Generalized Hurst exponent and multifractal function of original and translated texts mapped into frequency and length time series

A nonlinear dynamics approach can be used in order to quantify complexity in written texts. As a first step, a one-dimensional system is examined: two written texts by one author (Lewis Carroll) are considered, together with one translation into an artificial language (i.e., Esperanto) are mapped into time series. Their corresponding shuffled versions are used for obtaining a baseline. Two different one-dimensional time series are used here: one based on word lengths (LTS), the other on word frequencies (FTS). It is shown that the generalized Hurst exponent $h\left(q\right)$ and the derived $f\left(\alpha \right)$ curves of the original and translated texts show marked differences. The original texts are far from giving a parabolic $f\left(\alpha \right)$ function, in contrast to the shuffled texts. Moreover, the Esperanto text has more extreme values. This suggests cascade model-like, with multiscale time-asymmetric features as finally written texts. A discussion of the difference and complementarity of mapping into a LTS or FTS is presented. The FTS $f\left(\alpha \right)$ curves are more opened than the LTS ones.

Figure 1

The so-called partition function $\chi (s,q)$ vs $s$, the subseries size in Eq. (1), on log-log plot graphs, in order to obtain $\tau \left(q\right)$ [Eq. (3)] in the best possible power-law regime (see text) and subsequently the generalized Hurst exponent $h\left(q\right)$ [Eq. (4)] or the generalized fractal dimension $D\left(q\right)$ [Eq. (5)] in the case of FTS, for the (a) original ($o$) and (b) shuffled ($s$) texts. Only three representative $q$ values ($-$10, $+$2, $+$20) in each case are shown for space savings. Shorthand notations to understand the illustrating data are on the left axis; F $\to$ FTS; E$\to$ESP, T$\to$TLG, and A$\to$AWL, respectively. In the display, the data has been arbitrarily displaced along the $y$ axis since only the slope from a linear fit is relevant.

Figure 2

The so-called partition function $\chi (s,q)$ vs $s$, the subseries size in Eq. (1), on log-log plot graphs, in order to obtain $\tau \left(q\right)$ [Eq. (3)] in the best possible power-law regime (see text) and subsequently the generalized Hurst exponent $h\left(q\right)$ [Eq. (4)] or the generalized fractal dimension $D\left(q\right)$ [Eq. (5)] in the case of LTS, for the (a) original and (b) shuffled texts. Only 3 representative $q$-values ($-$10, $+$2, $+$20) in each case are shown for space savings. Only three representative $q$ values ($-$10, $+$2, $+$20) in each case are shown for space savings. Shorthand notations to understand the illustrating data are on the left axis; F $\to$ FTS; E$\to$ESP, T$\to$TLG, and A$\to$AWL, respectively. In the display, the data has been arbitrarily displaced along the $y$ axis since only the slope from a linear fit is relevant.

Figure 3

Generalized Hurst $h\left(q\right)$ exponent of three original ($o$) texts, AWL, ESP, and TLG, analyzed through FTS (a) and LTS (b) mapping. Lines are smooth guides to the eyes.

Figure 4

Generalized Hurst exponent $h\left(q\right)$ of three shuffled ($s$) texts, AWL, ESP, and TLG, analyzed through FTS (a) and LTS (b) mapping. Lines are smooth guides to the eyes.

Figure 5

$f\left(\alpha \right)$ for AWL, original (o) or shuffled (s) text along FTS or LTS mapping. Lines are smooth guides to the eyes.

Figure 6

$f\left(\alpha \right)$ for ESP, original (o) or shuffled (s) text along FTS or LTS mapping. Lines are smooth guides to the eyes.

Figure 7

$f\left(\alpha \right)$ for TLG, original (o) or shuffled (s) text along FTS or LTS mapping. Lines are smooth guides to the eyes.