Which is why I said it applies when the system is "diagonalizable". It won't
work for non-diagonalizable matrix A, because T (eigenvector matrix) is
singular.
Ravi.
________________________________________
From: peter dalgaard [pda...@gmail.com]
Sent: Thursday, August 18, 2011 6:37 PM
To: Ravi Varadhan
Cc: 'cbe...@tajo.ucsd.edu'; r-de...@stat.math.ethz.ch; 'nas...@uottawa.ca'
Subject: Re: [Rd] An example of very slow computation
On Aug 17, 2011, at 23:24 , Ravi Varadhan wrote:
> A principled way to solve this system of ODEs is to use the idea of
> "fundamental matrix", which is the same idea as that of diagonalization of a
> matrix (see Boyce and DiPrima or any ODE text).
>
> Here is the code for that:
>
>
> nlogL2 <- function(theta){
> k <- exp(theta[1:3])
> sigma <- exp(theta[4])
> A <- rbind(
> c(-k[1], k[2]),
> c( k[1], -(k[2]+k[3]))
> )
> eA <- eigen(A)
> T <- eA$vectors
> r <- eA$values
> x0 <- c(0,100)
> Tx0 <- T %*% x0
>
> sol <- function(t) 100 - sum(T %*% diag(exp(r*t)) %*% Tx0)
> pred <- sapply(dat[,1], sol)
> -sum(dnorm(dat[,2], mean=pred, sd=sigma, log=TRUE))
> }
> This is much faster than using expm(A*t), but much slower than "by hand"
> calculations since we have to repeatedly do the diagonalization. An obvious
> advantage of this fuunction is that it applies to *any* linear system of ODEs
> for which the eigenvalues are real (and negative).
I believe this is method 14 of the "19 dubious ways..." (Google for it) and
doesn't work for certain non-symmetric A matrices.
>
> Ravi.
>
> -------------------------------------------------------
> Ravi Varadhan, Ph.D.
> Assistant Professor,
> Division of Geriatric Medicine and Gerontology School of Medicine Johns
> Hopkins University
>
> Ph. (410) 502-2619
> email: rvarad...@jhmi.edu
>
>
> -----Original Message-----
> From: r-devel-boun...@r-project.org [mailto:r-devel-boun...@r-project.org] On
> Behalf Of Ravi Varadhan
> Sent: Wednesday, August 17, 2011 2:33 PM
> To: 'cbe...@tajo.ucsd.edu'; r-de...@stat.math.ethz.ch; 'nas...@uottawa.ca'
> Subject: Re: [Rd] An example of very slow computation
>
> Yes, the culprit is the evaluation of expm(A*t). This is a lazy way of
> solving the system of ODEs, where you save analytic effort, but you pay for
> it dearly in terms of computational effort!
>
> Even in this lazy approach, I am sure there ought to be faster ways to
> evaluating exponent of a matrix than that in "Matrix" package.
>
> Ravi.
>
> -------------------------------------------------------
> Ravi Varadhan, Ph.D.
> Assistant Professor,
> Division of Geriatric Medicine and Gerontology School of Medicine Johns
> Hopkins University
>
> Ph. (410) 502-2619
> email: rvarad...@jhmi.edu
>
> -----Original Message-----
> From: r-devel-boun...@r-project.org [mailto:r-devel-boun...@r-project.org] On
> Behalf Of cbe...@tajo.ucsd.edu
> Sent: Wednesday, August 17, 2011 1:08 PM
> To: r-de...@stat.math.ethz.ch
> Subject: Re: [Rd] An example of very slow computation
>
> John C Nash <nas...@uottawa.ca> writes:
>
>> This message is about a curious difference in timing between two ways of
>> computing the
>> same function. One uses expm, so is expected to be a bit slower, but "a bit"
>> turned out to
>> be a factor of >1000. The code is below. We would be grateful if anyone can
>> point out any
>> egregious bad practice in our code, or enlighten us on why one approach is
>> so much slower
>> than the other.
>
> Looks like A*t in expm(A*t) is a "matrix".
>
> 'getMethod("expm","matrix")' coerces it arg to a "Matrix", then calls
> expm(), whose method coerces its arg to a "dMatrix" and calls expm(),
> whose method coerces its arg to a 'dgeMatrix' and calls expm(), whose
> method finally calls '.Call(dgeMatrix_exp, x)'
>
> Whew!
>
> The time difference between 'expm( diag(10)+1 )' and 'expm( as( diag(10)+1,
> "dgeMatrix" ))' is a factor of 10 on my box.
>
> Dunno 'bout the other factor of 100.
>
> Chuck
>
>
>
>
>> The problem arose in an activity to develop guidelines for nonlinear
>> modeling in ecology (at NCEAS, Santa Barbara, with worldwide participants),
>> and we will be
>> trying to include suggestions of how to prepare problems like this for
>> efficient and
>> effective solution. The code for nlogL was the "original" from the worker
>> who supplied the
>> problem.
>>
>> Best,
>>
>> John Nash
>>
>> --------------------------------------------------------------------------------------
>>
>> cat("mineral-timing.R == benchmark MIN functions in R\n")
>> # J C Nash July 31, 2011
>>
>> require("microbenchmark")
>> require("numDeriv")
>> library(Matrix)
>> library(optimx)
>> # dat<-read.table('min.dat', skip=3, header=FALSE)
>> # t<-dat[,1]
>> t <- c(0.77, 1.69, 2.69, 3.67, 4.69, 5.71, 7.94, 9.67, 11.77, 17.77,
>> 23.77, 32.77, 40.73, 47.75, 54.90, 62.81, 72.88, 98.77, 125.92, 160.19,
>> 191.15, 223.78, 287.70, 340.01, 340.95, 342.01)
>>
>> # y<-dat[,2] # ?? tidy up
>> y<- c(1.396, 3.784, 5.948, 7.717, 9.077, 10.100, 11.263, 11.856, 12.251,
>> 12.699,
>> 12.869, 13.048, 13.222, 13.347, 13.507, 13.628, 13.804, 14.087, 14.185,
>> 14.351,
>> 14.458, 14.756, 15.262, 15.703, 15.703, 15.703)
>>
>>
>> ones<-rep(1,length(t))
>> theta<-c(-2,-2,-2,-2)
>>
>>
>> nlogL<-function(theta){
>> k<-exp(theta[1:3])
>> sigma<-exp(theta[4])
>> A<-rbind(
>> c(-k[1], k[2]),
>> c( k[1], -(k[2]+k[3]))
>> )
>> x0<-c(0,100)
>> sol<-function(t)100-sum(expm(A*t)%*%x0)
>> pred<-sapply(dat[,1],sol)
>> -sum(dnorm(dat[,2],mean=pred,sd=sigma, log=TRUE))
>> }
>>
>> getpred<-function(theta, t){
>> k<-exp(theta[1:3])
>> sigma<-exp(theta[4])
>> A<-rbind(
>> c(-k[1], k[2]),
>> c( k[1], -(k[2]+k[3]))
>> )
>> x0<-c(0,100)
>> sol<-function(tt)100-sum(expm(A*tt)%*%x0)
>> pred<-sapply(t,sol)
>> }
>>
>> Mpred <- function(theta) {
>> # WARNING: assumes t global
>> kvec<-exp(theta[1:3])
>> k1<-kvec[1]
>> k2<-kvec[2]
>> k3<-kvec[3]
>> # MIN problem terbuthylazene disappearance
>> z<-k1+k2+k3
>> y<-z*z-4*k1*k3
>> l1<-0.5*(-z+sqrt(y))
>> l2<-0.5*(-z-sqrt(y))
>> val<-100*(1-((k1+k2+l2)*exp(l2*t)-(k1+k2+l1)*exp(l1*t))/(l2-l1))
>> } # val should be a vector if t is a vector
>>
>> negll <- function(theta){
>> # non expm version JN 110731
>> pred<-Mpred(theta)
>> sigma<-exp(theta[4])
>> -sum(dnorm(dat[,2],mean=pred,sd=sigma, log=TRUE))
>> }
>>
>> theta<-rep(-2,4)
>> fand<-nlogL(theta)
>> fsim<-negll(theta)
>> cat("Check fn vals: expm =",fand," simple=",fsim," diff=",fand-fsim,"\n")
>>
>> cat("time the function in expm form\n")
>> tnlogL<-microbenchmark(nlogL(theta), times=100L)
>> tnlogL
>>
>> cat("time the function in simpler form\n")
>> tnegll<-microbenchmark(negll(theta), times=100L)
>> tnegll
>>
>> ftimes<-data.frame(texpm=tnlogL$time, tsimp=tnegll$time)
>> # ftimes
>>
>>
>> boxplot(log(ftimes))
>> title("Log times in nanoseconds for matrix exponential and simple MIN fn")
>>
>
> --
> Charles C. Berry cbe...@tajo.ucsd.edu
>
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--
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd....@cbs.dk Priv: pda...@gmail.com
"Døden skal tape!" --- Nordahl Grieg
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