/Fractal_Heart_Rate_Analysis

Analysis of Fractal Heart Rate

Primary LanguagePython

Fractals

  • A curve or geometrical figure, each part of which has the same statistical character as the whole.

  • In fractals, similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth and galaxy formation.

  • A fractal has a Hausdorff dimension that is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).

Hausdorff dimension:

  • The Hausdorff dimension of the fractal is "log N / log L"
  • N identical parts that are similar to the entire fractal with the scale factor of L and that the intersection between part is of the Lebesgue measure.

Techniques to generate Fractals:

Escape time fractals — These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal and the Lyapunov fractal.

Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.

Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.

Classification of fractals:

Exact self-similarity - The fractal appears identical at different scales Quasi-self-similarity - The fractal appears approximately (but not exactly) identical at different scales Statistical self-similarity - The fractal has numerical or statistical measures which are preserved across scales. statistically self-similar, but neither exactly nor quasi-self-similar.

Fractals in Heart rate:

  • Heart rate in nature is found to be a quasi self-similar fractal (fractals appears to be approximate but not exact) like a mandelbrot set.
  • They contain small copies of the entire fractal in distorted and degenerated forms.
  • A qualitative appreciation for the self-similar nature of fractal processes can be obtained by plotting their fluctuations at different temporal resolutions

Here, we can prove the HRV is fractal in nature by the following metrics:

  • Prove that the Heart Rate is a Quasi Fractal by segmenting the signals with time.
  • Use the methods that are used for a mandelbrot set to prove that it is a quasi fractal.
  • Relative Dispersion Analysis(RDA) and Detrended Fluctuation Analysis(DFA) of HRV signals are the methods used to prove quasi nature of a fractal.

References:

  1. A qualitative appreciation for the self-similar nature of fractal processes can be obtained by plotting their fluctuations at different temporal resolutions by Nadezhda V. Zudilina (Nadiya V. Zudilina). UDC [114:117]::530.1

  2. Fractals Analysis of Cardiac Arrhythmias by Mohammad Saeed. https://www.researchgate.net/publication/7606517_Fractals_Analysis_of_Cardiac_Arrhythmias