Topology from time series

Muldoon, Mark; MacKay, Robert S.; Huke, J. P.; Broomhead, D. S.

doi:10.1016/0167-2789(92)00026-u

Cited by 42 publications

(37 citation statements)

References 17 publications

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“…The original interest derived from explorations, during the 60's to the mid-80's, of behavior generated by nonlinear dynamical systems. The thread that focused especially on pattern and structural complexity originated, in effect, in attempts to reconstruct geometry [29], topology [30], equations of motion [31], periodic orbits [32], and stochastic processes [33] from observations of nonlinear processes. More recently, developing and using measures of complexity has been a concern of researchers studying neural computation [34,35], the clinical analysis of patterns from a variety of medical signals and imaging technologies [36,37,38], and machine learning and synchronization [39,40,41,42,43], to mention only a few contemporary applications.…”

Section: Introductionmentioning

confidence: 99%

The organization of intrinsic computation: Complexity-entropy diagrams and the diversity of natural information processing

Feldman

McTague

Crutchfield

2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

148

138

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Intrinsic computation refers to how dynamical systems store, structure, and transform historical and spatial information. By graphing a measure of structural complexity against a measure of randomness, complexity-entropy diagrams display the range and different kinds of intrinsic computation across an entire class of system. Here, we use complexity-entropy diagrams to analyze intrinsic computation in a broad array of deterministic nonlinear and linear stochastic processes, including maps of the interval, cellular automata and Ising spin systems in one and two dimensions, Markov chains, and probabilistic minimal finite-state machines. Since complexity-entropy diagrams are a function only of observed configurations, they can be used to compare systems without reference to system coordinates or parameters. It has been known for some time that in special cases complexity-entropy diagrams reveal that high degrees of information processing are associated with phase transitions in the underlying process space, the so-called "edge of chaos". Generally, though, complexity-entropy diagrams differ substantially in character, demonstrating a genuine diversity of distinct kinds of intrinsic computation. Discovering organization in the natural world is one of science's central goals. Recent innovations in nonlinear mathematics and physics, in concert with analyses of how dynamical systems store and process information, has produced a growing body of results on quantitative ways to measure natural organization. These efforts had their origin in earlier investigations of the origins of randomness. Eventually, however, it was realized that measures of randomness do not capture the property of organization. This led to the recent efforts to develop measures that are, on the one hand, as generally applicable as the randomness measures but which, on the other, capture a system's complexity-its organization, structure, memory, regularity, symmetry, and pattern. Here-analyzing processes from dynamical systems, statistical mechanics, stochastic processes, and automata theory-we show that measures of structural complexity are a necessary and useful complement to describing natural systems only in terms of their randomness. The result is a broad appreciation of the kinds of information processing embedded in nonlinear systems. This, in turn, * Electronic address:

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Section: Introductionmentioning

confidence: 99%

The organization of intrinsic computation: Complexity-entropy diagrams and the diversity of natural information processing

Feldman

McTague

Crutchfield

2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

148

138

View full text Add to dashboard Cite

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“…Our analysis extends those works by carefully analyzing the effects of sampling and noise on the multiscale singular values, by providing finite sample size bounds, and by extending these techniques to a setting far more general than the setting of smooth manifolds, which may be better suited for the analysis of data sets in truly high dimensional spaces. Also, M. Davies pointed us to work in the dynamical systems community, in particular [23,24,25] where local linear approximations to the trajectories of dynamic systems are considered in order to construct reduced models for the dynamics, and local singular values are used to decide on the dimension of the reduced system and/or on Lyapunov exponents. Finally, [78] discusses various tree constructions for fast nearest neighbor computations and studies how they adapt to the intrinsic dimension of the support of the probability measure from which points are drawn, and discusses and presents some examples of the behavior of the smallest k such that (in the notation of this paper)…”

Section: Overview Of Previous Work On Dimension Estimationmentioning

confidence: 99%

“…The inspiration for the current work originates from ideas in classical statistics (principal component analysis), dimension estimation of point clouds (see Section 2.1, 7 and references therein) and attractors of dynamical systems [23,24,25], and geometric measure theory [26,27,28], especially at its intersection with harmonic analysis. The ability of these tools to quantify and characterize geometric properties of rough sets of interest in harmonic analysis, suggests that they may be successfully adapted to the analysis of sampled noisy point clouds, where sampling and noise may be thought of as new types of (stochastic) perturbations not considered in the classical theory.…”

Section: Introductionmentioning

confidence: 99%

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

Little

Maggioni

Rosasco

2017

Applied and Computational Harmonic Analysis

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Large data sets are often modeled as being noisy samples from probability distributions µ in R D , with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be well-approximated by low-dimensional sets, perhaps even by manifolds. We shall consider sets that are locally well approximated by k-dimensional planes, with k ≪ D, with k-dimensional manifolds isometrically embedded in R D being a special case. Samples from µ are furthermore corrupted by D-dimensional noise. Certain tools from multiscale geometric measure theory and harmonic analysis seem well-suited to be adapted to the study of samples from such probability distributions, in order to yield quantitative geometric information about them. In this paper we introduce and study multiscale covariance matrices, i.e. covariances corresponding to the distribution restricted to a ball of radius r, with a fixed center and varying r, and under rather general geometric assumptions we study how their empirical, noisy counterparts behave. We prove that in the range of scales where these covariance matrices are most informative, the empirical, noisy covariances are close to their expected, noiseless counterparts. In fact, this is true as soon as the number of samples in the balls where the covariance matrices are computed is linear in the intrinsic dimension of M. As an application, we present an algorithm for estimating the intrinsic dimension of M.

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“…Those building blocks are cells, i.e., sets which are homeomorphic to the interiors of n disks. Hence, one basically needs to select a sufficiently large number of points which can be used as centers of disks enclosing their neighbors, so that every point in the invariant set under study is within at least one cell [5]. We arbitrarily chose the first point, and then sort the rest of them according to their distance to it.…”

mentioning

confidence: 99%

“…In this way, we assign a set of integers to each point of the set under study, which indicates the set of cells that the point belongs to. In order to decide whether a point belongs to a given cell, we inspect the singular values of the matrix [5],…”

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confidence: 99%