[R-sig-Geo] R-sig-Geo Digest, Vol 218, Issue 10

Babak Naimi n@|m|@b @end|ng |rom gm@||@com
Thu Oct 21 14:56:33 CEST 2021 

Hi, If by nan you mean NA pixels, the pixels with NA would not contribute
to the calculation so would not cause any problem for the calculation.



On Thu, Oct 21, 2021 at 12:44 PM Gabriel Cotlier <gabiklm01 using gmail.com>
wrote:

> Hello Babak,
> Thanks. a lot.
> Yes, elsa::moran() is indeed faster than raster::Moran() as I saw in your
> code; however I have nan values in my raster layer which are rivers of
> other geographic features which aren't relevant thus were converted to nan
> not to influence the results.
> Does there exist a way to make elsa::moran() to work with a raster layer
> with nan values?
> Thanks a lot again.
> Kind regards,
> Gabriel
>
> On Thu, Oct 21, 2021 at 1:52 PM Babak Naimi <naimi.b using gmail.com> wrote:
>
>> Hi Gabriel,
>>
>> You may also use the package elsa to calculate moran (and other spatial
>> autocorrelation statistics) for raster data that is quite fast given the
>> implementation of the functions in C.
>>
>> The equivalent function in the new package terra is also faster than the
>> one in the raster:
>>
>>
>> > r <- raster(xmn=0, nrows=100, ncols=100)
>> > values(r) <- 1:ncell(r)
>> > f <- matrix(c(0,1,0,1,0,1,0,1,0), nrow=3)
>>
>> > system.time(m1 <- raster::Moran(r, f))
>>    user  system elapsed
>>   0.142   0.001   0.143
>> > system.time(m2 <- elsa::moran(r, d1=0,d2=xres(r)))
>>    user  system elapsed
>>   0.002   0.000   0.002
>>
>> > rr <- terra::rast(r)
>> > system.time(m3 <- terra::autocor(rr, f))
>>    user  system elapsed
>>   0.022   0.001   0.022
>>
>> > 0.143 / 0.022
>> [1] 6.5.  # terra is 6.5X faster than raster
>> > 0.143 / 0.001
>> [1] 143 # elsa is 143X faster than raster
>> > 0.022 / 0.001
>> [1] 22 # elsa is 22X faster than terra
>>
>> > c(raster=m1, terra=m2, elsa=m3)
>>     raster      terra         elsa
>>   0.989899   0.989899   0.989899
>>
>> Hope this helps,
>> Cheers,
>> Babak
>>
>>
>>
>> On Thu, Oct 21, 2021 at 11:01 AM <r-sig-geo-request using r-project.org> wrote:
>>
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>>>
>>> Today's Topics:
>>>
>>>    1. question on raster Moran's I statistical significance
>>>       (Gabriel Cotlier)
>>>    2. Re:  question on raster Moran's I statistical significance
>>>       (Roger Bivand)
>>>    3. Re:  question on raster Moran's I statistical significance
>>>       (Roger Bivand)
>>>    4. Re:  question on raster Moran's I statistical significance
>>>       (Gabriel Cotlier)
>>>    5. Re:  question on raster Moran's I statistical significance
>>>       (Roger Bivand)
>>>
>>> ----------------------------------------------------------------------
>>>
>>> Message: 1
>>> Date: Wed, 20 Oct 2021 16:20:45 +0300
>>> From: Gabriel Cotlier <gabiklm01 using gmail.com>
>>> To: r-sig-geo <r-sig-geo using r-project.org>
>>> Subject: [R-sig-Geo] question on raster Moran's I statistical
>>>         significance
>>> Message-ID:
>>>         <CAAKwTDHO1rah2NX+z2p83kdypX+XHt=4fYSXT=SJixM=
>>> iGJUaA using mail.gmail.com>
>>> Content-Type: text/plain; charset="utf-8"
>>>
>>> Hello,
>>>
>>> I would like to estimate the Moran's *I* coefficient for raster data and
>>> together with the statical significance of the spatial autocorrelation
>>> obtained.
