Inhomogeneity and complexity measures for spatial patterns

Piasecki, R.; Martín, María Teresa; Plastino, A.

doi:10.1016/s0378-4371(01)00591-x

Cited by 18 publications

(16 citation statements)

References 31 publications

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“…As such, they are incapable of distinguishing structures that differ in their correlations over more than two variables, as we shall see below. Some recent work in this area, motivated in part by the need to characterize complex interfaces in surface science and geology [7,8,9,10,11,12,13], has suggested a set of approaches to these questions that are similar in spirit to fractal dimensions, in the sense that these approaches involve coarse-graining variables and then monitoring the changes that result as the coarse-graining scale is modulated. One can also use a multifractal approach, also known as the singularity spectrum, "f (α)", the thermodynamic formalism, and the fluctuation spectrum; for reviews, see, e.g., [14,15,16].…”

Section: Introductionmentioning

confidence: 99%

Structural information in two-dimensional patterns: Entropy convergence and excess entropy

2003

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We develop information-theoretic measures of spatial structure and pattern in more than one dimension. As is well known, the entropy density of a two-dimensional configuration can be efficiently and accurately estimated via a converging sequence of conditional entropies. We show that the manner in which these conditional entropies converge to their asymptotic value serves as a measure of global correlation and structure for spatial systems in any dimension. We compare and contrast entropy convergence with mutual-information and structure-factor techniques for quantifying and detecting spatial structure.

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Section: Introductionmentioning

confidence: 99%

Structural information in two-dimensional patterns: Entropy convergence and excess entropy

2003

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“…The quantitative evaluation of the spatial inhomogeneity degree can be obtained using a simple entropic measure for finite sized objects (see [15] for binary patterns and [16] for grey-scale ones), its q-extensions à la Tsallis [17] is given in Refs. [18,19]. The modified entropic measure can be also widely applied to statistical reconstructions of complex grey-scale patterns [20] and prototypical three-dimensional microstructures [21] with the usage of the decomposable multiphase entropic descriptor [22].…”

Section: A Possible Correlation Between the Degrees Of Spatial Disordmentioning

confidence: 99%

Sensitivity to Initial Conditions in an Extended Activator--Inhibitor Model for the Formation of Patterns

Piasecki¹,

Olchawa²,

Smaga³

2018

Acta Phys. Pol. B

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Despite simplicity, the synchronous cellular automaton [D.A. Young, Math. Biosci. 72, 51 (1984)] enables reconstructing basic features of patterns of skin. Our extended model allows studying the formatting of patterns and their temporal evolution also on the favourable and hostile environments. As a result, the impact of different types of an environment is accounted for the dynamics of patterns formation. The process is based on two diffusible morphogens, the short-range activator and the long-range inhibitor, produced by differentiated cells (DCs) represented as black pixels. For a neutral environment, the extended model reduces to the original one. However, even the reduced model is statistically sensitive to a type of the initial distribution of DCs. To compare the impact of the uniform random distribution of DCs (R-system) and the non-uniform distribution in the form of random Gaussian-clusters (G-system), we chose inhibitor as the control parameter. To our surprise, in the neutral environment, for the chosen inhibitorvalue that ensures stable final patterns, the average size of final G-populations is lower than in the R-case. In turn, when we consider the favourable environment, the relatively bigger shift toward higher final concentrations of DCs appears in the G. Thus, in the suitably favourable environment, this order can be reversed. Furthermore, the different critical values of the control parameter for the R and the G suggest some dissimilarities in temporal evolution of both systems. In particular, within the proper ranges of the critical values, their oscillatory behaviours are different. The respective temporal evolutions are illustrated by a few examples.

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“…In this contribution, we will consider only maximum entropies corresponding to the equiprobable distribution, the simplest case. Piasecki et al [38] and Abe and Plastino [2] have already presented the generalization for some special cases of the Tsallis entropy, and Bickel [11] has used it to quantify fractal intermittency in biological time series.…”

Section: Extended Entropies and "Disorders" -Definitionsmentioning

confidence: 99%

Extended Entropies and Disorder

Davison

Shiner

2005

Advs. Complex Syst.

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Landsberg's notion of disorder, entropy normalized to maximum entropy, was originally proposed for the Shannon information-theoretic entropy to overcome extensivity-based deficiencies of entropy as a measure of disorder. We generalize Landsberg's concept to three classes of extended entropies: Rényi, Tsallis and Landsberg-Vedral. We show an intimate connection between the Rényi disorders and the spectrum of dimensions known as multifractals. Three examples are treated, including one for power law distributions and one based on the logistic map. On the basis of the three examples, it is demonstrated that all three classes of extended disorder are required to fully characterize the corresponding properties of a system. We conjecture, and sketch a proof to support, that all three extended disorders are also sufficient to completely determine a dimension spectrum.

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Inhomogeneity and complexity measures for spatial patterns

Cited by 18 publications

References 31 publications

Structural information in two-dimensional patterns: Entropy convergence and excess entropy

Structural information in two-dimensional patterns: Entropy convergence and excess entropy

Sensitivity to Initial Conditions in an Extended Activator--Inhibitor Model for the Formation of Patterns

Extended Entropies and Disorder

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