60,465 research outputs found
Steady State Sensitivity Analysis of Continuous Time Markov Chains
In this paper we study Monte Carlo estimators based on the likelihood ratio
approach for steady-state sensitivity.
We first extend the result of Glynn and Olvera-Cravioto [doi:doi:
10.1287/stsy.2018.002] to the setting of continuous time Markov chains with a
countable state space which include models such as stochastic reaction kinetics
and kinetic Monte Carlo lattice system.
We show that the variance of the centered likelihood ratio estimators does
not grow in time. This result suggests that the centered likelihood ratio
estimators should be favored for sensitivity analysis when the mixing time of
the underlying continuous time Markov chain is large, which is typically the
case when systems exhibit multi-scale behavior. We demonstrate a practical
implication of this analysis on a numerical benchmark of two examples for the
biochemical reaction networks
Sensitivity analysis for multiscale stochastic reaction networks using hybrid approximations
We consider the problem of estimating parameter sensitivities for stochastic
models of multiscale reaction networks. These sensitivity values are important
for model analysis, and, the methods that currently exist for sensitivity
estimation mostly rely on simulations of the stochastic dynamics. This is
problematic because these simulations become computationally infeasible for
multiscale networks due to reactions firing at several different timescales.
However it is often possible to exploit the multiscale property to derive a
"model reduction" and approximate the dynamics as a Piecewise Deterministic
Markov process (PDMP), which is a hybrid process consisting of both discrete
and continuous components. The aim of this paper is to show that such PDMP
approximations can be used to accurately and efficiently estimate the parameter
sensitivity for the original multiscale stochastic model. We prove the
convergence of the original sensitivity to the corresponding PDMP sensitivity,
in the limit where the PDMP approximation becomes exact. Moreover we establish
a representation of the PDMP parameter sensitivity that separates the
contributions of discrete and continuous components in the dynamics, and allows
one to efficiently estimate both contributions.Comment: 34 pages, 4 figure
Sensitivity analysis for stochastic chemical reaction networks with multiple time-scales
Stochastic models for chemical reaction networks have become very popular in
recent years. For such models, the estimation of parameter sensitivities is an
important and challenging problem. Sensitivity values help in analyzing the
network, understanding its robustness properties and also in identifying the
key reactions for a given outcome. Most of the methods that exist in the
literature for the estimation of parameter sensitivities, rely on Monte Carlo
simulations using Gillespie's stochastic simulation algorithm or its variants.
It is well-known that such simulation methods can be prohibitively expensive
when the network contains reactions firing at different time-scales, which is a
feature of many important biochemical networks. For such networks, it is often
possible to exploit the time-scale separation and approximately capture the
original dynamics by simulating a "reduced" model, which is obtained by
eliminating the fast reactions in a certain way. The aim of this paper is to
tie these model reduction techniques with sensitivity analysis. We prove that
under some conditions, the sensitivity values of the reduced model can be used
to approximately recover the sensitivity values for the original model. Through
an example we illustrate how our result can help in sharply reducing the
computational costs for the estimation of parameter sensitivities for reaction
networks with multiple time-scales. To prove our result, we use coupling
arguments based on the random time change representation of Kurtz. We also
exploit certain connections between the distributions of the occupation times
of Markov chains and multi-dimensional wave equations
Parallel replica dynamics method for bistable stochastic reaction networks: simulation and sensitivity analysis
Stochastic reaction networks that exhibit bistability are common in many
fields such as systems biology and materials science. Sampling of the
stationary distribution is crucial for understanding and characterizing the
long term dynamics of bistable stochastic dynamical systems. However, this is
normally hindered by the insufficient sampling of the rare transitions between
the two metastable regions. In this paper, we apply the parallel replica
(ParRep) method for continuous time Markov chain to accelerate the stationary
distribution sampling of bistable stochastic reaction networks. The proposed
method uses parallel computing to accelerate the sampling of rare transitions
and it is very easy to implement. We combine ParRep with the path space
information bounds for parametric sensitivity analysis. We demonstrate the
efficiency and accuracy of the method by studying the Schl\"{o}gl model and the
genetic switches network.Comment: 7 figure
Unbiased estimation of parameter sensitivities for stochastic chemical reaction networks
Estimation of parameter sensitivities for stochastic chemical reaction
networks is an important and challenging problem. Sensitivity values are
important in the analysis, modeling and design of chemical networks. They help
in understanding the robustness properties of the system and also in
identifying the key reactions for a given outcome. In a discrete setting, most
of the methods that exist in the literature for the estimation of parameter
sensitivities rely on Monte Carlo simulations along with finite difference
computations. However these methods introduce a bias in the sensitivity
estimate and in most cases the size or direction of the bias remains unknown,
potentially damaging the accuracy of the analysis. In this paper, we use the
random time change representation of Kurtz to derive an exact formula for
parameter sensitivity. This formula allows us to construct an unbiased
estimator for parameter sensitivity, which can be efficiently evaluated using a
suitably devised Monte Carlo scheme. The existing literature contains only one
method to produce such an unbiased estimator. This method was proposed by
Plyasunov and Arkin and it is based on the Girsanov measure transformation. By
taking a couple of examples we compare our method to this existing method. Our
results indicate that our method can be much faster than the existing method
while computing sensitivity with respect to a reaction rate constant which is
small in magnitude. This rate constant could correspond to a reaction which is
