----------------------------------------
> From: marchy...@hotmail.com
> To: rvarad...@jhmi.edu; pda...@gmail.com; alex.ols...@gmail.com
> Date: Sat, 21 May 2011 20:40:44 -0400
> CC: r-help@r-project.org
> Subject: Re: [R] maximum likelihood convergence reproducing Anderson Blundell
> 1982 Econometrica R vs Stata
>
>
>
>
>
>
>
>
>
>
>
>
> ----------------------------------------
> > From: rvarad...@jhmi.edu
> > To: marchy...@hotmail.com; pda...@gmail.com; alex.ols...@gmail.com
> > CC: r-help@r-project.org
> > Date: Sat, 21 May 2011 17:26:29 -0400
> > Subject: RE: [R] maximum likelihood convergence reproducing Anderson
> > Blundell 1982 Econometrica R vs Stata
> >
> > Hi,
> >
> > I don't think the final verdict has been spoken. Peter's posts have hinted
> > at ill-conditioning as the crux of the problem. So, I decided to try a
> > couple of more things: (1) standardizing the covariates, (2) exact
> > gradient, and (3) both (1) and (2).
>
> Cool, I thought everyone lost interest. I'll get back to this then.
> Before launching into this, I was curious however if the STATA
> solution ( or whatever other proudct was used) was thought
> to reprsent a good solution or the actual global optimum.
>
> IIRC, your earlier post along these lines,
>
> > # exact solution as the starting value
> > start <- c(coef(lm(y1~x1+x2+x3)), coef(lm(y2~x1+x2+x3)))
>
> was what got me started.
>
>
> I guess my point was that the problem would not obviously
> have a nice surface to optimize and IIRC the title of the paper
> suggested the system was a bit difficult ( I won't pay for med papers,
> not going to pay for econ LOL). The function to be minimized, and this
> is stated as fact but only for sake of eliciting criticism,
> is the determinant of 2 unrelated residuals vectors. And, from memory,
>
> > e1 <- y1 - theta[1] - theta[2]*x1 - theta[3]*x2 - theta[4]*x3
> > e2 <- y2 - theta[5] - theta[6]*x1 - theta[7]*x2 - theta[8]*x3
>
>
>
> this gives something like E=|e1|^2*|e2|^2*(1-Cos^2(d)) with
> d being angle between residual vectors ( in space with dimension of number
> of data points). Or, E=F*Sin^2(d) and depending on data is would seem
> possible to move in such a way that F increases but just more slowly than
> Sin^2(d). Any solution for which they are colinear would seem to be optimal.
>
> I guess if my starting point is not too far off I may see if
> I can find some diagnostics to determine is the data set creates
> a condition as I have outlined and optimizer.
>
> It might, for example, be interesting to see what happens to |e1||e2|
> and Sin(d) at various points.
> ( T1 is just theta components 1-4 and T2 is 5-8, I guess presuming "x0"=1
> LOL),
> E1=Y1-T1*X , E2=Y2-T2*x
> E1.E2=Y1.Y2-Y1.(T2*X)-(T1*X).Y2+(T1*X).(T2*X)
>
> I guess I keep working on the above and see if it points
> to anything pathological or that doesn't play nice with
> optimizer.
