Re: [R] Hessian in box-constraint problem - concern OPTIM function

[Cleber Nogueira Borges]

Hello Mr. Graves,
Hello all useRs,

Many thanks for your attention.

     This is an interesting question.

What is the problem you are trying to solve and how do the boundary conditions function as part of this system?

One of the functions that I need to minimized is:

sum( ( ( hat_xi - xi )^2 )*uxi^-2 + ( ( ( a+b*hat_xi ) - yi)^2 )*uyi^-2)

the context: I need to considerate errors in regressors:  x[i] ~ N( x[i] ;
ux[i]^2 )

u = 'uncertainty' is the same of std, but this 'u' is because the
metrology terminology.

And, I would like to restrict the x[i] variables in ~95% CI.

My dirty code (test) follow below

Thanks again.
Cleber

###############


n <- 7 ### TODO: any number???

xi <- c(1:n) ; uxi <- round( abs( rnorm( n,0,1e-1)),6)
yi <- round(xi + uxi + rnorm(n,0,.9),6) ; uyi <- round(abs(rnorm(
n,0,1e-1)),6)

naive <- lm( yi ~ xi )
# p: parameters
p <- 2
plot( xi,yi )
abline( naive )


fobjetiva <- function( optPar=c(xi,a,b) , xi, uxi, yi, uyi )
{
   n <- length( xi )
   hat_xi <- optPar[1:n] ; a <- optPar[n+1] ; b <- optPar[n+2]
       sum( ( (hat_xi - xi)^2 )*uxi^-2 + ( ( (a+b*hat_xi) - yi)^2
)*uyi^-2 )
}

## testing
fobjetiva(c(xi, coef(naive)), xi,uxi,yi,uyi)

### box-constraints
seCoef <- sqrt(diag(vcov( naive )))
Linf <- as.numeric(c( xi-2*uxi, coef(naive)-5000*seCoef ))
Lsup <- as.numeric(c( xi+2*uxi, coef(naive)+5000*seCoef ))



metodo <-  4   #   1,2,3,4 ou 5

all_iterOptim <- capture.output(
 reportOptim <- optim(
   par=c(xi,coef(naive)),
   fn=fobjetiva,
   xi=xi,
   uxi=uxi,
   yi=yi,
   uyi=uyi,
   method=c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN")[metodo],
   lower=Linf,
   upper=Lsup,
   hessian=TRUE,
   control=list(
     REPORT=1,
     maxit=1e4,
     trace=6
   )
 )
)

### get all steps
all_steps <- grep("^X", all_iterOptim )
all_steps <- all_iterOptim[ all_steps ]
steps <- length(all_steps)
steps

matrix_steps <- matrix(0,nr=steps, nc=(n+p) )
for( i in 1:steps )
 {
 matrix_steps[i,] <- as.numeric(unlist(strsplit(all_steps[i], "
"))[3:(2+n+p)])
 }
windows(restore=T)

### view a animation of this otimization

for( i in 1:steps )
 {
 x <- matrix_steps[i,1:n]
 a <- matrix_steps[i,n+p-1]
 b <- matrix_steps[i,n+p]
 y <- a+b*x
 plot( xi, yi, pch=19, main=paste("Passo",i,"de",steps,sep=" "), cex=2 )
 abline(a,b, col='blue', lwd=3)
 abline(naive, lwd=2, lty=2, col='red')
 points( x, y, col='red', pch=19, cex=2 )
 segments(xi,yi,x,y, lwd=2)
 Sys.sleep(.11)
 ###file = paste("ISO_",(i+100),".png", sep="")
 ###savePlot(filename=file, type ="png", device=dev.cur() )
 }

###comando = "convert -dispose previous -adjoin -delay 35  ISO_*.png
-loop 0  ISO_animator.gif"
###shell(comando)
###unlink("ISO_*.png")


## view trajectory

windows(restore=T)
par( mfrow=c(4,2))
for( i in 1:7 ){
restri <- xi[i]+uxi[i]*c(-2,2)
interv <- range(matrix_steps[,i], restri )
plot(1:steps, matrix_steps[,i], t='l', xlab="Passos", ylab="x1",
ylim=interv, lwd=2, las=2 )
abline( h=restri, col='red', lwd=2)
titulo=c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN")[metodo]
title(titulo)
}

I ask because the asymptotic theory behind your formula for 's.e.' breaks down with parameters at boundaries. It assumes that you are minimizing the negative log(likelihood) AND the optimum is in the interior of the region AND the log(likelihood) is sufficiently close to being parabolic that a reasonable approximation for the distribution of the maximum likelihood estimates (MLEs) has a density adequately approximated by a second-order Taylor series expansion about the MLEs. In this case, transforming the parameters will not solve the problem. If the maximum is at a boundary and if you send the boundary to Inf with a transformation, then a second-order Taylor series expansion of the log(likelihood) about the MLEs will be locally flat in some direction(s), so the hessian can not be inverted. These days, the experts typically approach problems like this using Monte Carlo, often in the form of Markov Chain Monte Carlo (MCMC). One example of an analysis of this type of problem appears in section 2.4 of Pinheiro and Bates (2000) Mixed-Effects Models in S and S-Plus (Springer). Hope this helps. Spencer Graves

Cleber Nogueira Borges wrote:

Hello all useRs,

I am using the OPTIM function with particular interest in the method L-BFGS-B,

because it is a box-constraint method.
I have interest in the errors estimates too.
I make:

s.e. <- sqrt( diag( solve( optim(...,method='L-BFGS-B', hessian=TRUE)$hessian )))

but in help say:
"Note that this is the Hessian of the unconstrained problem even if the
box constraints are active."
My doubts is:
How to obtain a authentic hessian for a box-constraint problem?

How I should make a interpretation of this result (concern the hessian) ? Is possible make some transformation or so can I considerate this result a good approximation??

I am grateful for some help!
References are welcome! :-D
Cleber Borges

______________________________________________
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.