TY - JOUR A1 - Fackeldey, Konstantin A1 - Sikorski, Alexander A1 - Weber, Marcus T1 - Spectral Clustering for Non-Reversible Markov Chains JF - Computational and Applied Mathematics N2 - Spectral clustering methods are based on solving eigenvalue problems for the identification of clusters, e.g., the identification of metastable subsets of a Markov chain. Usually, real-valued eigenvectors are mandatory for this type of algorithms. The Perron Cluster Analysis (PCCA+) is a well-known spectral clustering method of Markov chains. It is applicable for reversible Markov chains, because reversibility implies a real-valued spectrum. We also extend this spectral clustering method to non-reversible Markov chains and give some illustrative examples. The main idea is to replace the eigenvalue problem by a real-valued Schur decomposition. By this extension non-reversible Markov chains can be analyzed. Furthermore, the chains do not need to have a positive stationary distribution. In addition to metastabilities, dominant cycles and sinks can also be identified. This novel method is called GenPCCA (i.e., generalized PCCA), since it includes the case of non-reversible processes. We also apply the method to real-world eye-tracking data. KW - Spectral clustering KW - Markov chain KW - Non-reversible KW - Schur decomposition KW - GenPCCA Y1 - 2018 U6 - https://doi.org/https://doi.org/10.1007/s40314-018-0697-0 VL - 37 IS - 5 SP - 6376 EP - 6391 ER - TY - JOUR A1 - Weber, Marcus A1 - Bujotzek, Alexander A1 - Haag, Rainer T1 - Quantifying the rebinding effect in multivalent chemical ligand-receptor systems JF - J. Chem. Phys. Y1 - 2012 VL - 137 IS - 5 SP - 054111 ER - TY - JOUR A1 - Röblitz, Susanna A1 - Weber, Marcus T1 - Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification JF - Advances in Data Analysis and Classification Y1 - 2013 U6 - https://doi.org/10.1007/s11634-013-0134-6 VL - 7 IS - 2 SP - 147 EP - 179 ER - TY - JOUR A1 - Haack, Fiete A1 - Fackeldey, Konstantin A1 - Röblitz, Susanna A1 - Scharkoi, Olga A1 - Weber, Marcus A1 - Schmidt, Burkhard T1 - Adaptive spectral clustering with application to tripeptide conformation analysis JF - The Journal of Chemical Physics Y1 - 2013 U6 - https://doi.org/10.1063/1.4830409 VL - 139 SP - 110 EP - 194 ER - TY - JOUR A1 - Nielsen, Adam T1 - The Monte Carlo Computation Error of Transition Probabilities JF - Statistics & Probability Letters N2 - In many applications one is interested to compute transition probabilities of a Markov chain. This can be achieved by using Monte Carlo methods with local or global sampling points. In this article, we analyze the error by the difference in the $L^2$ norm between the true transition probabilities and the approximation achieved through a Monte Carlo method. We give a formula for the error for Markov chains with locally computed sampling points. Further, in the case of reversible Markov chains, we will deduce a formula for the error when sampling points are computed globally. We will see that in both cases the error itself can be approximated with Monte Carlo methods. As a consequence of the result, we will derive surprising properties of reversible Markov chains. Y1 - 2016 U6 - https://doi.org/10.1016/j.spl.2016.06.011 VL - 118 SP - 163 EP - 170 PB - Elsevier ER - TY - JOUR A1 - Schütte, Christof A1 - Nielsen, Adam A1 - Weber, Marcus T1 - Markov State Models and Molecular Alchemy JF - Molecular Physics N2 - In recent years Markov State Models (MSMs) have attracted a consid- erable amount of attention with regard to modelling conformation changes and associated function of biomolecular systems. They have been used successfully, e.g., for peptides including time-resolved spectroscopic experiments, protein function and protein folding , DNA and RNA, and ligand-receptor interaction in drug design and more complicated multivalent scenarios. In this article a novel reweighting scheme is introduced that allows to construct an MSM for certain molecular system out of an MSM for a similar system. This permits studying how molecular properties on long timescales differ between similar molecular systems without performing full molecular dynamics simulations for each system under con- sideration. The performance of the reweighting scheme is illustrated for simple test cases including one where the main wells of the respective energy landscapes are located differently and an alchemical transformation of butane to pentane where the dimension of the state space is changed. KW - MSM KW - Reweighting KW - Girsanov Y1 - 2015 U6 - https://doi.org/10.1080/00268976.2014.944597 VL - 113 IS - 1 SP - 69 EP - 78 ER - TY - JOUR A1 - Weber, Marcus A1 - Fackeldey, Konstantin T1 - Computing the Minimal Rebinding Effect Included in a Given Kinetics JF - Multiscale Model. Simul. N2 - The rebinding effect is a phenomenon which occurs when observing a ligand-receptor binding process. On the macro scale this process comprises the Markov property. This Makovian view is spoiled when switching to the atomistic scale of a binding process. We therefore suggest a model which accurately describes the rebinding effect on the atomistic scale by allowing ''intermediate'' bound states. This allows us to define an indicator for the magnitude of rebinding and to formulate an optimization problem. The results form our examples show good agreement with data form laboratory. Y1 - 2014 U6 - https://doi.org/10.1137/13091124X VL - 12 IS - 1 SP - 318 EP - 334 ER - TY - JOUR A1 - Nielsen, Adam A1 - Weber, Marcus T1 - Computing the nearest reversible Markov chain JF - Numerical Linear Algebra with Applications N2 - Reversible Markov chains are the basis of many applications. However, computing transition probabilities by a finite sampling of a Markov chain can lead to truncation errors. Even if the original Markov chain is reversible, the approximated Markov chain might be non-reversible and will lose important properties, like the real valued spectrum. In this paper, we show how to find the closest reversible Markov chain to a given transition matrix. It turns out that this matrix can be computed by solving a convex minimization problem. KW - Reversible Markov Chain KW - Convex Optimization KW - MSM Y1 - 2015 U6 - https://doi.org/10.1002/nla.1967 VL - 22 IS - 3 SP - 483 EP - 499 ER - TY - JOUR A1 - Djurdjevac Conrad, Natasa A1 - Weber, Marcus A1 - Schütte, Christof T1 - Finding dominant structures of nonreversible Markov processes JF - Multiscale Modeling and Simulation Y1 - 2016 U6 - https://doi.org/10.1137/15M1032272 VL - 14 IS - 4 SP - 1319 EP - 1340 ER -