The Hurst Exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst exponent were originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river's volatile rain and drought conditions that had been observed over a long period of time. The name "Hurst exponent", or "Hurst coefficient", derives from Harold Edwin Hurst (1880 - 1978), who was the lead researcher in these studies; the use of the standard notation H for the coefficient relates to his name also.
The Hurst exponent is referred to as the "index of dependence" or "index of long-range dependence". It quantifies the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction.
Fractal Dimension from Mark Jurik is much smoother than the others but the general rule is the same: it is not a directional indicator, but is attempting to determine if there is a trend in the current market price changes or not, and it should be used bearing that in mind.Fractal dimension index (Sevcik/Matulich)
Mandelbrot describes the Fractal Dimension Index (FDI) as a way to measure "how convoluted and irregular" something is. The FDI can be used as a stock market indicator. The closer prices move in a one-dimensional straight line, the closer the FDI moves to 1.0. The more closely prices resemble a two-dimensional plane, the closer the FDI moves to 2.0.