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Fluctuation

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Statistical Universals of Language

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

Abstract

The previous two chapters presented analyses based on return intervals. Chapter 7 was about the distribution of returns, whereas Chap. 8 considered sequences of returns.

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Notes

  1. 1.

    The number of times an element appears in a segment l in an i.i.d. sequence follows a Poisson distribution, as deduced in Sect. 21.5. A Poisson distribution is characterized by its mean and variance being equal. If l is made k times larger, then its mean obviously becomes k times larger as well, and so does the variance. Therefore, ν = 1 for an i.i.d. sequence. Section 21.6 gives a more formal explanation.

  2. 2.

    For fitting Fig. 9.1, the least-squares method was applied (cf. Sect. 21.1) to the plots for l ∝ a k, with k = 1, 2, 3, …, a = 1.8. ε = 0.0797 for Moby Dick, whereas ε = 0.00936 for the shuffled text. As introduced in Sect. 3.5, the shuffled text used here is word shuffled, and therefore, the order of the characters within words is maintained. Nevertheless, ν ≈ 1.

  3. 3.

    ε = 0.0541.

  4. 4.

    The choice of l is arbitrary but must be sufficiently smaller than the text length to accurately calculate the mean and standard deviation. Among different values of l taken from logarithmic bins, a maximum l that could apply to all texts was adopted here. This resulted in a choice of l around 5000 words, specifically l = 5620 ≈ 103.75.

  5. 5.

    For fitting Fig. 9.3, the least-squares method was applied (cf. Sect. 21.1). ε = 0.0623 for Moby Dick, whereas ε = 0.0238 for the shuffled text.

  6. 6.

    Previously, the corresponding value for the EN method was 1. Taylor analysis uses the standard deviation, so the i.i.d. case has α = 0.5. Section 21.6 gives a more formal explanation.

  7. 7.

    The fourth plot shows the Taylor exponent for the Wall Street Journal in addition to the Taylor exponents for the Japanese and Chinese newspapers used in (Kobayashi and Tanaka-Ishii, 2018; Tanaka-Ishii and Kobayashi, 2018).

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Authors and Affiliations

  1. Research Center for Advanced Science and Technology (RCAST), The University of Tokyo, Tokyo, Japan

    Kumiko Tanaka-Ishii

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  1. Kumiko Tanaka-Ishii
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Tanaka-Ishii, K. (2021). Fluctuation. In: Statistical Universals of Language. Mathematics in Mind. Springer, Cham. https://doi.org/10.1007/978-3-030-59377-3_9

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