profiling function which has been optimized with optim - r

Question

I have the following set of functions.
funk <- function(x,b) { 10^b * exp(-x/10) }
lambda <- function(y,k) { exp(-k*y) }
funk1 <- function(y,x,xb,b,k) {
funk(x-xb-y,b) *exp(- integrate(lambda, lower=0, upper = y, k=k)$value) }
funk2 <-function(x,xb,b,k) {
integrate(funk1, lower= 0, upper=x-xb, x=x,xb=xb, b=b,k=k)$value }
funk2_vc <- Vectorize(funk2)
optim_funk2 <- function(param) {
b <-param[1]
k <- param[2]
R1 <- sum((y - funk2_vc(xx,xb,b,k))^2)
-log(R1) }
fit <- optim(par=c(5, 0.05), fn=optim_funk2)
and
xx <- seq(0,500,5)
xb <- seq(0,100,1)
y <- seq(1000,0,-10)
I wish to profile the function funk2 to figure out the path that optim has taken to estimate parameter values and if the function is optimizied for local or global minima.
I am a newbie to R and have no clue how to go about it. All suggestions welcome.

Answer 1

Here's a simple way to track the path of the parameters. I'll use linear regression as an example. Say our data is
x <- 1:10
y <- -3 + 2 * x + rnorm(length(x))
plot(x, y)
So y is a linear function of x plus some noise. Our goal is to find parameters a and b such that the sum of squared errors sum((y - (a + b * x))^2) is minimized. (This can of course be solved algebraically, but for illustration we'll solve it with optim().)
Here's the code to do the optimization and keep track of parameters:
par.path <- matrix(nrow=0, ncol=2, dimnames=list(NULL, c("a","b")))
funk <- function(par) {
a <- par[1]; b <- par[2]
par.path <<- rbind(par.path, par)
sum((y - (a + b * x))^2)
}
optim(par=c(0,0), fn=funk)
The first line creates a 0-row matrix called par.path to store the parameter path. Within the objective function funk, we add the current value of par to par.path. Note that we have to use <<- rather than <- to update par.path because it lives outside the scope of funk. (If we used <-, then funk would create a new local variable also called par.path, and the par.path outside the function wouldn't get updated.) Since optim calls funk repeatedly, par.path will get progressively longer (more rows).
There are various ways to plot the matrix par.path. Since in this case there are only two parameters, we can plot them against each other:
plot(par.path, type='l')
points(par.path[c(1,nrow(par.path)),], col=c("green","red"), cex=2, pch=16)
The 2nd line adds green and red dots to indicate the start and stop of the path. More flexibly, we can plot all the columns of par.path against the iteration number of optim:
matplot(par.path, type='l', col=c("black","red"), lty=1)
legend("bottomleft", c("a","b"), col=c("black","red"), lty=1)
Here are these two plots.