<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 =====================================================================
Spectro-temporal receptive field (STRF) estimation on continuous data
=====================================================================
This demonstrates how an encoding model can be fit with multiple continuous
inputs. In this case, we simulate the model behind a spectro-temporal receptive
field (or STRF). First, we create a linear filter that maps patterns in
spectro-temporal space onto an output, representing neural activity. We fit
a receptive field model that attempts to recover the original linear filter
that was used to create this data.
References
Estimation of spectro-temporal and spatio-temporal receptive fields using
modeling with continuous inputs is described in
Step1:  Load audio data
We'll read in the audio data from [3]_ in order to simulate a response.
In addition, we'll downsample the data along the time dimension in order to
speed up computation. Note that depending on the input values, this may
not be desired. For example if your input stimulus varies more quickly than
1/2 the sampling rate to which we are downsampling.
Step2:  Create a receptive field
We'll simulate a linear receptive field for a theoretical neural signal. This
defines how the signal will respond to power in this receptive field space.
Step3:  Simulate a neural response
Using this receptive field, we'll create an artificial neural response to
a stimulus.
To do this, we'll create a time-delayed version of the receptive field, and
then calculate the dot product between this and the stimulus. Note that this
is effectively doing a convolution between the stimulus and the receptive
field. See here &lt;https
Step4:  Fit a model to recover this receptive field
Finally, we'll use the 
Step5:  Visualize the effects of regularization
Above we fit a 
Step6:  Using different regularization types
In addition to the standard ridge regularization, the
Step7:  Compare model performance
Below we visualize the model performance of each regularization method
(ridge vs. Laplacian) for different levels of alpha. As you can see, the
Laplacian method performs better in general, because it imposes a smoothness
constraint along the time and feature dimensions of the coefficients.
This matches the "true" receptive field structure and results in a better
model fit.
Step8:  Plot the original STRF, and the one that we recovered with modeling.
<ASSISTANT_TASK:>
Python Code:
# Authors: Chris Holdgraf <choldgraf@gmail.com>
#          Eric Larson <larson.eric.d@gmail.com>
#
# License: BSD (3-clause)
# sphinx_gallery_thumbnail_number = 7
import numpy as np
import matplotlib.pyplot as plt
import mne
from mne.decoding import ReceptiveField, TimeDelayingRidge
from scipy.stats import multivariate_normal
from scipy.io import loadmat
from sklearn.preprocessing import scale
rng = np.random.RandomState(1337)  # To make this example reproducible
Explanation: =====================================================================
Spectro-temporal receptive field (STRF) estimation on continuous data
=====================================================================
This demonstrates how an encoding model can be fit with multiple continuous
inputs. In this case, we simulate the model behind a spectro-temporal receptive
field (or STRF). First, we create a linear filter that maps patterns in
spectro-temporal space onto an output, representing neural activity. We fit
a receptive field model that attempts to recover the original linear filter
that was used to create this data.
References
Estimation of spectro-temporal and spatio-temporal receptive fields using
modeling with continuous inputs is described in:
.. [1] Theunissen, F. E. et al. Estimating spatio-temporal receptive
       fields of auditory and visual neurons from their responses to
       natural stimuli. Network 12, 289-316 (2001).
.. [2] Willmore, B. & Smyth, D. Methods for first-order kernel
       estimation: simple-cell receptive fields from responses to
       natural scenes. Network 14, 553-77 (2003).
.. [3] Crosse, M. J., Di Liberto, G. M., Bednar, A. & Lalor, E. C. (2016).
       The Multivariate Temporal Response Function (mTRF) Toolbox:
       A MATLAB Toolbox for Relating Neural Signals to Continuous Stimuli.
       Frontiers in Human Neuroscience 10, 604.
       doi:10.3389/fnhum.2016.00604
.. [4] Holdgraf, C. R. et al. Rapid tuning shifts in human auditory cortex
       enhance speech intelligibility. Nature Communications, 7, 13654 (2016).
       doi:10.1038/ncomms13654
.. [5] Crosse, M. J., Di Liberto, G. M., Bednar, A. & Lalor, E. C. (2016).
