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 (@math{x},@math{y}) falls in the range of that bin. For a two-dimensional histogram the probability distribution takes the form @math{p(x,y) dx dy} where,

In this equation @math{n_{ij}} is the number of events in the bin which contains @math{(x,y)}, @math{A_{ij}} is the area of the bin and @math{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.