I was too lazy to continue the above on paper but empirically this seems to
be part of symptoms. For example,
e1m=sum(e1*e1);
e2m=sum(e2*e2);
e12=sum(e1*e2);
phit=e12/sqrt(e1m*e2m);
dete=det(sigma);
xxx=paste("e1m*e2m=",e1m*e2m," aphi=",phit, " dete=",dete,sep="");
print(xxx);
on this call,
print("doing BGFS from zero");
p <- optim(0*c(1:8), lnl, method="BFGS", hessian=TRUE, y1=data$y1, y2=data$y2, x
1=data$x1, x2=data$x2, x3=data$x3)
shows that the the product of e1 and e2 explodes with the error vectors become
parallel,
( since I got an answer I like, I haven't bothered to check for typos
or coding errors LOL note in particular things like sqrt() etc )
[1] "e1m*e2m=41.1669479546532 aphi=0.960776458479452 dete=3.16609220303528"
[1] "e1m*e2m=41.0289397617336 aphi=0.960734505863182 dete=3.15878562836847"
[1] "e1m*e2m=41.3051898313859 aphi=0.960818247708638 dete=3.17340730403131"
[1] "e1m*e2m=39.7867410499191 aphi=0.960397122114274 dete=3.08893784983331"
[1] "e1m*e2m=42.5708634641545 aphi=0.961141202950479 dete=3.24422282329344"
[1] "e1m*e2m=39.952815544519 aphi=0.960383767149008 dete=3.10285630456706"
[1] "e1m*e2m=42.3994556719457 aphi=0.961155360764465 dete=3.23000632577225"
[1] "e1m*e2m=40.6077431942131 aphi=0.960502159746792 dete=3.14448500444145"
[1] "e1m*e2m=41.7300849780045 aphi=0.96104579811445 dete=3.18780183350788"
[1] "e1m*e2m=40.8461025929321 aphi=0.960565688832973 dete=3.15795749888956"
[1] "e1m*e2m=41.4890962339491 aphi=0.960985035909235 dete=3.17423784875441"
[1] "e1m*e2m=38.0026479775113 aphi=0.958424486959574 dete=3.09427071121483"
[1] "e1m*e2m=44.4634366247632 aphi=0.962889636822005 dete=3.23887444893151"
[1] "e1m*e2m=38.3679152097842 aphi=0.958864175571152 dete=3.09166714763783"
[1] "e1m*e2m=44.0684332841423 aphi=0.962518076859826 dete=3.24162775623887"
[1] "e1m*e2m=39.8519719942113 aphi=0.960141707858905 dete=3.11355091583515"
[1] "e1m*e2m=42.5038484736989 aphi=0.961384528172175 dete=3.21923251471042"
[1] "e1m*e2m=21375239704009490432 aphi=0.999977697068197 dete=953450394271031"
[1] "e1m*e2m=34177731168061536 aphi=0.999977905000934 dete=1510297191323.93"
[1] "e1m*e2m=54503403593545 aphi=0.999978923036089 dete=2297508328.68597"
[1] "e1m*e2m=85767789487.156 aphi=0.999983466344755 dete=2836086.67942723"
[1] "e1m*e2m=126124258.261637 aphi=0.999991704142303 dete=2092.60911721655"
[1] "e1m*e2m=127695.808241124 aphi=0.999539670386456 dete=117.537264947809"
[1] "e1m*e2m=1.79314748193414 aphi=0.458869014722204 dete=1.41558096262301"
[1] "e1m*e2m=1.82076399719734 aphi=0.469891415967073 dete=1.41874305229269"
[1] "e1m*e2m=1.76630916851626 aphi=0.447604317206231 dete=1.41242978935563"
[1] "e1m*e2m=2.10765446888618 aphi=0.559484707051066 dete=1.44790985442968"
[1] "e1m*e2m=1.55759367420566 aphi=0.33387034624138 dete=1.38396962928268"
[1] "e1m*e2m=2.07345917255291 aphi=0.547539248673881 dete=1.45183771159372"
[1] "e1m*e2m=1.57402829646691 aphi=0.351106401124671 dete=1.37998884867053"
[1] "e1m*e2m=1.92570387464855 aphi=0.500413807670452 dete=1.44348070520072"
[1] "e1m*e2m=1.67368610942138 aphi=0.413157386356889 dete=1.38798952087868"
[1] "e1m*e2m=1.80091875441907 aphi=0.458998972457491 dete=1.42150108909529"
[1] "e1m*e2m=1.78539325145298 aphi=0.458738462509367 dete=1.40967335131397"
[1] "e1m*e2m=1.87239816244392 aphi=0.46015903715983 dete=1.4759247054976"
[1] "e1m*e2m=1.71562581348667 aphi=0.457519283487653 dete=1.3565043362516"
[1] "e1m*e2m=1.86276444029005 aphi=0.460322529304816 dete=1.46805055851996"
[1] "e1m*e2m=1.72487059095196 aphi=0.457356617377108 dete=1.36407065493321"
[1] "e1m*e2m=1.82498046816382 aphi=0.460003447183359 dete=1.43880881331975"
[1] "e1m*e2m=1.76160126627035 aphi=0.457713483547811 dete=1.39254292425402"
[1] "e1m*e2m=1351553015129414 aphi=-0.999994920076885 dete=13731535927.7031"
[1] "e1m*e2m=2152851612889.67 aphi=-0.999994409734947 dete=24069954.9936224"
[1] "e1m*e2m=3367962636.07941 aphi=-0.999991168164692 dete=59490.3199496781"
[1] "e1m*e2m=4794712.10051354 aphi=-0.999956405381663 dete=418.038175816592"
[1] "e1m*e2m=3699.85578987196 aphi=-0.998908076827754 dete=8.07550521781124"
[1] "e1m*e2m=12.6686499822451 aphi=0.956985935569274 dete=1.06642059359798"
[1] "e1m*e2m=12.7330364607788 aphi=0.957056380194584 dete=1.07012366722186"
[1] "e1m*e2m=12.6044297074752 aphi=0.95691501854723 dete=1.0627254405059"
but what is interesting is that each time it emerges, it seems to get a better
plateau.
