Linearize the spectral coherence
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Questions to David Rotermund
Let us assume we have two time series (white in spectrum) $x_1(t)$ and $x_2(t)$. Both are linearly mixed together via a mixing coefficent $\alpha$:
\[y(t) = (1- \alpha) x_1(t) + \alpha * x_2(t)\]Wouldn’t it to be nice if the spectral coherence would be $\alpha$?
For white times series with the length of infinity this can be achived via the transformation
coherence_scaled = 1.0 / (1.0 + np.sqrt((1.0 / coherence) - 1.0))
see Attention Selectively Gates Afferent Signal Transmission to Area V4 for details
The emphesis lies on infinity and a white spectrum. For shorter time series the results might vary.
import numpy as np
import matplotlib.pyplot as plt
import pywt # type: ignore
from tqdm import trange # type: ignore
# Calculate the wavelet scales we requested
def calculate_wavelet_scale(
number_of_frequences: int,
frequency_range_min: float,
frequency_range_max: float,
dt: float,
) -> np.ndarray:
s_spacing: np.ndarray = (1.0 / (number_of_frequences - 1)) * np.log2(
frequency_range_max / frequency_range_min
)
scale: np.ndarray = np.power(2, np.arange(0, number_of_frequences) * s_spacing)
frequency_axis_request: np.ndarray = frequency_range_min * np.flip(scale)
return 1.0 / (frequency_axis_request * dt)
def get_y_ticks(
reduction_to_ticks: int, frequency_axis: np.ndarray, round: int
) -> tuple[np.ndarray, np.ndarray]:
output_ticks = np.arange(
0,
frequency_axis.shape[0],
int(np.floor(frequency_axis.shape[0] / reduction_to_ticks)),
)
if round < 0:
output_freq = frequency_axis[output_ticks]
else:
output_freq = np.round(frequency_axis[output_ticks], round)
return output_ticks, output_freq
def get_x_ticks(
reduction_to_ticks: int, dt: float, number_of_timesteps: int, round: int
) -> tuple[np.ndarray, np.ndarray]:
time_axis = dt * np.arange(0, number_of_timesteps)
output_ticks = np.arange(
0, time_axis.shape[0], int(np.floor(time_axis.shape[0] / reduction_to_ticks))
)
if round < 0:
output_time_axis = time_axis[output_ticks]
else:
output_time_axis = np.round(time_axis[output_ticks], round)
return output_ticks, output_time_axis
def calculate_cone_of_influence(dt: float, frequency_axis: np.ndarray):
wave_scales = 1.0 / (frequency_axis * dt)
cone_of_influence: np.ndarray = np.ceil(np.sqrt(2) * wave_scales).astype(np.int64)
return cone_of_influence
def mask_cone_of_influence(
complex_spectrum: np.ndarray,
cone_of_influence: np.ndarray,
fill_value: float = np.NaN,
) -> np.ndarray:
assert complex_spectrum.shape[0] == cone_of_influence.shape[0]
for frequency_id in range(0, cone_of_influence.shape[0]):
# Front side
start_id: int = 0
end_id: int = int(
np.min((cone_of_influence[frequency_id], complex_spectrum.shape[1]))
)
complex_spectrum[frequency_id, start_id:end_id] = fill_value
start_id = np.max(
(
complex_spectrum.shape[1] - cone_of_influence[frequency_id] - 1,
0,
)
)
end_id = complex_spectrum.shape[1]
complex_spectrum[frequency_id, start_id:end_id] = fill_value
return complex_spectrum
def calculate_wavelet_tf_complex_coeffs(
data: np.ndarray,
number_of_frequences: int = 25,
frequency_range_min: float = 15,
frequency_range_max: float = 200,
dt: float = 1.0 / 1000,
) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
assert data.ndim == 1
t: np.ndarray = np.arange(0, data.shape[0]) * dt
# The wavelet we want to use
mother = pywt.ContinuousWavelet("cmor1.5-1.0")
wave_scales = calculate_wavelet_scale(
number_of_frequences=number_of_frequences,
frequency_range_min=frequency_range_min,
frequency_range_max=frequency_range_max,
dt=dt,
)
complex_spectrum, frequency_axis = pywt.cwt(
