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<h2>DESCRIPTION</h2>
<em>v.kernel</em> generates a raster density map from vector points
data using a moving
kernel. Available <a href="https://en.wikipedia.org/wiki/Kernel_(statistics)#Kernel_functions_in_common_use">kernel
density functions</a> are <em>uniform, triangular, epanechnikov,
quartic, triweight, gaussian, cosine</em>, default
is <em>gaussian</em>.
<p>The module can also generate a vector density map on a vector network.
Conventional kernel functions produce biased estimates by overestimating
the densities around network nodes, whereas the equal split method of
Okabe et al. (2009) produces unbiased density estimates. The equal split
method uses the kernel function selected with the <b>kernel</b> option
and can be enabled with <b>node=split</b>.
<h2>NOTES</h2>
The <b>multiplier</b> option is needed to overcome the limitation that
the resulting density in case of a vector map output is stored as category
(integer). The density result stored as category may be multiplied by this number.
<p>
For the <em>gaussian</em> kernel, standard deviation for the
<a href="https://en.wikipedia.org/wiki/Kernel_(statistics)#Kernel_functions_in_common_use">gaussian function</a>
is set to 1/4 of the radius.
<p>
With the <b>-o</b> flag (experimental) the command tries to calculate an
optimal radius. The value of <em>radius</em> is taken
as maximum value. The radius is calculated based on the gaussian function,
using ALL points, not just those in the current region.
<h2>EXAMPLES</h2>
Compute density of points (using vector map of schools from North Carolina sample dataset):
<div class="code"><pre>
g.region region=wake_30m
v.kernel input=schools_wake output=schools_density radius=5000 multiplier=1000000
r.colors map=schools_density color=bcyr
</pre></div>
<center>
<img src="v_kernel.png" alt="Density of schools" border="0"><br>
School density
</center>
<h2>KNOWN ISSUES</h2>
The module only considers the presence of points, but not
(yet) any attribute values.
<h2>REFERENCES</h2>
<ul>
<li>Okabe, A., Satoh, T., Sugihara, K. (2009). <i>A kernel density estimation
method for networks, its computational method and a GIS-based tool</i>.
<b>International Journal of Geographical Information Science</b>, Vol 23(1),
pp. 7-32.<br>
DOI: <a href="https://doi.org/10.1080/13658810802475491">10.1080/13658810802475491</a></li>
</ul>
<h2>SEE ALSO</h2>
<em><a href="v.surf.rst.html">v.surf.rst</a></em>
<p>
Overview: <a href="https://grasswiki.osgeo.org/wiki/Interpolation">Interpolation and Resampling</a> in GRASS GIS
<h2>AUTHORS</h2>
Stefano Menegon, <a href="http://mpa.itc.it/">ITC-irst</a>, Trento, Italy<br>
Radim Blazek (additional kernel density functions and network part)
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