# Binning¶

This page explains the mathematics behind the binning functionality found in bins. The binning functions are purely mathematical in nature though they do have application in other parts of pyne that deal more directly physics with physics.

## Pointwise Linear Collapse¶

The pointwise_linear_collapse() function takes an array of pointwise data $$y$$ that has the independent variable $$x$$ and collapses this into $$G$$ groups as defined by the bin boundaries $$x_g$$ where $$g$$ indexes $$G$$. Both $$x$$ and $$x_g$$ must be monotonic and in the same direction. Say that there are $$N$$ points in $$x$$ and $$y$$. Let $$n$$ index $$N$$. Then for all points internal to a group, the collapsed value $$y_g$$ is:

$y_g = \frac{1}{x_{g+1} - x_g} \sum_{n|x_g \le x_n}^{x_n \le x_{g+1}} \frac{y_{n+1} + y_n}{2} * (x_{n+1} - x_n)$

The term $$(y_{n+1} + y_n)/2$$ is the center (average) value of a linear interpolation between the two points. Therefore, $$y_g$$ is the $$x$$-weighted average of $$y$$ over the entire group.

In the event that the line between $$y_n$$ and $$y_{n+1}$$ crosses either the lower or upper bin boundary (or both) then their values are adjusted via a linear interpolation to the value at the bin boundary.

For a lower boundary crossing, the following substitutions are made to the equation above:

$x_n \to x_g$
$y_n \to \frac{y_{n+1} - y_n}{x_{n+1} - x_n} (x_g - x_n) + y_n$

For an upper boundary crossing:

$x_{n+1} \to x_{g+1}$
$y_{n+1} \to \frac{y_{n+1} - y_n}{x_{n+1} - x_n} (x_{g+1} - x_n) + y_n$