Introduction of Error Estimation for Clarifying Fractal Phonomena

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The purpose of this study is to introduce the concept of reliability in the measurement concerning fractal phenomena, by proposing techniques of error estimation. The following outcomes are obtained :1. In the measurement of the "fractal dimension" of a realization "self-similar recurrent event process" proposed by Mandelbrot, two approaches are considered to be important. This study derived analytical error estimations of the obtained values by these approaches. The results suggest that a relatively small number of data are enough for obtaining accurate values for an ideal process.2. A new approach for testing the randomness of heteroskedastic time series data is proposed, and is successfully applied to finacial data. This approach detects existence of "mean reversion, " which is a feature of fractal caused by long term memory. This result also detects other types of serial correlations, which is important from financial point of view.3. In the analysis of the mechanism of "mean reversion, " it was found that stock market indices may behave in substantially different ways in the macroeconomic analysis. One must be careful in selecting them in their analysis.4. The error of observed "length" is analytically calculated for a polygonal approximation to a path of Brownian motion. This result is expected to show how the bias of observed "fractal dimension" convergers to the true value when the number of sampling points increases.5. It was found that the "digital straightness" and "digital convexity" are characterized in terms of the "length" and the "absolute curvature" proposed in the author. This fact shows that a relation of "digital geometry" and the "length" used in this approach is very close.