[R] How to find a split point in a curve?

[Gabor Grothendieck]

We can use nls2 to try each value in 10:100 as a possible split point
picking the one with lowest residual sum of squares:

library(nls2)
fm <- nls2(Y ~ cbind(1, pmin(X, X0)), start = data.frame(X0 = 10:100),
  algorithm = "plinear-brute")
plot(Y ~ X)
lines(fitted(fm) ~ X, col = "red")

> fm
Nonlinear regression model
  model: Y ~ cbind(1, pmin(X, X0))
   data: parent.frame()
     X0   .lin1   .lin2
18.0000  6.5570  0.2616
 residual sum-of-squares: 4.999

Number of iterations to convergence: 91
Achieved convergence tolerance: NA

On Thu, May 14, 2020 at 11:13 AM Luigi Marongiu
<marongiu.luigi using gmail.com> wrote:
>
> Dear all,
> I am trying to find a turning point in some data. In the initial phase, the
> data increases more or less exponentially (thus it is linear in a nat log
> transform), then reaches a plateau. I would like to find the point that
> marks the end of the exponential phase.
> I understand that the function spline can build a curve; is it possible
> with it to find the turning points? I have no idea of how to use spline
> though.
> Here is a working example.
> Thank you
>
> ```
> Y = c(259, 716, 1404, 2173, 3944, 5403, 7140, 9121,
>       11220, 13809, 16634, 19869, 23753, 27447,
>       30590, 33975, 36627, 39600, 42067, 44082,
>       58190, 63280, 65921, 67929, 69977, 71865,
>       73614, 74005, 74894, 75717, 76365, 76579,
>       77087, 77493, 77926, 78253, 78680, 79253,
>       79455, 79580, 79699, 79838, 79981, 80080,
>       80124, 80164, 80183, 80207, 80222, 80230,
>       80241, 80261, 80261, 80277, 80290, 80303,
>       80337, 80376, 80422, 80461, 80539, 80586,
>       80653, 80708, 80762, 80807, 80807, 80886,
>       80922, 80957, 80988, 81007, 81037, 81076,
>       81108, 81108, 81171, 81213, 81259, 81358,
>       81466, 81555, 81601, 81647, 81673, 81998,
>       82025, 82041, 82053, 82064, 82094, 82104,
>       82110, 82122, 82133, 82136, 82142, 82164,
>       82168, 82180, 82181, 82184, 82187, 82188,
>       82190, 82192, 82193, 82194)
> Y = log(Y)
> X = 1:length(Y)
> plot(Y ~ X, ylab = "Ln(Y)", xlim=c(0,10, main="zoomed in"))
> abline(lm(Y[1:3] ~ X[1:3]))
> abline(lm(Y[1:5] ~ X[1:5]), lty=2)
> text(7, 6, "After third or fifth point, there is deviance", pos=3)
> text(2.5, 10, "Solid line: linear model points 1:3", pos =3)
> text(2.5, 9, "Dashed line: linear model points 1:5", pos =3)
> plot(Y ~ X, ylab = "Ln(Y)", xlim=c(0,10, main="overall"))
> abline(lm(Y[1:3] ~ X[1:3]))
> ```
>
>         [[alternative HTML version deleted]]
>
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