The common patterns of abundance: the log series and Zipf's law.

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ID: 49850
2019
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Abstract
In a language corpus, the probability that a word occurs times is often proportional to 1/ . Assigning rank, , to words according to their abundance, log vs log typically has a slope of minus one. That simple Zipf's law pattern also arises in the population sizes of cities, the sizes of corporations, and other patterns of abundance. By contrast, for the abundances of different biological species, the probability of a population of size is typically proportional to 1/ , declining exponentially for larger , the log series pattern. This article shows that the differing patterns of Zipf's law and the log series arise as the opposing endpoints of a more general theory. The general theory follows from the generic form of all probability patterns as a consequence of conserved average values and the associated invariances of scale. To understand the common patterns of abundance, the generic form of probability distributions plus the conserved average abundance is sufficient. The general theory includes cases that are between the Zipf and log series endpoints, providing a broad framework for analyzing widely observed abundance patterns.
Reference Key
frank2019thef1000research Use this key to autocite in the manuscript while using SciMatic Manuscript Manager or Thesis Manager
Authors Frank, Steven A;
Journal F1000Research
Year 2019
DOI
10.12688/f1000research.18681.1
URL
Keywords
Ecology demography linguistics probability theory scaling patterns

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