Local Prediction in musical time series

Applications in modern nonlinear data analysis techniques indicate that chaotic dynamics are quite common, and that in many cases random behaviour is due to low dimensional chaos rather than complicated dynamics involving many irreducible degrees of freedom. The great promise of chaos lies in the hope that randomness might become predictable. Although chaotic dynamics puts limits on longterm prediction, it implies predictability over the short term. The monitoring of a single scalar observable is sufficient for characterizing and understanding the dynamics of a finite-dimensional attractor. The first step in analyzing such experimental data is a reconstruction of the observed dynamics. A new procedure to forecast the next value in a chaotic time series has been developed by using a modified version of Oi's work. The basic idea of the proposed approach is to embed the data in a state space and then use a straightforward numerical technique to build a nonlinear dynamic model. Practically, the authors' procedure can be considered as a local linearization of the generating law approximated by the data relationship in the reconstructed embedding space. It has been tested on many well-known synthetic chaotic systems, giving very good results. The procedure is used to predict the behaviour of time series extracted from a rhythmic pattern database composed by patterns from six different rhythms, three different tempos, three different instruments, and three different degrees of complexity for a total number of 162 time series. The average absolute percentage error over 100 predicted values is computed to evaluate the precision of the prediction.