Linear graph turning exponential at a particular point For a line graph, it behaves linearly upto a particular point and varies exponentially after it. Please suggest me a statistical approach/test to know this threshold point.


When I plot a scatter plot I see a linear trend as seen in the plot.

For same data when I plot a line graph I see a linear trend upto a certain threshold point and then an exponential trend. So basically I am trying to find out this point from where the behaviour changes.
The actual dataset is huge around 500 observations( therefore can't share) 
 A: Edited for changes in question:

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*you still don't specify if the model should be smooth at the join (which reduces the model degrees of freedom), or can have a kink in it.


*it looks like the variation in the second plot increases with whatever the variable on the $x$-axis is. You should also explain something about how you think the variation of the data about the model is expected to work.


*one approach which is suitable for a constant (or reasonably so) variability about the model is nonlinear least squares, though there are other ways to approach this sort of problem.
There are some posts on site outlining this or other approaches to this sort of model on CV. For example this answer gives a minimal changepoint example using nonlinear least squares done in R; a simple example is also in this one. The exponential-linear case can be done in just the same fashion.


*It might perhaps make some sense to work on the log-scale -- it's still going to be a combination of nonlinear and linear but it might help stabilize variance. On the other hand, if your data are count-percentages, a different scale might do better.
With some additional clarification, some more detail may be possible.
A: You can set the threshold as a parameter in your model. The model will have these components: (1) linear part up to the threshold, (2) exponential part above the threshold, (3) the condition that the value of the two "sub-models" is identical at the threshold.
You can then find the parameter values by likelihood optimization, or by MCMC. Of course, in order to calculate the likelihood, you will need to come up with the appropriate error structure (the stochastic part of the model), which will depend on the nature of your data.
