# NAME

Math::Histogram - N-dimensional histogramming library

# SYNOPSIS

```
use Math::Histogram;
my @dimensions = (
Math::Histogram::Axis->new(10, 0., 1.), # x: 10 bins between 0 and 1
Math::Histogram::Axis->new([1, 2, 4, 8, 16]), # y: 5 bins of variable size
Math::Histogram::Axis->new(2, -1., 1.), # z: 2 bins: [-1, 0) and [0, 1)
);
my $hist = Math::Histogram->new(\@dimensions);
# FIXME cover make_histogram here, too
# Fill some primitive data
while (<>) {
chomp;
my @cols = split /\s+/, $_;
die "Invalid number of columns: " . scalar(@cols)
if @cols != 3;
# Insert new datum into histogram
$hist->fill(\@cols);
}
# Dump histogram content to screen (excluding overflow)
for my $iz (1 .. $hist->get_axis(2)->nbins) {
for my $iy (1 .. $hist->get_axis(1)->nbins) {
for my $ix (1 .. $hist->get_axis(0)->nbins) {
print $hist->get_bin_content([$ix, $iy, $iz]), " ";
}
print "\n";
}
print "\n";
}
```

# DESCRIPTION

This Perl module wraps an n-dimensional histogramming library written in C.

**Beware, this is an early release. While the basic functionality is rather well tested, the library has not been used in production. If you intend to adopt it for production, please test your application well and get in touch with the author.**

## On N-Dimensional Histogramming

If all you are looking for is a regular one dimensional histogram, then consider other libraries such as Math::SimpleHisto::XS first for simplicity and performance. Some care has been taken to optimize the library for performance given a variable number of dimensions, but not knowing the number of dimensions statically makes for both somewhat inefficient algorithmic implementation as well as occasionally awkward APIs. For example, simply iterating through all bins of a 2D histogram -- a matrix -- is as simple as

```
# Pseudo-code
foreach my $ix (0..$nx-1) {
foreach my $iy (0..$ny-1) {
my $z = $matrix->get_bin_content([$ix, $iy]);
}
}
```

If you don't know the number of dimensions statically, you need to do something like this (there are other ways to do it, too):

```
# Pseudo-code
my $coords = [(0) x $ndims];
foreach my $i (0..$unrolled_total_nbins-1) {
my $z = $ndimhisto->get_bin_content($coords);
my $i = 0;
++$coords->[$i];
while ($i < $ndims
&& $coords->[$i] >= $ndimhisto->get_axis($i)->nbins)
{
$coords->[$i] = 0;
++$coords->[++$i];
}
}
```

Not pretty, eh? Not fast either. So keep that in mind: Your application knows the number of dimensions that you care about, this histogramming library does not.

## Overview

Generally speaking, a histogram object in the context of this library contains N axis objects (axises 0 to N-1) that define the binning of each dimension. Below and above its coordinate range, each axis has an under- and an overflow bin. When you fill a histogram with data using the `fill()`

method, and the provided coordinates are outside the range of the histogram, then the data will be filled into the correct under- or overflow bin. For example, if you create a 2D histogram with the following axises:

```
my $h = Math::Histogram->new([
Math::Histogram::Axis->new(2, 0., 1.),
Math::Histogram::Axis->new(3, 0., 3.),
]);
# Worst ASCII drawing ever:
# +-+-+-+-+
# |:|.|.|:|
# +-+-+-+-+
# |.| | |.|
# +-+-+-+-+
# |.| | |.| ^
# +-+-+-+-+ |
# |.| | |.| |
# +-+-+-+-+ dimension 1
# |:|.|.|:|
# +-+-+-+-+
# ---> dimension 0
#
# Bins marked with . are under- or overflow in one dimension.
# Bins marked with : are under- or overflow in BOTH dimensions.
```

Then you created a histogram with six regular bins: two bins in the X direction and three bins in the Y direction for a total of `2 * 3 = 6`

. On top of that, you get a ring of over- and underflow bins around your ordinary bins. In this case, there are a grand total of 14 such over- and underflow bins. As you increase the number of bins in your actual histogram, the relative number of over- and underflow bins goes down.

You can access histogram content both by the N-dimensional bin numbers (so, in the 2D example, an array reference containing two integers) or by user coordinates (eg. an array reference of two floating point numbers). The module provides facilities to determine the bin in which a particular set of coordinates falls. The lower boundary of a bin is always considered part of the bin, whereas the upper boundary is not. Internally, the histogram data is stored in a flat array since the dimensionality is unknown at compile time. The linear index into this array is what may be referred to as the "flat" or "linear" bin number. In a 1D histogram, it corresponds to the bin numbers of the only axis in the histogram.

