Finite Representation

The next phase in my research was to learn about Conley Index Theory, with the intention of applying it to time series analysis. In my research, I approach time series as an incomplete map. “Maps” are deterministic, discrete-time dynamical systems of the form,

xn+1 = f ( xn )

Maps have dynamical characteristics such as fixed points and periodic points. Fixed points are points where the point maps to itself  f ( x ) = x. Period-k points are points that map to them selves after k iterations. For example a period-4 point is one that satisfies this equation, f ( f ( f ( f ( x ) ) ) ) = x. A period-3 point implies that the map is chaotic!

A common way to compute the dynamics of a map is to divide the map into closed intervals, which creates an outer representation. This interval can then be mapped forward. This forms a directed graph, where each edge  marks the mapping of an interval to another or itself. For example, dividing the tent map, defined as

f ( x ) = { r x   x < 0.5

                       { r (1 – x)    x>0.5

uniformly into four intervals yields the following,

grid

The tent map divided into four uniform intervals

digraph

The directed graph of the outer approximation

If we view time series as incomplete maps, then we can try the same techniques to gain information about the dynamics of the time series. This will give us a finite representation of the time series. We should expect that this finite representation will be a subset of the true outer approximation.