are a result of the underlying statistical characteristics of the data, the tiling used to divide the domain and the limited sample taken from the population. Therefore, ragged patterns might appear where a continuous transition should take place. It is then usual to apply a smoothing over the binning counts to avoid this.hexbin: Hexagonal Binning Routines in Rhttp://cran.r-project.org/web/packages/hexbin/Hexagon Binning of Word FrequencyAnalyzing the relation between word frequency and its rank has been a key object of study in quantitative linguistics for almost 80 years. It is well known that words occur according to a famously systematic frequency distribution known as Zipf's or Zipf-Mandelbrot law. The generalization proposed by Mandelbrot starts that the relation between rank ( $r$ ) and frequency ( $f$ ) is given by$ f(r) = \frac{C}{(r + \beta)^\alpha} $where $C$, $\beta$ and $\alpha$ are a constants.The standard method to compute the word frequency distribution is to count the number of occurrences of each word and sort them afterwards according to their decreasing frequency of occurrence. The frequency $f(r)$ of the $r$ most frequent word is plotted against its rank $r$, yielding a roughly linear curve in a log-log plot. The frequency and rank are both estimated from the very same corpus, what could lead to correlated errors between them. Analyzing the example proposed by Wentian Li (1992), and also previously by George A. Miller (1957), we might observe that a problem might incur from the method described above to count and rank words. Words that are equally probable will, by chance, appear with different frequency count and therefore they will appear as a strikingly decreasing curve, suggesting an interesting relation between frequency and rank, that turns out to be more problematic for low-frequency words, whose frequencies are measured with lack of precision. It might be a spurious association created between an observed pattern and an underlying structure.This unwelcome situation might be mitigated by using an extremely large corpus or by using two independent corpora to estimate both variables: frequency and rank.Steven T. Piantadosi proposes:Fortunately, the problem is easily fixed: We may use two independent corpora to estimate the frequency and frequency rank. In the above case where all words are equally probable, use of independent corpora will lead to no apparent structure -- just a roughly flat frequency-rank relationship. In general, we need not have two independent corpora from the start; we can imagine splitting our initial corpus into two subcorpora before any text processing takes place. This creates two corpora that are independent bodies of text (conditioned on the general properties of the starting corpus) and, so, from which we can independently estimate r and f(r). A convenient technique to perform this split is to perform a binomial split on observed frequency of each word: If we observe a word, say, 100 times, we may sample from a binomial (N = 100, p = .5) and arrive at a frequency of, say, 62 used to estimate its true frequency and a frequency of N - 62 = 38 to estimate its true frequency rank. This exactly mirrors randomly