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I have function looks like this:

enter image description here

So, the point is after it comes to steady state, it slowly goes down. How can I describe such a behaviour?

The datapoints are:

x        y 1        10 2        60 3        72 4        70 5        69,8 6        69,6 7        69,4 8        69,2 9        69 10        68,8 11        68,6 12        68,4 13        68,2 14        68 15        67,8 16        67,6 

1 Answers 1

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I don't think there's a one-word name for such a situation.

Depending on exactly what you want to do with the data when you are done, you might be looking for Segmented (Piecewise) Regression. It is clear that after a "breakpoint" in your model1, you experience completely linear behavior ($x > 3$). The first part looks quadratic.

Let's split our database into two separate lists:

Quadratic     1        10     2        60     3        72  Linear     4         70     5         69,8     6         69,6     7         69,4     8         69,2     9         69     10        68,8     11        68,6     12        68,4     13        68,2     14        68     15        67,8     16        67,6 

Run regression on the first data set, receiving $f_1(x) = -19x^2 + 107x - 78$.

Run regression on the second data set, receiving $f_2(x) = -0.2x + 70.8$.

Now create the piecewise function: $ f(x) = \left\{ \begin{array}{lr} -19x^2 + 107x - 78 & (x \le 3)\\ -0.2x + 70.8 & (x > 3)\\ \end{array} \right. $

Perhaps we want such a function to be continuous. If you want, you can set the two function equal to each other, and you will find they intersect2 at $3 \le \dfrac{268+2\sqrt{286}}{95} \approx 3.17708 \le 4 $. So you could write: $ f(x) = \left\{ \begin{array}{lr} -19x^2 + 107x - 78 & (x \le 3.17708)\\ -0.2x + 70.8 & (x > 3.17708)\\ \end{array} \right. $

Now your function is continuous as well.


I would definitely recommend the piecewise regression method, but by using symbolic regression and the software Eureqa, I was able to find a quite interesting formula:

$ 70.8 \cdot \mathrm{logistic}(3.541x - 5.323) - 0.2x $

Where $\mathrm{logistic}(x) = \dfrac{1}{1 + e^{-x}}$, which is a quite common function, so you should have no trouble describing it.

If I was describing it to a friend, I would probably say "a logistic function followed by a linear decrease" or perhaps "a logistic function".

Some stats:

  • $ R^2 \approx 0.999 $
  • Maximum error at $x=3$ with $\approx 2.151$
  • $\mathrm{MSE} \approx 0.309 $

1 There are varying algorithms for determining breakpoints, one such algorithm is listed briefly here.

2 Is this allowed? Will this always be the case? Perhaps an algorithmic approach would be better.

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    @George-V-Williams, that is wonderful! I appreciate your help. This is what I was looking for. And thank you for the links - they will be useful for me!2013-01-04