If you actually look at something like Poincare map, it is interesting that
aphi heads to +/-1 as the error magnitude product gets large which is about what
I motivated. There seem to be a cluster of points around aphi=-.5 with minimal
product. Using scatterplot3d on the above with log for magnitudes there seem
to be "paths" but interpreation is not immediately clear. I admit at this point
an interactive viewer would be helpful to rotate the thing, and in fact
rgl.surface
may be useful ...
>
>
>
>
> >
> > I compute the "exact" gradient using a complex-step derivative approach.
> > This works just like the standard first-order, forward differencing. The
> > only (but, essential) difference is that an imaginary increment, i*dx, is
> > used. This, incredibly, gives exact gradients (up to machine precision).
> >
> > Here are the code and the results of my experiments:
> >
> > data <- read.table("h:/computations/optimx_example.dat", header=TRUE,
> > sep=",")
> > attach(data)
> > require(optimx)
> >
> > ## model 18
> > lnl <- function(theta,y1, y2, x1, x2, x3) {
> > n <- length(y1)
> > beta <- theta[1:8]
> > e1 <- y1 - theta[1] - theta[2]*x1 - theta[3]*x2 - theta[4]*x3
> > e2 <- y2 - theta[5] - theta[6]*x1 - theta[7]*x2 - theta[8]*x3
> > e <- cbind(e1, e2)
> > sigma <- t(e)%*%e
> > det.sigma <- sigma[1,1] * sigma[2,2] - sigma[1,2] * sigma[2,1]
> > logl <- -1*n/2*(2*(1+log(2*pi)) + log(det.sigma))
> > return(-logl)
> > }
> >
> > csd <- function(fn, x, ...) {
> > # Complex step derivatives; yields exact derivatives
> > h <- .Machine$double.eps
> > n <- length(x)
> > h0 <- g <- rep(0, n)
> > for (i in 1:n) {
> > h0[i] <- h * 1i
> > g[i] <- Im(fn(x+h0, ...))/h
> > h0[i] <- 0
> > }
> > g
> > }
> >
> > gr.csd <- function(theta,y1, y2, x1, x2, x3) {
> > csd(lnl, theta, y1=y1, y2=y2, x1=x1, x2=x2, x3=x3)
> > }
> >
> > # exact solution as the starting value
> > start <- c(coef(lm(y1~x1+x2+x3)), coef(lm(y2~x1+x2+x3)))
> > p1 <- optimx(start, lnl, y1=y1, y2=y2, x1=x1, x2=x2, x3=x3,
> > control=list(all.methods=TRUE, maxit=1500))
> >
> > # numerical gradient
> > p2 <- optimx(rep(0,8), lnl, y1=y1, y2=y2, x1=x1, x2=x2, x3=x3,
> > control=list(all.methods=TRUE, maxit=1500))
> >
> > # exact gradient
> > p2g <- optimx(rep(0,8), lnl, gr.csd, y1=y1, y2=y2, x1=x1, x2=x2, x3=x3,
> > control=list(all.methods=TRUE, maxit=1500))
> >
> > # comparing p2 and p2g, we see the dramatic improvement in BFGS when exact
> > gradient is used, we also see a major difference for L-BFGS-B
> > # Exact gradient did not affect the gradient methods, CG and spg, much.
