In this chapter we will discuss the notion of the predictability of a system evolving over time or, strictly speaking, of a signal emitted by such a system. Forecasting future values of some quantity is a classical problem in time series analysis but the conceptual importance of the prediction problem is not limited to those who want to get rich by knowing tomorrow's exchange rates. Even if, instead, you are interested in describing, understanding or classifying signals, stay with us for a few pages.

In this book we are concerned with the detection and quantification of possibly complicated structures in a signal. We want to be able to convince others that the structures we find are real and not just fluctuations. The most convincing argument for the presence of some pattern is if it can be used to give an improved prediction. It is a necessary condition for a theory to be compatible with the known data but it is not sufficient. In order to become accepted, a theory must successfully predict something which can be verified subsequently. In time series analysis, we can take this requirement of predictive quality quite literally.

Most concepts, which we will introduce later in order to describe time series data, can be interpreted to some extent as indirect measures of predictability. Due to their indirect nature, some conclusions will remain controversial, especially if the structures are rather faint. The statistically significant ability to predict the signal better than other techniques do will then be a more convincing affirmation of nonlinear and deterministic structure than several dubious digits of the fractal dimension.