@misc{WeberFackeldey2013, author = {Weber, Marcus and Fackeldey, Konstantin}, title = {Computing the Minimal Rebinding Effect Included in a Given Kinetics}, issn = {1438-0064}, doi = {10.1137/13091124X}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-17796}, year = {2013}, abstract = {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.}, language = {en} } @misc{NielsenFackeldeyWeber2013, author = {Nielsen, Adam and Fackeldey, Konstantin and Weber, Marcus}, title = {On a Generalized Transfer Operator}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-43162}, year = {2013}, abstract = {We introduce a generalized operator for arbitrary stochastic processes by using a pre-kernel, which is a generalization of the Markov kernel. For deterministic processes, such an operator is already known as the Frobenius-Perron operator, which is defined for a large class of measures. For Markov processes, there exists transfer operators being only well defined for stationary measures in \$L^2\$. Our novel generalized transfer operator is well defined for arbitrary stochastic processes, in particular also for deterministic ones. We can show that this operator is acting on \$L^1\$. For stationary measures, this operator is also an endomorphism of \$L^2\$ and, therefore, allows for a mathematical analysis in Hilbert spaces.}, language = {en} } @article{FackeldeySikorskiWeber2018, author = {Fackeldey, Konstantin and Sikorski, Alexander and Weber, Marcus}, title = {Spectral Clustering for Non-Reversible Markov Chains}, volume = {37}, journal = {Computational and Applied Mathematics}, number = {5}, doi = {https://doi.org/10.1007/s40314-018-0697-0}, pages = {6376 -- 6391}, year = {2018}, abstract = {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.}, language = {en} } @article{WeberBujotzekHaag2012, author = {Weber, Marcus and Bujotzek, Alexander and Haag, Rainer}, title = {Quantifying the rebinding effect in multivalent chemical ligand-receptor systems}, volume = {137}, journal = {J. Chem. Phys.}, number = {5}, pages = {054111}, year = {2012}, language = {en} } @misc{Berg2012, type = {Master Thesis}, author = {Berg, Mascha}, title = {Laufzeitoptimierung der Robusten Perron Cluster Analyse (PCCA+)}, year = {2012}, language = {de} } @article{RoeblitzWeber2013, author = {R{\"o}blitz, Susanna and Weber, Marcus}, title = {Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification}, volume = {7}, journal = {Advances in Data Analysis and Classification}, number = {2}, doi = {10.1007/s11634-013-0134-6}, pages = {147 -- 179}, year = {2013}, language = {en} } @article{HaackFackeldeyRoeblitzetal.2013, author = {Haack, Fiete and Fackeldey, Konstantin and R{\"o}blitz, Susanna and Scharkoi, Olga and Weber, Marcus and Schmidt, Burkhard}, title = {Adaptive spectral clustering with application to tripeptide conformation analysis}, volume = {139}, journal = {The Journal of Chemical Physics}, doi = {10.1063/1.4830409}, pages = {110 -- 194}, year = {2013}, language = {en} } @misc{Nielsen2016, author = {Nielsen, Adam}, title = {The Monte Carlo Computation Error of Transition Probabilities}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59933}, year = {2016}, abstract = {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.}, language = {en} } @article{Nielsen2016, author = {Nielsen, Adam}, title = {The Monte Carlo Computation Error of Transition Probabilities}, volume = {118}, journal = {Statistics \& Probability Letters}, publisher = {Elsevier}, doi = {10.1016/j.spl.2016.06.011}, pages = {163 -- 170}, year = {2016}, abstract = {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.}, language = {en} } @misc{WeberFackeldey2015, author = {Weber, Marcus and Fackeldey, Konstantin}, title = {G-PCCA: Spectral Clustering for Non-reversible Markov Chains}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-55505}, year = {2015}, abstract = {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 extend this spectral clustering method also 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 need not have a positive stationary distribution. And additionally to metastabilities, dominant cycles and sinks can be identified, too.}, language = {en} } @misc{DjurdjevacConradWeberSchuette2015, author = {Djurdjevac Conrad, Natasa and Weber, Marcus and Sch{\"u}tte, Christof}, title = {Finding dominant structures of nonreversible Markov processes}, issn = {1438-0064}, doi = {10.1137/15M1032272}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-55739}, year = {2015}, abstract = {Finding metastable sets as dominant structures of Markov processes has been shown to be especially useful in modeling interesting slow dynamics of various real world complex processes. Furthermore, coarse graining of such processes based on their dominant structures leads to better understanding and dimension reduction of observed systems. However, in many cases, e.g. for nonreversible Markov processes, dominant structures are often not formed by metastable sets but by important cycles or mixture of both. This paper aims at understanding and identifying these different types of dominant structures for reversible as well as nonreversible ergodic Markov processes. Our algorithmic approach generalizes spectral based methods for reversible process by using Schur decomposition techniques which can tackle also nonreversible cases. We illustrate the mathematical construction of our new approach by numerical experiments.}, language = {en} }