# Histograms

This chapter describes functions for creating histograms. Histograms provide a convenient way of summarizing the distribution of a set of data. A histogram consists of a set of bins which count the number of events falling into a given range of a continuous variable x. In GSL the bins of a histogram contain floating-point numbers, so they can be used to record both integer and non-integer distributions. The bins can use arbitrary sets of ranges (uniformly spaced bins are the default). Both one and two-dimensional histograms are supported.

Once a histogram has been created it can also be converted into a probability distribution function. The library provides efficient routines for selecting random samples from probability distributions. This can be useful for generating simulations based real data.

The functions are declared in the header files `gsl_histogram.h' and `gsl_histogram2d.h'.

## The histogram struct

A histogram is defined by the following struct,

Data Type: gsl_histogram
`size_t n`
This is the number of histogram bins
`double * range`
The ranges of the bins are stored in an array of n+1 elements pointed to by range.
`double * bin`
The counts for each bin are stored in an array of n elements pointed to by bin. The bins are floating-point numbers, so you can increment them by non-integer values if necessary.

The range for bin[i] is given by range[i] to range[i+1]. For n bins there are n+1 entries in the array range. Each bin is inclusive at the lower end and exclusive at the upper end. Mathematically this means that the bins are defined by the following inequality,

```bin[i] corresponds to range[i] <= x < range[i+1]
```

Here is a diagram of the correspondence between ranges and bins on the number-line for x,

```     [ bin )[ bin )[ bin )[ bin )[ bin )
---|---------|---------|---------|---------|---------|---  x
r      r      r      r      r      r

```

In this picture the values of the range array are denoted by r. On the left-hand side of each bin the square bracket "`[`" denotes an inclusive lower bound ( r <= x), and the round parentheses "`)`" on the right-hand side denote an exclusive upper bound (x < r). Thus any samples which fall on the upper end of the histogram are excluded. If you want to include this value for the last bin you will need to add an extra bin to your histogram.

The `gsl_histogram` struct and its associated functions are defined in the header file `gsl_histogram.h'.

## Histogram allocation

The functions for allocating memory to a histogram follow the style of `malloc` and `free`. In addition they also perform their own error checking. If there is insufficient memory available to allocate a histogram then the functions call the error handler (with an error number of `GSL_ENOMEM`) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every histogram `alloc`.

Function: gsl_histogram * gsl_histogram_alloc (size_t n)
This function allocates memory for a histogram with n bins, and returns a pointer to a newly created `gsl_histogram` struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of `GSL_ENOMEM`. The bins and ranges are not initialized, and should be prepared using one of the range-setting functions below in order to make the histogram ready for use.

Function: int gsl_histogram_set_ranges (gsl_histogram * h, const double range[], size_t size)
This function sets the ranges of the existing histogram h using the array range of size size. The values of the histogram bins are reset to zero. The `range` array should contain the desired bin limits. The ranges can be arbitrary, subject to the restriction that they are monotonically increasing.

The following example shows how to create a histogram with logarithmic bins with ranges [1,10), [10,100) and [100,1000).

```gsl_histogram * h = gsl_histogram_alloc (3);

/* bin covers the range 1 <= x < 10 */
/* bin covers the range 10 <= x < 100 */
/* bin covers the range 100 <= x < 1000 */

double range = { 1.0, 10.0, 100.0, 1000.0 };

gsl_histogram_set_ranges (h, range, 4);
```

Note that the size of the range array should be defined to be one element bigger than the number of bins. The additional element is required for the upper value of the final bin.

Function: int gsl_histogram_set_ranges_uniform (gsl_histogram * h, double xmin, double xmax)
This function sets the ranges of the existing histogram h to cover the range xmin to xmax uniformly. The values of the histogram bins are reset to zero. The bin ranges are shown in the table below,

```bin corresponds to xmin <= x < xmin + d
bin corresponds to xmin + d <= x < xmin + 2 d
......
bin[n-1] corresponds to xmin + (n-1)d <= x < xmax
```

where d is the bin spacing, d = (xmax-xmin)/n.

