Simulating data for a Gompertz curve

Question

I have a set of data that I have collected which consists of a time series, where each y-value is found by taking the mean of 30 samples of grape cluster weight.

The growth follows a Gompertz curve with formula y = a*exp(-exp(-(x-x0)/b)), with

• a = 88.8
• b = 11.7
• x0 = 15.1.

The data:

x = c(0, 28, 36, 42, 50, 58, 63, 71, 79, 85, 92, 99, 106, 112)
y = c(0, 15, 35, 55, 62, 74, 80, 96, 127, 120, 146, 160, 177, 165).

• x refers to the days from fruit set (i.e. 0 is when the time series starts)
• x values correspond to the days in which the measurements are taken (which depends on certain growth stages of grapes)

I want to simulate more data from this, with the same number of x and y values, so that I can carry out some Bayesian analysis to find the posterior distribution of the data.

Effectively what I need is:

• to simulate data which follows the Gompertz curve to create the posterior distribution. This data would technically be for "previous years" time series data.
• to construct and test the fit of the predictive time-series model based on the distribution

If there is some skeleton code where it is possible to change around the parameters, then this could potentially be very helpful for me too.

Thanks

Answer 1

Let's inspect your data

x <- c(0, 28, 36, 42, 50, 58, 63, 71, 79, 85, 92, 99, 106, 112)
y <- c(0, 15, 35, 55, 62, 74, 80, 96, 127, 120, 146, 160, 177, 165)

and fitted Gompertz curve

gFun <- function(x){
 a <- 88.8
 b <- 11.7
 x0 <- 15.1
 est <- a*exp(-exp(-(x-x0)/b))
  return(est)
}

by visualisation:

library(ggplot2)
 ggplot(ggData, aes(x=x, y=y) ) +
  geom_point() + 
  stat_function(fun=gFun, colour="blue") + 
  theme_bw()

This doesn't look as a good fit. However, simulating data y|x at fixed x as in the vector above can be done by adding error term. I've used normal distribution with sd=4 for illustration.

nSim <- 10

simData <- data.frame(x=c(0, rep(x[-1], each=nSim)) ) # x[-1] removes 0 from simulation
simData$y <- gFun(simData$x) + rnorm(n=nrow(simData), sd=4)

ggplot(simData, aes(x=x, y=y) ) +
 geom_point(alpha=0.4) + 
 stat_function(fun=gFun, colour="blue") + 
 scale_x_continuous(limits=c(0, max(x)) ) +
 theme_bw()