       The Multivariate Temporal Response Function (mTRF) Toolbox:
       A MATLAB Toolbox for Relating Neural Signals to Continuous Stimuli.
       Frontiers in Human Neuroscience 10, 604. doi:10.3389/fnhum.2016.00604
End of explanation
# Read in audio that's been recorded in epochs.
path_audio = mne.datasets.mtrf.data_path()
data = loadmat(path_audio + '/speech_data.mat')
audio = data['spectrogram'].T
sfreq = float(data['Fs'][0, 0])
n_decim = 2
audio = mne.filter.resample(audio, down=n_decim, npad='auto')
sfreq /= n_decim
Explanation: Load audio data
We'll read in the audio data from [3]_ in order to simulate a response.
In addition, we'll downsample the data along the time dimension in order to
speed up computation. Note that depending on the input values, this may
not be desired. For example if your input stimulus varies more quickly than
1/2 the sampling rate to which we are downsampling.
End of explanation
n_freqs = 20
tmin, tmax = -0.1, 0.4
# To simulate the data we'll create explicit delays here
delays_samp = np.arange(np.round(tmin * sfreq),
                        np.round(tmax * sfreq) + 1).astype(int)
delays_sec = delays_samp / sfreq
freqs = np.linspace(50, 5000, n_freqs)
grid = np.array(np.meshgrid(delays_sec, freqs))
# We need data to be shaped as n_epochs, n_features, n_times, so swap axes here
grid = grid.swapaxes(0, -1).swapaxes(0, 1)
# Simulate a temporal receptive field with a Gabor filter
means_high = [.1, 500]
means_low = [.2, 2500]
cov = [[.001, 0], [0, 500000]]
gauss_high = multivariate_normal.pdf(grid, means_high, cov)
gauss_low = -1 * multivariate_normal.pdf(grid, means_low, cov)
weights = gauss_high + gauss_low  # Combine to create the "true" STRF
kwargs = dict(vmax=np.abs(weights).max(), vmin=-np.abs(weights).max(),
              cmap='RdBu_r', shading='gouraud')
fig, ax = plt.subplots()
ax.pcolormesh(delays_sec, freqs, weights, **kwargs)
ax.set(title='Simulated STRF', xlabel='Time Lags (s)', ylabel='Frequency (Hz)')
plt.setp(ax.get_xticklabels(), rotation=45)
plt.autoscale(tight=True)
mne.viz.tight_layout()
Explanation: Create a receptive field
We'll simulate a linear receptive field for a theoretical neural signal. This
defines how the signal will respond to power in this receptive field space.
End of explanation
# Reshape audio to split into epochs, then make epochs the first dimension.
n_epochs, n_seconds = 16, 5
audio = audio[:, :int(n_seconds * sfreq * n_epochs)]
X = audio.reshape([n_freqs, n_epochs, -1]).swapaxes(0, 1)
n_times = X.shape[-1]
# Delay the spectrogram according to delays so it can be combined w/ the STRF
# Lags will now be in axis 1, then we reshape to vectorize
delays = np.arange(np.round(tmin * sfreq),
                   np.round(tmax * sfreq) + 1).astype(int)
# Iterate through indices and append
X_del = np.zeros((len(delays),) + X.shape)
for ii, ix_delay in enumerate(delays):
    # These arrays will take/put particular indices in the data
    take = [slice(None)] * X.ndim
    put = [slice(None)] * X.ndim
    if ix_delay > 0:
        take[-1] = slice(None, -ix_delay)
        put[-1] = slice(ix_delay, None)
    elif ix_delay < 0:
        take[-1] = slice(-ix_delay, None)
        put[-1] = slice(None, ix_delay)
    X_del[ii][put] = X[take]
# Now set the delayed axis to the 2nd dimension
X_del = np.rollaxis(X_del, 0, 3)
X_del = X_del.reshape([n_epochs, -1, n_times])
n_features = X_del.shape[1]
weights_sim = weights.ravel()
# Simulate a neural response to the sound, given this STRF
y = np.zeros((n_epochs, n_times))
for ii, iep in enumerate(X_del):
    # Simulate this epoch and add random noise
    noise_amp = .002
    y[ii] = np.dot(weights_sim, iep) + noise_amp * rng.randn(n_times)
# Plot the first 2 trials of audio and the simulated electrode activity
X_plt = scale(np.hstack(X[:2]).T).T
y_plt = scale(np.hstack(y[:2]))
time = np.arange(X_plt.shape[-1]) / sfreq
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(6, 6), sharex=True)
ax1.pcolormesh(time, freqs, X_plt, vmin=0, vmax=4, cmap='Reds')
ax1.set_title('Input auditory features')
ax1.set(ylim=[freqs.min(), freqs.max()], ylabel='Frequency (Hz)')
ax2.plot(time, y_plt)
ax2.set(xlim=[time.min(), time.max()], title='Simulated response',
        xlabel='Time (s)', ylabel='Activity (a.u.)')
