In Markov examples, the forbidden pairs tell the whole story: any forbidden string must contain a forbidden pair. This can be interpreted as placing a limit on the effects of history: only the immediately previous step is important.
We can test this for driven IFS by noting the empty length 2 address squares, those corresponding to forbidden pairs, and then looking at the empty length 3, length 4, ... address squares. If each of these contains one of the forbidden pairs, then the underlying process has a one-step memory. If some do not, the memory is longer. For example, the driven IFS below has
| empty length 2 squares | forbidden transitions |
| 12 | 2 -> 1 |
| 22 | 2 -> 2 |
| 13 | 3 -> 1 |
| 23 | 3 -> 2 |
| 32 | 2 -> 3 |
| 44 | 4 -> 4 |
Note the square 241 is empty. (The 1/8 by 1/8 square
outlined in the picture.) This corresponds to the transitions
|
|
Of course, any measured time series is of finite length. Consequently, when we see an empty subsquare, we should ask whether it is empty because its address is excluded in the dynamical process driving the IFS, or because the time series is not long enough. That is, if we had more data, would the subsquare eventually be visited. Obviously, the smaller the subsquare, the more data is needed. Here is a simple calculation, illustrating how certain we are that the empty square 241 represents a real exclusion.
More detailed analyses can estimate the length of memory of a system, thus placing bounds on how far into the future we can predict.
Return to Data-Driven IFS