Re: [Rd] An example of very slow computation

From: Michael Lachmann <lachmann_at_eva.mpg.de>
Date: Fri, 19 Aug 2011 02:08:48 +0200

On my trials, after eliminating all the extra matrix<->dgeMatrix conversions, using expm() and the method below were equally fast.

Michael

On 19 Aug 2011, at 1:32AM, Ravi Varadhan wrote:

> 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 [pdalgd_at_gmail.com]
> Sent: Thursday, August 18, 2011 6:37 PM
> To: Ravi Varadhan
> Cc: 'cberry_at_tajo.ucsd.edu'; r-devel_at_stat.math.ethz.ch; 'nashjc_at_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: rvaradhan_at_jhmi.edu
>>
>>
>> -----Original Message-----
>> From: r-devel-bounces_at_r-project.org [mailto:r-devel-bounces_at_r-project.org] On Behalf Of Ravi Varadhan
>> Sent: Wednesday, August 17, 2011 2:33 PM
>> To: 'cberry_at_tajo.ucsd.edu'; r-devel_at_stat.math.ethz.ch; 'nashjc_at_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: rvaradhan_at_jhmi.edu
>>
>> -----Original Message-----
>> From: r-devel-bounces_at_r-project.org [mailto:r-devel-bounces_at_r-project.org] On Behalf Of cberry_at_tajo.ucsd.edu
>> Sent: Wednesday, August 17, 2011 1:08 PM
>> To: r-devel_at_stat.math.ethz.ch
>> Subject: Re: [Rd] An example of very slow computation
>>
>> John C Nash <nashjc_at_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 cberry_at_tajo.ucsd.edu
>>
>> ______________________________________________
>> R-devel@r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-devel
>>
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>>
>> ______________________________________________
>> R-devel_at_r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-devel
> 
> --
> Peter Dalgaard, Professor,
> Center for Statistics, Copenhagen Business School
> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> Phone: (+45)38153501
> Email: pd.mes_at_cbs.dk  Priv: PDalgd_at_gmail.com
> "Døden skal tape!" --- Nordahl Grieg
> 
> 
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