Predict based on time using a weibull

Question

Example data with weibull decay for growth:

library(tidyverse)

# example weibull
example_df <- data.frame(
  day = 1:30,
  growth = dweibull(seq(1, 30, by = 1), shape = 0.1)
)

example_df |> ggplot(aes(x = day, y = growth)) + 
  geom_point()

Data looks like this:

example_df |> head()
  day      growth
1   1 0.036787944
2   2 0.018348802
3   3 0.012186100
4   4 0.009104847
5   5 0.007257658
6   6 0.006027639

Each day, growth decreases a little - a decay.

I would like to predict decay on a given day. Here's where I get confused.

Suppose I'm asked what decay is on day 20? Or day 25? How can I use a weibull regression to tell me? Tried something:

library(fitdistrplus)
fit.weibull <- fitdist(example_df$growth, distr = "weibull", method = "mle", lower = c(0, 0))

fit.weibull |> summary()
Fitting of the distribution ' weibull ' by maximum likelihood 
Parameters : 
         estimate Std. Error
shape 0.962954378         NA
scale 0.004707553         NA
Loglikelihood:  130.1494   AIC:  -256.2989   BIC:  -253.4965 
Correlation matrix:
[1] NA

This seems to fit a weibull curve to my data. But I also need to 'tell it' that growth is dependent on day.

I looked at survreg and survival packages but they want binary 1/0 data for survived/not survived. For my problem, I just want to associate day with growth using a weibull to be able to predict growth on a given day index.

How can I use a weibull to predict growth on a given day?

Answer 1

I think you are looking for something like this:

df <- structure(list(day = 1:6, 
                     growth = c(0.036787944, 0.018348802, 0.0121861, 0.009104847, 0.007257658, 0.006027639)), 
                class = "data.frame", row.names = c(NA, -6L))

#fit the model
model<-nls(growth ~ a*exp(-b*day), data=df, 
    start=list(a=1, b=0.1))

#Set up the prediction
test <- data.frame(day=1:10)
test$y<-predict(model, test)

#Plot
plot(df$day, df$growth, ylim=c(0, 0.04), xlim=c(0, 10))
lines(test$day, test$y, col="blue")

Here we are using the simple exponential decay formula $growth = a*e^{-b*day}$. If this fit is not satisfactory, then maybe try a polynomial or another function of your choosing.