gstat now uses Lapack / failing cokriging
Hi Gerard, Just a small comment on the use of pseudo- vs. normal- cross-semivariograms. Normal cross-semivariograms may yield nonoptimal weights for cokriging, as shown in the example of Ver Hoef and Cressie (1993), and I believe it's a nomenclature problem that the 'pseudo' one is that which should be used. A quick look at the gstat documentation I see that the pseudo cross-semivariogram is what is used but maybe Edzer can confirm. Best, Andrew ----------------------------------------------------------------------------- Andrew Zammit Mangion, School of Mathematics and Applied Statistics, University of Wollongong NSW 2522 ------------------------------------------------------------------------------
From: R-sig-Geo <r-sig-geo-bounces at r-project.org> on behalf of Heuvelink, Gerard <gerard.heuvelink at wur.nl>
Sent: 12 December 2015 01:29
To: 'r-sig-geo at r-project.org'
Cc: Bruin, Sytze de
Subject: [R-sig-Geo] FW: gstat now uses Lapack / failing cokriging
Sent: 12 December 2015 01:29
To: 'r-sig-geo at r-project.org'
Cc: Bruin, Sytze de
Subject: [R-sig-Geo] FW: gstat now uses Lapack / failing cokriging
Hi Edzer,
I checked the Ver Hoef and Cressie (1993) paper but could not find a reference to deriving the cross-covariance from the cross-variogram as "C(h)=C(0)-gamma(h) with C(0)=max(0, sum of the positive sill values)". I did notice that their paper is mostly about the pseudo cross-variogram (defined in Eq. 9), while we and presumably gstat as well work with the 'normal' cross-variogram (Eq. 8).
I think the proper relation is Cij(h)=Cij(0)-Vij(h), where Cij(h)=E[(Zi(s)-E[Zi(s)])*(Zj(s+h)-E[Zj(s+h)])] is the cross-covariance and Vij(h)=0.5*E[(Zi(s)-Zi(s+h))*(Zj(s)-Zj(s+h))] is the normal cross-variogram. This relation is valid if Z is second-order stationary (conditions 1 and 2 on page 220 of VH&C) and satisfies the symmetric cross-covariance condition (i.e. Cij(h)=Cij(-h), condition 3 on the same page). I would recommend that you implement this relation instead of "C(h)=C(0)-gamma(h) with C(0)=max(0, sum of the positive sill values)".
What I still do not understand is that the 'old' version of gstat did not complain about our Allier example, because using the alternative derivation of the cross-covariance from the cross-variogram produces a kriging system that is not positive-definite.
Gerard
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Today's Topics:
1. Re: gstat now uses Lapack / failing cokriging (Bruin, Sytze de)
2. Re: gstat now uses Lapack / failing cokriging (Edzer Pebesma)
3. issues with NAs when using resample function (Beth Forrestel)
4. Re: issues with NAs when using resample function (Stephen Stewart)
5. grass ascii support in the raster package (Cornel Pop)
----------------------------------------------------------------------
Message: 1
Date: Tue, 17 Nov 2015 17:14:22 +0000
From: "Bruin, Sytze de" <sytze.debruin at wur.nl>
To: "'r-sig-geo at r-project.org'" <r-sig-geo at r-project.org>
Subject: Re: [R-sig-Geo] gstat now uses Lapack / failing cokriging
Message-ID: <4d812a6858d942f589a5502e117b1234 at scomp5295.wurnet.nl>
Content-Type: text/plain; charset="us-ascii"
Hi Edzer, thanks for your prompt reply. I tried reproducing your matrix r but got a different result, i.e. a valid covariance matrix. The cross-covariances are different. Using the same example as above, my code is as follows:
