18,379 research outputs found
Likelihood based observability analysis and confidence intervals for predictions of dynamic models
Mechanistic dynamic models of biochemical networks such as Ordinary
Differential Equations (ODEs) contain unknown parameters like the reaction rate
constants and the initial concentrations of the compounds. The large number of
parameters as well as their nonlinear impact on the model responses hamper the
determination of confidence regions for parameter estimates. At the same time,
classical approaches translating the uncertainty of the parameters into
confidence intervals for model predictions are hardly feasible.
In this article it is shown that a so-called prediction profile likelihood
yields reliable confidence intervals for model predictions, despite arbitrarily
complex and high-dimensional shapes of the confidence regions for the estimated
parameters. Prediction confidence intervals of the dynamic states allow a
data-based observability analysis. The approach renders the issue of sampling a
high-dimensional parameter space into evaluating one-dimensional prediction
spaces. The method is also applicable if there are non-identifiable parameters
yielding to some insufficiently specified model predictions that can be
interpreted as non-observability. Moreover, a validation profile likelihood is
introduced that should be applied when noisy validation experiments are to be
interpreted.
The properties and applicability of the prediction and validation profile
likelihood approaches are demonstrated by two examples, a small and instructive
ODE model describing two consecutive reactions, and a realistic ODE model for
the MAP kinase signal transduction pathway. The presented general approach
constitutes a concept for observability analysis and for generating reliable
confidence intervals of model predictions, not only, but especially suitable
for mathematical models of biological systems
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
This work derives a residual-based a posteriori error estimator for reduced
models learned with non-intrusive model reduction from data of high-dimensional
systems governed by linear parabolic partial differential equations with
control inputs. It is shown that quantities that are necessary for the error
estimator can be either obtained exactly as the solutions of least-squares
problems in a non-intrusive way from data such as initial conditions, control
inputs, and high-dimensional solution trajectories or bounded in a
probabilistic sense. The computational procedure follows an offline/online
decomposition. In the offline (training) phase, the high-dimensional system is
judiciously solved in a black-box fashion to generate data and to set up the
error estimator. In the online phase, the estimator is used to bound the error
of the reduced-model predictions for new initial conditions and new control
inputs without recourse to the high-dimensional system. Numerical results
demonstrate the workflow of the proposed approach from data to reduced models
to certified predictions
Levy stable noise induced transitions: stochastic resonance, resonant activation and dynamic hysteresis
A standard approach to analysis of noise-induced effects in stochastic
dynamics assumes a Gaussian character of the noise term describing interaction
of the analyzed system with its complex surroundings. An additional assumption
about the existence of timescale separation between the dynamics of the
measured observable and the typical timescale of the noise allows external
fluctuations to be modeled as temporally uncorrelated and therefore white.
However, in many natural phenomena the assumptions concerning the
abovementioned properties of "Gaussianity" and "whiteness" of the noise can be
violated. In this context, in contrast to the spatiotemporal coupling
characterizing general forms of non-Markovian or semi-Markovian L\'evy walks,
so called L\'evy flights correspond to the class of Markov processes which
still can be interpreted as white, but distributed according to a more general,
infinitely divisible, stable and non-Gaussian law. L\'evy noise-driven
non-equilibrium systems are known to manifest interesting physical properties
and have been addressed in various scenarios of physical transport exhibiting a
superdiffusive behavior. Here we present a brief overview of our recent
investigations aimed to understand features of stochastic dynamics under the
influence of L\'evy white noise perturbations. We find that the archetypal
phenomena of noise-induced ordering are robust and can be detected also in
systems driven by non-Gaussian, heavy-tailed fluctuations with infinite
variance.Comment: 7 pages, 8 figure
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