Regression with exponential decay

Question

I have the data from Li et al. 2003 paper "Belowground biomass dynamics in the Carbon Budget Model of the Canadian Forest Sector". I am trying to recreate equation 6 in R so that I can produce the same parameter estimates. (eq. 6), however, Pf has a maximum value of 0.426, as values greater will produce a 0. The paper doesn't state whether a linear or non linear model is used so I am having trouble figuring out how to go about coding this is R.

At first I was thinking something like model3<-lm(log(fine_prop)~log(total_roots),data=Li2003_root_proportion) But this isn't correct as it won't match the parameters stated in the table 3 for equation number 6.

I know this isn't much to go on but I would appreciate any help. Thanks!

Answer 1

At first I was thinking something like model3<-lm(log(fine_prop)~log(total_roots),data=Li2003_root_proportion). But this isn't correct as it won't match the parameters

What you fitted was

$$\log(P_f) = a + b \log(RB)$$

which relates to a power law instead of an exponential distribution

$$P_f = e^{\log P_f} = e^{a + b \log(RB)} = (e^a) RB^b$$

To fit the exponential function $Pf = a+b \exp(c RB)$ you use a non-linear fitting method instead.

however, Pf has a maximum value of 0.426, as values greater will produce a 0.

To impose a maximum for the function you can impose limits on the parameters. The easiest is if you can reparameterize the function in terms of the maximum. In the case that $RB>0$ and the exponential term is decaying, we have for the function $P_f(RB) = a+b \exp(c RB)$ the maximum

$$m = \max(P_f) = a+b$$

So we could rewrite $a$ as

$$a = m-b$$ and fit the equation

$$P_f(RB) = m-b+b \exp(c RB) = m(1-b \exp(c RB))$$

while imposing a maximum restriction on $m$. For these restrictions see the lower and upper parameters of the function nls in R (https://stat.ethz.ch/R-manual/R-devel/library/stats/html/nls.html).