Construction of Symbolic Dynamics from Experimental Time Series

Mischaikow, Konstantin; Mrożek, Marian; Reiss, Joshua D.; Szymczak, Andrzej

doi:10.1103/physrevlett.82.1144

Cited by 140 publications

(118 citation statements)

References 5 publications

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“…Such applications include vision and recognition by Allili, Mischaikow and Tannenbaum [1], hybrid systems and control theory by Ames [2], rigorous verification of dynamics from experimental data by Mischaikow, Mrozek, Reiss and Szymczak [19], and global analysis of large data sets by the first author and Carlsson [5]. In the last example, there is a growing literature on the importance of persistent homologyhomology classes which persist as one changes a parameter in the system.…”

Section: Resultsmentioning

confidence: 99%

Coverage in sensor networks via persistent homology

Silva

Ghrist

2007

Algebr. Geom. Topol.

369

280

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We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks with minimal sensing capabilities). In particular, we give a homological criterion to rigorously guarantee that a collection of balls covers a bounded domain based on the homology of a certain simplicial pair. This pair of (Vietoris-Rips) complexes is derived from graphs representing a coarse form of distance estimation between nodes and a proximity sensor for the boundary of the domain. The methods we introduce come from persistent homology theory and are applicable to nonlocalized sensor networks with ad hoc wireless communications. 55M25, 93A15; 55N35

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Section: Resultsmentioning

confidence: 99%

Coverage in sensor networks via persistent homology

Silva

Ghrist

2007

Algebr. Geom. Topol.

369

280

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“…Many procedures have been proposed for the election of P ∈ Ω. We can mention techniques based on symbolic dynamics [7], Fourier analysis [8], and wavelet transform [9] (among others). The applicability of these approaches depends on the data-characteristics, i.e., stationarity, length of the series, parameter-variations, levels of noise-contamination, etc.…”

Section: Introductionmentioning

confidence: 99%

Generalized Complexity and Classical-Quantum Transition

Kowalski

Plastino

Casas

2009

Entropy

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We investigate the classical limit of the dynamics of a semiclassical system that represents the interaction between matter and a given field. On using as a quantifier the qComplexity, we find that it describes appropriately the quantum-classical transition, detecting the most salient details of the changeover. Additionally the q-Complexity results a better quantifier of the problem than the q-entropy, in the sense that the q-range is enlarged, describing the q-Complexity, the most important characteristics of the transition for all q-value.

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“…We can mention, among others: (a) frequency count [35] (b) procedures based on amplitude statistics (histograms) [36], (c) binary symbolic dynamics [37], (d) Fourier analysis [38] and, (e) wavelet transform [39]. Their applicability depends on particular characteristics of the data such as stationarity, length of the time series, variation of the parameters, level of noise contamination, etc.…”

Section: Methodologies For Selecting Pdfsmentioning

confidence: 99%

Distances in Probability Space and the Statistical Complexity Setup

Kowalski

Martín

Plastino

et al. 2011

Entropy

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Statistical complexity measures (SCM) are the composition of two ingredients: (i) entropies and (ii) distances in probability-space. In consequence, SCMs provide a simultaneous quantification of the randomness and the correlational structures present in the system under study. We address in this review important topics underlying the SCM structure, viz., (a) a good choice of probability metric space and (b) how to assess the best distance-choice, which in this context is called a "disequilibrium" and is denoted with the letter Q. Q, indeed the crucial SCM ingredient, is cast in terms of an associated distance D. Since out input data consists of time-series, we also discuss the best way of extracting from the time series a probability distribution P . As an illustration, we show just how these issues affect the description of the classical limit of quantum mechanics.

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Construction of Symbolic Dynamics from Experimental Time Series

Cited by 140 publications

References 5 publications

Coverage in sensor networks via persistent homology

Coverage in sensor networks via persistent homology

Generalized Complexity and Classical-Quantum Transition

Distances in Probability Space and the Statistical Complexity Setup

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