# neuropsychology/NeuroKit

neurokit2/complexity/fractal_katz.py

### Summary

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###### Test Coverage
``````# -*- coding: utf-8 -*-
import numpy as np
import pandas as pd

def fractal_katz(signal):

"""**Katz's Fractal Dimension (KFD)**

Computes Katz's Fractal Dimension (KFD). The euclidean distances between successive points in
the signal are summed and averaged, and the maximum distance between the starting point and any
other point in the sample.

Here, fractal dimensions range from 1.0 for straight lines, through
approximately 1.15 for random-walk waveforms, to approaching 1.5 for the most
convoluted waveforms.

Parameters
----------
signal : Union[list, np.array, pd.Series]
The signal (i.e., a time series) in the form of a vector of values.

Returns
-------
kfd : float
Katz's fractal dimension of the single time series.
info : dict
A dictionary containing additional information (currently empty, but returned nonetheless
for consistency with other functions).

Examples
----------
* **Step 1.** Simulate different kinds of signals

.. ipython:: python

import neurokit2 as nk
import numpy as np

# Simulate straight line
straight = np.linspace(-1, 1, 2000)

# Simulate random
random = nk.complexity_simulate(duration=2, method="randomwalk")
random = nk.rescale(random, [-1, 1])

# Simulate simple
simple = nk.signal_simulate(duration=2, frequency=[5, 10])

# Simulate complex
complex = nk.signal_simulate(duration=2,
frequency=[1, 3, 6, 12],
noise = 0.1)

@savefig p_katz.png scale=100%
nk.signal_plot([straight, random, simple, complex])

* **Step 2.** Compute KFD for each of them

.. ipython:: python

KFD, _ = nk.fractal_katz(straight)
KFD
KFD, _ = nk.fractal_katz(random)
KFD
KFD, _ = nk.fractal_katz(simple)
KFD
KFD, _ = nk.fractal_katz(complex)
KFD

References
----------
* Katz, M. J. (1988). Fractals and the analysis of waveforms.
Computers in Biology and Medicine, 18(3), 145-156. doi:10.1016/0010-4825(88)90041-8.

"""

# Sanity checks
if isinstance(signal, (np.ndarray, pd.DataFrame)) and signal.ndim > 1:
raise ValueError(
"Multidimensional inputs (e.g., matrices or multichannel data) are not supported yet."
)

# Force to array
signal = np.array(signal)

# Drop missing values
signal = signal[~np.isnan(signal)]

# Define total length of curve
dists = np.abs(np.diff(signal))
length = np.sum(dists)

# Average distance between successive points
a = np.mean(dists)

# Compute farthest distance between starting point and any other point
d = np.max(np.abs(signal - signal))

kfd = np.log10(length / a) / (np.log10(d / a))

return kfd, {}``````