Characterization of complex data functions through local persistence of increments

Abstract

The local persistence of increments is a new concept defined in the present paper by the quantity of consecutive ups or consecutive downs in a function until it reverses. Each degree of persistence occurs in a given quantity for a given complex data function, namely higher consecutive ups/downs occur less often than lower consecutive ones. Moreover the decay of the frequency distribution of local persistences is exponential. The slope of the associated log plot leads to the definition of the local persistence coefficient which characterizes and differs one data from another. The method was tested in Weierstrass-Mandelbrot deterministic functions as well as in random fractional Brownian motions with varying fractal dimensions. An empiric expression relating the corresponding fractal dimensions and the defined persistence coefficient is proposed where the latter coincides with the corresponding Hurst exponent. Thus it is conjectured that the frequency distribution of the persistences of increments is related with long-range correlations. An empiric application of the method in data from S&P 500 index is presented. In this case the coefficient associated with ups is higher than that associated with downs. Thus while increasing S&P index persists more than while decreasing and the quantity of unitary negative persistences is higher than the positive ones. The method does not take into account the intensity of the increments and then the fractal dimensions may have no relation with the intensities of the increments. The method may be useful for (1) characterization of complex data; (2) estimation of Hurst coefficient or fractal dimension of data functions and (3) as a tool to predict the probability of the next behavior (up/down) of a complex data function.