Fractal dimension for fractal structures: A Hausdorff approach

Fernández–Martínez, M.; Sánchez-Granero, M.A.

doi:10.1016/j.topol.2011.04.023

Cited by 26 publications

(57 citation statements)

References 15 publications

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“…It is an application of fractal dimension III defined in [32], and is based both in the box-counting and Hausdorff dimensions (see [31], end of Sect. 3.1 for the relation between both classical dimensions, which is also the motivation for this definition).…”

Section: The Fractal Dimension Of a Curvementioning

confidence: 99%

Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series

2012

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Abstract. In this paper, three new algorithms are introduced in order to explore long memory in financial time series. They are based on a new concept of fractal dimension of a curve. A mathematical support is provided for each algorithm and its accuracy is tested for different length time series by Monte Carlo simulations. In particular, in the case of short length series, the introduced algorithms perform much better than the classical methods. Finally, an empirical application for some stock market indexes as well as some individual stocks is presented.

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Section: The Fractal Dimension Of a Curvementioning

confidence: 99%

Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series

2012

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“…It is worth pointing out that such a dimension leads to fruitful applications and results for those interested in the calculation the self-similarity exponent. The main references this work is based on are [1][2][3][4][5]. Next, we brie y describe how the present article is organized.…”

Section: Introductionmentioning

confidence: 99%

After notes on self-similarity exponent for fractal structures

Fernández–Martínez¹,

Garratón²

2017

Open Physics

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Previous works have highlighted the suitability of the concept of fractal structure, which derives from asymmetric topology, to propound generalized de nitions of fractal dimension. The aim of the present article is to collect some results and approaches allowing to connect the self-similarity index and the fractal dimension of a broad spectrum of random processes. To tackle with, we shall use the concept of induced fractal structure on the image set of a sample curve. The main result in this paper states that given a sample function of a random process endowed with the induced fractal structure on its image, it holds that the self-similarity index of that function equals the inverse of its fractal dimension.

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“…In the present work, we apply the fractal dimension definition for a curve introduced in [23] (based on a fractal dimension model contributed previously in [7]) to provide an algorithm which allows its estimation.…”

Section: Introductionmentioning

confidence: 99%

“…In this way, some definitions of fractal dimension for a fractal structure have been already studied in [7][8][9]11] and applied in [10,23], to quote some of them.…”

Section: Introductionmentioning

confidence: 99%

“…One of these concepts, known as fractal dimension III, was introduced in [7], where some of its theoretical properties were analyzed, as well as some connections it presents within other fractal dimension definitions (including the classical ones, namely both the Hausdorff dimension and the box-counting dimension). Moreover, in [23], fractal dimension III was used to define the fractal dimension for a curve with respect to a fractal structure induced on its image set.…”

Section: Introductionmentioning

confidence: 99%

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A new topological indicator for chaos in mechanical systems

et al. 2015

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The main goal in this paper was to provide a novel chaos indicator based on a topological model which allows to calculate the fractal dimension of any curve. A fractal structure is a topological tool whose recursiveness becomes ideal to generalize the concept of fractal dimension. In this paper, we provide an algorithm to calculate a new fractal dimension specially designed for a parametrization of a curve or a random process, whose definition is made by means of fractal structures. As an application, we explore the use of this new concept of fractal dimension as a chaos indicator for dynamical systems, in a similar way to the classical maximal Lyapunov exponent. To illustrate it, we apply the new fractal dimension as an indicator to model the chaotic behavior of a satellite which is moving around a planet whose gravity field is approximated by the field of a point mass.

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Fractal dimension for fractal structures: A Hausdorff approach

Cited by 26 publications

References 15 publications

Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series

Introducing fractal dimension algorithms to calculate the Hurst exponent of financial time series

After notes on self-similarity exponent for fractal structures

A new topological indicator for chaos in mechanical systems

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