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

A Variational Approach to Parameter Estimation in Ordinary Differential Equations

Ordinary differential equations are widely-used in the field of systems biology and chemical engineering to model chemical reaction networks. Numerous techniques have been developed to estimate parameters like rate constants, initial conditions or steady state concentrations from time-resolved data. In contrast to this countable set of parameters, the estimation of entire courses of network components corresponds to an innumerable set of parameters. The approach presented in this work is able to deal with course estimation for extrinsic system inputs or intrinsic reactants, both not being constrained by the reaction network itself. Our method is based on variational calculus which is carried out analytically to derive an augmented system of differential equations including the unconstrained components as ordinary state variables. Finally, conventional parameter estimation is applied to the augmented system resulting in a combined estimation of courses and parameters. The combined estimation approach takes the uncertainty in input courses correctly into account. This leads to precise parameter estimates and correct confidence intervals. In particular this implies that small motifs of large reaction networks can be analysed independently of the rest. By the use of variational methods, elements from control theory and statistics are combined allowing for future transfer of methods between the two fields

Networks : On the relation of bi- and multivariate measures

Date of Acceptance: 28/04/2015 Acknowledgement The article processing charge was funded by the German Research Foundation (DFG) and the Albert Ludwigs University Freiburg in the funding programme Open Access PublishingPeer reviewedPublisher PD

A numerically efficient implementation of the expectation maximization algorithm for state space models

Peer reviewedPostprin

Analysis of phase-resolved partial discharge patterns of voids based on a stochastic process approach

A method is presented for the determination of physical discharge parameters for partial discharges (PD) of voids in solid insulation. Based on a recently developed stochastic theory of PD processes, a statistical analysis of a measured Phase-Resolved Partial Discharge (PRPD) pattern allows the determination of the relevant physical parameters like first electron availability or decay time constants for deployed charge carriers. These parameters can be estimated directly from the measured patterns without the need of performing simulations. Furthermore, error bounds for the parameter values can be given. The parameter estimation algorithm is based on the analysis of a contiguous region of the PRPD pattern where this region can be chosen nearly arbitrarily. Thus, even patterns with several active PD defects or patterns which are corrupted by noise can be analyzed. The method is applied to a sequence of patterns of a void in epoxy resin. The change in first electron availability in the course of a day can be determined quantitatively from the data while the other physical parameters remain constant

Reconstructing gene-regulatory networks from time series, knock-out data, and prior knowledge

BACKGROUND: Cellular processes are controlled by gene-regulatory networks. Several computational methods are currently used to learn the structure of gene-regulatory networks from data. This study focusses on time series gene expression and gene knock-out data in order to identify the underlying network structure. We compare the performance of different network reconstruction methods using synthetic data generated from an ensemble of reference networks. Data requirements as well as optimal experiments for the reconstruction of gene-regulatory networks are investigated. Additionally, the impact of prior knowledge on network reconstruction as well as the effect of unobserved cellular processes is studied. RESULTS: We identify linear Gaussian dynamic Bayesian networks and variable selection based on F-statistics as suitable methods for the reconstruction of gene-regulatory networks from time series data. Commonly used discrete dynamic Bayesian networks perform inferior and this result can be attributed to the inevitable information loss by discretization of expression data. It is shown that short time series generated under transcription factor knock-out are optimal experiments in order to reveal the structure of gene regulatory networks. Relative to the level of observational noise, we give estimates for the required amount of gene expression data in order to accurately reconstruct gene-regulatory networks. The benefit of using of prior knowledge within a Bayesian learning framework is found to be limited to conditions of small gene expression data size. Unobserved processes, like protein-protein interactions, induce dependencies between gene expression levels similar to direct transcriptional regulation. We show that these dependencies cannot be distinguished from transcription factor mediated gene regulation on the basis of gene expression data alone. CONCLUSION: Currently available data size and data quality make the reconstruction of gene networks from gene expression data a challenge. In this study, we identify an optimal type of experiment, requirements on the gene expression data quality and size as well as appropriate reconstruction methods in order to reverse engineer gene regulatory networks from time series data