mne.viz.tight_layout()
Explanation: Simulate a neural response
Using this receptive field, we'll create an artificial neural response to
a stimulus.
To do this, we'll create a time-delayed version of the receptive field, and
then calculate the dot product between this and the stimulus. Note that this
is effectively doing a convolution between the stimulus and the receptive
field. See here &lt;https://en.wikipedia.org/wiki/Convolution&gt;_ for more
information.
End of explanation
# Create training and testing data
train, test = np.arange(n_epochs - 1), n_epochs - 1
X_train, X_test, y_train, y_test = X[train], X[test], y[train], y[test]
X_train, X_test, y_train, y_test = [np.rollaxis(ii, -1, 0) for ii in
                                    (X_train, X_test, y_train, y_test)]
# Model the simulated data as a function of the spectrogram input
alphas = np.logspace(-3, 3, 7)
scores = np.zeros_like(alphas)
models = []
for ii, alpha in enumerate(alphas):
    rf = ReceptiveField(tmin, tmax, sfreq, freqs, estimator=alpha)
    rf.fit(X_train, y_train)
    # Now make predictions about the model output, given input stimuli.
    scores[ii] = rf.score(X_test, y_test)
    models.append(rf)
times = rf.delays_ / float(rf.sfreq)
# Choose the model that performed best on the held out data
ix_best_alpha = np.argmax(scores)
best_mod = models[ix_best_alpha]
coefs = best_mod.coef_[0]
best_pred = best_mod.predict(X_test)[:, 0]
# Plot the original STRF, and the one that we recovered with modeling.
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(6, 3), sharey=True, sharex=True)
ax1.pcolormesh(delays_sec, freqs, weights, **kwargs)
ax2.pcolormesh(times, rf.feature_names, coefs, **kwargs)
ax1.set_title('Original STRF')
ax2.set_title('Best Reconstructed STRF')
plt.setp([iax.get_xticklabels() for iax in [ax1, ax2]], rotation=45)
plt.autoscale(tight=True)
mne.viz.tight_layout()
# Plot the actual response and the predicted response on a held out stimulus
time_pred = np.arange(best_pred.shape[0]) / sfreq
fig, ax = plt.subplots()
ax.plot(time_pred, y_test, color='k', alpha=.2, lw=4)
ax.plot(time_pred, best_pred, color='r', lw=1)
ax.set(title='Original and predicted activity', xlabel='Time (s)')
ax.legend(['Original', 'Predicted'])
plt.autoscale(tight=True)
mne.viz.tight_layout()
Explanation: Fit a model to recover this receptive field
Finally, we'll use the :class:mne.decoding.ReceptiveField class to recover
the linear receptive field of this signal. Note that properties of the
receptive field (e.g. smoothness) will depend on the autocorrelation in the
inputs and outputs.