library(sp)
library(gstat)
# some data
x <- c(215, 330, 410, 470, 545)
y <- c(230, 310, 330, 340, 365)
fc <- c(0.211, 0.251, 0.281, 0.262, 0.242) por <- c(0.438, 0.457, 0.419, 0.430, 0.468) Allier <- data.frame(x, y, fc, por)
coordinates(Allier) = ~x+y
# gstat object for co-kriging using linear model of co-regionalization g <- gstat(id=c("fc"), formula=fc~1, data=Allier,
model=vgm(0.00247, "Sph", 480, 0.00166)) g <- gstat(g, id="por", formula=por~1, data=Allier,
model=vgm(0.00239, "Sph", 480, 0.00118)) g <- gstat(g, id=c("fc", "por"),
model=vgm(0.00151, "Sph", 480, -0.00124))
dists <- spDists(Allier)
r11 <- variogramLine(g$model$fc, dist_vector = dists, covariance = T)
r12 <- variogramLine(g$model$fc.por, dist_vector = dists, covariance = T)
r22 <- variogramLine(g$model$por, dist_vector = dists, covariance = T) r <- cbind(r11, r12) r <- rbind(r, cbind(r12, r22))
> r
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 0.0041300000 0.0014193874 0.0008959951 0.0005655821 0.0002240733 0.0002700000 0.0008677227 [2,] 0.0014193874 0.0041300000 0.0018397575 0.0013976206 0.0008790828 0.0008677227 0.0002700000 [3,] 0.0008959951 0.0018397575 0.0041300000 0.0020030001 0.0014238096 0.0005477541 0.0011247100 [4,] 0.0005655821 0.0013976206 0.0020030001 0.0041300000 0.0018652970 0.0003457607 0.0008544158 [5,] 0.0002240733 0.0008790828 0.0014238096 0.0018652970 0.0041300000 0.0001369841 0.0005374150 [6,] 0.0002700000 0.0008677227 0.0005477541 0.0003457607 0.0001369841 0.0035700000 0.0013734154 [7,] 0.0008677227 0.0002700000 0.0011247100 0.0008544158 0.0005374150 0.0013734154 0.0035700000 [8,] 0.0005477541 0.0011247100 0.0002700000 0.0012245061 0.0008704261 0.0008669750 0.0017801702 [9,] 0.0003457607 0.0008544158 0.0012245061 0.0002700000 0.0011403233 0.0005472637 0.0013523535 [10,] 0.0001369841 0.0005374150 0.0008704261 0.0011403233 0.0002700000 0.0002168159 0.0008506105
[,8] [,9] [,10]
[1,] 0.0005477541 0.0003457607 0.0001369841 [2,] 0.0011247100 0.0008544158 0.0005374150 [3,] 0.0002700000 0.0012245061 0.0008704261 [4,] 0.0012245061 0.0002700000 0.0011403233 [5,] 0.0008704261 0.0011403233 0.0002700000 [6,] 0.0008669750 0.0005472637 0.0002168159 [7,] 0.0017801702 0.0013523535 0.0008506105 [8,] 0.0035700000 0.0019381256 0.0013776943 [9,] 0.0019381256 0.0035700000 0.0018048825 [10,] 0.0013776943 0.0018048825 0.0035700000
> eigen(r)$values
[1] 0.0124609506 0.0055040132 0.0046425761 0.0035980843 0.0031064016 0.0028989486 0.0028042439 [8] 0.0018107335 0.0010254131 0.0006486352
------------------------------
Message: 2
Date: Tue, 17 Nov 2015 20:38:33 +0100
From: Edzer Pebesma <edzer.pebesma at uni-muenster.de>
To: r-sig-geo at r-project.org
Cc: "Heuvelink, Gerard" <Gerard.Heuvelink at wur.nl>
Subject: Re: [R-sig-Geo] gstat now uses Lapack / failing cokriging
Message-ID: <564B8239.6080401 at uni-muenster.de>
Content-Type: text/plain; charset="windows-1252"
On 17/11/15 18:14, Bruin, Sytze de wrote:
> Hi Edzer, thanks for your prompt reply. I tried reproducing your matrix r but got a different result, i.e. a valid covariance matrix. The cross-covariances are different. Using the same example as above, my code is as follows:
>
> library(sp)
> library(gstat)
>
> # some data
> x <- c(215, 330, 410, 470, 545)
> y <- c(230, 310, 330, 340, 365)
> fc <- c(0.211, 0.251, 0.281, 0.262, 0.242) por <- c(0.438, 0.457,
