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\subsection{Approximation of probability distribution based on trajectories}  The approximation of probability distribution of positions or accelerations is a key part of mutual information computing. Both are estimated with the same methods.  First we have to discretize the feature space (i.e. the space of values that positions and accelerations can reach).  Basically for position we take 13x17 bins and for acceleration we take 23x23 bins.  Besides we consider sample of 3 minutes of the soccer match.  Those values result of a trade off between sparsity and precision of the model.  \subsection{A first approach: histogram}  The easiest way to approximate the probability distribution i.e. all $p(x,y)$ for $(x,y)$ the bin's coordinates of the discretized field (we will come to that later) is to use histogram. $p(x,y)$ is then the occurence ratio of $(x,y)$ among the whole set of positions traveled by the trajectory $T$.  in another words $p(x,y)$ is given by:  \[P_{hist}((x,y),T) = \sum{(x',y')\in T}\mathbb{1}_{x=x',y=y'}\]  \subsubsection{sparsity problem}