Function: void gsl_histogram_free (gsl_histogram * h)
This function frees the histogram h and all of the memory associated with it.

## Copying Histograms

Function: int gsl_histogram_memcpy (gsl_histogram * dest, const gsl_histogram * src)
This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copy of src. The two histograms must be of the same size.

Function: gsl_histogram * gsl_histogram_clone (const gsl_histogram * src)
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.

## Updating and accessing histogram elements

There are two ways to access histogram bins, either by specifying an x coordinate or by using the bin-index directly. The functions for accessing the histogram through x coordinates use a binary search to identify the bin which covers the appropriate range.

Function: int gsl_histogram_increment (gsl_histogram * h, double x)
This function updates the histogram h by adding one (1.0) to the bin whose range contains the coordinate x.

If x lies in the valid range of the histogram then the function returns zero to indicate success. If x is less than the lower limit of the histogram then the function returns `GSL_EDOM`, and none of bins are modified. Similarly, if the value of x is greater than or equal to the upper limit of the histogram then the function returns `GSL_EDOM`, and none of the bins are modified. The error handler is not called, however, since it is often necessary to compute histogram for a small range of a larger dataset, ignoring the values outside the range of interest.

Function: int gsl_histogram_accumulate (gsl_histogram * h, double x, double weight)
This function is similar to `gsl_histogram_increment` but increases the value of the appropriate bin in the histogram h by the floating-point number weight.

Function: double gsl_histogram_get (const gsl_histogram * h, size_t i)
This function returns the contents of the ith bin of the histogram h. If i lies outside the valid range of indices for the histogram then the error handler is called with an error code of `GSL_EDOM` and the function returns 0.

Function: int gsl_histogram_get_range (const gsl_histogram * h, size_t i, double * lower, double * upper)
This function finds the upper and lower range limits of the ith bin of the histogram h. If the index i is valid then the corresponding range limits are stored in lower and upper. The lower limit is inclusive (i.e. events with this coordinate are included in the bin) and the upper limit is exclusive (i.e. events with the coordinate of the upper limit are excluded and fall in the neighboring higher bin, if it exists). The function returns 0 to indicate success. If i lies outside the valid range of indices for the histogram then the error handler is called and the function returns an error code of `GSL_EDOM`.

Function: double gsl_histogram_max (const gsl_histogram * h)
Function: double gsl_histogram_min (const gsl_histogram * h)
Function: size_t gsl_histogram_bins (const gsl_histogram * h)
These functions return the maximum upper and minimum lower range limits and the number of bins of the histogram h. They provide a way of determining these values without accessing the `gsl_histogram` struct directly.

Function: void gsl_histogram_reset (gsl_histogram * h)
This function resets all the bins in the histogram h to zero.

## Searching histogram ranges

The following functions are used by the access and update routines to locate the bin which corresponds to a given x coordinate.

Function: int gsl_histogram_find (const gsl_histogram * h, double x, size_t * i)
This function finds and sets the index i to the bin number which covers the coordinate x in the histogram h. The bin is located using a binary search. The search includes an optimization for histograms with uniform range, and will return the correct bin immediately in this case. If x is found in the range of the histogram then the function sets the index i and returns `GSL_SUCCESS`. If x lies outside the valid range of the histogram then the function returns `GSL_EDOM` and the error handler is invoked.

## Histogram Statistics

Function: double gsl_histogram_max_val (const gsl_histogram * h)
This function returns the maximum value contained in the histogram bins.

Function: size_t gsl_histogram_max_bin (const gsl_histogram * h)
This function returns the index of the bin containing the maximum value. In the case where several bins contain the same maximum value the smallest index is returned.

Function: double gsl_histogram_min_val (const gsl_histogram * h)
This function returns the minimum value contained in the histogram bins.

