On autocorrelation based on Hermann Weyl´s discrepancy norm for time series analysis

Hermann Weyl´s concept of a discrepancy measure is discussed in the context of time series analysis. A concept for autocorrelation based on this discrepancy notion is introduced. It is shown that in particular for high frequent signals as they, for example, are typically encountered in a financial context, the introduced autocorrelation concept stands out by a better discriminative power than its classical counterpart. While the computational complexity of this novel autocorrelation is of quadratic order in terms of the number of given time steps an approximation based on L_p-norms is introduced which can be computed by convolution, and therefore reduces the order of complexity to that of its classical counterpart. It is shown that the proposed approximation can be tuned to be arbitrarily close to the original discrepancy based version, and that it shows similar desirable behavior.