# aliquot

## < a quantity that can be divided into another a whole number of time />

This post is a rough attempt at constructing a proper histogram for a sequence of real values, usually stored as a list or vector in Common Lisp. With minor adaptation, it should work with Scheme dialects too.

Histograms allow to visually depict the distribution of a continuous variable, and they should not be confused with other graphical displays that use bar-like representation, like bar charts. This kind of statistical plot expect a discretized variable as input, and in this post we are going to focus on this very specific aspect of binning for a continuous variable. From a purely statistical and technical viewpoint, the probability that a random variable (r.v.) $X$ lies in a specific interval can be represented using the cumulative distribution function, defined as $F(x)=\Pr(X\le x)$; its inverse corresponds to the quantile function. Another well-known estimator of the probability law of an r.v. is, of course, the density function, defined as $\Pr(a< X\le b)=\int_a^bf(t)dt$, and an histogram is nothing else than yet another estimator of the density of a continuous r.v.. More discussion is available on Arthur Charpentier’s excellent blog.

Gnuplot has no histogram facilities, so you have to run your own implementation, as I did in one of my previous posts, but see also these SO answers of mine. However, R or Stata provide ready-to-use implementations of this statistical estimator as hist (or better, MASS:truehist) and histogram. In this post, I am going to explore Common Lisp utilities to work on sequences of integer or real numbers. Indeed, Lisp comes with a class of objects, called sequences, which can be lists, vectors or strings. The iterate or Alexandria packages are two among many of the packages that provide utilities to manipulate such objects. For instance, using Alexandria it is quite easy to get a permuted series of values stored in a list:

(ql:quickload "alexandria")
(defparameter s '(1 2 3 4 5 6 7 8 9))
(alexandria:shuffle s)


Note that it also works with strings (try replacing the previous list of integers with "random string"). Now, we need to design a little helper function that will discretize a continuous variable in $k$ buckets of data, if the list is non-empty of course. Here it is

(defun emptyp (lst)
(eq nil lst))

(defun slice (a b xs &optional (ys '()))
"Take a subsequence out of a sequence xs'"
(cond ((> a 0) (slice (1- a) b (cdr xs) ys))
((> b 0) (slice a (1- b) (cdr xs) (cons (car xs) ys)))
((reverse ys))))

(defun split (predicate xs)
"Split a sequence xs' into two sublists according to a predicate [https://stackoverflow.com/a/18117017]."
(loop for x in xs
if (funcall predicate x) collect x into in
else collect x into out
finally (return (values in out))))

(defun binning (xs &key (len 10))
"Convert a sequence of numbers xs' into packets of size len' or bins' packets."
(unless (<= len 0)
(let* ((lower (apply #'min xs))
(upper (apply #'max xs))
(step (+ lower (/ (- upper lower) len))))
(multiple-value-bind (bin rest)
(split (lambda (x) (<= x step)) xs)
(cons (length bin)
(binning rest :len (1- len)))))))


There may be better alternatives to split the data into buckets, e.g. arnesi:partition or cl21:partition-if, as well as solutions to split list into sublists discussed on SO. Note also that Alexandria provides an emptyp function. Finally, the slice function (adapted from code available in Scheme) could be used to restrict the values to a specific range in the list, as when computing a trimmed mean, for instance. Even if it stands minimalistic the above code works, at least the function returns the correct counts. Here is an example of use:

> (binning '(1 2 3 4 5) :binsize 2)
(3 2)
> (binning '(1 2 3 4 5) :binsize 4)
(2 1 1 1)


Please note that we are lacking some critical information, namely the labels for each bucket counts, i.e. the range of values of xs corresponding to each count. It would also be nice to compute an optimal number of bins automagically, e.g., using Sturge’s formula or Scott’s normal reference rule.1 Here is one way to apply the latter (outside the function body, though), using a more realistic sample of size $n=100$ of random uniform or gaussian draws from the GSLL package:

(ql:quickload "gsll")
(defvar *rng* (gsl:make-random-number-generator gsl:+mt19937+ 0))
(defparameter σ (coerce 1 'double-float))
(defparameter xu (loop repeat 100 collect (gsl:sample *rng* :uniform)))
(defparameter xn (loop repeat 100 collect (gsl:sample *rng* :gaussian :sigma σ)))
(defparameter bins (1+ (ceiling (/ (log (length xu)) (log 2)))))
(binning xu :len bins)
(binning xn :len bins)


And here are the results I got when running this code in CCL REPL:

> (11 14 20 13 7 11 10 14)
> (3 6 20 28 21 17 3 2)
`

1. The R statistical package uses Sturge’s formula, meaning the default number of bins is computed as $\lceil \log_2(n)\rceil + 1$. Stata considers $\text{min}\left\{\sqrt{N},10\ln(N)/\ln(10)\right\}$, which is close. The Freedman-Diaconis rule is also quite robust, where the bin width is chosen as $h=2\times\text{IQR}\times n^{-\frac{1}{3}}$ (IQR, interquartile range), so that the number of bins is $(\text{max}-\text{min})/h$. ↩︎