Function: size_t gsl_histogram_min_bin (const gsl_histogram * h)
This function returns the index of the bin containing the minimum value. In the case where several bins contain the same maximum value the smallest index is returned.

Function: double gsl_histogram_mean (const gsl_histogram * h)
This function returns the mean of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.

Function: double gsl_histogram_sigma (const gsl_histogram * h)
This function returns the standard deviation of the histogrammed variable, where the histogram is regarded as a probability distribution. Negative bin values are ignored for the purposes of this calculation. The accuracy of the result is limited by the bin width.

## Histogram Operations

Function: int gsl_histogram_equal_bins_p (const gsl_histogram *h1, const gsl_histogram *h2)
This function returns 1 if the all of the individual bin ranges of the two histograms are identical, and 0 otherwise.

Function: int gsl_histogram_add (gsl_histogram *h1, const gsl_histogram *h2)
This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e. h'_1(i) = h_1(i) + h_2(i). The two histograms must have identical bin ranges.

Function: int gsl_histogram_sub (gsl_histogram *h1, const gsl_histogram *h2)
This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1, i.e. h'_1(i) = h_1(i) - h_2(i). The two histograms must have identical bin ranges.

Function: int gsl_histogram_mul (gsl_histogram *h1, const gsl_histogram *h2)
This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i) = h_1(i) * h_2(i). The two histograms must have identical bin ranges.

Function: int gsl_histogram_div (gsl_histogram *h1, const gsl_histogram *h2)
This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i) = h_1(i) / h_2(i). The two histograms must have identical bin ranges.

Function: int gsl_histogram_scale (gsl_histogram *h, double scale)
This function multiplies the contents of the bins of histogram h by the constant scale, i.e. h'_1(i) = h_1(i) * scale.

Function: int gsl_histogram_shift (gsl_histogram *h, double offset)
This function shifts the contents of the bins of histogram h by the constant offset, i.e. h'_1(i) = h_1(i) + offset.

The library provides functions for reading and writing histograms to a file as binary data or formatted text.

Function: int gsl_histogram_fwrite (FILE * stream, const gsl_histogram * h)
This function writes the ranges and bins of the histogram h to the stream stream in binary format. The return value is 0 for success and `GSL_EFAILED` if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

Function: int gsl_histogram_fread (FILE * stream, gsl_histogram * h)
This function reads into the histogram h from the open stream stream in binary format. The histogram h must be preallocated with the correct size since the function uses the number of bins in h to determine how many bytes to read. The return value is 0 for success and `GSL_EFAILED` if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

Function: int gsl_histogram_fprintf (FILE * stream, const gsl_histogram * h, const char * range_format, const char * bin_format)
This function writes the ranges and bins of the histogram h line-by-line to the stream stream using the format specifiers range_format and bin_format. These should be one of the `%g`, `%e` or `%f` formats for floating point numbers. The function returns 0 for success and `GSL_EFAILED` if there was a problem writing to the file. The histogram output is formatted in three columns, and the columns are separated by spaces, like this,

```range range bin
range range bin
range range bin
....
range[n-1] range[n] bin[n-1]
```

The values of the ranges are formatted using range_format and the value of the bins are formatted using bin_format. Each line contains the lower and upper limit of the range of the bins and the value of the bin itself. Since the upper limit of one bin is the lower limit of the next there is duplication of these values between lines but this allows the histogram to be manipulated with line-oriented tools.

Function: int gsl_histogram_fscanf (FILE * stream, gsl_histogram * h)
This function reads formatted data from the stream stream into the histogram h. The data is assumed to be in the three-column format used by `gsl_histogram_fprintf`. The histogram h must be preallocated with the correct length since the function uses the size of h to determine how many numbers to read. The function returns 0 for success and `GSL_EFAILED` if there was a problem reading from the file.

