[R-sig-Geo] Intensity Estimation Methods

[Adrian Baddeley]

Cenk ©ÇÖZ <cicoz at anadolu.edu.tr> writes:


 I have a spatial point pattern . I am trying to estimate its intensity
> both with a fixed bandwidth and  with an adaptive bandwidth. How could
> I compare the goodness of these two fits? I mean are there any things
> like mse, aic or any other criteria??? I want to compare the
> difference between the estimated intensity and the original pattern's
> intensity.


If you know the 'true' intensity then you could compute, for example, the integrated squared error.

In the spatstat package, if 'lamtrue' is the true intensity and 'lamest' the estimated intensity, given as pixel images (class 'im') then you can just do

     ISE <- integral((lamtrue-lamest)^2)


Alternatives include the Kulback-Leibler divergence

     KL <- integral(log(lamest/lamtrue) * lamtrue)

and the total variation distance

     TV <- integral(abs(lamtrue-lamest))/2


However if you have two competing estimates of the intensity of an observed point pattern, you're probably best to use the point process likelihood. Suppose lam1 and lam2 are pixel images giving  two competing estimates of the intensity of the same point pattern X. Then you could do


      lik1 <- sum(log(lam1[X])) - integral(lam1)

      lik2 <- sum(log(lam2[X])) - integral(lam2)


and compare the likelihoods.


Adrian Baddeley


________________________________
From: Cenk ©ÇÖZ <cicoz at anadolu.edu.tr>
Sent: Friday, 15 December 2017 8:23 PM
To: Rolf Turner
Cc: r-sig-geo at r-project.org; Adrian Baddeley; Ege Rubak
Subject: RE: [R-sig-Geo] Intensity Estimation Methods

Thanks a lot.
I did superimpose the original pattern to pixel images of intensities. With adaptive smothing intensity higher zones are too narrowed.
Also the differences of intensities are getting higher in the study region. The fixed bandwidth choosen with bw.ppl( ) in spatstat give me a better Picture. This is my opinion and also this is in my case . I use this bandwidth as a global bandwidth for adaptive smoothing.


-----Original Message-----
From: Rolf Turner [mailto:r.turner at auckland.ac.nz]
Sent: Thursday, December 14, 2017 11:14 PM
To: Cenk ©ÇÖZ <cicoz at anadolu.edu.tr>
Cc: r-sig-geo at r-project.org; Adrian.Baddeley at curtin.edu.au; Ege Rubak <rubak at math.aau.dk>
Subject: Re: [R-sig-Geo] Intensity Estimation Methods

On 15/12/17 01:01, Cenk ©ÇÖZ via R-sig-Geo wrote:
> Hello list,,
>
> I have a spatial point pattern . I am trying to estimate its intensity
> both with a fixed bandwidth and  with an adaptive bandwidth. How could
> I compare the goodness of these two fits? I mean are there any things
> like mse, aic or any other criteria??? I want to compare the
> difference between the estimated intensity and the original pattern's
> intensity.

I think that the following fortune (fortunes::fortune(340)) might be
relevant:

> Bandwidth selection is an unresolved (and possibly unsolvable) problem
> in smoothing, so you're perfectly justified in trying/choosing an
> arbitrary value if it produces good pictures!
>    -- Adrian Baddeley (answering a user's question about the choice of
>       smoothing parameter when using the density.ppp() function from the
>       spatstat package)
>       private communication (March 2013)

I am cc-ing to the man himself to see if he has further comment.

cheers,

Rolf Turner

--
Technical Editor ANZJS
Department of Statistics
University of Auckland
Phone: +64-9-373-7599 ext. 88276


________________________________

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