> > However, convergence of Rcgmin improved when exact gradient was used
> >
> > x1s <- scale(x1)
> > x2s <- scale(x2)
> > x3s <- scale(x3)
> >
> > p3 <- optimx(rep(0,8),lnl, y1=y1, y2=y2, x1=x1s, x2=x2s, x3=x3s,
> > control=list(all.methods=TRUE, maxit=1500))
> >
> > # both scaling and exact gradient
> > p3g <- optimx(rep(0,8),lnl, gr.csd, y1=y1, y2=y2, x1=x1s, x2=x2s, x3=x3s,
> > control=list(all.methods=TRUE, maxit=1500))
> >
> > # Comparing p3 and p3g, use of exact gradient improved spg
> > dramatically[[elided Hotmail spam]]
> >
> > # Of course, derivative-free methods newuoa, and Nelder-Mead are improved
> > by scaling, but not by the availability of exact gradients. I don't know
> > what is wrong with bobyqa in this example.
> >
> > In short, even with scaling and exact gradients, this optimization problem
> > is recalcitrant.
> >
> > Best,
> > Ravi.
> >
> > ________________________________________
> > From: Mike Marchywka [marchy...@hotmail.com]
> > Sent: Thursday, May 12, 2011 8:30 AM
> > To: Ravi Varadhan; pda...@gmail.com; alex.ols...@gmail.com
> > Cc: r-help@r-project.org
> > Subject: RE: [R] maximum likelihood convergence reproducing Anderson
> > Blundell 1982 Econometrica R vs Stata
> >
> > So what was the final verdict on this discussion? I kind of
> > lost track if anyone has a minute to summarize and critique my summary
> > below.
> >
> >
> > Apparently there were two issues, the comparison between R and Stata
> > was one issue and the "optimum" solution another. As I understand it,
> > there was some question about R numerical gradient calculation. This would
> > suggest some features of the function may be of interest to consider.
> >
> > The function to be optimized appears to be, as OP stated,
> > some function of residuals of two ( unrelated ) fits.
> > The residual vectors e1 and e2 are dotted in various combinations
> > creating a matrix whose determinant is (e1.e1)(e2.e2)-(e1.e2)^2 which
> > is the result to be minimized by choice of theta. Theta it seems is
> > an 8 component vector, 4 components determine e1 and the other 4 e2.
> > Presumably a unique solution would require that e1 and e2, both n-component
> > vectors,
> > point in different directions or else both could become aribtarily large
> > while keeping the error signal at zero. For fixed magnitudes, colinearity
> > would reduce the "Error." The intent would appear to be to
> > keep the residuals distributed similarly in the two ( unrelated) fits.
> > I guess my question is,
> > " did anyone determine that there is a unique solution?" or
> > am I totally wrong here ( I haven't used these myself to any
> > extent and just try to run some simple teaching examples, asking
> > for my own clarification as much as anything).
> >
> > Thanks.
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> > ----------------------------------------
> > > From: rvarad...@jhmi.edu
> > > To: pda...@gmail.com; alex.ols...@gmail.com
> > > Date: Sat, 7 May 2011 11:51:56 -0400
> > > CC: r-help@r-project.org
> > > Subject: Re: [R] maximum likelihood convergence reproducing Anderson
> > > Blundell 1982 Econometrica R vs Stata
> > >
> > > There is something strange in this problem. I think the log-likelihood is
> > > incorrect. See the results below from "optimx". You can get much larger
> > > log-likelihood values than for the exact solution that Peter provided.
> > >
> > > ## model 18
> > > lnl <- function(theta,y1, y2, x1, x2, x3) {
> > > n <- length(y1)
> > > beta <- theta[1:8]
> > > e1 <- y1 - theta[1] - theta[2]*x1 - theta[3]*x2 - theta[4]*x3
> > > e2 <- y2 - theta[5] - theta[6]*x1 - theta[7]*x2 - theta[8]*x3
> > > e <- cbind(e1, e2)
> > > sigma <- t(e)%*%e
> > > logl <- -1*n/2*(2*(1+log(2*pi)) + log(det(sigma))) # it looks like there
> > > is something wrong here
> > > return(-logl)
> > > }
> > >
> > > data <- read.table("e:/computing/optimx_example.dat", header=TRUE,
> > > sep=",")
> > >
> > > attach(data)
> > >
> > > require(optimx)
> > >
> > > start <- c(coef(lm(y1~x1+x2+x3)), coef(lm(y2~x1+x2+x3)))
> > >
> > > # the warnings can be safely ignored in the "optimx" calls
> > > p1 <- optimx(start, lnl, hessian=TRUE, y1=y1, y2=y2,
> > > + x1=x1, x2=x2, x3=x3, control=list(all.methods=TRUE, maxit=1500))
> > >
> > > p2 <- optimx(rep(0,8), lnl, hessian=TRUE, y1=y1, y2=y2,
> > > + x1=x1, x2=x2, x3=x3, control=list(all.methods=TRUE, maxit=1500))
> > >
> > > p3 <- optimx(rep(0.5,8), lnl, hessian=TRUE, y1=y1, y2=y2,
> > > + x1=x1, x2=x2, x3=x3, control=list(all.methods=TRUE, maxit=1500))
> > >
> > > Ravi.
