deletions | additions       diff --git a/subsubsection_sparsity_problem_However_with__.tex b/subsubsection_sparsity_problem_However_with__.tex  index 95ded20..8d2ff41 100644  --- a/subsubsection_sparsity_problem_However_with__.tex  +++ b/subsubsection_sparsity_problem_However_with__.tex 
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\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 ($p(x,y)$) and 4-dimensional ($p(x_{1},y_{1},x_{2},y_{2})$) 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 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.