slow in the reference time-scale of the system. Since many biological systems
have such slow reactions, our method can be a useful tool for sensitivity
analysis.Comment: Submitte
An efficient and unbiased method for sensitivity analysis of stochastic reaction networks
We consider the problem of estimating parameter sensitivity for Markovian
models of reaction networks. Sensitivity values measure the responsiveness of
an output to the model parameters. They help in analyzing the network,
understanding its robustness properties and identifying the important reactions
for a specific output. Sensitivity values are commonly estimated using methods
that perform finite-difference computations along with Monte Carlo simulations
of the reaction dynamics. These methods are computationally efficient and easy
to implement, but they produce a biased estimate which can be unreliable for
certain applications. Moreover the size of the bias is generally unknown and
hence the accuracy of these methods cannot be easily determined. There also
exist unbiased schemes for sensitivity estimation but these schemes can be
computationally infeasible, even for simple networks. Our goal in this paper is
to present a new method for sensitivity estimation, which combines the
computational efficiency of finite-difference methods with the accuracy of
unbiased schemes. Our method is easy to implement and it relies on an exact
representation of parameter sensitivity that we recently proved in an earlier
paper. Through examples we demonstrate that the proposed method can outperform
the existing methods, both biased and unbiased, in many situations
Efficient estimators for likelihood ratio sensitivity indices of complex stochastic dynamics
We demonstrate that centered likelihood ratio estimators for the sensitivity
indices of complex stochastic dynamics are highly efficient with low, constant
in time variance and consequently they are suitable for sensitivity analysis in
long-time and steady-state regimes. These estimators rely on a new covariance
formulation of the likelihood ratio that includes as a submatrix a Fisher
Information Matrix for stochastic dynamics and can also be used for fast
screening of insensitive parameters and parameter combinations. The proposed
methods are applicable to broad classes of stochastic dynamics such as chemical
reaction networks, Langevin-type equations and stochastic models in finance,
including systems with a high dimensional parameter space and/or disparate
decorrelation times between different observables. Furthermore, they are simple
to implement as a standard observable in any existing simulation algorithms
without additional modifications.Comment: Revision of the paper. Added a new estimato
Path-space information bounds for uncertainty quantification and sensitivity analysis of stochastic dynamics
Uncertainty quantification is a primary challenge for reliable modeling and
simulation of complex stochastic dynamics. Such problems are typically plagued
with incomplete information that may enter as uncertainty in the model
parameters, or even in the model itself. Furthermore, due to their dynamic
nature, we need to assess the impact of these uncertainties on the transient
and long-time behavior of the stochastic models and derive corresponding
uncertainty bounds for observables of interest. A special class of such
challenges is parametric uncertainties in the model and in particular
sensitivity analysis along with the corresponding sensitivity bounds for
stochastic dynamics. Moreover, sensitivity analysis can be further complicated
in models with a high number of parameters that render straightforward
approaches, such as gradient methods, impractical. In this paper, we derive
uncertainty and sensitivity bounds for path-space observables of stochastic
dynamics in terms of new goal-oriented divergences; the latter incorporate both
observables and information theory objects such as the relative entropy rate.
These bounds are tight, depend on the variance of the particular observable and
are computable through Monte Carlo simulation. In the case of sensitivity
analysis, the derived sensitivity bounds rely on the path Fisher Information
Matrix, hence they depend only on local dynamics and are gradient-free. These
features allow for computationally efficient implementation in systems with a
high number of parameters, e.g., complex reaction networks and molecular
simulations.Comment: 29 pages, 2 figure
Unbiased estimation of second-order parameter sensitivities for stochastic reaction networks
This paper deals with the problem of estimating second-order parameter
sensitivities for stochastic reaction networks, where the reaction dynamics is
modeled as a continuous time Markov chain over a discrete state space.
Estimation of such second-order sensitivities (the Hessian) is necessary for
implementing the Newton-Raphson scheme for optimization over the parameter
space. To perform this estimation, Wolf and Anderson have proposed an efficient
finite-difference method, that uses a coupling of perturbed processes to reduce
the estimator variance. The aim of this paper is to illustrate that the same
coupling can be exploited to derive an exact representation for second-order
parameter sensitivity. Furthermore with this representation one can construct
an unbiased estimator which is easy to implement. The ideas contained in this
paper are extensions of the ideas presented in our recent papers on first-order
parameter sensitivity estimation.Comment: Work in progress. Any feedback from the readers is welcom
Stochastic Averaging and Sensitivity Analysis for Two Scale Reaction Networks
In the presence of multiscale dynamics in a reaction network, direct
simulation methods become inefficient as they can only advance the system on
the smallest scale. This work presents stochastic averaging techniques to
accelerate computations for obtaining estimates of expected values and
sensitivities with respect to the steady state distribution. A two-time-scale
formulation is used to establish bounds on the bias induced by the averaging
method. Further, this formulation provides a framework to create an accelerated
`averaged' version of most single-scale sensitivity estimation method. In
particular, we propose a new lower-variance ergodic likelihood ratio type
estimator for steady-state estimation and show how one can adapt it to
accelerated simulations of multiscale systems.Lastly, we develop an adaptive
"batch-means" stopping rule for determining when to terminate the
micro-equilibration process.Comment: 20 pages, 6 figures, 2 tables, in this version: corrigendum of
Proposition III.1 - only convergence is established, at present no rate
available for general cas
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