RELATIONSHIP BETWEEN THE FRACTAL DIMENSION AND THE POWER LAW INDEX FOR A TIME-SERIES - A NUMERICAL INVESTIGATION

A time series showing self-affine over all time scales follows a power law spectrum; when such a time series, X(j) (j = 1, 2,..., N), is transformed into the discrete Fourier transformation (i.e. X(j) = ΣN/2k = 0Ck cos(2πjkN - θk), where Ck and θk are the amplitude and phase for wavenumber k, respectively), the power spectrum density for wavenumber k is given by P(k) = C2k/N α k-α. The relationship between the power law index α and the fractal dimension D for a time series following a power law spectrum is investigated by using a numerical experiment. It should be noted that the time series with the same power law index shows various behaviors in a time domain, depending on distribution of αk, and gives different values of the fractal dimension accordingly. In short, the phase distribution strongly affects the irregularity represented in terms of the fractal dimension. In addition, the relationships between α and D are also examined both for the differenced and for the integrated time series. © 1990.