> > > ________________________________________
> > > From: r-help-boun...@r-project.org [r-help-boun...@r-project.org] On
> > > Behalf Of peter dalgaard [pda...@gmail.com]
> > > Sent: Saturday, May 07, 2011 4:46 AM
> > > To: Alex Olssen
> > > Cc: r-help@r-project.org
> > > Subject: Re: [R] maximum likelihood convergence reproducing Anderson
> > > Blundell 1982 Econometrica R vs Stata
> > >
> > > On May 6, 2011, at 14:29 , Alex Olssen wrote:
> > >
> > > > Dear R-help,
> > > >
> > > > I am trying to reproduce some results presented in a paper by Anderson
> > > > and Blundell in 1982 in Econometrica using R.
> > > > The estimation I want to reproduce concerns maximum likelihood
> > > > estimation of a singular equation system.
> > > > I can estimate the static model successfully in Stata but for the
> > > > dynamic models I have difficulty getting convergence.
> > > > My R program which uses the same likelihood function as in Stata has
> > > > convergence properties even for the static case.
> > > >
> > > > I have copied my R program and the data below. I realise the code
> > > > could be made more elegant - but it is short enough.
> > > >
> > > > Any ideas would be highly appreciated.
> > >
> > > Better starting values would help. In this case, almost too good values
> > > are available:
> > >
> > > start <- c(coef(lm(y1~x1+x2+x3)), coef(lm(y2~x1+x2+x3)))
> > >
> > > which appears to be the _exact_ solution.
> > >
> > > Apart from that, it seems that the conjugate gradient methods have
> > > difficulties with this likelihood, for some less than obvious reason.
> > > Increasing the maxit gets you closer but still not satisfactory.
> > >
> > > I would suggest trying out the experimental optimx package. Apparently,
> > > some of the algorithms in there are much better at handling this
> > > likelihood, notably "nlm" and "nlminb".
> > >
> > >
> > >
> > >
> > > >
> > > > ## model 18
> > > > lnl <- function(theta,y1, y2, x1, x2, x3) {
> > > > n <- length(y1)
> > > > beta <- theta[1:8]
> > > > e1 <- y1 - theta[1] - theta[2]*x1 - theta[3]*x2 - theta[4]*x3
> > > > e2 <- y2 - theta[5] - theta[6]*x1 - theta[7]*x2 - theta[8]*x3
> > > > e <- cbind(e1, e2)
> > > > sigma <- t(e)%*%e
> > > > logl <- -1*n/2*(2*(1+log(2*pi)) + log(det(sigma)))
> > > > return(-logl)
> > > > }
> > > > p <- optim(0*c(1:8), lnl, method="BFGS", hessian=TRUE, y1=y1, y2=y2,
> > > > x1=x1, x2=x2, x3=x3)
> > > >
> > > > "year","y1","y2","x1","x2","x3"
> > > > 1929,0.554779,0.266051,9.87415,8.60371,3.75673
> > > > 1930,0.516336,0.297473,9.68621,8.50492,3.80692
> > > > 1931,0.508201,0.324199,9.4701,8.27596,3.80437
> > > > 1932,0.500482,0.33958,9.24692,7.99221,3.76251
> > > > 1933,0.501695,0.276974,9.35356,7.98968,3.69071
> > > > 1934,0.591426,0.287008,9.42084,8.0362,3.63564
> > > > 1935,0.565047,0.244096,9.53972,8.15803,3.59285
> > > > 1936,0.605954,0.239187,9.6914,8.32009,3.56678
> > > > 1937,0.620161,0.218232,9.76817,8.42001,3.57381
> > > > 1938,0.592091,0.243161,9.51295,8.19771,3.6024
> > > > 1939,0.613115,0.217042,9.68047,8.30987,3.58147
> > > > 1940,0.632455,0.215269,9.78417,8.49624,3.57744
