Predictability

Incremental redundancies quantify how well several sequential measurements of x predict the next x, at that lag and embedding dimension. For instance, say incremental redundancy at the optimum embedding dimension stays roughly constant with lag. That means predictability about the next x remains constant at some positive value, regardless of the number of measurements included in the vector. In contrast, a downward-sloping relation means that our ability to predict the next x from D measurements decreases with lag.
A typical goal in regard to predictability is to find the smallest embedding dimension that provides optimum predictive power, for the given accuracy of the measurements and lag. That dimension is indicated by the proximity of the relations for successive embedding dimensions on the graph of incremental redundancy versus lag, as mentioned earlier. When those relations get close to the asymptotic accumulation line, predictive ability has nearly reached the noise scale. Further increases in embedding dimension then are fruitless; no more predictive power can be gained.
The same types of graphs also indicate approximate limits of predictability in terms of lag. If the general trend of the ΔR-versus-m relation is roughly horizontal, then lag doesn't affect predictive power. If, instead, incremental redundancy decreases with lag at a large embedding dimension, then we're losing predictive power as we increase lag. That decay of ability to predict typifies chaos, although it's not unique to chaotic systems (Ellner 1991). Predictability at that resolution becomes essentially zero when the relation falls to the vicinity of the abscissa (an incremental redundancy near zero).
In addition to the possible uses of mutual information and redundancy discussed above, other applications are being looked at as well. For instance, (1993) proposes a way to use redundancies to test for nonlinearity in a time series.