## Resampling from histograms

A histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of an event where the value of x falls in the range of that bin. The probability distribution function has the one-dimensional form p(x)dx where,

```p(x) = n_i/ (N w_i)
```

In this equation n_i is the number of events in the bin which contains x, w_i is the width of the bin and N is the total number of events. The distribution of events within each bin is assumed to be uniform.

## The histogram probability distribution struct

The probability distribution function for a histogram consists of a set of bins which measure the probability of an event falling into a given range of a continuous variable x. A probability distribution function is defined by the following struct, which actually stores the cumulative probability distribution function. This is the natural quantity for generating samples via the inverse transform method, because there is a one-to-one mapping between the cumulative probability distribution and the range [0,1]. It can be shown that by taking a uniform random number in this range and finding its corresponding coordinate in the cumulative probability distribution we obtain samples with the desired probability distribution.

Data Type: gsl_histogram_pdf
`size_t n`
This is the number of bins used to approximate the probability distribution function.
`double * range`
The ranges of the bins are stored in an array of n+1 elements pointed to by range.
`double * sum`
The cumulative probability for the bins is stored in an array of n elements pointed to by sum.

The following functions allow you to create a `gsl_histogram_pdf` struct which represents this probability distribution and generate random samples from it.

Function: gsl_histogram_pdf * gsl_histogram_pdf_alloc (size_t n)
This function allocates memory for a probability distribution with n bins and returns a pointer to a newly initialized `gsl_histogram_pdf` struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of `GSL_ENOMEM`.

Function: int gsl_histogram_pdf_init (gsl_histogram_pdf * p, const gsl_histogram * h)
This function initializes the probability distribution p with with the contents of the histogram h. If any of the bins of h are negative then the error handler is invoked with an error code of `GSL_EDOM` because a probability distribution cannot contain negative values.

Function: void gsl_histogram_pdf_free (gsl_histogram_pdf * p)
This function frees the probability distribution function p and all of the memory associated with it.

Function: double gsl_histogram_pdf_sample (const gsl_histogram_pdf * p, double r)
This function uses r, a uniform random number between zero and one, to compute a single random sample from the probability distribution p. The algorithm used to compute the sample s is given by the following formula,

```s = range[i] + delta * (range[i+1] - range[i])
```

where i is the index which satisfies sum[i] <= r < sum[i+1] and delta is (r - sum[i])/(sum[i+1] - sum[i]).

## Example programs for histograms

The following program shows how to make a simple histogram of a column of numerical data supplied on `stdin`. The program takes three arguments, specifying the upper and lower bounds of the histogram and the number of bins. It then reads numbers from `stdin`, one line at a time, and adds them to the histogram. When there is no more data to read it prints out the accumulated histogram using `gsl_histogram_fprintf`.

```#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_histogram.h>

int
main (int argc, char **argv)
{
double a, b;
size_t n;

if (argc != 4)
{
printf ("Usage: gsl-histogram xmin xmax n\n"
"Computes a histogram of the data "
"on stdin using n bins from xmin "
"to xmax\n");
exit (0);
}

a = atof (argv);
b = atof (argv);
n = atoi (argv);

{
int status;
double x;

gsl_histogram * h = gsl_histogram_alloc (n);

gsl_histogram_set_uniform (h, a, b);

while (fscanf(stdin, "%lg", &x) == 1)
{
gsl_histogram_increment(h, x);
}

gsl_histogram_fprintf (stdout, h, "%g", "%g");

gsl_histogram_free (h);
}

exit (0);
}
```

Here is an example of the program in use. We generate 10000 random samples from a Cauchy distribution with a width of 30 and histogram them over the range -100 to 100, using 200 bins.

```\$ gsl-randist 0 10000 cauchy 30
| gsl-histogram -100 100 200 > histogram.dat
```

A plot of the resulting histogram shows the familiar shape of the Cauchy distribution and the fluctuations caused by the finite sample size.

```\$ awk '{print \$1, \$3 ; print \$2, \$3}' histogram.dat
| graph -T X
```

## Two dimensional histograms

A two dimensional histogram consists of a set of bins which count the number of events falling in a given area of the (x,y) plane. The simplest way to use a two dimensional histogram is to record two-dimensional position information, n(x,y). Another possibility is to form a joint distribution by recording related variables. For example a detector might record both the position of an event (x) and the amount of energy it deposited E. These could be histogrammed as the joint distribution n(x,E).

