New approach to automatic mode detection in stochastic models of complex real-world systems
Final Report Abstract
During the DFG Project we developed a set of novel data clustering and statistical inference techniques for detecting metastable and spectral modes of stochastic processes, which are applicable to a lot of scientific and technical fields for understanding and predicting long-time behavior of complex systems. 1. The first technique is the maximum margin metastable clustering (M3 C), which exploit the widely used maximum margin principle to solve the problem of metastable mode detection. The key idea of this technique is to replace the original data space by the space of “transition pairs” when searching for the optimal classification boundary, which implies that a large number of static data analysis techniques developed based on the maximum margin principle has also the potential to be utilized in dynamical analysis. 2. The second technique is the Markov transition model (MTM) based spectral mode detection. The MTM is a continuous extension of the traditional MSM, which allows people to use more flexible and “soft” coarse-graining of phase spaces for modeling Markov processes and can achieve more accurate spectral approximation. (The traditional MSMs can only accept “hard” partitioning of phase spaces.) 3. The third technique is the infinite-state hidden Markov model (HMM) based spectral mode detection. We discovered for the first time that the spectral approximation is an ill-posed problem for the traditional HMMs and the infinite-state HMMs developed in recent years, and presented the corresponding mathematical explanation. Our work shows that the sparse spectrum priors are necessary for spectral estimation of HMMs. More general theory and methods of sparse spectrum HMMs will be a focus of our future research. 4. The last technique is the observable operator model (OOM) based spectral mode detection. The OOM is an algebraic representation of stochastic dynamics and studied as a generalization of the HMM in the field of machine learning, but it is not so widely applied as the HMM because of its poor intelligibility. Our work shows that the OOM can be applied to modeling and spectral analysis of complex systems without any modeling error under the framework of projected Markov model. This result was very surprising, and highly praised by Prof. H. Jaeger (Jacobs University Bremen), the inventor of the OOM. As he said, it was the first time that the OOM made its way into an applied scenario. The OOM has the potential to be a standard spectral analysis tool especially in the field of biophysics, and we will investigate the OOM based mode detection and analysis methods for more complicated cases in the future.
Publications
- “A Bayesian nonparametric model for spectral estimation of metastable systems,” in Proceedings of the conference on Uncertainty in artificial intelligence (UAI), Quebec City, Canada, 2014, pp. 878-887
H. Wu
- “Gaussian Markov transition models of molecular kinetics,” Journal of Chemical Physics, 2015, 142(8): 084104
H. Wu and F. Noe
(See online at https://doi.org/10.1063/1.4913214) - “Maximum margin clustering for state decomposition of metastable systems,” Neurocomputing, 2015, 164(21): 5-22
H. Wu
(See online at https://dx.doi.org/10.1016/j.neucom.2014.12.093) - “Projected metastable Markov processes and their estimation with observable operator models,” Journal of Chemical Physics, 2015, 143(14): 144101
H. Wu, J. -H. Prinz and F. Noe
(See online at https://doi.org/10.1063/1.4932406)
