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Related Statistical Universals

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Statistical Universals of Language

Part of the book series: Mathematics in Mind ((MATHMIN))

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Abstract

This last chapter of Part II further considers the nature of a vocabulary population in terms of two related properties that have a mathematical relation with Zipf’s law: the density function and the vocabulary growth. Similarly to Zipf’s law, both nearly indicate power-law behavior but are subject to some deviations.

The original version of this chapter was revised. The correction to this chapter is available at https://doi.org/10.1007/978-3-030-59377-3_23

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Change history

  • 29 November 2022

    The original version of the chapter “Language as a Complex System” was previously published without updating missing reference in footnote 4, page 22. This change has now been included and the chapter and the book have been updated with the change.

Notes

  1. 1.

    More precisely, the density function represents the distribution that defines the information source of the author of Moby Dick, Herman Melville. Such a true function is never known, however, so a guess is deduced from the word counts, as mentioned as P in the main text.

  2. 2.

    LL = 2.092.

  3. 3.

    Precisely, this depends on the value of ζ. For the values of ζ acquired from texts, the result of the integral becomes a power function.

  4. 4.

    The proof is based on a very simple, elegant consideration by Lü et al. (2010).

  5. 5.

    LL = 2.093.

  6. 6.

    LL = 0.432.

  7. 7.

    LL = 3.376, LL = 3.180, and LL = 7.718 for the Wall Street Journal, Thomas, and Chinese characters, respectively.

  8. 8.

    a = 1.3 for the type-token relations shown in this book.

  9. 9.

    For fitting Fig. 6.3, the least-squares method was applied (cf. Sect. 21.1 ). As defined in Sect. 21.1 , the goodness of fit was measured by the residual ε. For Moby Dick, the shuffled text, and the monkey text, respectively, the error ε = 105.322,  175.808,  45.538. The values are very large because of the ranges of the axes. The comparison of residuals at least shows how the monkey text fits well to the fit line.

  10. 10.

    LL = 2.092.

  11. 11.

    Among these are Baeza-Yates and Navarro (2000); Montemurro and Zanette (2002); van Leijenhorst and van der Weide (2005); Zanette and Montemurro (2005); Serrano et al. (2009); Lü et al. (2010).

References

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Authors and Affiliations

  1. University of Tokyo Research Center for Advanced Sci and Tech, Research Center for Advanced Sci and Tech, Tokyo, Japan

    Kumiko Tanaka-Ishii

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  1. Kumiko Tanaka-Ishii
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Correspondence to Kumiko Tanaka-Ishii .

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Tanaka-Ishii, K. (2021). Related Statistical Universals. In: Statistical Universals of Language. Mathematics in Mind. Springer, Cham. https://doi.org/10.1007/978-3-030-59377-3_6

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