# Quadratic Term Correction on Heaps' Law

arXiv: https://arxiv.org/abs/2511.14683 · HTML: https://mlsanity.com/paper/2511.14683/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 48 references that are in the map.
- First posted on arXiv 18 Nov 2025.

## Builds on (most cited first, 9 of 9)

- [Statistical laws in linguistics](https://mlsanity.com/paper/1502.03296/) (2015)
- [Complex systems approach to natural language](https://mlsanity.com/paper/2401.02772/) (2024)
- [In narrative texts punctuation marks obey the same statistics as words](https://mlsanity.com/paper/1604.00834/) (2016)
- [Heaps' law and Heaps functions in tagged texts: Evidences of their linguistic relevance](https://mlsanity.com/paper/2001.02178/) (2020)
- [Linguistic laws in biology](https://mlsanity.com/paper/2310.07387/) (2023)
- [Types, Tokens, and Hapaxes: A New Heap's Law](https://mlsanity.com/paper/1901.00521/) (2018)
- [Corrections of Zipf's and Heaps' Laws Derived from Hapax Rate Models](https://mlsanity.com/paper/2307.12896/) (2023)
- [Beta Rank Function: A Smooth Double-Pareto-Like Distribution](https://mlsanity.com/paper/1910.05364/) (2019)
- [A Simple Derivation of the Heap's Law from the Generalized Zipf's Law](https://mlsanity.com/paper/1711.03066/) (2017)

## Built on by (most cited first, 1 of 1)

- [Non-Zipfian Distribution of Stopwords or Function Words and Subset Selection Models](https://mlsanity.com/paper/2603.04691/) (2026)

## Abstract (arXiv)

Heaps' or Herdan's law characterizes the word-type vs. word-token relation by a power-law function, which is concave in linear-linear scale but a straight line in log-log scale. However, it has been observed that even in log-log scale, the type-token curve is still slightly concave, invalidating the power-law relation. At the next-order approximation, we have shown, by twenty English novels or writings (some are translated from another language to English), that quadratic functions in log-log scale fit the type-token data perfectly. Regression analyses of log(type)-log(token) data with both a linear and quadratic term consistently lead to a linear coefficient of slightly larger than 1, and a quadratic coefficient around -0.02. Using the ``random drawing colored ball from the bag with replacement" model, we have shown that the curvature of the log-log scale is identical to a ``pseudo-variance" which is negative. Although a pseudo-variance calculation may encounter numeric instability when the number of tokens is large, due to the large values of pseudo-weights, this formalism provides a rough estimation of the curvature when the number of tokens is small.