Molecular dynamics simulation generates large quantities of data that must be interpreted using physically meaningful analysis. A common approach is to describe the system dynamics in terms of transitions between coarse partitions of conformational space. In contrast to previous work that partitions the space according to geometric proximity, the authors examine here clustering based on kinetics, merging configurational microstates together so as to identify long-lived, i.e., dynamically metastable, states. As test systems microsecond molecular dynamics simulations of the polyalanines Ala(8) and Ala(12) are analyzed. Both systems clearly exhibit metastability, with some kinetically distinct metastable states being geometrically very similar. Using the backbone torsion rotamer pattern to define the microstates, a definition is obtained of metastable states whose lifetimes considerably exceed the memory associated with interstate dynamics, thus allowing the kinetics to be described by a Markov model. This model is shown to be valid by comparison of its predictions with the kinetics obtained directly from the molecular dynamics simulations. In contrast, clustering based on the hydrogen-bonding pattern fails to identify long-lived metastable states or a reliable Markov model. Finally, an approach is proposed to generate a hierarchical model of networks, each having a different number of metastable states. The model hierarchy yields a qualitative understanding of the multiple time and length scales in the dynamics of biomolecules.

In mixed quantum-classical molecular dynamics few but important degrees of freedom of a molecular system are modeled quantum mechanically while the remaining degrees of freedom are treated within the classical approximation. Such models can be systematically derived as a first-order approximation to the partial Wigner transform of the quantum Liouville-von Neumann equation. The resulting adiabatic quantum-classical Liouville equation ͑QCLE͒ can be decomposed into three individual propagators by means of a Trotter splitting: ͑1͒ phase oscillations of the coherences resulting from the time evolution of the quantum-mechanical subsystem, ͑2͒ exchange of densities and coherences reflecting non adiabatic effects in quantum-classical dynamics, and ͑3͒ classical Liouvillian transport of densities and coherences along adiabatic potential energy surfaces or arithmetic means thereof. A novel stochastic implementation of the QCLE is proposed in the present work. In order to substantially improve the traditional algorithm based on surface hopping trajectories ͓J. C. Tully, J. Chem. Phys. 93, 1061 ͑1990͔͒, we model the evolution of densities and coherences by a set of surface hopping Gaussian phase-space packets ͑GPPs͒ with variable width and with adjustable real or complex amplitudes, respectively. The dense sampling of phase space offers two main advantages over other numerical schemes to solve the QCLE. First, it allows us to perform a quantum-classical simulation employing a constant number of particles; i.e., the generation of new trajectories at each surface hop is avoided. Second, the effect of nonlocal operators on the exchange of densities and coherences can be treated beyond the momentum jump approximation. For the example of a single avoided crossing we demonstrate that convergence towards fully quantum-mechanical dynamics is much faster for surface hopping GPPs than for trajectory-based methods. For dual avoided crossings the Gaussian-based dynamics correctly reproduces the quantum-mechanical result even when trajectory-based methods not accounting for the transport of coherences fail qualitatively.

We give an alternative and unified derivation of the general framework developed in the last few years for analyzing nonstationary time series. A different approach for handling the resulting variational problem numerically is introduced. We further expand the framework by employing adaptive finite element algorithms and ideas from information theory to solve the problem of finding the most adequate model based on a maximum-entropy ansatz, thereby reducing the number of underlying probabilistic assumptions. In addition, we formulate and prove the result establishing the link between the optimal parametrizations of the direct and the inverse problems and compare the introduced algorithm to standard approaches like Gaussian mixture models, hidden Markov models, artificial neural networks and local kernel methods. Furthermore, based on the introduced general framework, we show how to create new data analysis methods for specific practical applications. We demonstrate the application of the framework to data samples from toy models as well as to real-world problems such as biomolecular dynamics, DNA sequence analysis and financial applications.

We present a new approach to clustering of time series based on a minimization of the averaged clustering functional. The proposed functional describes the mean distance between observation data and its representation in terms of K abstract models of a certain predefined class (not necessarily given by some probability distribution). For a fixed time series x(t) this functional depends on K sets of model parameters Θ = (θ1,. .. , θ K) and K functions of cluster affiliations Γ = (γ1(t),. .. , γ K (t)) (characterizing the affiliation of any element x(t) of the analyzed time series to one of the K clusters defined by the considered model parameters). We demonstrate that for a fixed set of model parameters Θ the appropriate Tykhonov-type regularization of this functional with some regularization factor ǫ 2 results in a minimization problem similar to a variational problem usually associated with one-dimensional non-homogeneous partial differential equation. This analogy allows us to apply the finite element framework to the problem of time series analysis and to propose a numerical scheme for time series clustering. We investigate the conditions under which the proposed scheme allows a monotone improvement of the initial parameter guess wrt. the minimization of the discretized version of the regularized functional. We also discuss the interpretation of the regularization factor in the Markovian case and show its connection to metastability and exit times. The computational performance of the resulting method is investigated numerically on multidimensional test data and is applied to the analysis of multidimensional historical stock market data.

A numerical framework for data-based identification of nonstationary linear factor models is presented. The approach is based on the extension of the recently developed method for identification of persistent dynamical phases in multidimensional time series, permitting the identification of discontinuous temporal changes in underlying model parameters. The finite element method (FEM) discretization of the resulting variational functional is applied to reduce the dimensionality of the resulting problem and to construct the numerical iterative algorithm. The presented method results in the sparse sequential linear minimization problem with linear constrains. The performance of the framework is demonstrated for the following two application examples: (i) in the context of subgrid-scale parameterization for the Lorenz model with external forcing and (ii) in an analysis of climate impact factors acting on the blocking events in the upper troposphere. The importance of accounting for the nonstationarity issue is demonstrated in the second application example: modeling the 40-yr ECMWF Re-Analysis (ERA-40) geopotential time series via a single best stochastic model with time-independent coefficients leads to the conclusion that all of the considered external factors are found to be statistically insignificant, whereas considering the nonstationary model (which is demonstrated to be more appropriate in the sense of information theory) identified by the methodology presented in the paper results in identification of statistically significant external impact factor influences.