Ronald Coifman

Publications (3)3.33 Total impact

• Chapter: Coarse Collective Dynamics of Animal Groups

  Thomas A. Frewen, Iain D. Couzin, Allison Kolpas, Jeff Moehlis, Ronald Coifman, Ioannis G. Kevrekidis
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  ABSTRACT: The coarse-grained, computer-assisted analysis of models of collective dynamics in animal groups involves (a) identifying appropriate observables that best describe the state of these complex systems and (b) characterizing the dynamics of such observables. We devise “equation-free” simulation protocols for the analysis of a prototypical individual-based model of collective group dynamics. Our approach allows the extraction of information at the macroscopic level via parsimonious usage of the detailed, “microscopic” computational model. Identification of meaningful coarse observables (“reduction coordinates”) is critical to the success of such an approach, and we use a recently-developed dimensionality-reduction approach (diffusion maps) to detect good observables based on data generated by local model simulation bursts. This approach can be more generally applicable to the study of coherent behavior in a broad class of collective systems (e.g., collective cell migration).
  10/2010: pages 299-309;
• Source

  Available from: citeseerx.ist.psu.edu

  Chapter: Diffusion Maps - a Probabilistic Interpretation for Spectral Embedding and Clustering Algorithms

  Boaz Nadler, Stephane Lafon, Ronald Coifman, Ioannis G. Kevrekidis
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  ABSTRACT: Spectral embedding and spectral clustering are common methods for non-linear dimensionality reduction and clustering of complex high dimensional datasets. In this paper we provide a diffusion based probabilistic analysis of algorithms that use the normalized graph Laplacian. Given the pairwise adjacency matrix of all points in a dataset, we define a random walk on the graph of points and a diffusion distance between any two points. We show that the diffusion distance is equal to the Euclidean distance in the embedded space with all eigenvectors of the normalized graph Laplacian. This identity shows that characteristic relaxation times and processes of the random walk on the graph are the key concept that governs the properties of these spectral clustering and spectral embedding algorithms. Specifically, for spectral clustering to succeed, a necessary condition is that the mean exit times from each cluster need to be significantly larger than the largest (slowest) of all relaxation times inside all of the individual clusters. For complex, multiscale data, this condition may not hold and multiscale methods need to be developed to handle such situations.
  11/2007: pages 238-260;
• Source

  Available from: arxiv.org

  Article: Variable-free exploration of stochastic models: a gene regulatory network example.

  Radek Erban, Thomas A Frewen, Xiao Wang, Timothy C Elston, Ronald Coifman, Boaz Nadler, Ioannis G Kevrekidis
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  ABSTRACT: Finding coarse-grained, low-dimensional descriptions is an important task in the analysis of complex, stochastic models of gene regulatory networks. This task involves (a) identifying observables that best describe the state of these complex systems and (b) characterizing the dynamics of the observables. In a previous paper [R. Erban et al., J. Chem. Phys. 124, 084106 (2006)] the authors assumed that good observables were known a priori, and presented an equation-free approach to approximate coarse-grained quantities (i.e., effective drift and diffusion coefficients) that characterize the long-time behavior of the observables. Here we use diffusion maps [R. Coifman et al., Proc. Natl. Acad. Sci. U.S.A. 102, 7426 (2005)] to extract appropriate observables ("reduction coordinates") in an automated fashion; these involve the leading eigenvectors of a weighted Laplacian on a graph constructed from network simulation data. We present lifting and restriction procedures for translating between physical variables and these data-based observables. These procedures allow us to perform equation-free, coarse-grained computations characterizing the long-term dynamics through the design and processing of short bursts of stochastic simulation initialized at appropriate values of the data-based observables.
  The Journal of Chemical Physics 05/2007; 126(15):155103. · 3.33 Impact Factor