A Robust Algorithm for Optimisation and Customisation of Fractal Dimensions of Time Series Modified by Nonlinearly Scaling Their Time Derivatives: Mathematical Theory and Practical Applications

Franz Konstantin Fuss

doi:10.1155/2013/178476

Abstract

Standard methods for computing the fractal dimensions of time series are usually tested with continuous nowhere differentiable functions, but not benchmarked with actual signals. Therefore they can produce opposite results in extreme signals. These methods also use different scaling methods, that is, different amplitude multipliers, which makes it difficult to compare fractal dimensions obtained from different methods. The purpose of this research was to develop an optimisation method that computes the fractal dimension of a normalised (dimensionless) and modified time series signal with a robust algorithm and a running average method, and that maximises the difference between two fractal dimensions, for example, a minimum and a maximum one. The signal is modified by transforming its amplitude by a multiplier, which has a non-linear effect on the signal's time derivative. The optimisation method identifies the optimal multiplier of the normalised amplitude for targeted decision making based on fractal dimensions. The optimisation method provides an additional filter effect and makes the fractal dimensions less noisy. The method is exemplified by, and explained with, different signals, such as human movement, EEG, and acoustic signals.

Highlights

The Hausdorff-Besicovitch dimension, DH, is defined by most efficient covering [1] of irregular curves and surface profiles, usually approximated by the box-counting, circlecounting, or yardstick methods
This paper introduces a method for optimising the fractal dimension DHm of a signal for improved decision making by scaling the signal amplitude after normalising amplitude and DH 31 Hz
Standard methods for computing fractal dimensions use different scaling methods, that is, different amplitude multipliers, which makes it difficult to compare fractal dimensions obtained from different methods

Summary

Introduction

The Hausdorff-Besicovitch dimension, DH, is defined by most efficient covering [1] of irregular curves and surface profiles, usually approximated by the box-counting, circlecounting, or yardstick methods. This method dates back to Felix Hausdorff who coined the term “fractal dimension” (“gebrochene Dimension,” [2]) by extending Caratheodory’s [3] p-dimensional measure to noninteger values of p. It was reinvented by Richardson [4], for investigating the complexity and ruggedness of coastlines with yardstick methods, who empirically found the following equation: Σl ∝ l−α, (2). It was Mandelbrot [5] who recognised that this exponent corresponds to a fractal dimension, to be calculated from the gradient of log N against log 1/r.

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