## The 2D histogram struct

Two dimensional histograms are defined by the following struct,

Data Type: gsl_histogram2d
`size_t nx, ny`
This is the number of histogram bins in the x and y directions.
`double * xrange`
The ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by xrange.
`double * yrange`
The ranges of the bins in the y-direction are stored in an array of ny + 1 pointed to by yrange.
`double * bin`
The counts for each bin are stored in an array pointed to by bin. The bins are floating-point numbers, so you can increment them by non-integer values if necessary. The array bin stores the two dimensional array of bins in a single block of memory according to the mapping `bin(i,j)` = `bin[i * ny + j]`.

The range for `bin(i,j)` is given by `xrange[i]` to `xrange[i+1]` in the x-direction and `yrange[j]` to `yrange[j+1]` in the y-direction. Each bin is inclusive at the lower end and exclusive at the upper end. Mathematically this means that the bins are defined by the following inequality,

```bin(i,j) corresponds to xrange[i] <= x < xrange[i+1]
and yrange[j] <= y < yrange[j+1]
```

Note that any samples which fall on the upper sides of the histogram are excluded. If you want to include these values for the side bins you will need to add an extra row or column to your histogram.

The `gsl_histogram2d` struct and its associated functions are defined in the header file `gsl_histogram2d.h'.

## 2D Histogram allocation

The functions for allocating memory to a 2D histogram follow the style of `malloc` and `free`. In addition they also perform their own error checking. If there is insufficient memory available to allocate a histogram then the functions call the error handler (with an error number of `GSL_ENOMEM`) in addition to returning a null pointer. Thus if you use the library error handler to abort your program then it isn't necessary to check every 2D histogram `alloc`.

Function: gsl_histogram2d * gsl_histogram2d_alloc (size_t nx, size_t ny)
This function allocates memory for a two-dimensional histogram with nx bins in the x direction and ny bins in the y direction. The function returns a pointer to a newly created `gsl_histogram2d` struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of `GSL_ENOMEM`. The bins and ranges must be initialized with one of the functions below before the histogram is ready for use.

Function: int gsl_histogram2d_set_ranges (gsl_histogram2d * h, const double xrange[], size_t xsize, const double yrange[], size_t ysize)
This function sets the ranges of the existing histogram h using the arrays xrange and yrange of size xsize and ysize respectively. The values of the histogram bins are reset to zero.

Function: int gsl_histogram2d_set_ranges_uniform (gsl_histogram2d * h, double xmin, double xmax, double ymin, double ymax)
This function sets the ranges of the existing histogram h to cover the ranges xmin to xmax and ymin to ymax uniformly. The values of the histogram bins are reset to zero.

Function: void gsl_histogram2d_free (gsl_histogram2d * h)
This function frees the 2D histogram h and all of the memory associated with it.

## Copying 2D Histograms

Function: int gsl_histogram2d_memcpy (gsl_histogram2d * dest, const gsl_histogram2d * src)
This function copies the histogram src into the pre-existing histogram dest, making dest into an exact copy of src. The two histograms must be of the same size.

Function: gsl_histogram2d * gsl_histogram2d_clone (const gsl_histogram2d * src)
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.

## Updating and accessing 2D histogram elements

You can access the bins of a two-dimensional histogram either by specifying a pair of (x,y) coordinates or by using the bin indices (i,j) directly. The functions for accessing the histogram through (x,y) coordinates use binary searches in the x and y directions to identify the bin which covers the appropriate range.

