deletions | additions       diff --git a/subsubsection_sparsity_problem_However_with__.tex b/subsubsection_sparsity_problem_However_with__.tex  index f15ecd5..67e4b0e 100644  --- a/subsubsection_sparsity_problem_However_with__.tex  +++ b/subsubsection_sparsity_problem_However_with__.tex 
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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. similarly using a 3x3x3x3 gaussian filter.