Manual least-squares (using optim)

[ashiklom]

x <- c(60,100,330,15000) 
y <- c(0.429, 0.423, 0.4146, 0.127)
log_x <- log(x)
test_data <- data.frame(y, log_x)

head(test_data)
plot(y ~ log_x, data = test_data)

## ls_m <- nls(
##   y ~ r + (s-r)/ (1 + (alpha * log_x)^n) ^ (1 - 1/n) ,
##   nls.control(maxiter = 5000) ,
##   data = test_data,
##   start = list(r = 0.1, s = 0.5, n = 1.2, alpha = 0.015),
##   lower = list(r = 0, s = 0, n = 0, alpha = 0),
##   trace = TRUE,
##   algorithm = "port"
## )

f <- function(log_x, r, s, n, alpha) r + (s-r)/ (1 + (alpha * log_x)^n) ^ (1 - 1/n) 

likelihood <- function(params) {
  pred <- f(test_data$log_x, params[1], params[2], params[3], params[4])
  sum((pred - test_data$y) ^ 2)
}

opt <- optim(c(r = 0.1, s = 0.5, n = 1.2, alpha = 0.015),
             fn = likelihood,
             lower = c(r = 0, s = 0, n = 0, alpha = 0),
             method = "L-BFGS-B")

vals <- opt$par

plot(y ~ log_x, data = test_data)
lines(seq(3, 10, 0.1), f(seq(3, 10, 0.1), vals[1], vals[2], vals[3], vals[4]))