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 <https://en.wikipedia.org/wiki/Pseudometric_space>`_. They must have the following properties:

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

Here :math:`A` and :math:`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 :math:`d(A, B) \gg 0`. If the datasets have similar physical properties, the distance should be small :math:`d(A, B) \approx 0`.

Additionally, the statistics should ideally be insensitive to spatial shifts :math:`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) :math:`d\left( A + \mathcal{N}\left(0, \sigma_1^2 \right), A + \mathcal{N}\left(0, \sigma_2^2 \right) \right) \approx 0`.

Source Code
-----------
.. automodapi:: turbustat.statistics
    :no-heading:
    :no-inheritance-diagram:

.. autoclass:: turbustat.statistics.base_statistic.BaseStatisticsMixIn
.. autoclass:: turbustat.statistics.base_pspec2.StatisticBase_PSpec2D