Measure of predictability

Many techniques have been developed to measure the difficulty of forecasting data from an observed time series. This paper introduces a measure which we call the "forecast entropy" designed to measure the predictability of a time series. We use attractors reconstructed from the time series and the distributions in the regular and tangent spaces of the data which comprise the attractor. We then consider these distributions on different scales. We present a formula for calculating the forecast entropy. To provide a standard of predictability, we define an idealized random system whose forecast entropy will be maximal; we then use this measure to rescale the forecast entropy to lie in the range [0,1]. The time series obtained from several chaotic systems as well as from a pseudorandom system are studied using this measure. We present evidence that the forecast entropy can be used as a tool for determining optimal delays and embedding dimensions used for reconstructing better attractors. We also show that the forecast entropy of a random system has completely different characteristics from that of a deterministic one.