def compute_rescaled_range(sig, win_len):
"""Compute rescaled range of a given time series at a given scale.
Parameters
----------
sig : 1d array
Time series.
win_len : int
Window length for each rescaled range computation, in samples.
Returns
-------
rs : float
Average rescaled range over windows.
"""
# Demean signal
sig = sig - np.mean(sig)
# Calculate cumulative sum of the signal & split the signal into segments
segments = split_signal(sp.cumsum(sig), win_len).T
# Calculate rescaled range as range divided by standard deviation (of non-cumulative signal)
rs_win = np.ptp(segments, axis=0) / np.std(split_signal(sig, win_len).T,
axis=0)
# Take the mean across windows
rs = np.mean(rs_win)
return rs
def compute_detrended_fluctuation(sig, win_len, deg=1):
"""Compute detrended fluctuation of a time series at the given window length.
Parameters
----------
sig : 1d array
Time series.
win_len : int
Window length for each detrended fluctuation fit, in samples.
deg : int, optional, default=1
Polynomial degree for detrending.
Returns
-------
det_fluc : float
Measured detrended fluctuation, as the average error fits of the window.
"""
# Calculate cumulative sum of the signal & split the signal into segments
segments = split_signal(sp.cumsum(sig - np.mean(sig)), win_len).T
# Calculate local trend, as the line of best fit within the time window
_, fluc, _, _, _ = np.polyfit(np.arange(win_len),
segments,
deg=deg,
full=True)
# Convert to root-mean squared error, from squared error
det_fluc = np.mean((fluc / win_len))**0.5
return det_fluc
def compute_rescaled_range(sig, win_len):
"""Compute rescaled range of a given time series at a given scale.
Parameters
----------
sig : 1d array
Time series.
win_len : int
Window length for each rescaled range computation, in samples.
Returns
-------
rs : float
Average rescaled range over windows.
Notes
-----
- Rescaled range was introduced as a measure of time series variability, by Harold Hurst [1]_.
References
----------
.. [1] https://en.wikipedia.org/wiki/Rescaled_range
"""
# Demean signal
sig = sig - np.mean(sig)
# Calculate cumulative sum of the signal & split the signal into segments
segments = split_signal(np.cumsum(sig), win_len).T
# Calculate rescaled range as range divided by standard deviation (of non-cumulative signal)
rs_win = np.ptp(segments, axis=0) / np.std(split_signal(sig, win_len).T,
axis=0)
# Take the mean across windows
rs = np.mean(rs_win)
return rs
def lagged_coherence_1freq(sig, fs, freq, n_cycles):
"""Compute the lagged coherence of a frequency using the hanning-taper FFT method.
Parameters
----------
sig : 1d array
Time series.
fs : float
Sampling rate, in Hz.
freq : float
The frequency at which to estimate lagged coherence.
n_cycles : float
Number of cycles at the examined frequency to use to compute lagged coherence.
Returns
-------
float
The computed lagged coherence value.
"""
# Determine number of samples to be used in each window to compute lagged coherence
n_samps = int(np.ceil(n_cycles * fs / freq))
# Split the signal into chunks
chunks = split_signal(sig, n_samps)
n_chunks = len(chunks)
# For each chunk, calculate the Fourier coefficients at the frequency of interest
hann_window = hann(n_samps)
fft_freqs = np.fft.fftfreq(n_samps, 1 / float(fs))
fft_freqs_idx = np.argmin(np.abs(fft_freqs - freq))
fft_coefs = np.zeros(n_chunks, dtype=complex)
for ind, chunk in enumerate(chunks):
fourier_coef = np.fft.fft(chunk * hann_window)
fft_coefs[ind] = fourier_coef[fft_freqs_idx]
# Compute the lagged coherence value
lcs_num = 0
for ind in range(n_chunks - 1):
lcs_num += fft_coefs[ind] * np.conj(fft_coefs[ind + 1])
lcs_denom = np.sqrt(
np.sum(np.abs(fft_coefs[:-1])**2) * np.sum(np.abs(fft_coefs[1:])**2))
return np.abs(lcs_num / lcs_denom)
def compute_detrended_fluctuation(sig, win_len, deg=1):
"""Compute detrended fluctuation of a time series at the given window length.
Parameters
----------
sig : 1d array
Time series.
win_len : int
Window length for each detrended fluctuation fit, in samples.
deg : int, optional, default=1
Polynomial degree for detrending.
Returns
-------
det_fluc : float
Measured detrended fluctuation, as the average error fits of the window.
Notes
-----
- DFA was originally proposed in [1]_.
- There is a relationship between DFA measures and 1/f exponent, as detailed in [2]_.
References
----------
.. [1] Peng, C.-K., Buldyrev, S. V., Havlin, S., Simons, M., Stanley, H. E., &
Goldberger, A. L. (1994). Mosaic organization of DNA nucleotides.
Physical Review E, 49(2), 1685–1689.
DOI: https://doi.org/10.1103/PhysRevE.49.1685
.. [2] https://en.wikipedia.org/wiki/Detrended_fluctuation_analysis
"""
# Calculate cumulative sum of the signal & split the signal into segments
segments = split_signal(np.cumsum(sig - np.mean(sig)), win_len).T
# Calculate local trend, as the line of best fit within the time window
_, fluc, _, _, _ = np.polyfit(np.arange(win_len),
segments,
deg=deg,
full=True)
# Convert to root-mean squared error, from squared error
det_fluc = np.mean((fluc / win_len))**0.5
return det_fluc