deletions | additions       diff --git a/subsubsection_sparsity_problem_However_with__.tex b/subsubsection_sparsity_problem_However_with__.tex  index 8d2ff41..f15ecd5 100644  --- a/subsubsection_sparsity_problem_However_with__.tex  +++ b/subsubsection_sparsity_problem_However_with__.tex 
...
\subsubsection{sparsity problem}  However with the histogram method the empirical distribution remain too sparse, which lead to inconsistant empirical mutual information values. Indeed the feature space of positions or accelerations are 2-dimensional ($p(x,y)$) \Big($p(x,y)$\Big)  and 4-dimensional ($p(x_{1},y_{1},x_{2},y_{2})$) \Big($p(x_{1},y_{1},x_{2},y_{2})$\Big)  whereas our data is in both case a 1-dimensional path in the feature space. So histogram of trajectories are liable to be sparse. To cope with sparsity we used Parzen windowing \textbf{Parzen windowing}  as it is described in \cite{Pluim_2003} for instance. Given a trajectory $T$, the probability $p(x,y)$ of $(x,y)$ is the sum of the contribution of each $(x',y')$ in $T$. The contributions are functions of a Gaussian kernel.  Hence the following definition of the probability of $(x,y)$ given $T$: