Self-similarity Analysis of Time Series

Self-similarity is a typical feature for fractal and chaos. Regular fractals in theory have strict self-similarity, but for irregular fractals in nature, their self-similarity could be seen only within a certain scale-invariant region. Time series acquired by sampling are commonly used for studying objects in nature, and they could be treated as curves on plane. Fractal analysis could be used to discuss the self-similarity of time series. Based on the fractal dimension calculating method by continuous wavelet transform, a novel scale-invariant extent parameter is proposed to evaluate the level of self-similarity of time series. The longer the scale-invariant region length is, the higher level of the self-similarity is. Otherwise, short scale-invariant region length corresponding to low self-similarity level. Time series with different self-similarity levels could be classified directly using this evaluation parameter.