Potential Functions as Superposition of Scoring Functions (PF)
==============================================================

Description
-----------

The general concept of the PF metric is to define a potential function for each static or dynamic object considered by the metric [Wolf2018]_.
This includes potentials for lane markings, the road geometry, other vehicles, or, in more urban areas, pedestrians and bicyclists.
Once a potential function for each object in the scene, denoted by :math:`U_i(A, S)`, is chosen, one can apply e.g. gradient descent for a given scene :math:`S` to the combined potential function :math:`U(A, S) = U_1(A, S) + \dots + U_k(A, S)`, where :math:`A` is an actor and :math:`k` denotes the number of objects.
A simple example of how to evaluate this metric for an actor :math:`A_1` and a given scene :math:`S'` is by inserting the values into :math:`U`, i.e.

.. math::
		\mathit{PF}(A_1, S') = U(A_1, S') = U_1(A_1, S') + \dots + U_k(A_1, S') \,.

However, methods involving the mentioned gradient descent to assess the criticality can improve precision and also provide a suggestion for criticality-reducing vehicle movement. 

Due to the way this metric is defined, almost all properties depend on the specified potential functions. 
Furthermore, while ethical questions play a role when defining any safety surrogate, it becomes more evident for potential functions, as an active decision making in the definition of the potentials is required.

Properties
----------


Run-time capability
~~~~~~~~~~~~~~~~~~~

Yes

Target values
~~~~~~~~~~~~~

None found, also highly dependent on the used potential functions

Subject type
~~~~~~~~~~~~

Any, but requires a potential function for each considered subject type

Scenario type
~~~~~~~~~~~~~

Depends on specified potential functions

Inputs
~~~~~~

Potential function for each static/dynamic object in the scene that is supposed to be considered, other inputs depend entirely on said potential functions

Output scale
~~~~~~~~~~~~

:math:`[-\infty, \infty]`, number (negative values are possible if goal locations are defined), ratio scale

Reliability
~~~~~~~~~~~

Largely depends on the used potential functions

Validity
~~~~~~~~

Largely depends on the used potential functions; no empirical analysis identified

Sensitivity
~~~~~~~~~~~

Largely depends on the used potential functions

Specificity
~~~~~~~~~~~

Largely depends on the used potential functions

Prediction model
~~~~~~~~~~~~~~~~

Time window
^^^^^^^^^^^
Depends on quality of potential functions and reliability of computation of the solution to the potential equation problem

Time mode
^^^^^^^^^
Branching time