>>>
>>> I found that the raster package function Moran() although calculates the
>>> spatial autocorrelation index it apparently does not give directly the
>>> statical significance of the results obtained :
>>> https://search.r-project.org/CRAN/refmans/raster/html/autocor.html
>>>
>>> Could it be be possible to obtain the statistical significance of the
>>> results with either raster package or similar one?
>>>
>>> Thanks a lot.
>>> Kind regards,
>>> Gabriel
>>>
>>>         [[alternative HTML version deleted]]
>>>
>>>
>>>
>>>
>>> ------------------------------
>>>
>>> Message: 2
>>> Date: Wed, 20 Oct 2021 19:42:28 +0200 (CEST)
>>> From: Roger Bivand <Roger.Bivand using nhh.no>
>>> To: Gabriel Cotlier <gabiklm01 using gmail.com>
>>> Cc: r-sig-geo <r-sig-geo using r-project.org>
>>> Subject: Re: [R-sig-Geo]  question on raster Moran's I statistical
>>>         significance
>>> Message-ID: <91dcba60-909f-ed29-1f4-f39c63a60 using reclus2.nhh.no>
>>> Content-Type: text/plain; charset="us-ascii"; Format="flowed"
>>>
>>> On Wed, 20 Oct 2021, Gabriel Cotlier wrote:
>>>
>>> > Hello,
>>> >
>>> > I would like to estimate the Moran's *I* coefficient for raster data
>>> and
>>> > together with the statical significance of the spatial autocorrelation
>>> > obtained.
>>> >
>>> > I found that the raster package function Moran() although calculates
>>> the
>>> > spatial autocorrelation index it apparently does not give directly the
>>> > statical significance of the results obtained :
>>> > https://search.r-project.org/CRAN/refmans/raster/html/autocor.html
>>> >
>>> > Could it be be possible to obtain the statistical significance of the
>>> > results with either raster package or similar one?
>>>
>>> fortunes::fortune("This is R")
>>>
>>>
>>> library(raster)
>>> r <- raster(nrows=10, ncols=10)
>>> values(r) <- 1:ncell(r)
>>> f <- matrix(c(0,1,0,1,0,1,0,1,0), nrow=3)
>>> (rI <- Moran(r, f))
>>> r1 <- r
>>> nsim <- 499
>>> res <- numeric(nsim)
>>> set.seed(1)
>>> for (i in 1:nsim) {
>>>    values(r1) <- values(r)[sample(prod(dim(r)))]
>>>    res[i] <- Moran(r1, f)
>>> }
>>>
>>> Hope-type tests date back to Cliff and Ord; they are permutation
>>> bootstraps.
>>>
>>> r_g <- as(r, "SpatialPixelsDataFrame")
>>> library(spdep)
>>> nb <- poly2nb(as(r_g, "SpatialPolygons"), queen=FALSE)
>>> set.seed(1)
>>> o <- moran.mc(r_g$layer, nb2listw(nb, style="B"), nsim=nsim,
>>>    return_boot=TRUE)
>>> x_a <- range(c(o$t, o$t0, res, rI))
>>> plot(density(o$t), xlim=x_a)
>>> abline(v=o$t0)
>>> lines(density(res), lty=2)
>>> abline(v=rI, lty=2)
>>>
>>> It is not immediately obvious from the code of raster::Moran() why it is
>>> different, possibly because of padding the edges of the raster and
>>> thus increasing the cell count.
>>>
>>> For added speed, the bootstrap can be parallelized in both cases;
>>> polygon
>>> boundaries are perhaps not ideal.
>>>
>>> Hope this clarifies. Always provide a reproducible example, never post
>>> HTML mail.
>>>
>>> Roger Bivand
>>>
>>> >
>>> > Thanks a lot.
