linted

Author: BalzaniEdoardo

docs/examples/plot_1D_basis_function.py

@@ -65,8 +65,10 @@
 # -----------------
 # Each basis type may necessitate specific hyperparameters for instantiation. For a comprehensive description,
 # please refer to the  [Code References](../../../reference/neurostatslib/basis). After instantiation, all classes
-# share the same syntax for basis evaluation. The following is an example of how to instantiate and
-# evaluate a log-spaced cosine raised function basis.
+# share the same syntax for basis evaluation.
+#
+# ### The Log-Spaced Raised Cosine Basis
+# The following is an example of how to instantiate and  evaluate a log-spaced cosine raised function basis.
 
 # Instantiate the basis noting that the `RaisedCosineBasisLog` does not require an `order` parameter
 raised_cosine_log = nsl.basis.RaisedCosineBasisLog(n_basis_funcs=10)
@@ -81,3 +83,84 @@
 plt.plot(samples, eval_basis)
 plt.show()
 
+# %%
+# ### The Fourier Basis
+# Another type of basis available is the Fourier Basis. Fourier basis are ideal to capture periodic and
+# quasi-periodic patterns. Such oscillatory, rhythmic behavior is a common signature of many neural signals.
+# Additionally, the Fourier basis has the advantage of being orthogonal, which simplifies the estimation and
+# interpretation of the model parameters, each of which will represent the relative contribution of a specific
+# oscillation frequency to the overall signal.
+
+
+# A Fourier basis can be instantiated with the usual syntax.
+# The user can pass the desired frequencies for the basis or
+# the frequencies will be set to np.arange(n_basis_funcs//2).
+# The number of basis function is required to be even.
+fourier_basis = nsl.basis.FourierBasis(n_freqs=4)
+
+# evaluate on equi-spaced samples
+samples, eval_basis = fourier_basis.evaluate_on_grid(1000)
+
+# plot the `sin` and `cos` separately
+plt.figure(figsize=(6, 3))
+plt.subplot(121)
+plt.title("Cos")
+plt.plot(samples, eval_basis[:, :4])
+plt.subplot(122)
+plt.title("Sin")
+plt.plot(samples, eval_basis[:, 4:])
+plt.tight_layout()
+
+# %%
+# !!! note "Fourier basis convolution and Fourier transform"
+#     The Fourier transform of a signal $ s(t) $ restricted to a temporal window $ [t_0,\;t_1] $ is
+#     $$ \\hat{x}(\\omega) = \\int_{t_0}^{t_1} s(\\tau) e^{-j\\omega \\tau} d\\tau. $$
+#     where $ e^{-j\\omega \\tau} = \\cos(\\omega \\tau) - j \\sin (\\omega \\tau) $.
+#
+#     When computing the cross-correlation of a signal with the Fourier basis functions,
+#     we essentially measure how well the signal correlates with sinusoids of different frequencies,
+#     within a specified temporal window. This process mirrors the operation performed by the Fourier transform.
+#     Therefore, it becomes clear that computing the cross-correlation of a signal with the Fourier basis defined here
+#     is equivalent to computing the discrete Fourier transform on a sliding window of the same size
+#     as that of the basis.
+
+n_samples = 1000
+n_freqs = 20
+
+# define a signal
+signal = np.random.normal(size=n_samples)
+
+# evaluate the basis
+_, eval_basis = nsl.basis.FourierBasis(n_freqs=n_freqs).evaluate_on_grid(n_samples)
+
+# compute the cross-corr with the signal and the basis
+# Note that we are inverting the time axis of the basis because we are aiming
+# for a cross-correlation, while np.convolve compute a convolution which would flip the time axis.
+xcorr = np.array(
+    [
+        np.convolve(eval_basis[::-1, k], signal, mode="valid")[0]
+        for k in range(2 * n_freqs - 1)
+    ]
+)
+
+# compute the power (add back sin(0 * t) = 0)
+fft_complex = np.fft.fft(signal)
+fft_amplitude = np.abs(fft_complex[:n_freqs])
+fft_phase = np.angle(fft_complex[:n_freqs])
+# compute the phase and amplitude from the convolution
+xcorr_phase = np.arctan2(np.hstack([[0], xcorr[n_freqs:]]), xcorr[:n_freqs])
+xcorr_aplitude = np.sqrt(xcorr[:n_freqs] ** 2 + np.hstack([[0], xcorr[n_freqs:]]) ** 2)
+
+fig, ax = plt.subplots(1, 2)
+ax[0].set_aspect("equal")
+ax[0].set_title("Signal amplitude")
+ax[0].scatter(fft_amplitude, xcorr_aplitude)
+ax[0].set_xlabel("FFT")
+ax[0].set_ylabel("cross-correlation")
+
+ax[1].set_aspect("equal")
+ax[1].set_title("Signal phase")
+ax[1].scatter(fft_phase, xcorr_phase)
+ax[1].set_xlabel("FFT")
+ax[1].set_ylabel("cross-correlation")
+plt.tight_layout()