Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps

Abstract

Nonlinear independent component analysis is combined with diffusion-map data analysis techniques to detect good observables in high-dimensional dynamic data. These detections are achieved by integrating local principal component analysis of simulation bursts by using eigenvectors of a Markov matrix describing anisotropic diffusion. The widely applicable procedure, a crucial step in model reduction approaches, is illustrated on stochastic chemical reaction network simulations.slow manifold | dimensionality reduction | chemical reactions E volution of dynamical systems often occurs on two or more time scales. A simple deterministic example is given by the coupled system of ordinary differential equations (ODEs)with the small parameter 0 < τ 1, where α(u, v) and β(u, v) are O(1). For any given initial condition (u 0 , v 0 ), already at t = O(τ) the system approaches a new value (u 0 , v), where v satisfies the asymptotic relation β(u 0 , v) = 0. Although the system is fully described by two coordinates, the relation β(u, v) = 0 defines a slow one-dimensional manifold which approximates the slow dynamics for t τ. In this example, it is clear that v is the fast variable whereas u is the slow one. Projecting onto the slow manifold here is rather easy: The fast foliation is simply "vertical", i.e. u = const. However, when we observe the system in terms of the variables x = x(u, v) and y = y (u, v) which are unknown nonlinear functions of u and v, then the "observables" x and y have both fast and slow dynamics. Projecting onto the slow manifold becomes nontrivial, because the transformation from (x, y) to (u, v) is unknown. Detecting the existence of an intrinsic slow manifold under these conditions and projecting onto it are important in any model reduction technique. Knowledge of a good parametrization of such a slow manifold is a crucial component of the equation-free framework for modeling and computation of complex/multiscale systems (1-3).Principal component analysis (PCA, also known as POD) (4-6) has traditionally been used for data and model reduction in contexts ranging from meteorology (7) and transitional flows (8) to protein folding (9, 10); in these contexts the PCA procedure is used to detect good global reduced coordinates that best capture the data variability. In recent years, diffusion maps (11-17) have been used in a similar spirit to detect low-dimensional, nonlinear manifolds underlying high-dimensional datasets.In this paper, we integrate ensembles of local PCA analyses in the diffusion-map framework to enable the detection of slow variables in high-dimensional data arising from dynamic model simulations. The proposed algorithm is built along the lines of the nonlinear independent component analysis method recently introduced in ref. 18. The approach takes into account the time dependence of the data, whereas in the diffusion-map approach the time labeling of the data points is not included. We demonstrate our algorithm for stochastic simulators arising in the context of chemical/biochemical react...

A Study of the Morphology, Dynamics, and Folding Pathways of Ring Polymers with Supramolecular Topological Constraints Using Molecular Simulation and Nonlinear Manifold Learning

“…37,38,[51][52][53][54][55][56] Geometrically, this can be understood as the emergence of a small number of collective variables governing the long-time evolution of the system to which the remaining degrees of freedom are slaved. 37,38,[57][58][59] We and others have previously demonstrated that these collective variables can be determined from molecular simulation trajectories using nonlinear manifold learning, [37][38][39]51,[60][61][62][63][64][65][66][67][68][69][70][71][72][73] the particular variant of which we use here are diffusion maps. 60,[65][66][67][68] In a nutshell, the diffusion map approach constructs a random walk over the high-dimensional simulation trajectory with hopping probabilities based on the structural similarity of the constituent snapshots, then performs a spectral analysis of the harmonics of the resultant discrete Markov process to ascertain the effective dimensionality of the underlying "intrinsic manifold" and nonlinear collective variables with which to parameterize it.…”

Systematic determination of order parameters for chain dynamics using diffusion maps

“…Isomap (5,22) and LLE (23) have been successfully applied to peptide systems, and, although diffusion maps have been used to study phenomena as diverse as chemical reaction networks (24) and defect mobility at an interface (25), they have not been previously applied to systems of biophysical significance.…”

“…In the case of C 8 we find the dynamics to be governed by torsional motions. For C 16 and C 24 we extract three global order parameters with which we characterize the fundamental dynamics, and determine that the low free-energy pathway of globular collapse proceeds by a "kink and slide" mechanism, whereby a bend near the end of the linear chain migrates toward the middle to form a hairpin and, ultimately, a coiled helix. The low-dimensional representation is subtly perturbed in the solvated phase relative to the ideal gas, and its geometric structure is conserved between C 16 and C 24 .…”

Multiscale geometric methods for data sets I: Multiscale SVD, noise and curvature

“…(d) the vectors of (resVar l ) l for Q 10 (1000, 0), S 9 (1000, 0), Q 10 (1000, 0.1), S 9 (1000, 0.1): it seems hard to see a difference between the intrinsic dimensions 10 and 9, in both the noiseless and noisy cases. (e) The dimension of S 9 (2000, 0.01) in R 100 estimated according to the heuristic in [74] yields the wrong dimension (∼ 8) even for small amounts of noise; this is of course not a rigorous test, and it is a heuristic procedure, not an algorithm, as described in [74].…”

“…We expect similar phenomena to be common to other manifold learning algorithms, and leave a complete investigation to future work. In [74] it is suggested that diffusion maps [7] may be used in order to estimate intrinsic dimension as well as a scale parameter, for example in the context of dynamical systems where a small number of slow variables are present. Rather than an automatic algorithm for dimension estimation, [74] suggests a criterion that involves eyeballing the function i,j e − ||x i −x j || 2 ǫ 2 , as a function of ǫ, to find a region of linear growth, whose slope is an estimate of the intrinsic dimension.…”