Simplifications used in jackHI binning routines¶

jackHI assumes that the bins are of equal width in linear or log space. Moreover, in this version, no weights can be assigned.

(see the theoretical underpinnings page for definitions of most of the parameters)

Implementation of (linear) constant-width binning¶

In this case, all \(\Delta_i\) are equal and set to \(\Delta\), and it is assumed that the parameter values \(x\) vary in a range \(x_{min} < x < x_{max}\). Therefore, only 2 hyperparameters remain important: the number of bins \(N\), and the smoothing \(\alpha\).

jackHI uses Eq.(7) of Hogg (2008) to construct the binning function:

\[\begin{equation} i(x) = \operatorname{floor}\left(\dfrac{x - x_{min}}{\Delta} - \delta\right)\,, \end{equation}\]

where \(\delta\) is the binning phase parameter, and where \(\Delta\) is the bin width, which can be calculated as

\[\begin{equation} \Delta \equiv \dfrac{x_{max} - x_{min}}{N}\,. \end{equation}\]

Implementation of logarithmic binning¶

In this case, all \(\Delta_i\) are set to be equally wide in log-space. They are thus not of constant width. The log-transformed data are then defined to be \(\mathbf{l} = \log(\mathbf{x})\), and the binning function becomes

\[\begin{equation} i(l) = \operatorname{floor}\left(\dfrac{l - l_{min}}{\Delta_{i(l)}}\right)\,. \end{equation}\]

Note the lack of a logarithmic equivalent of the binning phase parameter \(\delta\).