R: nls2 misses the solution - r

Question

I'm trying to fit a bi exponential function:
t = seq(0, 30, by = 0.1)
A = 20 ; B = 10 ; alpha = 0.25 ; beta = 0.01
y = A*exp(-alpha*t) + B*exp(-beta*(t))
df = as.data.frame(cbind(t,y))
ggplot(df, aes(t, y)) + geom_line() + scale_y_continuous(limits=c(0, 50))
This problem can't be solve by a simple transformation like log so I wanted to use the nls2 package:
library(nls2)
fo <- y ~ Ahat*exp(-alphahat*t) + Bhat*exp(-betahat*t)
fit <- nls2(fo,
start = list(Ahat=5, Bhat=5, alphahat=0.5,betahat=0.5),
algorithm = "brute-force",
trace = TRUE,
lower = c(Ahat=0, Bhat=0, alphahat=0, betahat=0),
upper = c(Ahat=50, Bhat=50, alphahat=10,betahat=10))
fit
Here is the result:
Nonlinear regression model
model: y ~ Ahat * exp(-alphahat * t) + Bhat * exp(-betahat * t)
data: NULL
Ahat Bhat alphahat betahat
5.0 5.0 0.5 0.5
residual sum-of-squares: 37910
Number of iterations to convergence: 4
Achieved convergence tolerance: NA
I assume something is wrong in my code because :
data: NULL ?
Why only 4 iterations ?
Hard to think nls2 didn't find a better solution than the starting point.
The result is far from the solution

Answer 1

From the documentation, the start parameter should be a data.frame of two rows that define the grid to search in, or a data.frame with more rows corresponding to parameter combinations to test if you are using brute-force. Also, nls will have trouble with your fit because it is a perfect curve, there is no noise. The brute-force method is slow, so here is an example where the search space is decreased for nls2. The result of the brute-force nls2 is then used as the starting values with nls default algorithm (or you could use nls2), after adding a tiny bit of noise to the data.
## Data
t = seq(0, 30, by = 0.1)
A = 20 ; B = 10 ; alpha = 0.25 ; beta = 0.01
y = A*exp(-alpha*t) + B*exp(-beta*(t))
df = as.data.frame(cbind(t,y))
library(nls2)
fo <- y ~ Ahat*exp(-alphahat*t) + Bhat*exp(-betahat*t)
## Define the grid to search in,
## Note: decreased the grid size
grd <- data.frame(Ahat=c(10,30),
Bhat=c(10, 30),
alphahat=c(0,2),
betahat=c(0,1))
## Do the brute-force
fit <- nls2(fo,
data=df,
start = grd,
algorithm = "brute-force",
control=list(maxiter=100))
coef(fit)
# Ahat Bhat alphahat betahat
# 10.0000000 23.3333333 0.0000000 0.3333333
## Now, run through nls:
## Fails, because there is no noise
final <- nls(fo, data=df, start=as.list(coef(fit)))
## Add a little bit of noise
df$y <- df$y+rnorm(nrow(df),0,0.001)
coef((final <- nls(fo, data=df, start=as.list(coef(fit)))))
# Ahat Bhat alphahat betahat
# 10.00034000 19.99956016 0.01000137 0.25000966
## Plot
plot(df, col="steelblue", pch=16)
points(df$t, predict(final), col="salmon", type="l")

Answer 2

Your data is null because you didn't add in any data into the nls2 statement.
This is how nls2 needs to be set up:
nls2(formula, data = parent.frame(), start, control = nls.control(),
algorithm = c("default", "plinear", "port", "brute-force",
"grid-search", "random-search", "plinear-brute", "plinear-random"),
trace = FALSE, weights, ..., all = FALSE)
Take a look at the official documentation for a full example.