estimate the fractal dimension with the algorithm of higuchi

资 源 简 介

estimate the fractal dimension with the algorithm of higuchi

详 情 说 明

Estimating fractal dimension (FD) is a common technique for quantifying the complexity or irregularity of signals, time series, or geometric patterns. Among the various algorithms, Higuchi, Petrosian, and Katz are widely used due to their computational efficiency and adaptability to different types of data. Here’s an overview of these methods:

### 1. Higuchi Algorithm The Higuchi method calculates the fractal dimension of a time series by analyzing its self-similarity at different scales. The core idea involves: Dividing the time series into multiple subsequences with varying segment lengths. Computing the average length of these segments for each scale. Using a logarithmic relationship between segment lengths and scales to estimate the FD. This algorithm is particularly robust for noisy or non-stationary signals, making it popular in biomedical signal processing (e.g., EEG, ECG analysis).

### 2. Petrosian Algorithm The Petrosian approach provides a fast estimate of FD by leveraging binary signal characteristics. Key steps include: Converting the signal into a binary sequence (e.g., using thresholding or derivative-based methods). Counting the number of sign changes or transitions in the binary sequence. Relating these transitions to the fractal dimension through a simple formula. While computationally lightweight, Petrosian’s method may sacrifice accuracy for speed and is best suited for preliminary analyses.

### 3. Katz Algorithm Katz’s method estimates FD based on the geometric properties of the signal’s trajectory in a multi-dimensional space. The process involves: Calculating the total length of the signal path and the maximum distance between any two points. Deriving FD from the ratio of these geometric measures, adjusted by the average step size. This method is intuitive but sensitive to signal scaling; normalization is often required for consistent results.

### Practical Considerations Data Type: Higuchi works well for continuous signals, Petrosian for binary/thresholded data, and Katz for path-based geometric analysis. Computational Cost: Petrosian is the fastest, while Higuchi offers a balance between accuracy and complexity. Noise Sensitivity: Higuchi and Katz are more resilient to noise compared to Petrosian’s simplistic approach.

These algorithms are foundational in fields like neuroscience, geophysics, and image analysis, where quantifying complexity helps uncover underlying patterns or anomalies.