Generating signals with multiscale time irreversibility: The asymmetric weierstrass function

Time irreversibility (asymmetry with respect to time reversal) is an important property of many time series derived from processes in nature. Some time series (e.g., healthy heart rate dynamics) demonstrate even more complex, multiscale irreversibility, such that not only the original but also coarse-grained time series are asymmetric over a wide range of scales. Several indices to quantify multiscale asymmetry have been introduced. However, there has been no simple generator of model time series with "tunable" multiscale asymmetry to test such indices. We introduce an asymmetric Weierstrass function W(A) (constructed from asymmetric sawtooth functions instead of cosine waves) that can be used to construct time series with any given value of the multiscale asymmetry. We show that multiscale asymmetry appears to be independent of other multiscale complexity indices, such as fractal dimension and multiscale entropy. We further generalize the concept of multiscale asymmetry by introducing time-dependent (local) multiscale asymmetry and provide examples of such time series. The W(A) function combines two essential features of complex fluctuations, namely fractality (self-similarity) and irreversibility (multiscale time asymmetry); moreover, each of these features can be tuned independently. The proposed family of functions can be used to compare and refine multiscale measures of time series asymmetry.