Statistics¶

Along with acting as the base classes for the distance metrics, the statistics can also return useful information about single datasets.

Distance Metrics¶

The distance metrics are computed using certain outputs contained in the related statistics class.

Nearly all of the distance metrics are actually “pseudo” - distance metrics. They must have the following properties:

1. \(d(A, A) = 0\)
2. Symmetric \(d(A, B) = d(B, A)\)
3. Triangle Inequality \(d(A, B) \leq d(A, C) + d(B, C)\)

Here \(A\) and \(B\) represent two datasets (either a PPV datacube, column density map, or an associated moment of the cube).

For two datasets with different physical properties, a good statistic will return a large value \(d(A, B) \gg 0\). If the datasets have similar physical properties, the distance should be small \(d(A, B) \approx 0\).

Additionally, the statistics should ideally be insensitive to spatial shifts \(d\left( A\left[ x,y,v \right], A\left[ x+\delta x,y,v \right] \right)=0\) and independent of the noise level (for observational data) \(d\left( A + \mathcal{N}\left(0, \sigma_1^2 \right), A + \mathcal{N}\left(0, \sigma_2^2 \right) \right) \approx 0\).