deletions | additions       diff --git a/subsection_Approximation_of_probability_distribution__.tex b/subsection_Approximation_of_probability_distribution__.tex  index c92b485..1bac2be 100644  --- a/subsection_Approximation_of_probability_distribution__.tex  +++ b/subsection_Approximation_of_probability_distribution__.tex 
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\subsection{Approximation of probability distribution based on trajectories}  The approximation of probability distribution (and joint distributions)  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. 
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Whereas the simple histogram method places a spike function (i.e. $K = \delta$) at the bin corresponding to $(x,y)$ and update only a single bin, Parzen windowing places a kernel at the bin of $(x,y)$ and updates all bins falling under the kernel with the corresponding kernel value.  As a result using a gaussian filter, the estimated distributions are more smooth and less sparse.  The previous formula stand for distribution of position but we approximate distribution and joint distributions of position or acceleration similarly.