>>> > Kind regards,
>>> > Gabriel
>>> >
>>> >       [[alternative HTML version deleted]]
>>> >
>>> > _______________________________________________
>>> > R-sig-Geo mailing list
>>> > R-sig-Geo using r-project.org
>>> > https://stat.ethz.ch/mailman/listinfo/r-sig-geo
>>> >
>>>
>>> --
>>> Roger Bivand
>>> Emeritus Professor
>>> Department of Economics, Norwegian School of Economics,
>>> Postboks 3490 Ytre Sandviken, 5045 Bergen, Norway.
>>> e-mail: Roger.Bivand using nhh.no
>>> https://orcid.org/0000-0003-2392-6140
>>> https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>>>
>>>
>>>
>>>
>>> ------------------------------
>>>
>>> Message: 3
>>> Date: Wed, 20 Oct 2021 19:45:01 +0200 (CEST)
>>> From: Roger Bivand <Roger.Bivand using nhh.no>
>>> To: Gabriel Cotlier <gabiklm01 using gmail.com>
>>> Cc: r-sig-geo <r-sig-geo using r-project.org>
>>> Subject: Re: [R-sig-Geo]  question on raster Moran's I statistical
>>>         significance
>>> Message-ID: <3e639398-e137-5546-1848-5a3afb809240 using reclus2.nhh.no>
>>> Content-Type: text/plain; charset="us-ascii"; Format="flowed"
>>>
>>> On Wed, 20 Oct 2021, Gabriel Cotlier wrote:
>>>
>>> > Hello,
>>> >
>>> > I would like to estimate the Moran's *I* coefficient for raster data
>>> and
>>> > together with the statical significance of the spatial autocorrelation
>>> > obtained.
>>> >
>>> > I found that the raster package function Moran() although calculates
>>> the
>>> > spatial autocorrelation index it apparently does not give directly the
>>> > statical significance of the results obtained :
>>> > https://search.r-project.org/CRAN/refmans/raster/html/autocor.html
>>> >
>>> > Could it be be possible to obtain the statistical significance of the
>>> > results with either raster package or similar one?
>>>
>>>
>>> fortunes::fortune("This is R")
>>>
>>>
>>> library(raster)
>>> r <- raster(nrows=10, ncols=10)
>>> values(r) <- 1:ncell(r)
>>> f <- matrix(c(0,1,0,1,0,1,0,1,0), nrow=3)
>>> (rI <- Moran(r, f))
>>> r1 <- r
>>> nsim <- 499
>>> res <- numeric(nsim)
>>> set.seed(1)
>>> for (i in 1:nsim) {
>>>    values(r1) <- values(r)[sample(prod(dim(r)))]
>>>    res[i] <- Moran(r1, f)
>>> }
>>>
>>> Hope-type tests date back to Cliff and Ord; they are permutation
>>> bootstraps.
>>>
>>> r_g <- as(r, "SpatialPixelsDataFrame")
>>> library(spdep)
>>> nb <- poly2nb(as(r_g, "SpatialPolygons"), queen=FALSE)
>>> set.seed(1)
>>> o <- moran.mc(r_g$layer, nb2listw(nb, style="B"), nsim=nsim,
>>>    return_boot=TRUE)
>>> x_a <- range(c(o$t, o$t0, res, rI))
>>> plot(density(o$t), xlim=x_a)
>>> abline(v=o$t0)
>>> lines(density(res), lty=2)
>>> abline(v=rI, lty=2)
>>>
>>> It is not immediately obvious from the code of raster::Moran() why it is
>>> different, possibly because of padding the edges of the raster and thus
>>> increasing the cell count.
>>>
>>> For added speed, the bootstrap can be parallelized in both cases; polygon
>>> boundaries are perhaps not ideal.
>>>
>>> Hope this clarifies. Always provide a reproducible example, never post
>>> HTML
>>> mail.
>>>
>>> Roger Bivand
>>>
>>>
>>> >
>>> > Thanks a lot.
