Combined state and parameter estimation in level-set methods

Reduced-order models based on level-set methods are widely used tools to qualitatively capture and track the nonlinear dynamics of an interface. The aim of this paper is to develop a physics-informed, data-driven, statistically rigorous learning algorithm for state and parameter estimation with level-set methods. A Bayesian approach based on data assimilation is introduced. Data assimilation is enabled by the ensemble Kalman filter and smoother, which are used in their probabilistic formulations. The level-set data assimilation framework is verified in onedimensional and two-dimensional test cases, where state estimation, parameter estimation and uncertainty quantification are performed. The statistical performance of the proposed ensemble Kalman filter and smoother is quantified by twin experiments. In the twin experiments, the combined state and parameter estimation fully recovers the reference solution, which validates the proposed algorithm. The level-set data assimilation framework is then applied to the prediction of the nonlinear dynamics of a forced premixed flame, which exhibits the formation of sharp cusps and intricate topological changes, such as pinch-off events. The proposed physics-informed statistical learning algorithm opens up new possibilities for making reduced-order models of interfaces quantitatively predictive, any time that reference data is available

Systematic identifiability testing for unambiguous mechanistic modeling – application to JAK-STAT, MAP kinase, and NF-κB signaling pathway models

<p>Abstract</p> <p>Background</p> <p>When creating mechanistic mathematical models for biological signaling processes it is tempting to include as many known biochemical interactions into one large model as possible. For the JAK-STAT, MAP kinase, and NF-<it>κ</it>B pathways a lot of biological insight is available, and as a consequence, large mathematical models have emerged. For large models the question arises whether unknown model parameters can uniquely be determined by parameter estimation from measured data. Systematic approaches to answering this question are indispensable since the uniqueness of model parameter values is essential for predictive mechanistic modeling.</p> <p>Results</p> <p>We propose an eigenvalue based method for efficiently testing identifiability of large ordinary differential models and compare this approach to three existing ones. The methods are benchmarked by applying them to models of the signaling pathways mentioned above. In all cases the eigenvalue method proposed here and the orthogonal method find the largest set of identifiable parameters, thus clearly outperforming the other approaches. The identifiability analysis shows that the pathway models are not identifiable, even under the strong assumption that all system state variables are measurable. We demonstrate how the results of the identifiability analysis can be used for model simplification.</p> <p>Conclusion</p> <p>While it has undoubtedly contributed to recent advances in systems biology, mechanistic modeling by itself does not guarantee unambiguous descriptions of biological processes. We show that some recent signal transduction pathway models have reached a level of detail that is not warranted. Rigorous identifiability tests reveal that even if highly idealized experiments could be carried out to measure all state variables of these signaling pathways, some unknown parameters could still not be estimated. The identifiability tests therefore show that the level of detail of the investigated models is too high <it>in principle</it>, not just because too little experimental information is available. We demonstrate how the proposed method can be combined with biological insight, however, to simplify these models.</p

Investigation of the Hammerstein hypothesis in the modeling of electrically stimulated muscle

To restore functional use of paralyzed muscles by automatically controlled stimulation, an accurate quantitative model of the stimulated muscles is desirable. The most commonly used model for isometric muscle has had a Hammerstein structure, in which a linear dynamic block is preceded by a static nonlinear function, To investigate the accuracy of the Hammerstein model, the responses to a pseudo-random binary sequence (PRBS) excitation of normal human plantarflexors, stimulated with surface electrodes, were used to identify a Hammerstein model but also four local models which describe the responses to small signals at different mean levels of activation. Comparison of the local models with the Linearized Hammerstein model showed that the Hammerstein model concealed a fivefold variation in the speed of response. Also, the small-signal gain of the Hammerstein model was in error by factors up to three. We conclude that, despite the past widespread use of the Hammerstein model, it is not an accurate representation of isometric muscle. On the other hand, local models, which are more accurate predictors, can be identified from the responses to short PRBS sequences. The utility of local models for controller design is discussed

Fast inference in nonlinear dynamical systems using gradient matching

Parameter inference in mechanistic models of coupled differential equations is a topical problem. We propose a new method based on kernel ridge regression and gradient matching, and an objective function that simultaneously encourages goodness of fit and penalises inconsistencies with the differential equations. Fast minimisation is achieved by exploiting partial convexity inherent in this function, and setting up an iterative algorithm in the vein of the EM algorithm. An evaluation of the proposed method on various benchmark data suggests that it compares favourably with state-of-the-art alternatives

Determination of mass of IGR J17091-3624 from "Spectro-Temporal" variations during onset-phase of the 2011 outburst

The 2011 outburst of the black hole candidate IGR J17091-3624 followed the canonical track of state transitions along with the evolution of Quasi-Periodic Oscillation (QPO) frequencies before it began exhibiting various variability classes similar to GRS 1915+105. We use this canonical evolution of spectral and temporal properties to determine the mass of IGR J17091-3624, using three different methods, viz : Photon Index ($\Gamma$) - QPO frequency ($\nu$) correlation, QPO frequency ($\nu$) - Time (day) evolution and broadband spectral modelling based on Two Component Advective Flow. We provide a combined mass estimate for the source using a Naive Bayes based joint likelihood approach. This gives a probable mass range of 11.8 M$_{\odot}$ - 13.7 M$_{\odot}$. Considering each individual estimate and taking the lowermost and uppermost bounds among all three methods, we get a mass range of 8.7 M$_{\odot}$ - 15.6 M$_{\odot}$ with 90% confidence. We discuss the probable implications of our findings in the context of two component accretion flow.Comment: 10 pages, 5 figures (4 in colour), 2 tables. Accepted for publication in Ap

Controlling for the effects of information in a public goods discrete choice model

This paper develops a reduced form method of controlling for differences in information sets of subjects in public good discrete choice models, using stated preference data. The main contribution of our method comes from accounting for the effect of information provided during a survey on the mean and the variance of individual-specific scale parameters. In this way we incorporate both scale heterogeneity as well as observed and unobserved preference heterogeneity to investigate differences across and within information treatments. Our approach will also be useful to researchers who want to combine stated preference data sets while controlling for scale differences. We illustrate our approach using the data from a discrete choice experiment study of a biodiversity conservation program and find that the mean of individual-specific scale parameters and its variance in the sample is sensitive to the information set provided to the respondents