  In football game analysis, space evaluation is an important issue because it
is directly related to the quality of ball passing or player formations.
Previous studies have primarily focused on a field division approach wherein a
field is divided into dominant regions in which a certain player can arrive
prior to any other players. However, the field division approach is
oversimplified because all locations within a region are regarded as uniform
herein. The objective of the current study is to propose a fundamental
framework for space evaluation based on field weighting. In particular, we
employed the motion model and calculated a minimum arrival time $ \tau $ for
each player to all locations on the football field. Our main contribution is
that two variables $ \tau_{\textrm{of}} $ and $ \tau_{\textrm{df}} $
corresponding to the minimum arrival time for offense and defense teams are
considered; using $ \tau_{\textrm{of}} $ and $ \tau_{\textrm{df}} $, new
orthogonal variables $ z_{1} $ and $ z_{2} $ are defined. In particular, based
on real datasets comprising of data from 45 football games of the J1 League in
2018, we provide a detailed characterization of $ z_{1} $ and $ z_{2} $ in
terms of ball passing. By using our method, we found that $ z_{1}(\vec{x}, t) $
and $ z_{2}(\vec{x}, t) $ represent the degree of safety for a pass made to $
\vec{x} $ at $ t $ and degree of sparsity of $ \vec{x} $ at $ t $,
respectively; the success probability of passes could be well-fitted using a
sigmoid function. Moreover, a new type of field division approach and
evaluation of ball passing just before shoots using real game data are
discussed.