>>> > Kind regards,
>>> > Gabriel
>>> >
>>> >       [[alternative HTML version deleted]]
>>> >
>>> > _______________________________________________
>>> > R-sig-Geo mailing list
>>> > R-sig-Geo using r-project.org
>>> > https://stat.ethz.ch/mailman/listinfo/r-sig-geo
>>> >
>>>
>>> --
>>> Roger Bivand
>>> Emeritus Professor
>>> Department of Economics, Norwegian School of Economics,
>>> Postboks 3490 Ytre Sandviken, 5045 Bergen, Norway.
>>> e-mail: Roger.Bivand using nhh.no
>>> https://orcid.org/0000-0003-2392-6140
>>> https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>>>
>>>
>>>
>>>
>>> ------------------------------
>>>
>>> Message: 4
>>> Date: Thu, 21 Oct 2021 08:36:51 +0300
>>> From: Gabriel Cotlier <gabiklm01 using gmail.com>
>>> To: Roger.Bivand using nhh.no
>>> Cc: r-sig-geo <r-sig-geo using r-project.org>
>>> Subject: Re: [R-sig-Geo]  question on raster Moran's I statistical
>>>         significance
>>> Message-ID:
>>>         <
>>> CAAKwTDHmsathQr3nZLZPwtE++t91oOD_bvdV8DpG-4uO4hjxyA using mail.gmail.com>
>>> Content-Type: text/plain; charset="utf-8"
>>>
>>> Hello
>>>
>>> Thank you very much.
>>> I have large raster layers and would like to ask, in order to reduce the
>>> processing time of the simulation, choosing a smaller nsim value could
>>> help
>>> ? and if so, what could be the minimum nsim value recommended?
>>> Thanks a lot again.
>>> Kind regards,
>>>
>>>
>>> On Wed, Oct 20, 2021 at 8:45 PM Roger Bivand <Roger.Bivand using nhh.no>
>>> wrote:
>>>
>>> > On Wed, 20 Oct 2021, Gabriel Cotlier wrote:
>>> >
>>> > > Hello,
>>> > >
>>> > > I would like to estimate the Moran's *I* coefficient for raster data
>>> and
>>> > > together with the statical significance of the spatial
>>> autocorrelation
>>> > > obtained.
>>> > >
>>> > > I found that the raster package function Moran() although calculates
>>> the
>>> > > spatial autocorrelation index it apparently does not give directly
>>> the
>>> > > statical significance of the results obtained :
>>> > > https://search.r-project.org/CRAN/refmans/raster/html/autocor.html
>>> > >
>>> > > Could it be be possible to obtain the statistical significance of the
>>> > > results with either raster package or similar one?
>>> >
>>> >
>>> > fortunes::fortune("This is R")
>>> >
>>> >
>>> > library(raster)
>>> > r <- raster(nrows=10, ncols=10)
>>> > values(r) <- 1:ncell(r)
>>> > f <- matrix(c(0,1,0,1,0,1,0,1,0), nrow=3)
>>> > (rI <- Moran(r, f))
>>> > r1 <- r
>>> > nsim <- 499
>>> > res <- numeric(nsim)
>>> > set.seed(1)
>>> > for (i in 1:nsim) {
>>> >    values(r1) <- values(r)[sample(prod(dim(r)))]
>>> >    res[i] <- Moran(r1, f)
>>> > }
>>> >
>>> > Hope-type tests date back to Cliff and Ord; they are permutation
>>> > bootstraps.
>>> >
>>> > r_g <- as(r, "SpatialPixelsDataFrame")
>>> > library(spdep)
>>> > nb <- poly2nb(as(r_g, "SpatialPolygons"), queen=FALSE)
>>> > set.seed(1)
>>> > o <- moran.mc(r_g$layer, nb2listw(nb, style="B"), nsim=nsim,
>>> >    return_boot=TRUE)
>>> > x_a <- range(c(o$t, o$t0, res, rI))
>>> > plot(density(o$t), xlim=x_a)
>>> > abline(v=o$t0)
>>> > lines(density(res), lty=2)
>>> > abline(v=rI, lty=2)
>>> >
>>> > It is not immediately obvious from the code of raster::Moran() why it
>>> is
>>> > different, possibly because of padding the edges of the raster and thus
>>> > increasing the cell count.
