deletions | additions       diff --git a/subsection_Approximation_of_probability_distribution__.tex b/subsection_Approximation_of_probability_distribution__.tex  index 6234f16..5bb6d52 100644  --- a/subsection_Approximation_of_probability_distribution__.tex  +++ b/subsection_Approximation_of_probability_distribution__.tex 
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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 $T$:  \[P((x,y),T) = \sum_{(x',y')\in T}K((x,y),(x',y'))\]  where $K$ is a gaussian kernel. in practice we take a discrete gaussian kernel  filter for $K$. Whereas the simple histogram method places a spike function (i.e. $K = \delta_{(x,y)}$) 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.