Function: int gsl_histogram2d_increment (gsl_histogram2d * h, double x, double y)
This function updates the histogram h by adding one (1.0) to the bin whose x and y ranges contain the coordinates (x,y).

If the point (x,y) lies inside the valid ranges of the histogram then the function returns zero to indicate success. If (x,y) lies outside the limits of the histogram then the function returns `GSL_EDOM`, and none of bins are modified. The error handler is not called, since it is often necessary to compute histogram for a small range of a larger dataset, ignoring any coordinates outside the range of interest.

Function: int gsl_histogram2d_accumulate (gsl_histogram2d * h, double x, double y, double weight)
This function is similar to `gsl_histogram2d_increment` but increases the value of the appropriate bin in the histogram h by the floating-point number weight.

Function: double gsl_histogram2d_get (const gsl_histogram2d * h, size_t i, size_t j)
This function returns the contents of the (i,j)th bin of the histogram h. If (i,j) lies outside the valid range of indices for the histogram then the error handler is called with an error code of `GSL_EDOM` and the function returns 0.

Function: int gsl_histogram2d_get_xrange (const gsl_histogram2d * h, size_t i, double * xlower, double * xupper)
Function: int gsl_histogram2d_get_yrange (const gsl_histogram2d * h, size_t j, double * ylower, double * yupper)
These functions find the upper and lower range limits of the ith and jth bins in the x and y directions of the histogram h. The range limits are stored in xlower and xupper or ylower and yupper. The lower limits are inclusive (i.e. events with these coordinates are included in the bin) and the upper limits are exclusive (i.e. events with the value of the upper limit are not included and fall in the neighboring higher bin, if it exists). The functions return 0 to indicate success. If i or j lies outside the valid range of indices for the histogram then the error handler is called with an error code of `GSL_EDOM`.

Function: double gsl_histogram2d_xmax (const gsl_histogram2d * h)
Function: double gsl_histogram2d_xmin (const gsl_histogram2d * h)
Function: size_t gsl_histogram2d_nx (const gsl_histogram2d * h)
Function: double gsl_histogram2d_ymax (const gsl_histogram2d * h)
Function: double gsl_histogram2d_ymin (const gsl_histogram2d * h)
Function: size_t gsl_histogram2d_ny (const gsl_histogram2d * h)
These functions return the maximum upper and minimum lower range limits and the number of bins for the x and y directions of the histogram h. They provide a way of determining these values without accessing the `gsl_histogram2d` struct directly.

Function: void gsl_histogram2d_reset (gsl_histogram2d * h)
This function resets all the bins of the histogram h to zero.

## Searching 2D histogram ranges

The following functions are used by the access and update routines to locate the bin which corresponds to a given (x\,y) coordinate.

Function: int gsl_histogram2d_find (const gsl_histogram2d * h, double x, double y, size_t * i, size_t * j)
This function finds and sets the indices i and j to the to the bin which covers the coordinates (x,y). The bin is located using a binary search. The search includes an optimization for histogram with uniform ranges, and will return the correct bin immediately in this case. If (x,y) is found then the function sets the indices (i,j) and returns `GSL_SUCCESS`. If (x,y) lies outside the valid range of the histogram then the function returns `GSL_EDOM` and the error handler is invoked.

## 2D Histogram Statistics

Function: double gsl_histogram2d_max_val (const gsl_histogram2d * h)
This function returns the maximum value contained in the histogram bins.

Function: void gsl_histogram2d_max_bin (const gsl_histogram2d * h, size_t * i, size_t * j)
This function returns the indices (i,j) of the bin containing the maximum value in the histogram h. In the case where several bins contain the same maximum value the first bin found is returned.

Function: double gsl_histogram2d_min_val (const gsl_histogram2d * h)
This function returns the minimum value contained in the histogram bins.

