Eigenvalue Bounds on Restrictions of Reversible Nearly Uncoupled Markov Chains

AbstractIn this paper we analyze decompositions of reversible nearly uncoupled Markov chains into rapidly mixing subchains. We state upper bounds on the 2nd eigenvalue for restriction and stochastic complementation chains of reversible Markov chains, as well as a relation between them. We illustrate the obtained bounds analytically for bunkbed graphs, and furthermore apply them to restricted Markov chains that arise when analyzing conformation dynamics of a small biomolecule

Sequential Change Point Detection in Molecular Dynamics Trajectories

Motivated from a molecular dynamics context we propose a sequential change point detection algorithm for vector-valued autoregressive models based on Bayesian model selection. The algorithm does not rely on any sampling procedure or assumptions underlying the dynamics of the transitions and is designed to cope with high-dimensional data. We show the applicability of the algorithm on a time series obtained from numerical simulation of a penta-peptide molecule

Illustration of Transition Path Theory on a Collection of Simple Examples

Transition path theory (TPT) has been recently introduced as a theoretical framework to describe the reaction pathways of rare events between long lived states in complex systems. TPT gives detailed statistical information about the reactive trajectories involved in these rare events, which are beyond the realm of transition state theory or transition path sampling. In this paper the TPT approach is outlined, its distinction from other approaches is discussed, and, most importantly, the main insights and objects provided by TPT are illustrated in detail via a series of low dimensional test problems

Impacts of soil carbon on hydrological responses – a sensitivity study of scenarios across diverse climatic zones in South Africa

Soil organic carbon (SOC) content and the water holding capacity of soils are two properties which link the carbon and hydrological cycles. Hydrological model inputs seldom include soil carbon as a parameter even though soil carbon content is known to influence soil water retention capacities. This study is a sensitivity analysis of changes in hydrological responses when the model inputs include different soil carbon percentages for the topsoil horizon. Sensitivities of hydrological responses such as transpiration, runoff volumes, the stormflow component of runoff and extreme runoff events to SOC content were quantified under various climatic conditions in South Africa. The soil water holding capacities at the drained upper limit (i.e. field capacity), permanent wilting point and saturation were calculated for the topsoil horizon, using SOC dependent pedo (soil)-transfer functions for different soil carbon scenarios and locations in South Africa. These variables, together with other pre-determined soil- and locationrelated inputs, as well as 50 years of daily climate, were then used as inputs in a process-based hydrological model. Overall, it was found that increased SOC content in the topsoil horizon leads to an increase in transpiration, a reduction in runoff, especially in its stormflow component, and a reduction of extreme runoff events. However, these changes are relatively small compared to the influence of climate, particularly of rainfall amount and distribution.Significance: Organic carbon content of the soil and the water holding capacity of soils link the carbon and hydrological cycles. Management interventions that increase SOC lead to win-win situations because, in addition to climate change mitigation, plant water availability improves, and overall surface runoff ‘flashiness’ becomes more regulated. While rainfall amount and distribution over space and time remain the most critical determinants of hydrological responses, increased SOC in the topsoil horizon leads to increases in transpiration and thus plant growth, and to a reduction in runoff, especially in its stormflow component, and hence to a small reduction of severe flooding events

Transition Path Theory for Markov Jump Processes

The framework of transition path theory (TPT) is developed in the context of continuous-time Markov chains on discrete state-spaces. Under assumption of ergodicity, TPT singles out any two subsets in the state-space and analyzes the statistical properties of the associated reactive trajectories, i.e., those trajectories by which the random walker transits from one subset to another. TPT gives properties such as the probability distribution of the reactive trajectories, their probability current and flux, and their rate of occurrence and the dominant reaction pathways. In this paper the framework of TPT for Markov chains is developed in detail, and the relation of the theory to electric resistor network theory and data analysis tools such as Laplacian eigenmaps and diffusion maps is discussed as well. Various algorithms for the numerical calculation of the various objects in TPT are also introduced. Finally, the theory and the algorithms are illustrated in several examples

Optimal Fuzzy Aggregation of Networks

This paper is concerned with the problem of fuzzy aggregation of a network with non-negative weights on its edges into a small number of clusters. Specifically we want to optimally define a probability of affiliation of each of the n nodes of the network to each of m < n clusters or aggregates. We take a dynamical perspective on this problem by analyzing the discrete-time Markov chain associated with the network and mapping it onto a Markov chain describing transitions between the clusters. We show that every such aggregated Markov chain and affiliation function can be lifted again onto the full network to define the so-called lifted transition matrix between the nodes of the network. The optimal aggregated Markov chain and affiliation function can then be determined by minimizing some appropriately defined distance between the lifted transition matrix and the transition matrix of the original chain. In general, the resulting constrained nonlinear minimization problem comes out to have continuous level sets of minimizers. We exploit this fact to devise an algorithm for identification of the optimal cluster number by choosing specific minimizers from the level sets. Numerical minimization is performed by some appropriately adapted version of restricted line search using projected gradient descent. The resulting algorithmic scheme is shown to perform well on several test examples

Automated Model Reduction for Complex Systems exhibiting Metastability

We present a novel method for the identification of the most important metastable states of a system with complicated dynamical behavior from time series information. The novel approach represents the effective dynamics of the full system by a Markov jump process between metastable states and the dynamics within each of these metastable states by rather simple stochastic differential equations (SDEs). Its algorithmic realization exploits the concept of hidden Markov models with output behavior given by SDEs. The numerical effort of the method is linear in the length of the given time series and quadratic in terms of the number of metastable states. The performance of the resulting method is illustrated by numerical tests and by application to molecular dynamics time series of a trialanine molecule