1,465 research outputs found
Filters and smoothers for self-exciting Markov modulated counting processes
We consider a self-exciting counting process, the parameters of which depend
on a hidden finite-state Markov chain. We derive the optimal filter and
smoother for the hidden chain based on observation of the jump process. This
filter is in closed form and is finite dimensional. We demonstrate the
performance of this filter both with simulated data, and by analysing the
`flash crash' of 6th May 2010 in this framework
EM algorithm for Markov chains observed via Gaussian noise and point process information: Theory and case studies
In this paper we study parameter estimation via the Expectation Maximization (EM) algorithm for a continuous-time hidden Markov model with diffusion and point process observation. Inference problems of this type arise for instance in credit risk modelling. A key step in the application of the EM algorithm is the derivation of finite-dimensional filters for the quantities that are needed in the E-Step of the algorithm. In this context we obtain exact, unnormalized and robust filters, and we discuss their numerical implementation. Moreover, we propose several goodness-of-fit tests for hidden Markov models with Gaussian noise and point process observation. We run an extensive simulation study to test speed and accuracy of our methodology. The paper closes with an application to credit risk: we estimate the parameters of a hidden Markov model for credit quality where the observations consist of rating transitions and credit spreads for US corporations
Uncoupled Analysis of Stochastic Reaction Networks in Fluctuating Environments
The dynamics of stochastic reaction networks within cells are inevitably
modulated by factors considered extrinsic to the network such as for instance
the fluctuations in ribsome copy numbers for a gene regulatory network. While
several recent studies demonstrate the importance of accounting for such
extrinsic components, the resulting models are typically hard to analyze. In
this work we develop a general mathematical framework that allows to uncouple
the network from its dynamic environment by incorporating only the
environment's effect onto the network into a new model. More technically, we
show how such fluctuating extrinsic components (e.g., chemical species) can be
marginalized in order to obtain this decoupled model. We derive its
corresponding process- and master equations and show how stochastic simulations
can be performed. Using several case studies, we demonstrate the significance
of the approach. For instance, we exemplarily formulate and solve a marginal
master equation describing the protein translation and degradation in a
fluctuating environment.Comment: 7 pages, 4 figures, Appendix attached as SI.pdf, under submissio
An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions
Stochastic differential equations (SDEs) or diffusions are continuous-valued
continuous-time stochastic processes widely used in the applied and
mathematical sciences. Simulating paths from these processes is usually an
intractable problem, and typically involves time-discretization approximations.
We propose an exact Markov chain Monte Carlo sampling algorithm that involves
no such time-discretization error. Our sampler is applicable to the problem of
prior simulation from an SDE, posterior simulation conditioned on noisy
observations, as well as parameter inference given noisy observations. Our work
recasts an existing rejection sampling algorithm for a class of diffusions as a
latent variable model, and then derives an auxiliary variable Gibbs sampling
algorithm that targets the associated joint distribution. At a high level, the
resulting algorithm involves two steps: simulating a random grid of times from
an inhomogeneous Poisson process, and updating the SDE trajectory conditioned
on this grid. Our work allows the vast literature of Monte Carlo sampling
algorithms from the Gaussian process literature to be brought to bear to
applications involving diffusions. We study our method on synthetic and real
datasets, where we demonstrate superior performance over competing methods.Comment: 37 pages, 13 figure
The Hitchhiker's Guide to Nonlinear Filtering
Nonlinear filtering is the problem of online estimation of a dynamic hidden
variable from incoming data and has vast applications in different fields,
ranging from engineering, machine learning, economic science and natural
sciences. We start our review of the theory on nonlinear filtering from the
simplest `filtering' task we can think of, namely static Bayesian inference.
From there we continue our journey through discrete-time models, which is
usually encountered in machine learning, and generalize to and further
emphasize continuous-time filtering theory. The idea of changing the
probability measure connects and elucidates several aspects of the theory, such
as the parallels between the discrete- and continuous-time problems and between
different observation models. Furthermore, it gives insight into the
construction of particle filtering algorithms. This tutorial is targeted at
scientists and engineers and should serve as an introduction to the main ideas
of nonlinear filtering, and as a segway to more advanced and specialized
literature.Comment: 64 page
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