Create README.md

Signed-off-by: David Rotermund <54365609+davrot@users.noreply.github.com>

data_analysis/spectral_coherence_scale/README.md

@@ -0,0 +1,266 @@
+# Linearize the spectral coherence 
+{:.no_toc}
+
+<nav markdown="1" class="toc-class">
+* TOC
+{:toc}
+</nav>
+
+## Top
+
+Questions to [David Rotermund](mailto:davrot@uni-bremen.de)
+
+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
+
+```python
+coherence_scaled = 1.0 / (1.0 + np.sqrt((1.0 / coherence) - 1.0))
+```
+see [Attention Selectively Gates Afferent Signal Transmission to Area V4](https://www.jneurosci.org/content/38/14/3441) for details 
+
+The emphesis lies on infinity and a white spectrum. For shorter time series the results might vary. 
+
+![image0.png](image0.png)
+
+```python
+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()
+```
+
+
+