Create README.md

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

data_analysis/spectral_coherence/README.md

@@ -0,0 +1,175 @@
+# Spectral Coherence
+{:.no_toc}
+
+<nav markdown="1" class="toc-class">
+* TOC
+{:toc}
+</nav>
+
+## Top
+
+Questions to [David Rotermund](mailto:davrot@uni-bremen.de)
+
+## Test data
+
+```python
+import numpy as np
+import matplotlib.pyplot as plt
+
+f_base: float = 50
+f_delta: float = 50
+
+rng = np.random.default_rng(1)
+n = 10000
+dt = 1.0 / 1000.0
+x = dt * 2.0 * np.pi * (f_base + f_delta * 2 * (rng.random((n,)) - 0.5))
+x = np.cumsum(x, axis=0)
+
+y: np.ndarray = np.sin(x)
+t: np.ndarray = np.arange(0, n) * dt
+
+np.savez("testdata.npz", y=y, t=t)
+
+plt.plot(t, y)
+plt.xlabel("Time [s]")
+plt.xlim(0, 0.5)
+plt.show()
+```
+!{image0.png](image0.png)
+
+Let us look at wavelet power of the time series:
+
+```python
+import numpy as np
+import matplotlib.pyplot as plt
+import pywt
+
+
+# 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
+
+
+t = np.load("testdata.npz")["t"]
+y = np.load("testdata.npz")["y"]
+dt = t[1] - t[0]
+
+# The wavelet we want to use
+mother = pywt.ContinuousWavelet("cmor1.5-1.0")
+
+# Parameters for the wavelet transform
+number_of_frequences: int = 25  # frequency bands
+frequency_range_min: float = 15  # Hz
+frequency_range_max: float = 200  # Hz
+
+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=y, scales=wave_scales, wavelet=mother, sampling_period=dt
+)
+
+cone_of_influence = calculate_cone_of_influence(dt, frequency_axis)
+
+complex_spectrum = mask_cone_of_influence(
+    complex_spectrum=complex_spectrum,
+    cone_of_influence=cone_of_influence,
+    fill_value=np.NaN,
+)
+
+plt.imshow(abs(complex_spectrum) ** 2, cmap="hot", aspect="auto")
+plt.colorbar()
+
+y_ticks, y_labels = get_y_ticks(
+    reduction_to_ticks=10, frequency_axis=frequency_axis, round=1
+)
+
+x_ticks, x_labels = get_x_ticks(
+    reduction_to_ticks=10, dt=dt, number_of_timesteps=complex_spectrum.shape[1], round=2
+)
+
+plt.yticks(y_ticks, y_labels)
+plt.xticks(x_ticks, x_labels)
+
+plt.xlabel("Time [sec]")
+plt.ylabel("Frequency [Hz]")
+plt.show()
+```
+
+!{image1.png](image1.png)