Function: void gsl_histogram2d_min_bin (const gsl_histogram2d * h, size_t * i, size_t * j)
This function returns the indices (i,j) of the bin containing the minimum value in the histogram h. In the case where several bins contain the same maximum value the first bin found is returned.

## 2D Histogram Operations

Function: int gsl_histogram2d_equal_bins_p (const gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function returns 1 if the all of the individual bin ranges of the two histograms are identical, and 0 otherwise.

Function: int gsl_histogram2d_add (gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function adds the contents of the bins in histogram h2 to the corresponding bins of histogram h1, i.e. h'_1(i,j) = h_1(i,j) + h_2(i,j). The two histograms must have identical bin ranges.

Function: int gsl_histogram2d_sub (gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function subtracts the contents of the bins in histogram h2 from the corresponding bins of histogram h1, i.e. h'_1(i,j) = h_1(i,j) - h_2(i,j). The two histograms must have identical bin ranges.

Function: int gsl_histogram2d_mul (gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function multiplies the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i,j) = h_1(i,j) * h_2(i,j). The two histograms must have identical bin ranges.

Function: int gsl_histogram2d_div (gsl_histogram2d *h1, const gsl_histogram2d *h2)
This function divides the contents of the bins of histogram h1 by the contents of the corresponding bins in histogram h2, i.e. h'_1(i,j) = h_1(i,j) / h_2(i,j). The two histograms must have identical bin ranges.

Function: int gsl_histogram2d_scale (gsl_histogram2d *h, double scale)
This function multiplies the contents of the bins of histogram h by the constant scale, i.e. h'_1(i,j) = h_1(i,j) scale.

Function: int gsl_histogram2d_shift (gsl_histogram2d *h, double offset)
This function shifts the contents of the bins of histogram h by the constant offset, i.e. h'_1(i,j) = h_1(i,j) + offset.

## Reading and writing 2D histograms

The library provides functions for reading and writing two dimensional histograms to a file as binary data or formatted text.

Function: int gsl_histogram2d_fwrite (FILE * stream, const gsl_histogram2d * h)
This function writes the ranges and bins of the histogram h to the stream stream in binary format. The return value is 0 for success and `GSL_EFAILED` if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

Function: int gsl_histogram2d_fread (FILE * stream, gsl_histogram2d * h)
This function reads into the histogram h from the stream stream in binary format. The histogram h must be preallocated with the correct size since the function uses the number of x and y bins in h to determine how many bytes to read. The return value is 0 for success and `GSL_EFAILED` if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

Function: int gsl_histogram2d_fprintf (FILE * stream, const gsl_histogram2d * h, const char * range_format, const char * bin_format)
This function writes the ranges and bins of the histogram h line-by-line to the stream stream using the format specifiers range_format and bin_format. These should be one of the `%g`, `%e` or `%f` formats for floating point numbers. The function returns 0 for success and `GSL_EFAILED` if there was a problem writing to the file. The histogram output is formatted in five columns, and the columns are separated by spaces, like this,

```xrange xrange yrange yrange bin(0,0)
xrange xrange yrange yrange bin(0,1)
xrange xrange yrange yrange bin(0,2)
....
xrange xrange yrange[ny-1] yrange[ny] bin(0,ny-1)

xrange xrange yrange yrange bin(1,0)
xrange xrange yrange yrange bin(1,1)
xrange xrange yrange yrange bin(1,2)
....
xrange xrange yrange[ny-1] yrange[ny] bin(1,ny-1)

....

xrange[nx-1] xrange[nx] yrange yrange bin(nx-1,0)
xrange[nx-1] xrange[nx] yrange yrange bin(nx-1,1)
xrange[nx-1] xrange[nx] yrange yrange bin(nx-1,2)
....
xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)
```

Each line contains the lower and upper limits of the bin and the contents of the bin. Since the upper limits of the each bin are the lower limits of the neighboring bins there is duplication of these values but this allows the histogram to be manipulated with line-oriented tools.