data=data, scales=wave_scales, wavelet=mother, sampling_period=dt
)
return (complex_spectrum, frequency_axis, t)
def calculate_spectral_coherence(
n_trials: int,
y_a: np.ndarray,
y_b: np.ndarray,
number_of_frequences: int,
frequency_range_min: float,
frequency_range_max: float,
dt: float,
) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
for trial_id in range(0, n_trials):
wave_data_a, frequency_axis, t = calculate_wavelet_tf_complex_coeffs(
data=y_a[..., trial_id],
number_of_frequences=number_of_frequences,
frequency_range_min=frequency_range_min,
frequency_range_max=frequency_range_max,
dt=dt,
)
wave_data_b, frequency_axis, t = calculate_wavelet_tf_complex_coeffs(
data=y_b[..., trial_id],
number_of_frequences=number_of_frequences,
frequency_range_min=frequency_range_min,
frequency_range_max=frequency_range_max,
dt=dt,
)
cone_of_influence = calculate_cone_of_influence(dt, frequency_axis)
wave_data_a = mask_cone_of_influence(
complex_spectrum=wave_data_a,
cone_of_influence=cone_of_influence,
fill_value=np.NaN,
)
wave_data_b = mask_cone_of_influence(
complex_spectrum=wave_data_b,
cone_of_influence=cone_of_influence,
fill_value=np.NaN,
)
if trial_id == 0:
calculation = wave_data_a * np.conj(wave_data_b)
norm_data_a = np.abs(wave_data_a) ** 2
norm_data_b = np.abs(wave_data_b) ** 2
else:
calculation += wave_data_a * np.conj(wave_data_b)
norm_data_a += np.abs(wave_data_a) ** 2
norm_data_b += np.abs(wave_data_b) ** 2
calculation /= float(n_trials)
norm_data_a /= float(n_trials)
norm_data_b /= float(n_trials)
coherence = np.abs(calculation) ** 2 / ((norm_data_a * norm_data_b) + 1e-20)
return np.nanmean(coherence, axis=-1), frequency_axis, t
# Parameters for the wavelet transform
number_of_frequences: int = 3 # frequency bands
frequency_range_min: float = 5 # Hz
frequency_range_max: float = 200 # Hz
dt: float = 1.0 / 1000.0
# Test data ->
n_t: int = 10000
n_trials: int = 100
# We select one frequency because all look the same for this white random signal
frequency_select: int = 1
rng = np.random.default_rng(1)
mother_time_series_a: np.ndarray = rng.random((n_t, n_trials))
mother_time_series_a -= mother_time_series_a.mean(axis=0, keepdims=True)
mother_time_series_a /= mother_time_series_a.std(axis=0, keepdims=True)
mother_time_series_b: np.ndarray = rng.random((n_t, n_trials))
mother_time_series_b -= mother_time_series_b.mean(axis=0, keepdims=True)
mother_time_series_b /= mother_time_series_b.std(axis=0, keepdims=True)
# <- Test data
alpha_vector: np.ndarray = np.linspace(0.0, 1.0, 11, endpoint=True)
for alpha_id in trange(0, alpha_vector.shape[0]):
alpha: float = alpha_vector[alpha_id]
y_a = mother_time_series_a.copy()
y_b = (1.0 - alpha) * mother_time_series_a + alpha * mother_time_series_b
y_b -= y_b.mean(axis=0, keepdims=True)
y_b /= y_b.std(axis=0, keepdims=True)
temp, frequency_axis, t = calculate_spectral_coherence(
n_trials=n_trials,
y_a=y_a,
y_b=y_b,
number_of_frequences=number_of_frequences,
frequency_range_min=frequency_range_min,
frequency_range_max=frequency_range_max,
dt=dt,
)
if alpha_id == 0:
coherence: np.ndarray = np.zeros((temp.shape[0], alpha_vector.shape[0]))
coherence[:, alpha_id] = temp
coherence_scaled = 1.0 / (1.0 + np.sqrt((1.0 / coherence) - 1.0))
plt.plot(alpha_vector, coherence[frequency_select, :], label="unscaled")
plt.plot(alpha_vector, coherence_scaled[frequency_select, :], label="scaled")
plt.plot([0.5, 0.5], [0, 1], "k--")
plt.plot([0, 1], [0.5, 0.5], "k--")
plt.ylabel("Spectral Coherence")
plt.xlabel("Mixture Coefficent")
plt.legend()
plt.show()
The source code is Open Source and can be found on GitHub.