1,433 research outputs found
Zipf's law, 1/f noise, and fractal hierarchy
Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic
events are typical ubiquitous general empirical observations across the
individual sciences which cannot be understood within the set of references
developed within the specific scientific domains. All these observations are
associated with scaling laws and have caused a broad research interest in the
scientific circle. However, the inherent relationships between these scaling
phenomena are still pending questions remaining to be researched. In this
paper, theoretical derivation and mathematical experiments are employed to
reveal the analogy between fractal patterns, 1/f noise, and the Zipf
distribution. First, the multifractal process follows the generalized Zipf's
law empirically. Second, a 1/f spectrum is identical in mathematical form to
Zipf's law. Third, both 1/f spectra and Zipf's law can be converted into a
self-similar hierarchy. Fourth, fractals, 1/f spectra, Zipf's law, and the
occurrence of large catastrophic events can be described with similar
exponential laws and power laws. The self-similar hierarchy is a more general
framework or structure which can be used to encompass or unify different
scaling phenomena and rules in both physical and social systems such as cities,
rivers, earthquakes, fractals, 1/f noise, and rank-size distributions. The
mathematical laws on the hierarchical structure can provide us with a holistic
perspective of looking at complexity such as self-organized criticality (SOC).Comment: 20 pages, 9 figures, 3 table
The Mathematical Relationship between Zipf's Law and the Hierarchical Scaling Law
The empirical studies of city-size distribution show that Zipf's law and the
hierarchical scaling law are linked in many ways. The rank-size scaling and
hierarchical scaling seem to be two different sides of the same coin, but their
relationship has never been revealed by strict mathematical proof. In this
paper, the Zipf's distribution of cities is abstracted as a q-sequence. Based
on this sequence, a self-similar hierarchy consisting of many levels is defined
and the numbers of cities in different levels form a geometric sequence. An
exponential distribution of the average size of cities is derived from the
hierarchy. Thus we have two exponential functions, from which follows a
hierarchical scaling equation. The results can be statistically verified by
simple mathematical experiments and observational data of cities. A theoretical
foundation is then laid for the conversion from Zipf's law to the hierarchical
scaling law, and the latter can show more information about city development
than the former. Moreover, the self-similar hierarchy provides a new
perspective for studying networks of cities as complex systems. A series of
mathematical rules applied to cities such as the allometric growth law, the 2^n
principle and Pareto's law can be associated with one another by the
hierarchical organization.Comment: 30 pages, 5 figures, 5 tables, Physica A: Statistical Mechanics and
its Applications, 201
Point Information Gain and Multidimensional Data Analysis
We generalize the Point information gain (PIG) and derived quantities, i.e.
Point information entropy (PIE) and Point information entropy density (PIED),
for the case of R\'enyi entropy and simulate the behavior of PIG for typical
distributions. We also use these methods for the analysis of multidimensional
datasets. We demonstrate the main properties of PIE/PIED spectra for the real
data on the example of several images, and discuss possible further utilization
in other fields of data processing.Comment: 16 pages, 6 figure
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