Time series prediction is a major task in many areas of research, e.g., biology, physics, business and engineering. The ability to forecast the behaviour of a system hinges, generally, on the knowledge of the laws underlying a given phenomenon. When this knowledge is expressed as a solvable equation, one can predict the behaviour along the future once the initial condition is given. However, phenomenological models are often unknown or extremely time consuming.
Nevertheless, it is also possible to predict the dynamic behaviour of the system along the future by extracting knowledge from the past. In this case, the time series behaviour can be captured by expressing the value x(k+1) as a function of the d previous values of the time series, x(k),...,x(k-d), that is:
x(k+1) = F(x(k),...,x(k-d))
In many time series applications, one-step prediction schemes are used to predict the next sample of data, x(k+1), based on previous samples. However, one-step prediction may not provide enough information, especially in situations where a broader knowledge of the time series behaviour can be very useful or in situations where it is desirable to anticipate the behaviour of the time series process.
In the context of time series, the long-term or multi-step prediction problem is also an interesting problem. It obtains predictions several steps ahead into the future -x(k+1), x(k+2), x(k+3),...- starting from information at current instant k.