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Im analyzing time-series to detect when the y-value is so flat that one can assume there is an underlying factor limiting y from being higher. Is there a methodology or statistical discipline that do this kind of analyis automatically? Im after a name I can put in Google that points me to books, articles etc.

In the first example below, the eyes can easily detect that there is a cutoff at y=8. It looks like the curve should go higher but something is blocking it from doing that.

The second curve, although having the same max of y=8, does not seem to have a bottleneck or limitation.

I suppose our eyes and brain do a lot of post-processing, like looking at the variance before and after the plateau etc.

library(ggplot2)
the.data.1 <- data.frame(x.vals=seq(1:40),y.vals=c(1,2,3,2,3,2,3,4,5,4,5,6,6,5,6,7,8,8,8,8,7,8,8,8,7,8,8,8,7,6,7,6,6,5,4,5,4,3,2,1))
p <- ggplot(the.data.1, aes(x=x.vals, y=y.vals))
p <- p + geom_line()
p
the.data.2 <- data.frame(x.vals=seq(1:40),y.vals=c(1,2,3,2,3,2,3,4,5,4,5,6,6,5,6,7,8,7,6,5,6,6,7,8,7,8,7,6,5,6,7,6,6,5,4,5,4,3,2,1))
p <- ggplot(the.data.2, aes(x=x.vals, y=y.vals))
p <- p + geom_line()
p

Any suggestions for methods are welcome.

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