arXiv subjects

Susanna Röblitz

Publications and source records attributed to Susanna Röblitz.

4 recordsLinked to original sources

arXiv2022

Spectral clustering of Markov chain transition matrices with complex eigenvalues

The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral clustering algorithm for coarse-graining transition matrices of nearly decomposable Markov chains with transition states. Originally developed for reversible Markov chains, the algorithm only worked for transition matrices with real eigenvalues. In this paper, we therefore extend the theoretical framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show that by replacing a complex conjugate pair of eigenvectors by their real and imaginary components, a real representation of the same subspace is obtained, which is suitable for the cluster analysis. We show that our approach leads to the same results as the generalized PCCA+ (GenPCCA), which replaces the complex eigen-decomposition by a conceptually more difficult real Schur decomposition. We apply the method on non-reversible Markov chains, including circular chains,and demonstrate its efficiency compared to GenPCCA. The experiments are performed in the Matlab programming language and codes are provided.

stat.ME↗

arXiv2020

Objective Priors in the Empirical Bayes Framework

When dealing with Bayesian inference the choice of the prior often remains a debatable question. Empirical Bayes methods offer a data-driven solution to this problem by estimating the prior itself from an ensemble of data. In the nonparametric case, the maximum likelihood estimate is known to overfit the data, an issue that is commonly tackled by regularization. However, the majority of regularizations are ad hoc choices which lack invariance under reparametrization of the model and result in inconsistent estimates for equivalent models. We introduce a non-parametric, transformation invariant estimator for the prior distribution. Being defined in terms of the missing information similar to the reference prior, it can be seen as an extension of the latter to the data-driven setting. This implies a natural interpretation as a trade-off between choosing the least informative prior and incorporating the information provided by the data, a symbiosis between the objective and empirical Bayes methodologies.

stat.ME↗

arXiv2016

Empirical Bayes Methods for Prior Estimation in Systems Medicine

One of the main goals of mathematical modeling in systems medicine related to medical applications is to obtain patient-specific parameterizations and model predictions. In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Therefore, before applying Bayes' rule \emph{separately} to the data of each patient (which is typically performed using a non-informative prior), it is meaningful to use empirical Bayes methods in order to construct an informative prior from all available data. We compare the performance of four priors -- a non-informative prior and priors chosen by nonparametric maximum likelihood estimation (NPMLE), by maximum penalized likelihood estimation (MPLE) and by doubly-smoothed maximum likelihood estimation (DS-MLE) -- by applying them to a low-dimensional parameter estimation problem in a toy model as well as to a high-dimensional ODE model of the human menstrual cycle, which represents a typical example from systems biology modeling.

stat.ME↗

arXiv2013

Parameter identification in large kinetic networks with BioPARKIN

Modelling, parameter identification, and simulation play an important role in systems biology. Usually, the goal is to determine parameter values that minimise the difference between experimental measurement values and model predictions in a least-squares sense. Large-scale biological networks, however, often suffer from missing data for parameter identification. Thus, the least-squares problems are rank-deficient and solutions are not unique. Many common optimisation methods ignore this detail because they do not take into account the structure of the underlying inverse problem. These algorithms simply return a "solution" without additional information on identifiability or uniqueness. This can yield misleading results, especially if parameters are co-regulated and data are noisy.

cs.MS↗