Function: int gsl_histogram2d_fscanf (FILE * stream, gsl_histogram2d * h)
This function reads formatted data from the stream stream into the histogram h. The data is assumed to be in the five-column format used by `gsl_histogram_fprintf`. The histogram h must be preallocated with the correct lengths since the function uses the sizes of h to determine how many numbers to read. The function returns 0 for success and `GSL_EFAILED` if there was a problem reading from the file.

## Resampling from 2D histograms

As in the one-dimensional case, a two-dimensional histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of an event where (x,y) falls in the range of that bin. For a two-dimensional histogram the probability distribution takes the form p(x,y) dx dy where,

```p(x,y) = n_{ij}/ (N A_{ij})
```

In this equation n_{ij} is the number of events in the bin which contains (x,y), A_{ij} is the area of the bin and N is the total number of events. The distribution of events within each bin is assumed to be uniform.

Data Type: gsl_histogram2d_pdf
`size_t nx, ny`
This is the number of histogram bins used to approximate the probability distribution function in the x and y directions.
`double * xrange`
The ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by xrange.
`double * yrange`
The ranges of the bins in the y-direction are stored in an array of ny + 1 pointed to by yrange.
`double * sum`
The cumulative probability for the bins is stored in an array of nx*ny elements pointed to by sum.

The following functions allow you to create a `gsl_histogram2d_pdf` struct which represents a two dimensional probability distribution and generate random samples from it.

Function: gsl_histogram2d_pdf * gsl_histogram2d_pdf_alloc (size_t nx, size_t ny)
This function allocates memory for a two-dimensional probability distribution of size nx-by-ny and returns a pointer to a newly initialized `gsl_histogram2d_pdf` struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of `GSL_ENOMEM`.

Function: int gsl_histogram2d_pdf_init (gsl_histogram2d_pdf * p, const gsl_histogram2d * h)
This function initializes the two-dimensional probability distribution calculated p from the histogram h. If any of the bins of h are negative then the error handler is invoked with an error code of `GSL_EDOM` because a probability distribution cannot contain negative values.

Function: void gsl_histogram2d_pdf_free (gsl_histogram2d_pdf * p)
This function frees the two-dimensional probability distribution function p and all of the memory associated with it.

Function: int gsl_histogram2d_pdf_sample (const gsl_histogram2d_pdf * p, double r1, double r2, double * x, double * y)
This function uses two uniform random numbers between zero and one, r1 and r2, to compute a single random sample from the two-dimensional probability distribution p.

## Example programs for 2D histograms

This program demonstrates two features of two-dimensional histograms. First a 10 by 10 2d-histogram is created with x and y running from 0 to 1. Then a few sample points are added to the histogram, at (0.3,0.3) with a height of 1, at (0.8,0.1) with a height of 5 and at (0.7,0.9) with a height of 0.5. This histogram with three events is used to generate a random sample of 1000 simulated events, which are printed out.

```#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_histogram2d.h>

int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;

gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10)

gsl_histogram2d_set_ranges_uniform (h,
0.0, 1.0,
0.0, 1.0);

gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);

gsl_rng_env_setup();

T = gsl_rng_default;
r = gsl_rng_alloc(T);

{
int i;
gsl_histogram2d_pdf * p
= gsl_histogram2d_pdf_alloc (h->n);

gsl_histogram2d_pdf_init (p, h);

for (i = 0; i < 1000; i++) {
double x, y;
double u = gsl_rng_uniform (r);
double v = gsl_rng_uniform (r);

int status
= gsl_histogram2d_pdf_sample (p, u, v, &x, &y);

printf("%g %g\n", x, y);
}
}

return 0;
}
```

The following plot shows the distribution of the simulated events. Using a higher resolution grid we can see the original underlying histogram and also the statistical fluctuations caused by the events being uniformly distributed over the the area of the original bins.