> 0.419, 0.430, 0.468) Allier <- data.frame(x, y, fc, por)
> coordinates(Allier) = ~x+y
>
> # gstat object for co-kriging using linear model of co-regionalization
> g <- gstat(id=c("fc"), formula=fc~1, data=Allier,
> model=vgm(0.00247, "Sph", 480, 0.00166)) g <- gstat(g,
> id="por", formula=por~1, data=Allier,
> model=vgm(0.00239, "Sph", 480, 0.00118)) g <- gstat(g,
> id=c("fc", "por"),
> model=vgm(0.00151, "Sph", 480, -0.00124))
>
>
> dists <- spDists(Allier)
> r11 <- variogramLine(g$model$fc, dist_vector = dists, covariance = T)
> r12 <- variogramLine(g$model$fc.por, dist_vector = dists, covariance =
> T)
> r22 <- variogramLine(g$model$por, dist_vector = dists, covariance = T)
> r <- cbind(r11, r12) r <- rbind(r, cbind(r12, r22))
>
>> r
> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
> [1,] 0.0041300000 0.0014193874 0.0008959951 0.0005655821 0.0002240733
> 0.0002700000 0.0008677227 [2,] 0.0014193874 0.0041300000 0.0018397575
> 0.0013976206 0.0008790828 0.0008677227 0.0002700000 [3,] 0.0008959951
> 0.0018397575 0.0041300000 0.0020030001 0.0014238096 0.0005477541
> 0.0011247100 [4,] 0.0005655821 0.0013976206 0.0020030001 0.0041300000
> 0.0018652970 0.0003457607 0.0008544158 [5,] 0.0002240733 0.0008790828
> 0.0014238096 0.0018652970 0.0041300000 0.0001369841 0.0005374150 [6,]
> 0.0002700000 0.0008677227 0.0005477541 0.0003457607 0.0001369841
> 0.0035700000 0.0013734154 [7,] 0.0008677227 0.0002700000 0.0011247100
> 0.0008544158 0.0005374150 0.0013734154 0.0035700000 [8,] 0.0005477541 0.0011247100 0.0002700000 0.0012245061 0.0008704261 0.0008669750 0.0017801702 [9,] 0.0003457607 0.0008544158 0.0012245061 0.0002700000 0.0011403233 0.0005472637 0.0013523535 [10,] 0.0001369841 0.0005374150 0.0008704261 0.0011403233 0.0002700000 0.0002168159 0.0008506105
> [,8] [,9] [,10]
> [1,] 0.0005477541 0.0003457607 0.0001369841 [2,] 0.0011247100
> 0.0008544158 0.0005374150 [3,] 0.0002700000 0.0012245061 0.0008704261
> [4,] 0.0012245061 0.0002700000 0.0011403233 [5,] 0.0008704261
> 0.0011403233 0.0002700000 [6,] 0.0008669750 0.0005472637 0.0002168159
> [7,] 0.0017801702 0.0013523535 0.0008506105 [8,] 0.0035700000
> 0.0019381256 0.0013776943 [9,] 0.0019381256 0.0035700000 0.0018048825
> [10,] 0.0013776943 0.0018048825 0.0035700000
>
>> eigen(r)$values
> [1] 0.0124609506 0.0055040132 0.0046425761 0.0035980843 0.0031064016
> 0.0028989486 0.0028042439 [8] 0.0018107335 0.0010254131 0.0006486352
>
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>
Building upon your earlier example,
predict(g, SpatialPoints(data.frame(x=450, y=350)), debug = 32)
gives you the generalized covariance matrix that is used for the cokriging, which I looked at. gstat computes generalized covariances as
C(0)-gamma(h) with C(0) = max(0, sum of the positive sill values), instead of the sill of all sill values. If one of the sill components is negative, this matters.
I looked in the Ver Hoef & Cressie 1993 paper, but couldn't find out which one is right. Maybe Gerard can also take a look at it; the fix would be trivial.
--
Edzer Pebesma
Institute for Geoinformatics (ifgi), University of M?nster, Heisenbergstra?e 2, 48149 M?nster, Germany; +49 251 83 33081
Journal of Statistical Software: http://www.jstatsoft.org/
Computers & Geosciences: http://elsevier.com/locate/cageo/
Spatial Statistics Society http://www.spatialstatistics.info
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