Supported Methods

This page provides background on each histogram method supported through the rule argument. Our presentation is intended to be rather brief, and we do as such not cover the theoretical underpinnings of each method in great detail. For some further background on automatic histogram procedures and the theory behind them, we recommend the excellent reviews contained in the articles of Birgé and Rozenholc (2006) and Davies et al. (2009).

For ease of exposition, we present all methods covered here in the context of estimating the density of a sample $\boldsymbol{x} = (x_1, x_2, \ldots, x_n)$ on the unit interval, but note that extending the procedures presented here to other compact intervals is possible through a suitable affine transformation. In particular, if a density estimate with support $[a,b]$ is desired, we can scale the data to the unit interval through $z_i = (x_i - a)/(b-a)$, and apply the methods on this transformed sample and rescale the resulting density estimate to $[a,b]$. In cases where the support of the density is unknown, a natural choice is $a = x_{(1)}$ and $b = x_{(n)}$. Cases where only the lower or upper bound is known can be handled similarly. The transformation used to construct the histogram can be controlled through the support keyword, where the default argument support=(-Inf, Inf) uses the order statistics-based approach described above.

Notation

Before we describe the methods included here in more detail, we introduce some notation. We let $\mathcal{I} = (\mathcal{I}_1, \mathcal{I}_2, \ldots, \mathcal{I}_k)$ denote a partition of $[0,1]$ into $k$ intervals and write $|\mathcal{I}_j|$ for the length of interval $\mathcal{I}_j$. The intervals in the partition $\mathcal{I}$ can be either right- or left-closed. Whether a left- or right-closed partition is used to draw the histogram is controlled by the keyword argument closed, with options :left and :right (default). This choice is somewhat arbitrary, but is unlikely to matter much in practical applications.

Based on a partition $\mathcal{I}$, we can write down the corresponding histogram density estimate by

\[\widehat{f}(x) = \sum_{j=1}^k \frac{\widehat{\theta}_j}{|\mathcal{I}_j|}\mathbf{1}_{\mathcal{I}_j}(x), \quad x\in [0,1],\]

where $\mathbf{1}_{\mathcal{I}_j}$ is the indicator function, $\widehat{\theta}_j \geq 0$ for all $j$ and $\sum_{j=1}^k \widehat{\theta}_j = 1$.

For most of the methods considered here, the estimated bin probabilities are the maximum likelihood estimates $\widehat{\theta}_j = N_j/n$, where $N_j = \sum_{i=1}^n \mathbb{1}_{\mathcal{I}_j}(x_i)$ is number of observations landing in interval $\mathcal{I}_j$ . The exception to this rule is are the two Bayesian approaches, which uses the Bayes estimator $\widehat{\theta}_j = (a_j + N_j)/(a+n)$ for $(a_1, \ldots, a_k) \in (0,\infty)^k$ and $a = \sum_{j=1}^k a_j$ instead.

The goal of an automatic histogram procedure is to find a partition $\mathcal{I}$ based on the sample alone which produces a reasonable density estimate. Regular histogram procedures only consider regular partitions, where all intervals in the partition are of equal length, so that one only needs to determine the number $k$ of bins. Irregular histograms allow for partitions with intervals of unequal length, and try to determine both the number of bins and the locations of the cutpoints between the intervals.

A short note on using different rules

In order to fit a histogram using a specific rule, we call fit(AutomaticHistogram, x, rule), where x is the data vector. For many of the rules discussed below, the user can specify additional rule-specific keywords to rule, providing additional control over the supplied method when desired. We also provide a set of default values for these parameters, so that the user may for instance call fit(AutomaticHistogram, x, AIC()) to fit a regular histogram using the AIC criterion without having to worry about explicitly passing any keyword arguments.

Irregular histograms

The following section provides a description of all the irregular histogram rules that have been implemented in AutoHist.jl. In each case, the best partition is selected among the subset of interval partitions of the unit interval that have cut points belonging to a discrete set of cardinality $k_n-1$. In all the irregular procedures covered here, we attempt to find best partition according to a goodness-of-fit criterion among all partitions with endpoints belonging to a given discrete mesh $\{\tau_{j}\colon 0\leq j \leq k_n\}$.

Random irregular histogram

AutoHist.RIH — Type

RIH(;
    a::Real,
    logprior::Function:=k-> 0.0,
    grid::Symbol=:regular,
    maxbins::Union{Int, Symbol}=:default,
    alg::AbstractAlgorithm=SegNeig()
)

The random irregular histogram criterion.

Consists of maximizing the marginal log-posterior of the partition $\mathcal{I} = (\mathcal{I}_1, \ldots, \mathcal{I}_k)$,

\[ \sum_{j=1}^k \big\{\log \Gamma(a_j + N_j) - \log \Gamma(a_j) - N_j\log|\mathcal{I}_j|\big\} + \log p_n(k) - \log \binom{k_n-1}{k-1}\]

Keyword arguments

a: Specifies Dirichlet concentration parameter in the Bayesian histogram model. Must be a fixed positive number, and defaults to a=5.0.
logprior: Unnormalized logprior distribution on the number $k$ of bins. Defaults to a uniform prior, e.g. logprior(k) = 0 for all k.
grid: Symbol indicating how the finest possible mesh should be constructed. Options are :data, which uses each unique data point as a grid point, :regular (default) which constructs a fine regular grid, and :quantile which constructs the grid based on the sample quantiles.
maxbins: Maximal number of bins for which the above criterion is evaluated. Defaults to maxbins=:default, which sets maxbins to the ceil of min(1000, 4n/log(n)^2) if grid is regular or quantile. Ignored if grid=:data.
alg: Algorithm used to fit the model. Currently, only SegNeig is supported for this rule.

Examples

julia> x = (1.0 .- (1.0 .- LinRange(0.0, 1.0, 500)) .^(1/3)).^(1/3);

julia> rule = RIH(a = 5.0, logprior = k-> -log(k), grid = :data);

julia> fit(AutomaticHistogram, x, rule)
AutomaticHistogram
breaks: [0.0, 0.18220071105959446, 0.3587941358096334, 0.8722292888743843, 1.0]
density: [0.11858322346056327, 0.6490600487586273, 1.6066011289666577, 0.30436411114439915]
counts: [10, 57, 414, 19]
type: irregular
closed: right
a: 5.0

References

This approach to irregular histograms first appeared in Simensen et al. (2025).