Entropic measure of spatial disorder for systems of finite-sized objects

Piasecki, R.

doi:10.1016/s0378-4371(99)00458-6

Cited by 32 publications

(59 citation statements)

References 21 publications

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“…There has been some preliminary work in this direction [13,51,52,[73][74][75][76]; however, many questions remain. One of the central difficulties is that, unlike in one dimension where the various expressions for the excess entropy are equivalent, they yield different results when extended to two dimensions [77].…”

Section: Future Directionsmentioning

confidence: 99%

Regularities unseen, randomness observed: Levels of entropy convergence

Crutchfield

Feldman

2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

327

514

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We study how the Shannon entropy of sequences produced by an information source converges to the source's entropy rate. We synthesize several phenomenological approaches to applying information theoretic measures of randomness and memory to stochastic and deterministic processes by using successive derivatives of the Shannon entropy growth curve. This leads, in turn, to natural measures of apparent memory stored in a source and the amounts of information that must be extracted from observations of a source in order for it to be optimally predicted and for an observer to synchronize to it. One consequence of ignoring these structural properties is that the missed regularities are converted to apparent randomness. We demonstrate that this problem arises particularly for small data sets; e.g., in settings where one has access only to short measurement sequences.

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Section: Future Directionsmentioning

confidence: 99%

Regularities unseen, randomness observed: Levels of entropy convergence

Crutchfield

Feldman

2003

Chaos: An Interdisciplinary Journal of Nonlinear Science

327

514

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“…Now, to overcome the problem of incommensurate length scale it is enough to find a whole number m' such that m'L mod k = 0 and replace the initial arrangement of size L  L by the periodically created one of size m'L  m'L. Then we can define S  (k; L  L, n, )  S  (k; m'L  m'L, m' 2 n, m' 2 ); see the useful properties of the measure indicated in point (6) of [15].…”

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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“…This weakness has been recently overcome by introducing entropic average measure (per cell) of spatial inhomogeneity S ∆ (k) ≡ (S max − S)/χ, see Refs. [5,6]. Here S max and S describe respectively the highest possible configuration entropy and the real one for a given pattern; see also Refs.…”

Section: Modification Of the Point Measurementioning

confidence: 99%

“…In Section 2 we present the statistical measures h(PO) and h ∆ (FSO) in a slightly changed notation in comparison with Ref. [6]. Also a sliding cell-sampling approach used in this paper is briefly sketched.…”

Section: Introductionmentioning

confidence: 99%

A generalization of the inhomogeneity measure for point distributions to the case of finite size objects

Piasecki

2008

Physica A: Statistical Mechanics and its Applications

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The statistical measure of spatial inhomogeneity for n points placed in χ cells each of size k×k is generalized to incorporate finite size objects like black pixels for binary patterns of size L×L. As a function of length scale k, the measure is modified in such a way that it relates to the smallest realizable value for each considered scale. To overcome the limitation of pattern partitions to scales with k being integer divisors of L we use a sliding cell-sampling approach. For given patterns, particularly in the case of clusters polydispersed in size, the comparison between the statistical measure and the entropic one reveals differences in detection of the first peak while at other scales they well correlate. The universality of the two measures allows both a hidden periodicity traces and attributes of planar quasi-crystals to be explored. PACS: 05.90.+m

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Entropic measure of spatial disorder for systems of finite-sized objects

Cited by 32 publications

References 21 publications

Regularities unseen, randomness observed: Levels of entropy convergence

Regularities unseen, randomness observed: Levels of entropy convergence

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

A generalization of the inhomogeneity measure for point distributions to the case of finite size objects

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