Because of the observed data, a four group clustering emerges when you plot successive eruption durations, of the famous geyser. That is, the *x* axis is the first eruption and *y* is the next.

There appear to be four kinds of eruption pairs: 1. two short eruptions (the bottom-left cluster), 2. a short eruption followed by a long eruption, and the respective opposites (long-long and long-short).

The R code for this, is just a single line:

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plot( faithful$eruptions[1:length(faithful$eruptions)-1], faithful$eruptions[2:length(faithful$eruptions)], xlab="Previous Duration", ylab="Next Duration", main="Eruption Pairs" ) |

A line graph shows the state transitions. Notice that there are no bottom-left to top-right transitions. Curious!

**[UPDATE]** We can sample a subset of the Old Faithful eruption data and add arrows from previous to next observation. Here is the R code:

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z <- sample( faithful$eruptions, 50 ) x <- z[ 1 : length(z) - 1 ] y <- z[ 2 : length(z) ] plot( x, y, xlab = "Previous Duration", ylab = "Next Duration", main = "Eruption Pairs" ) s <- seq( length(x) - 1 ) # one shorter than data arrows( x[s], y[s], x[ s + 1 ], y[ s + 1 ], length = 0.2, angle = 20, col = "gray" ) |

And here is the plot: