Does the function y = at ^ b * exp (-ct) have a name? Can linearization? How can I rate a, b, c?

Question

I am trying to fit a non-linear model, but cannot find good examples online.

Does this function have a name?

Is it possible to linearize?

I tried to evaluate the parameters a , b and c with a random effect g (as in the group) as a function of time t below. I can fit the model with nls without a random effect, but I have problems getting the model to fit. Suggestions are welcome (preferably in R, but would any suitable package do)?

 ## time, repeated 16 times for 4 replicates from each of 4 groups t <- rep(1:20, 16) ## g, group g <- rep(1:4, each = 80) ## starting to create an example dataset, ## to see if I can recover known parameters a <- rep(c(3.5, 4, 4.1, 5), each = 80) b <- rep(c(1.1, 1.4, 1.8, 2.5), each = 80) c <- rep(c(0.125, 0.25), each = 160) ## error to add to above parameters set.seed(1) e_a <- runif(320, -0.5, 0.5) e_b <- runif(320, -0.1, -0.1) e_c <- runif(320, -0.02, 0.02) ## this is my function f <- function(t, a, b, c) a * (t^b) * exp(-c * t) ## simulate y y <- f(t = t, a + e_a, b + e_b, c + e_c) mydata <- data.frame(t = t, y = y, g = g) library(nlme) ## now fit the model to estimate a, b, c fm1 <- nlme(y ~ a * (t^b) * exp(-c * t), data = mydata, fixed = a + b + c~1, random = a + b + c ~ 1|g, start = c(a = 4, b = 1, c = 0.25), method = "REML") 

Answer 1

In physics (and some other areas) I saw this or its variants called the Hoerl curve or the Hoerl function for example. here , although he has other names. If c is negative and a and b are positive, this is a scaled gamma density.

When you ask about linearization, you have to be careful; equation y = at ^ b. exp (ct) does not really mean what you mean - the observations y (i) are not exactly equal to a. t (i) ^ b. exp (ct (i)) (otherwise almost any 3 observations will give you exact parameter values).

So the noise should somehow introduce your model for y. Is it additive? multiplicative or something else? (It is also important, but for other reasons: does its size change in some way with t or not? Are the noise conditions independent for different observations?)

If your actual model is y (i) = at (i) ^ b. exp (ct (i)) + ε (i), which is not linearizable.

If your actual model is y (i) = at (i) ^ b. exp (ct (i)). ε (i) and ε (i) = exp (η (i)) for some (I hope, zero mean) η (i), which is linearizable.

Taking the second form

log (y (i)) = log (a) + b log (t (i)) + ct (i) + log (ε (i))

or

y * (i) = a * + b.log (t (i)) + ct (i) + η (i)

which is linear in the parameters a * = log (a), b and c, and the error term η (i); therefore, if you are ready to make such an assumption about an error, you should take it into account by methods suitable for such linear models; in this case, you may wish to think about the brackets of the questions about the excess of the error, which may affect the way you model it.