I have a model where I assumed an exponential increase and then decrease with a beta distribution. I fitted curves to the sets of data using the following equation:
(a/b) * exp( -(t-c)/b-exp(-(t-c)/b) )
Even though the graph below looks similar to the expected results, the equation doesn't give a lot of mathematical meaning to my data I later realize.
I have an equation, which I am having trouble to code it into useable form, that fits better with my hypothesis:
P * (1 - exp(-aE/P)) * exp(-bE/P)
This equation contains an increasing and decreasing component, but the problem is that when I fit it to the graph, I ran into singularity problems because I don't have good start values.
I'm putting it here to see if anyone has a working equation that has similar properties.
Here is one of the nine sets of data that I have and what I already did.
df <- data.frame(PAR= c(0,6,18,35,61,93,121,195,268),
ETR = c(0.06, 1.63, 4.88, 7.70, 9.47, 9.47, 8.07, 6.55, 3.60))
nonlin <- function(t, a, b, c) { (a/b) * exp( -(t-c)/b-exp(-(t-c)/b) ) }
EQY24 <- data.frame(PAR=as.numeric(c(df$PAR)), ETR=c(df$ETR)) #24C
nlsfit <- nls(ETR ~ nonlin(PAR, a, b, c), data=df, start=list(a=800, b=50, c=75))
with(EQY24, plot(PAR, ETR, cex=0.5, ylim=c(0,12), col="red", xlab=xaxis, ylab=yaxis))
EQYseq <- seq(0,270,.1)
ETR <- coef(nlsfit)
lines(EQYseq, nonlin(EQYseq, ETR[1], ETR[2], ETR[3]), col="red", lwd=2)