Determinism and Sources of Predictability

In the article we find evidence that forecasts of music descriptor time series can still be performed at mid-term or relatively long horizons. In addition, we show that this ability becomes crucial for a real-world application, namely the automatic detection of cover songs. However, prediction errors are not as low as one would expect, even for short horizons. Thus, we here investigate in more detail which structures in our data are the sources of predictability. Prediction relies on the assumption that similar initial conditions are mapped to similar states after a certain time step h. If h is so large that this property gets lost, then inaccuracy of initial conditions, inaccuracies of the model, and the lack of smoothness of the h-step map make predictions impossible in practice [1]. We assess this property for our data using a technique to quantify the expansion or contraction of sample neighborhoods as h increases, and by comparing the behavior of music descriptor time series to well-known processes. Consider a time series S and its reconstructed state space S. For each point s∗n in S , we select a neighborhood Ωn of radius ǫ. As done with the locally constant predictor, we use a percentage of the average squared Euclidean norm between all points (ǫκ = 0.4 for descriptors and noise, ǫκ = 0.01 for the rest). In addition, samples with neighborhoods smaller than ν = 5 elements are discarded from further processing. Once we have the neighborhood Ωn, an average pairwise distance between all elements in Ωn is computed. This computation is done by taking into account the evolution of these elements as a function of the prediction horizon h: