I just asked another question (see here: How to calculate the shape of a curve given y coordinates and slope?) and was advised by the user who answered my question to ask a new question. I would appreciate any help on this topic. As in the previous thread, I apologise in advance if I'm not using the proper terminology, but I'm learning.
Although my problem is stated in the linked thread, I'll re-state it here with the details that required me to open a new thread.
I would like to figure out the shape of a curve given the information in the following graph:
On the y axis, I'm showing the slope of the curve whose shape I'm trying to find. On the x axis, I'm showing the y-coordinate of the curve corresponding to that slope. I am missing information about the x-coordinates of my curve. The different dots are different measurements I have made in an experiment.
First, I fit a non-parametric curve to my data (in this case, a loess regression curve):
This gives me a non-parametric description of the relationship between dy/dx of the curve whose shape I'm trying to find out, and its y coordinates.
In this case, I can intuitively understand that my mystery curve will have a sigmoidal shape because at low and high values of y, the slope is small, and at intermediate values of y, the slope is high.
I just learnt, in the previous question, that my problem involves solving a differential equation. However, there are times when I don't have an equation describing the relationship between dx/dy and y (as in the previous question that I asked), but instead I have a non-parametric curve like a loess (local regression) curve or a spline.
How could I solve this problem?
Many thanks in advance!