| | | | | | 2.5.12—
Problems:
The Embedding Theorems | | | | | | | | | We can always construct a plot from the points with coordinates X(t), X(t+Dt), X(t+2Dt), . . . . X(t+(N-l) Dt), where X(t) is the time series of one variable. Takens' theorem tells us that if certain conditions are met, then this plot is related to the phase space set. For example, the theorem assumes that the time series X(t) is "smooth," that is, that the derivatives dX(t)/dx and d2X(t)/dt2 exist. | | | | | | | | | However, in practice, experimental data may not have the properties assumed by Takens' theorem. For example, when the time series X(t) is fractal, then the finer we look in time, the more wiggles we find. Thus the derivatives dX(t)/dx and d2X(t)/dt2 do not exist. We can still plot the points with coordinates X(t), X(t+Dt), X(t+2Dt), . . .. X(t+(N-1) Dt), but because the data do not satisfy the assumptions of the theorem, we do not know how, if at all, this plot is related to the "real" phase space set. | | | | | | | | | Osborne and Provenzale showed that this embedding procedure does not produce the real phase space set from a time series of fractal data. They constructed the lag plot of points with coordinates X(t), X(t+Dt), X(t+2Dt), . . . . X(t+(N-l) Dt). Since their time series was generated by a random mechanism, the fractal dimension of the real phase space was infinite. However, they found that the fractal dimension of the lag plot was sometimes as low as 1, which is a lot less than infinity. Thus, for this type of data, the lag plot is not the "real" phase space set. | | | | | | | | | Experimental data do not always satisfy the assumptions of the embedding theorems. It would be very worthwhile for mathematicians to formulate and prove theorems that would tell us how to construct the phase space set from data that are not continuous or differentiable. | | | | |
