Non-linear independent component analysis with diffusion maps

Singer, Amit; Coifman, Ronald R.

doi:10.1016/j.acha.2007.11.001

Cited by 145 publications

(230 citation statements)

References 16 publications

(28 reference statements)

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“…18. Suppose u = u(x, y) = x + 2y (respectively, v = v(x, y) = −2x + y) are the slowly changing (respectively, the rapidly changing variables).…”

Section: Anisotropic Diffusion Mapsmentioning

confidence: 99%

“…18, which relates the anisotropic graph Laplacian in the observable space with the (isotropic) graph Laplacian in the inaccessible space. We formulate our method in a general setting.…”

Section: Anisotropic Diffusion Mapsmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

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

Singer

Erban²,

Kevrekidis³

et al. 2009

Proc. Natl. Acad. Sci. U.S.A.

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141

182

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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...

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“…18. Suppose u = u(x, y) = x + 2y (respectively, v = v(x, y) = −2x + y) are the slowly changing (respectively, the rapidly changing variables).…”

Section: Anisotropic Diffusion Mapsmentioning

confidence: 99%

“…18, which relates the anisotropic graph Laplacian in the observable space with the (isotropic) graph Laplacian in the inaccessible space. We formulate our method in a general setting.…”

Section: Anisotropic Diffusion Mapsmentioning

confidence: 99%

See 1 more Smart Citation

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

Singer

Erban²,

Kevrekidis³

et al. 2009

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

141

182

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show abstract

“…We will focus our attention here on the anisotropic version of these methods [9], which fits nicely to the problem we want to solve. The starting point is to assume that the sample is generated by a non linear function f acting on some d-dimensional parametric features l t that follow an Itô process…”

Section: Diffusion Methods Reviewmentioning

confidence: 99%

Diffusion Methods for Wind Power Ramp Detection

Fernández

Alaíz

González

et al. 2013

Advances in Computational Intelligence

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Esta es la versión de autor de la comunicación de congreso publicada en: This is an author produced version of a paper published in: Abstract. The prediction and management of wind power ramps is currently receiving large attention as it is a crucial issue for both system operators and wind farm managers. However, this is still a problem far from being solved and in this work we will address it as a classification problem working with delay vectors of the wind power time series and applying local Mahalanobis K-NN search with metrics derived from Anisotropic Diffusion methods. The resulting procedures clearly outperform a random baseline method and yield good sensitivity but more work is needed to improve on specificity and, hence, precision.

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“…. , T , we present an empirical method to construct a unique and consistent reduction coordinate set, represented here by x(t) 13 . Because the empirical method we will describe is independent of the observation function f , we refer to the coordinates of x(t) as Nonlinear Intrinsic Variables (NIV).…”

Section: A Overviewmentioning

confidence: 99%

Nonlinear intrinsic variables and state reconstruction in multiscale simulations

Dsilva¹,

Talmon²,

Rabin³

et al. 2013

The Journal of Chemical Physics

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Finding informative low-dimensional descriptions of high-dimensional simulation data (like the ones arising in molecular dynamics or kinetic Monte Carlo simulations of physical and chemical processes) is crucial to understanding physical phenomena, and can also dramatically assist in accelerating the simulations themselves. In this paper, we discuss and illustrate the use of nonlinear intrinsic variables (NIV) in the mining of high-dimensional multiscale simulation data. In particular, we focus on the way NIV allows us to functionally merge different simulation ensembles, and different partial observations of these ensembles, as well as to infer variables not explicitly measured.The approach relies on certain simple features of the underlying process variability to filter out measurement noise and systematically recover a unique reference coordinate frame. We illustrate the approach through two distinct sets of atomistic simulations: a stochastic simulation of an enzyme reaction network exhibiting both fast and slow time scales, and a molecular dynamics simulation of alanine dipeptide in explicit water.

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Non-linear independent component analysis with diffusion maps

Cited by 145 publications

References 16 publications

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

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

Diffusion Methods for Wind Power Ramp Detection

Nonlinear intrinsic variables and state reconstruction in multiscale simulations

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