Random sampling of the Zipf–Mandelbrot distribution as a representation of vocabulary growth

We develop a discrete model of type-token dynamics based on random type selection from the Zipf–Mandelbrot probability distribution, with a view to examining the relationships between the constants of Zipf’s and Heaps’ laws. Analysis of items randomly selected items from the Standardised Project Gutenberg Corpus (SPGC) reveal a significant low-frequency “droop” in the β-slope of the types vs. frequency distribution, inconsistent with the model when vocabulary is unlimited: when a finite vocabulary limit is imposed, optimal parameter selection allows the droop to be reproduced. We adjust the parameters of both the limited and unlimited vocabulary models to obtain optimal agreement with the vocabulary growth curves: the limited vocabulary model usually yields the best optimised agreement, but a sizeable minority of items are better represented by an unlimited vocabulary. While the optimised Zipf α indices correlate strongly with the corresponding values obtained directly from document statistics, the former are generally larger than the latter (though this is partially explained by the distorting effect of large values of the Mandelbrot parameter m). The β indices optimised from the limited vocabulary model are also compared with their directly measured equivalents, showing significant positive correlation. The relationship between optimised α and β agrees plausibly with the well-known continuum model, though the degree of agreement depends on how β is defined. The experiments yield repeatable results from each of three 100-item samples, demonstrating the statistical significance of the experiments.