>>> >
>>> > For added speed, the bootstrap can be parallelized in both cases;
>>> polygon
>>> > boundaries are perhaps not ideal.
>>> >
>>> > Hope this clarifies. Always provide a reproducible example, never post
>>> > HTML
>>> > mail.
>>> >
>>> > Roger Bivand
>>> >
>>> >
>>> > >
>>> > > Thanks a lot.
>>> > > Kind regards,
>>> > > Gabriel
>>> > >
>>> > >       [[alternative HTML version deleted]]
>>> > >
>>> > > _______________________________________________
>>> > > R-sig-Geo mailing list
>>> > > R-sig-Geo using r-project.org
>>> > > https://stat.ethz.ch/mailman/listinfo/r-sig-geo
>>> > >
>>> >
>>> > --
>>> > Roger Bivand
>>> > Emeritus Professor
>>> > Department of Economics, Norwegian School of Economics,
>>> > Postboks 3490 Ytre Sandviken, 5045 Bergen, Norway.
>>> > e-mail: Roger.Bivand using nhh.no
>>> > https://orcid.org/0000-0003-2392-6140
>>> > https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>>> >
>>>
>>>
>>> --
>>> Gabriel Cotlier, PhD
>>> Haifa Research Center for Theoretical Physics and Astrophysics (HCTPA)
>>> University of Haifa
>>> Israel
>>>
>>>         [[alternative HTML version deleted]]
>>>
>>>
>>>
>>>
>>> ------------------------------
>>>
>>> Message: 5
>>> Date: Thu, 21 Oct 2021 10:40:18 +0200 (CEST)
>>> From: Roger Bivand <Roger.Bivand using nhh.no>
>>> To: Gabriel Cotlier <gabiklm01 using gmail.com>
>>> Cc: r-sig-geo <r-sig-geo using r-project.org>
>>> Subject: Re: [R-sig-Geo]  question on raster Moran's I statistical
>>>         significance
>>> Message-ID: <895fb754-b644-5543-d213-228764873fc5 using reclus2.nhh.no>
>>> Content-Type: text/plain; charset="us-ascii"; Format="flowed"
>>>
>>> On Thu, 21 Oct 2021, Gabriel Cotlier wrote:
>>>
>>> > Hello
>>> >
>>> > Thank you very much.
>>> > I have large raster layers and would like to ask, in order to reduce
>>> the
>>> > processing time of the simulation, choosing a smaller nsim value could
>>> help
>>> > ? and if so, what could be the minimum nsim value recommended?
>>>
>>> If you examine the code, you see that raster::Moran() always performs
>>> multiple raster::focal() calls, effectively constructing the spatial
>>> weights on each run. The problem is not nsim, it is the way that
>>> raster::Moran() is constructed.
>>>
>>> Indeed, using a Hope-type test gives the same inferential outcome as
>>> using
>>> asymptotic methods for global tests for spatial autocorrelation, and is
>>> superfluous, but no other approach is feasible for raster::Moran()
>>> (which
>>> by the way only seems to use 4 neighbours although claiming it uses 8).
>>>
>>> 1. How large is large?
>>>
>>> It is possible to generate nb neighbour objects for large data sets,
>>> making the use of spdep::moran.test() feasible, certainly for tens of
>>> thousands of observations (all the census tracts in coterminous US, for
>>> example).