# METHODS

## new

Class method, constructor. Takes an array reference as first parameter. The array reference must contain one or more Math::Histogram::Axis objects that define the binning in one dimension each. The number of axises determines the dimensionality of the histogram.

## clone

Returns an exact clone of the histogram.

## new_alike

Returns a clone of the histogram, but without its content.

## get_axis

Given a dimension number (starting at 0), returns the axis object of that dimension.

## ndim

Returns the number of dimensions in the histogram.

## total

Returns the total content of the histogram. (The sum over all bins, except this is cached.)

## nfills

Returns the number of fill operations that have been performed on the histogram so far. This is not the same as total unless all fills have a weight of 1.

## fill

Given a reference to an array of coordinates, adds 1 to the content of the bin that the coordinates belong to.

## fill_w

Same as `fill()`

, except that the second argument needs to be a weight, the number to add to the bin content (instead of incrementing by 1).

## fill_n

Same as `fill()`

, except that the first parameter needs to be a reference to a nested array, each of the inner arrays containing a set of coordinates. In other words, this method works the same as calling `fill()`

repeatedly for each element in the outer array:

```
my @coords = (
[0.1, 0.2],
[3.8, -1.2],
...
);
$h->fill_n(\@coords);
# Is the same as:
$h->fill($_) for @coords;
# Except a teeny bit faster.
```

## fill_nw

This is to `fill_w(\@coord, $weight)`

what `fill_n(\@coords)`

is to `fill(\@coord)`

. In other words, the first argument is the same as for `fill_n()`

, the second is an array reference containing as many weights as the first had coordinate sets.

## fill_bin

Same as `fill()`

, but takes an array reference containing bin numbers as argument (instead of a reference to an array of coordinates).

## fill_bin_w

This is to `fill_w`

what `fill_bin`

is to `fill`

.

## fill_bin_n

This is to `fill_n`

what `fill_bin`

is to `fill`

.

## fill_bin_nw

This is to `fill_nw`

what `fill_bin`

is to `fill`

.

## get_bin_content

Given a reference to an array of bin numbers, returns the content of the specified bin. Throws an exception when out of bounds.

## find_bin_numbers

Given a reference to an array of coordinates, returns a reference to an array of (the same number of) bin numbers that correspond to the bin that the coordinates fall into.

## contract_dimension

Given a dimension number (starting at 0), creates an N-1 dimensional histogram that contains the original data, but with the specified dimension contracted. The original histogram is untouched. Throws an exception if the dimension is out of bounds.

## cumulate

Given a dimension number (starting at 0), cumulates along that dimension, modifying the input histogram. Throws an exception if the dimension is out of bounds. Example:

```
X ->
1 2 3 ^
4 5 6 |
7 8 9 Y
```

Cumulated along X, the result is:

```
1 3 6
4 9 15
7 15 24
```

Cumulated along Y instead, the result is (note direction of Y axis in example):

```
12 15 18
11 13 15
7 8 9
```

## data_equal_to

Given another histogram, returns true if the data content is equal to the invocant's data. Uses your machine `DBL_EPSILON`

for floating point comparisons.

## is_overflow_bin

Given a set of bin numbers, returns true if the bin is an under- or overflow bin, false otherwise. This is O(n) in the number of dimensions, but O(1) in the number of bins in the histogram.

## is_overflow_bin_linear

Given a linear bin number, returns true if the bin is an under- or overflow bin, false otherwise. This is O(1) in the number of dimensions and the number of bins in the histogram.

## serialize

Returns a JSON string that represents this histogram object.

## deserialize

Class method. Given a JSON string as generated by `serialize()`

, recreates the histogram object that it represents. Also accepts a scalar reference to a JSON string.

# SEE ALSO

Math::Histogram::Axis, which is part of this distribution, implements the binning for a histogram in a single dimension.

Math::SimpleHisto::XS is a fast 1D histogramming module.

SOOT is a dynamic wrapper around the ROOT C++ library which does histogramming and much more. Beware, it is experimental software.

# AUTHOR

Steffen Mueller, <smueller@cpan.org>

# COPYRIGHT AND LICENSE

Copyright (C) 2012 by Steffen Mueller

This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself, either Perl version 5.8.1 or, at your option, any later version of Perl 5 you may have available.