Fractal dimension for fractal structures: A Hausdorff approach revisited

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

doi:10.1016/j.jmaa.2013.07.011

Cited by 36 publications

(47 citation statements)

References 11 publications

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“…S(I) i is the cumulative sum sequences that can be built from the monitoring data sequence. 40,41 A detailed computation of fractal dimensions was performed by Wu et al 26 After the calculation of the fractal dimension, the mine slope deformation N i + 1 can be obtained as follows…”

Section: Fractal Modelmentioning

confidence: 99%

An improved fractal prediction model for forecasting mine slope deformation using GM (1, 1)

Dong

Shi

et al. 2015

Structural Health Monitoring

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The forecasting slope deformation potential is required to evaluate slope safety during open-pit mining, allowing us to formulate and promote effective emergency strategies in advance to prevent slope failure disasters. Although fractal models have been used to predict slope deformation, such limitations as low prediction accuracy, poor stability and the requirement for large amounts of data must be overcome. This article proposes an improved fractal model to forecast mine slope deformation using the grey system theory. The GM (1, 1) model is used in the improved fractal model to optimize the fitting function of the fractal dimension because of its high computational efficiency and strong fitting ability. Data sequences spanning 13 days from 11 global positioning system monitoring stations in the Jinduicheng open-pit mine in Shaanxi Province, China, were applied to forecast the slope deformation. The results from both the traditional fractal model and the improved fractal model can accurately forecast the slope deformation value fairly close to the actual field monitoring value, but the latter can make a more accurate prediction than the former. There is a significant relationship between the prediction accuracy and the data sequence dispersion. Further analysis revealed that our improved fractal model is more capable of resisting the volatility existing in the data sequences than the traditional fractal model. These findings assist in understanding the applicability of prediction models and the deformation trends of open-pit mine slopes.

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Section: Fractal Modelmentioning

confidence: 99%

An improved fractal prediction model for forecasting mine slope deformation using GM (1, 1)

Dong

Shi

et al. 2015

Structural Health Monitoring

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“…(9). In fact, notice that we assume here that the satellite is describing 50 revolutions following an elliptical orbit around a planet whose gravity field has been approximated by the field of a point mass.…”

Section: Figmentioning

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%

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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“…The new algorithm that the authors introduce in this paper is based on both [14, Definition 3.2] (for k = 4) and [14,Theorem 3.13]. Thus, while that definition provided a new discrete model to calculate the fractal dimension of any space with respect to a fractal structure by means of finite coverings (which is very suitable for computational purposes), the latter result implies that this new fractal dimension is equal to the Hausdorff dimension for compact Euclidean subsets.…”

Section: Introductionmentioning

confidence: 99%

“…This novel approach is based on both a new discrete model of fractal dimension for a fractal structure which considers finite coverings and a theoretical result that the authors contributed previously in [14]. This new procedure combines fractal techniques with tools from Machine Learning Theory.…”

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confidence: 99%