[R-sig-Geo] R-sig-Geo Digest, Vol 218, Issue 10
Gabriel Cotlier
g@b|k|m01 @end|ng |rom gm@||@com
Thu Oct 21 13:43:33 CEST 2021
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
>>
>> _______________________________________________
>> R-sig-Geo mailing list
>> R-sig-Geo using r-project.org
>> https://stat.ethz.ch/mailman/listinfo/r-sig-geo
>>
>>
>> ------------------------------
>>
>> 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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