> > > > 1941,0.663139,0.184409,10.0606,8.69868,3.6095
> > > > 1942,0.698179,0.164348,10.2892,8.84523,3.66664
> > > > 1943,0.70459,0.146865,10.4731,8.93024,3.65388
> > > > 1944,0.694067,0.161722,10.4465,8.96044,3.62434
> > > > 1945,0.674668,0.197231,10.279,8.82522,3.61489
> > > > 1946,0.635916,0.204232,10.1536,8.77547,3.67562
> > > > 1947,0.642855,0.187224,10.2053,8.77481,3.82632
> > > > 1948,0.641063,0.186566,10.2227,8.83821,3.96038
> > > > 1949,0.646317,0.203646,10.1127,8.82364,4.0447
> > > > 1950,0.645476,0.187497,10.2067,8.84161,4.08128
> > > > 1951,0.63803,0.197361,10.2773,8.9401,4.10951
> > > > 1952,0.634626,0.209992,10.283,9.01603,4.1693
> > > > 1953,0.631144,0.219287,10.3217,9.06317,4.21727
> > > > 1954,0.593088,0.235335,10.2101,9.05664,4.2567
> > > > 1955,0.60736,0.227035,10.272,9.07566,4.29193
> > > > 1956,0.607204,0.246631,10.2743,9.12407,4.32252
> > > > 1957,0.586994,0.256784,10.2396,9.1588,4.37792
> > > > 1958,0.548281,0.271022,10.1248,9.14025,4.42641
> > > > 1959,0.553401,0.261815,10.2012,9.1598,4.4346
> > > > 1960,0.552105,0.275137,10.1846,9.19297,4.43173
> > > > 1961,0.544133,0.280783,10.1479,9.19533,4.44407
> > > > 1962,0.55382,0.281286,10.197,9.21544,4.45074
> > > > 1963,0.549951,0.28303,10.2036,9.22841,4.46403
> > > > 1964,0.547204,0.291287,10.2271,9.23954,4.48447
> > > > 1965,0.55511,0.281313,10.2882,9.26531,4.52057
> > > > 1966,0.558182,0.280151,10.353,9.31675,4.58156
> > > > 1967,0.545735,0.294385,10.3351,9.35382,4.65983
> > > > 1968,0.538964,0.294593,10.3525,9.38361,4.71804
> > > > 1969,0.542764,0.299927,10.3676,9.40725,4.76329
> > > > 1970,0.534595,0.315319,10.2968,9.39139,4.81136
> > > > 1971,0.545591,0.315828,10.2592,9.34121,4.84082
> > > >
> > > > ______________________________________________
> > > > R-help@r-project.org mailing list
> > > > https://stat.ethz.ch/mailman/listinfo/r-help
> > > > PLEASE do read the posting guide
> > > > http://www.R-project.org/posting-guide.html
> > > > and provide commented, minimal, self-contained, reproducible code.
> > >
> > > --
> > > Peter Dalgaard
> > > Center for Statistics, Copenhagen Business School
> > > Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> > > Phone: (+45)38153501
> > > Email: pd....@cbs.dk Priv: pda...@gmail.com
> > >
> > > ______________________________________________
> > > R-help@r-project.org mailing list
> > > https://stat.ethz.ch/mailman/listinfo/r-help
> > > PLEASE do read the posting guide
> > > http://www.R-project.org/posting-guide.html
> > > and provide commented, minimal, self-contained, reproducible code.
> > >
> > > ______________________________________________
> > > R-help@r-project.org mailing list
> > > https://stat.ethz.ch/mailman/listinfo/r-help
> > > PLEASE do read the posting guide
> > > http://www.R-project.org/posting-guide.html
> > > and provide commented, minimal, self-contained, reproducible code.
>
> ______________________________________________
> R-help@r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
______________________________________________
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
