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The KDE Procedure |
To bin a set of weighted univariate data X1,X2, ... ,Xn to a grid x1, x2, ... ,xg, simply assign each sample Xi, together with its weight Wi, to the nearest grid point xj (also called the bin center). When binning is completed, each grid point xi has an associated number ci, which is the sum total of all the weights that correspond to sample points that have been assigned to xi. These cis are known as the "bin counts."
This procedure replaces the data (Xi,Wi), i = 1,2, ... ,n
with the smaller set (xi,ci), i = 1,2, ... ,g, and the
estimation is carried out with this new data. This is so-called
"simple binning," as versus the finer "linear binning" described
in Wand (1993). PROC KDE uses simple binning for the sake of faster
and easier implementation. Also, it is assumed that the bin centers
x1,x2, ... ,xg are equally spaced and in increasing
order. In addition, assume for notational convenience that
and, therefore,
.
If you replace the data (Xi,Wi), i = 1,2, ... ,n
with (xi,ci), i = 1,2, ... ,g, the weighted
estimator then becomes
The same idea of binning works similarly with bivariate data, where
you estimate over the grid matrix grid = gridX×gridY as follows.
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