I'm a marine spatial ecologist looking to remove land from some kernel density estimations of tagged sharks. I've seen numerous methods for this approach, with one such method described succinctly in Barry & McIntyre (2011):

A typical approach to remedying this [kernel density estimator boundary] problem is first to compute the estimator as if there were no boundaries, then to clip off inaccessible regions after the fact, and finally renormalize the density.

What does it mean to "renormalize" the density? Are they essentially saying that once you calculate the kernel density, you set your boundary areas (in my case, land) to zero, and then make every cell in your kernel range in value from 0 to 1?


1 Answer 1


Renormalizing here means that you are forcing the integral of your distribution to be equal to 1. This is a required property of any probability distribution.

Of course, any density estimation method, kernel one included, is made in a way that the result will be valid, which means that its integral will be 1. If you clipp off some regions from the domain of that fuction, its integral won't be 1 anymore. To fix this, you have to divide the function by its integral, so that the resulting function will have unit integral and therefore will be a valid probability density. This is the "normalization".

  • $\begingroup$ So essentially this is forcing a normal distribution ("bell curve") on all points in both dimensions of a 2D kernel? (This is the struggle of ecology grad students only knowing rudimentary statistics) $\endgroup$
    – spops
    Nov 19, 2019 at 16:36
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    $\begingroup$ +1. But note that it's likely that in this case the KDE does not estimate a probability: it will estimate the number of sharks per unit surface area of ocean. Your argument applies, mutatis mutandis: normalization means the total number of estimated sharks equals the total number of observed sharks. spops: there's no bell curve imposed. Normalization will not change a colored contour map of your density at all, for instance--it leaves the spatial pattern (within the water areas) unchanged. Normalization is a poor idea when many of the sharks are close to land: it downweights them. $\endgroup$
    – whuber
    Nov 19, 2019 at 16:36
  • $\begingroup$ @spops I didn't really understand your comment. What do you mean? $\endgroup$
    – carlo
    Nov 19, 2019 at 16:39
  • $\begingroup$ Thanks whuber & carlo for the clear explanations. I've been looking for the best method to estimate home ranges/density and have gotten the impression that cutting the land out is a bad idea - but I think I will post a separate question on this topic in the GIS forum! $\endgroup$
    – spops
    Nov 19, 2019 at 16:46
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    $\begingroup$ I don't know any package for doing that, but if it is a raster I think it would be very easy to do by hand (I mean by coding it, the integral would be evaluated as a sum over rester points). I never did it though, so I don't really know your workflow, sorry. $\endgroup$
    – carlo
    Nov 19, 2019 at 18:03

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