>>>
>>> This uses distances between raster cell centres to find neighbours:
>>>
>>> > library(spdep)
>>> Loading required package: sp
>>> Loading required package: spData
>>> Loading required package: sf
>>> Linking to GEOS 3.10.0, GDAL 3.3.2, PROJ 8.1.1
>>> > crds <- expand.grid(x=1:800, y=1:1200)
>>> > dim(crds)
>>> [1] 960000      2
>>> > grd <- st_as_sf(crds, coords=c("x", "y"))
>>> > grd$z <- runif(nrow(grd))
>>> > system.time(dnbr <- dnearneigh(grd, 0, 1.01))
>>>     user  system elapsed
>>>   30.065   0.235  30.381
>>> > dnbr
>>> Neighbour list object:
>>> Number of regions: 960000
>>> Number of nonzero links: 3836000
>>> Percentage nonzero weights: 0.0004162326
>>> Average number of links: 3.995833
>>> > system.time(dnbq <- dnearneigh(grd, 0, 1.42))
>>>     user  system elapsed
>>>   54.502   0.080  54.699
>>> > dnbq
>>> Neighbour list object:
>>> Number of regions: 960000
>>> Number of nonzero links: 7668004
>>> Percentage nonzero weights: 0.0008320317
>>> Average number of links: 7.987504
>>>
>>> Once the neighbour objects are ready, conversion to spatial weights
>>> objects takes some time, and computing I with a constant mean model
>>> depends on the numbers of neighbours:
>>>
>>> > system.time(lwr <- nb2listw(dnbr, style="B"))
>>>     user  system elapsed
>>>    6.694   0.000   6.722
>>> > system.time(Ir <- moran.test(grd$z, lwr))
>>>     user  system elapsed
>>>   24.371   0.000  24.470
>>> > Ir
>>>
>>>         Moran I test under randomisation
>>>
>>> data:  grd$z
>>> weights: lwr
>>>
>>> Moran I statistic standard deviate = -0.06337, p-value = 0.5253
>>> alternative hypothesis: greater
>>> sample estimates:
>>> Moran I statistic       Expectation          Variance
>>>      -4.679899e-05     -1.041668e-06      5.213749e-07
>>>
>>> > system.time(lwq <- nb2listw(dnbq, style="B"))
>>>     user  system elapsed
>>>    6.804   0.000   6.828
>>> > system.time(Iq <- moran.test(grd$z, lwq))
>>>     user  system elapsed
>>>   46.703   0.012  46.843
>>> > Iq
>>>
>>>         Moran I test under randomisation
>>>
>>> data:  grd$z
>>> weights: lwq
>>>
>>> Moran I statistic standard deviate = -0.70373, p-value = 0.7592
>>> alternative hypothesis: greater
>>> sample estimates:
>>> Moran I statistic       Expectation          Variance
>>>      -3.604417e-04     -1.041668e-06      2.608222e-07
>>>
>>> 2. The larger your N, the less likely that the test means anything at
>>> all,
>>> because the assumption is that the observed entities are not simply
>>> arbitrary products of, say, resolution.
>>>
>>> If you think of global Moran's I as a specification test of a regression
>>> of the variable of interest on the constant (the mean model is just the
>>> constant), for raster data the resolution controls the outcome
>>> (downscaling/upscaling will shift Moran's I). If you include covariates,
>>> patterning in the residuals of a richer model may well abate.
>>>
>>> Hope this clarifies,
>>>
>>> Roger
>>>
>>> > Thanks a lot again.
>>> > Kind regards,
>>> >
>>> >
>>> > On Wed, Oct 20, 2021 at 8:45 PM Roger Bivand <Roger.Bivand using nhh.no>
>>> wrote:
>>> >
>>> >> On Wed, 20 Oct 2021, Gabriel Cotlier wrote:
>>> >>
>>> >>> Hello,
>>> >>>
>>> >>> I would like to estimate the Moran's *I* coefficient for raster data
>>> and
>>> >>> together with the statical significance of the spatial
>>> autocorrelation
>>> >>> obtained.
>>> >>>
>>> >>> I found that the raster package function Moran() although calculates
>>> the
>>> >>> spatial autocorrelation index it apparently does not give directly
>>> the
>>> >>> statical significance of the results obtained :
>>> >>> https://search.r-project.org/CRAN/refmans/raster/html/autocor.html
>>> >>>
>>> >>> Could it be be possible to obtain the statistical significance of the
>>> >>> results with either raster package or similar one?
