Finding the elbow/knee in a curve

Question

I have these data:

x <- c(6.626,6.6234,6.6206,6.6008,6.5568,6.4953,6.4441,6.2186,6.0942,5.8833,5.702,5.4361,5.0501,4.744,4.1598,3.9318,3.4479,3.3462,3.108         


        

Answer 1

You can now find the knees/elbows with different methods by using the maxcurv function in the soilphysics package.

Answer 2

I think you want to find the points where the derivative of the function y=f(x) has a huge jump in value. you can try the following, as you can see there can be one or many such points depending on the threshold (for huge jump) we choose:

get.elbow.points.indices <- function(x, y, threshold) {
  d1 <- diff(y) / diff(x) # first derivative
  d2 <- diff(d1) / diff(x[-1]) # second derivative
  indices <- which(abs(d2) > threshold)  
  return(indices)
}

# first approximate the function, since we have only a few points
ap <- approx(x, y, n=1000, yleft=min(y), yright=max(y))
x <- ap$x
y <- ap$y

indices <- get.elbow.points.indices(x, y, 1e4) # threshold for huge jump = 1e4
x[indices]
#[1] 6.612851 # there is one such point
plot(x, y, pch=19)
points(x[indices], y[indices], pch=19, col='red')

 indices <- get.elbow.points.indices(x, y, 1e3) # threshold for huge jump = 1e3
 x[indices]
 #[1] 0.3409794 6.4353456 6.5931286 6.6128514 # there are 4 such points
 plot(x, y, pch=19)
 points(x[indices], y[indices], pch=19, col='red')