In the preceding two chapters we established algorithms to estimate the Lyapunov exponent and the correlation dimension from a time series. We tried to be very strict about the conditions which must be met in order to justify such estimates. The data quality and quantity had to be sufficient to observe clear scaling regions. The implied requirement that the data must be deterministic to a good approximation is also valid for successful nonlinear predictions (Chapter 4). If this were the whole story, the scope of these methods would be quite limited. In the main, well-controlled laboratory data from experiments which have been designed to show deterministic chaos would qualify. Although these include some very interesting signals, many other data sets for which classical, linear time series methods seem inappropriate do not fall into this class.

Indeed, there is a continuous stream of publications reporting more or less successful attempts to apply nonlinear algorithms, in particular the correlation dimension, to field data. Examples range from population dynamics in biology, stock exchange rates in economy, and time dependent hormone secretion or ECG and EEG signals in medicine to geophysical records of the earth's magnetic field or the variable luminosity of astronomical objects. In particular the interpretation of the results as measures of the “complexity” of the underlying systems has met with increasing criticism. It is now quite generally agreed that, in the absence of clear scaling behaviour, quantities derived from dimension or Lyapunov estimators can be at most relative measures of system properties. But even then it is not clear which properties are really measured.