>>> >>
>>> >>
>>> >> fortunes::fortune("This is R")
>>> >>
>>> >>
>>> >> library(raster)
>>> >> r <- raster(nrows=10, ncols=10)
>>> >> values(r) <- 1:ncell(r)
>>> >> f <- matrix(c(0,1,0,1,0,1,0,1,0), nrow=3)
>>> >> (rI <- Moran(r, f))
>>> >> r1 <- r
>>> >> nsim <- 499
>>> >> res <- numeric(nsim)
>>> >> set.seed(1)
>>> >> for (i in 1:nsim) {
>>> >>    values(r1) <- values(r)[sample(prod(dim(r)))]
>>> >>    res[i] <- Moran(r1, f)
>>> >> }
>>> >>
>>> >> Hope-type tests date back to Cliff and Ord; they are permutation
>>> >> bootstraps.
>>> >>
>>> >> r_g <- as(r, "SpatialPixelsDataFrame")
>>> >> library(spdep)
>>> >> nb <- poly2nb(as(r_g, "SpatialPolygons"), queen=FALSE)
>>> >> set.seed(1)
>>> >> o <- moran.mc(r_g$layer, nb2listw(nb, style="B"), nsim=nsim,
>>> >>    return_boot=TRUE)
>>> >> x_a <- range(c(o$t, o$t0, res, rI))
>>> >> plot(density(o$t), xlim=x_a)
>>> >> abline(v=o$t0)
>>> >> lines(density(res), lty=2)
>>> >> abline(v=rI, lty=2)
>>> >>
>>> >> It is not immediately obvious from the code of raster::Moran() why it
>>> is
>>> >> different, possibly because of padding the edges of the raster and
>>> thus
>>> >> increasing the cell count.
>>> >>
>>> >> For added speed, the bootstrap can be parallelized in both cases;
>>> polygon
>>> >> boundaries are perhaps not ideal.
>>> >>
>>> >> Hope this clarifies. Always provide a reproducible example, never post
>>> >> HTML
>>> >> mail.
>>> >>
>>> >> Roger Bivand
>>> >>
>>> >>
>>> >>>
>>> >>> Thanks a lot.
>>> >>> Kind regards,
>>> >>> Gabriel
>>> >>>
>>> >>>       [[alternative HTML version deleted]]
>>> >>>
>>> >>> _______________________________________________
>>> >>> R-sig-Geo mailing list
>>> >>> R-sig-Geo using r-project.org
>>> >>> https://stat.ethz.ch/mailman/listinfo/r-sig-geo
>>> >>>
>>> >>
>>> >> --
>>> >> Roger Bivand
>>> >> Emeritus Professor
>>> >> Department of Economics, Norwegian School of Economics,
>>> >> Postboks 3490 Ytre Sandviken, 5045 Bergen, Norway.
>>> >> e-mail: Roger.Bivand using nhh.no
>>> >> https://orcid.org/0000-0003-2392-6140
>>> >> https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>>> >>
>>> >
>>> >
>>> >
>>>
>>> --
>>> Roger Bivand
>>> Emeritus Professor
>>> Department of Economics, Norwegian School of Economics,
>>> Postboks 3490 Ytre Sandviken, 5045 Bergen, Norway.
>>> e-mail: Roger.Bivand using nhh.no
>>> https://orcid.org/0000-0003-2392-6140
>>> https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>>>
>>>
>>>
>>>
>>> ------------------------------
>>>
>>> Subject: Digest Footer
>>>
>>> _______________________________________________
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>>>
>>> ------------------------------
>>>
>>> End of R-sig-Geo Digest, Vol 218, Issue 10
>>> ******************************************
>>>
>>
>
> --
> Gabriel Cotlier, PhD
> Haifa Research Center for Theoretical Physics and Astrophysics (HCTPA)
> University of Haifa
> Israel
>
>
>

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