scholarly journals Accuracy of the box-counting algorithm for noisy fractals

2016 ◽  
Vol 27 (10) ◽  
pp. 1650112 ◽  
Author(s):  
A. Z. Górski ◽  
M. Stróż ◽  
P. Oświȩcimka ◽  
J. Skrzat
Keyword(s):  
Additive Noise ◽  
Fractal Dimensions ◽  
Growing Up ◽  
Embedding Dimension ◽  
Noise Effect ◽  
Fractal Sets ◽  
Considerable Error ◽  
Box Counting ◽  
Counting Algorithm

The box-counting (BC) algorithm is applied to calculate fractal dimensions of four fractal sets. The sets are contaminated with an additive noise with amplitude [Formula: see text]. The accuracy of calculated numerical values of the fractal dimensions is analyzed as a function of [Formula: see text] for different sizes of the data sample. In particular, it has been found that even in case of pure fractals ([Formula: see text]) as well as for tiny noise ([Formula: see text]) one has considerable error for the calculated exponents of order 0.01. For larger noise the error is growing up to 0.1 and more, with natural saturation limited by the embedding dimension. This prohibits the power-like scaling of the error. Moreover, the noise effect cannot be cured by taking larger data samples.

3D box-counting algorithm for calculating fractal dimension of cities

2010 ◽  
Vol 30 (8) ◽  
pp. 2070-2072
Author(s):  
Le-shan ZHANG ◽  
Ge CHEN ◽  
Yong HAN ◽  
Tao ZHANG
Keyword(s):  
Fractal Dimension ◽  
Box Counting ◽  
Counting Algorithm

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

Architectural Bias in Recurrent Neural Networks: Fractal Analysis

2003 ◽  
Vol 15 (8) ◽  
pp. 1931-1957 ◽  
Author(s):  
Peter Tiňo ◽  
Barbara Hammer
Keyword(s):  
Neural Networks ◽  
Fractal Analysis ◽  
Recurrent Neural Networks ◽  
Markov Models ◽  
Fractal Dimensions ◽  
Transition Diagram ◽  
Activation Patterns ◽  
Box Counting ◽  
Fractal Clusters ◽  
Finite State

We have recently shown that when initialized with “small” weights, recurrent neural networks (RNNs) with standard sigmoid-type activation functions are inherently biased toward Markov models; even prior to any training, RNN dynamics can be readily used to extract finite memory machines (Hammer & Tiňo, 2002; Tiňo, Čerňanský, &Beňušková, 2002a, 2002b). Following Christiansen and Chater (1999), we refer to this phenomenon as the architectural bias of RNNs. In this article, we extend our work on the architectural bias in RNNs by performing a rigorous fractal analysis of recurrent activation patterns. We assume the network is driven by sequences obtained by traversing an underlying finite-state transition diagram&a scenario that has been frequently considered in the past, for example, when studying RNN-based learning and implementation of regular grammars and finite-state transducers. We obtain lower and upper bounds on various types of fractal dimensions, such as box counting and Hausdorff dimensions. It turns out that not only can the recurrent activations inside RNNs with small initial weights be explored to build Markovian predictive models, but also the activations form fractal clusters, the dimension of which can be bounded by the scaled entropy of the underlying driving source. The scaling factors are fixed and are given by the RNN parameters.

scholarly journals Accuracy Analysis of the Box-Counting Algorithm

2012 ◽  
Vol 121 (2B) ◽  
pp. B-28-B-30 ◽  
Author(s):  
A.Z. Górski ◽  
S. Drożdż ◽  
A. Mokrzycka ◽  
J. Pawlik
Keyword(s):  
Accuracy Analysis ◽  
Box Counting ◽  
Counting Algorithm

scholarly journals The Meso-Analysis of the Rock-Burst Debris of Rock Similar Material Based on SEM

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Manqing Lin ◽  
Lan Zhang ◽  
Xiqi Liu ◽  
Yuanyou Xia ◽  
Jiaqi He ◽  
...  
Keyword(s):  
Rock Burst ◽  
Fractal Dimensions ◽  
Similar Material ◽  
Failure Characteristics ◽  
True Triaxial ◽  
Sem Image ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Combined Test ◽  
Experimental Process

In order to explore the specimen failure characteristics during rock-burst under different gradient stress conditions, in this paper, a novel experimental technique was proposed; a common series of tests under two gradient stress paths were conducted on rock similar material specimens using the true-triaxial gradient and hydraulic-pneumatic combined test apparatus. And plaster was used as the rock similar material. In the experimental process, several rock-burst debris with area sizes of 100 mm2 were collected, and the fractal dimensions of typical detrital section crystal contours were analyzed and calculated using a scanning electron microscopy (SEM) method. The results showed that the specimens’ failure characteristics which had been induced by the two gradient stress processes were various. Also, the mesoscopic morphology of the rock-burst detrital section had effectively reflected its macroscopic failure characteristics. It was found that the fractal dimensions of the crystal contours of the specimen’s detrital section had fractal characteristics, and the box-counting dimension based on the SEM image could quantitatively describe the rock-burst failure characteristics. Furthermore, under the same magnification, the fractal dimensions of the crystal contours of the splitting failures were found to be relatively smaller than those of the shearing failures.

Fast differential box-counting algorithm on GPU

2019 ◽  
Vol 76 (1) ◽  
pp. 204-225 ◽  
Author(s):  
Juan Ruiz de Miras
Keyword(s):  
Box Counting ◽  
Counting Algorithm
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