanding the relationships between simple and ordinary kriging methods, but also the relationships between interpolative kriging and factorial kriging. When used as a signal/noise-filtering method, factorial kriging is especially useful for multiscale modelling. Examples for general signal analysis and geophysical data signal filtering are given to illustrate the method.