Dimension of the minimal cover and fractal analysis of time series

We develop a new approach to the fractal analysis of time series of various natural, technological and social processes. To compute the fractal dimension, we introduce the sequence of the minimal covers associated with a decreasing scale δ. This results in new fractal characteristics: the dimension of minimal covers Dμ, the variation index μ related to Dμ, and the new multifractal spectrum ζ(q) defined on the basis of μ. Numerical computations performed for the financial series of companies entering Dow Jones Industrial Index show that the minimal scale τμ, which is necessary for determining μ with an acceptable accuracy, is almost two orders smaller than an analogous scale for the Hurst index H. This allows us to consider μ as a local fractal characteristic. The presented fractal analysis of the financial series shows that μ(t) is related to the stability of underlying processes. The results are interpreted in terms of the feedback.