Intrinsic Entropy: A Novel Adaptive Method for Measuring the Instantaneous Complexity of Time Series

The determination of appropriate parameters and an appropriate window size in most entropy-based measurements of time-series complexity is a challenging problem. Inappropriate settings can lead to the loss of intrinsic information within a time series. Therefore, two parameter-free methods, namely the intrinsic entropy (IE) and ensemble IE (eIE) methods, are proposed in this paper. The eIE method requires two parameters, which can be easily determined through an orthogonality test. The proposed approaches can measure instantaneous complexity; thus, they do not require a predetermined window size. White noise and three other varieties of colored noise were used to test the stability of the proposed methods, and five types of synthetic signals and logistic maps were applied for measuring instantaneous complexity and regularity. The results revealed that the IE and eIE methods exhibit satisfactory stability. Both methods provide point-by-point entropy measures for time series. The eIE method is useful for measuring the complexity of frequency and amplitude modulation. Furthermore, the periodicity of time series can be detected using the two proposed methods.