The visibility parameter for words and permutations

Cristea, Ligia-Loreta; Prodinger, Helmut

doi:10.2478/s11533-012-0135-2

Cited by 4 publications

(5 citation statements)

References 6 publications

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“…describes diagrams with an arbitrary number of hidden variables located between x 1 and x 2 , x 2 and x 3 , or x 3 and x 4 . The general [z (1) i z (2) j z (3) k ] approximant of this expansion (for i, j, k > 2) reads:…”

Section: Markovian Dynamics: a Constructive Solutionmentioning

confidence: 99%

“…Similarly, the correction of order α = 1 reduces to the intersection (H (x) < x) ∩ (H (2) (x) < H (x)) ∩ (H (3)…”

Section: Quasiperiodic Dynamicsmentioning

confidence: 99%

“…The set of graphs spanned by this mapping are called horizontal visibility graphs (HVGs). These are noncrossing outerplanar graphs with a Hamiltonian path [2,3], subgraphs of a more general mapping [4] that have been recently used in the context of time series analysis and signal processing [15] (see figure 1 for an illustration). The methodology proceeds by analysing the topological properties of G and, according to that information, characterizing the structure of {x t } and its underlying dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…where p 3 = P → (3). From previous sections, we have learned how to compute these terms, either in an exact or in a perturbative way.…”

mentioning

confidence: 99%

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On the degree distribution of horizontal visibility graphs associated with Markov processes and dynamical systems: diagrammatic and variational approaches

Lacasa¹

2014

Nonlinearity

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Dynamical processes can be transformed into graphs through a family of mappings called visibility algorithms, enabling the possibility of (i) making empirical data analysis and signal processing and (ii) characterising classes of dynamical systems and stochastic processes using the tools of graph theory. Recent works show that the degree distribution of these graphs encapsulates much information on the signals variability, and therefore constitutes a fundamental feature for statistical learning purposes. However, exact solutions for the degree distributions are only known in a few cases, such as for uncorrelated random processes. Here we analytically explore these distributions in a list of situations. We present a diagrammatic formalism which computes for all degrees their corresponding probability as a series expansion in a coupling constant which is the number of hidden variables. We offer a constructive solution for general Markovian stochastic processes and deterministic maps. As case tests we focus on Ornstein-Uhlenbeck processes, fully chaotic and quasiperiodic maps. Whereas only for certain degree probabilities can all diagrams be summed exactly, in the general case we show that the perturbation theory converges. In a second part, we make use of a variational technique to predict the complete degree distribution for special classes of Markovian dynamics with fast-decaying correlations. In every case we compare the theory with numerical experiments.

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Section: Markovian Dynamics: a Constructive Solutionmentioning

confidence: 99%

“…Similarly, the correction of order α = 1 reduces to the intersection (H (x) < x) ∩ (H (2) (x) < H (x)) ∩ (H (3)…”

Section: Quasiperiodic Dynamicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…where p 3 = P → (3). From previous sections, we have learned how to compute these terms, either in an exact or in a perturbative way.…”

mentioning

confidence: 99%

See 2 more Smart Citations

On the degree distribution of horizontal visibility graphs associated with Markov processes and dynamical systems: diagrammatic and variational approaches

Lacasa¹

2014

Nonlinearity

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“…The computations for the variance were already quite formidable in the previous paper [2]; here, it would be much worse, and we leave this to a younger person, perhaps within a Ph.D. project. In all these problems it is noticeable that the expected value is not too difficult to compute, while the variance is orders of magnitudes more complicated.…”

Section: Permutationsmentioning

confidence: 91%

The box parameter for words and permutations

Prodinger

2014

Open Mathematics

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The box parameter for words counts how often two letters and define a "box" such that all the letters +1 −1 fall into that box. It is related to the visibility parameter and other parameters on words. Three models are considered: Words over a finite alphabet, permutations, and words with letters following a geometric distribution. A typical result is: The average box parameter for words over an M letter alphabet is asymptotically given by 2 − 2 H M /M, for fixed M and → ∞. MSC:05A05, 05A30

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