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\subsection{Mutual Information: definition and properties}  Mutual Information is widely used for instance for registration of medical images as it is depicted in \cite{Pluim_2003}. The main idea is to introduce a feature space (or a joint probability) of the two trajectories we want to compare and to evaluate the quantity of information shared by the two trajectories based on this feature space. This quantity is calculated with Mutual Information.   In our case the feature space will be the distribution of the joint positions of two players' player  trajectories during a few minutes (we will come to that later). Thus Therfore,  it corresponds to a 4-dimension distribution. The Mutual Infomation of this distribution will be the lynchpin of our similarity measure metric  for trajectories. \subsubsection{Entropy}  Shannon introduced the entropy to be a measure of the quantity of information of embedded in  the distribution of a random variable. Let $X: P \rightarrow E$ be a random variable with $E$ a discrete probability space.  The entropy of the probability law distribution  of $X$, noted $S(X)$, is defined as: \[ S(X)=-\sum_{x \in E}P_{X}(x)log\big(P_{X}(x)\big) \]    It is noteworthy that, in particular, the entropy of the joint probability law distribution  of two random variables $X : P_{1} \rightarrow E_{1}$ and $Y : P_{2} \rightarrow E_{2}$ is defined as: \[ S(X,Y)=-\sum_{(x,y) \in E_1\times E_2}P_{(X,Y)}(x,y)log\big(P_{(X,Y)}(x,y)\big) \]  Besides the entropy of the probability law distribution  of $X$ conditionally to the probability law distribution  of $Y$ is defined as: \[ S(X|Y)=-\sum_{x \in E_1}P_{X|Y}(x)log\big(P_{X|Y}(x)\big) \]  Somewhat imprecisely, we used to designate the entropy of the probability law distribution  of a random variable $X$ as simply \textit{the entropy of $X$}. Hence the notation \textit{$S(X)$} for the entropy of $X$. The entropy of a probability law distribution  (or of a random variable) has three interpretations: \begin{itemize}  \item the amount of information of embedded in  a probability law distribution  \item the uncertainty about the outcome of a probability law distribution  \item the dispersion ofthe probability law of  a probability law distribution  \end{itemize}  For more information about entropy the reader can refer to \cite{Pluim_2003} or \href{http://www.yann-ollivier.org/entropie/entropie1}{\textit{La théorie de l'information : l'origine de l'entropie}}.