End of explanation
# Plot model score for each ridge parameter
fig = plt.figure(figsize=(10, 4))
ax = plt.subplot2grid([2, len(alphas)], [1, 0], 1, len(alphas))
ax.plot(np.arange(len(alphas)), scores, marker='o', color='r')
ax.annotate('Best parameter', (ix_best_alpha, scores[ix_best_alpha]),
            (ix_best_alpha, scores[ix_best_alpha] - .1),
            arrowprops={'arrowstyle': '->'})
plt.xticks(np.arange(len(alphas)), ["%.0e" % ii for ii in alphas])
ax.set(xlabel="Ridge regularization value", ylabel="Score ($R^2$)",
       xlim=[-.4, len(alphas) - .6])
mne.viz.tight_layout()
# Plot the STRF of each ridge parameter
for ii, (rf, i_alpha) in enumerate(zip(models, alphas)):
    ax = plt.subplot2grid([2, len(alphas)], [0, ii], 1, 1)
    ax.pcolormesh(times, rf.feature_names, rf.coef_[0], **kwargs)
    plt.xticks([], [])
    plt.yticks([], [])
    plt.autoscale(tight=True)
fig.suptitle('Model coefficients / scores for many ridge parameters', y=1)
mne.viz.tight_layout()
Explanation: Visualize the effects of regularization
Above we fit a :class:mne.decoding.ReceptiveField model for one of many
values for the ridge regularization parameter. Here we will plot the model
score as well as the model coefficients for each value, in order to
visualize how coefficients change with different levels of regularization.
These issues as well as the STRF pipeline are described in detail
in [1], [2], and [4]_.
End of explanation
scores_lap = np.zeros_like(alphas)
models_lap = []
for ii, alpha in enumerate(alphas):
    estimator = TimeDelayingRidge(tmin, tmax, sfreq, reg_type='laplacian',
                                  alpha=alpha)
    rf = ReceptiveField(tmin, tmax, sfreq, freqs, estimator=estimator)
    rf.fit(X_train, y_train)
    # Now make predictions about the model output, given input stimuli.
    scores_lap[ii] = rf.score(X_test, y_test)
    models_lap.append(rf)
ix_best_alpha_lap = np.argmax(scores_lap)
Explanation: Using different regularization types
In addition to the standard ridge regularization, the
:class:mne.decoding.TimeDelayingRidge class also exposes
Laplacian &lt;https://en.wikipedia.org/wiki/Laplacian_matrix&gt;_ regularization
term as:
\begin{align}\left[\begin{matrix}
        1 & -1 &   &   & & \
       -1 &  2 & -1 &   & & \
          & -1 & 2 & -1 & & \
          & & \ddots & \ddots & \ddots & \
          & & & -1 & 2 & -1 \
          & & &    & -1 & 1\end{matrix}\right]\end{align}
This imposes a smoothness constraint of nearby time samples and/or features.
Quoting [5]_:
Tikhonov [identity] regularization (Equation 5) reduces overfitting by
   smoothing the TRF estimate in a way that is insensitive to
   the amplitude of the signal of interest. However, the Laplacian
   approach (Equation 6) reduces off-sample error whilst preserving
   signal amplitude (Lalor et al., 2006). As a result, this approach
   usually leads to an improved estimate of the system’s response (as
   indexed by MSE) compared to Tikhonov regularization.
End of explanation
fig = plt.figure(figsize=(10, 6))
ax = plt.subplot2grid([3, len(alphas)], [2, 0], 1, len(alphas))
ax.plot(np.arange(len(alphas)), scores_lap, marker='o', color='r')
ax.plot(np.arange(len(alphas)), scores, marker='o', color='0.5', ls=':')
ax.annotate('Best Laplacian', (ix_best_alpha_lap,
                               scores_lap[ix_best_alpha_lap]),
            (ix_best_alpha_lap, scores_lap[ix_best_alpha_lap] - .1),
            arrowprops={'arrowstyle': '->'})
ax.annotate('Best Ridge', (ix_best_alpha, scores[ix_best_alpha]),
            (ix_best_alpha, scores[ix_best_alpha] - .1),
            arrowprops={'arrowstyle': '->'})
plt.xticks(np.arange(len(alphas)), ["%.0e" % ii for ii in alphas])
ax.set(xlabel="Laplacian regularization value", ylabel="Score ($R^2$)",
       xlim=[-.4, len(alphas) - .6])
mne.viz.tight_layout()
# Plot the STRF of each ridge parameter
xlim = times[[0, -1]]
for ii, (rf_lap, rf, i_alpha) in enumerate(zip(models_lap, models, alphas)):
    ax = plt.subplot2grid([3, len(alphas)], [0, ii], 1, 1)
    ax.pcolormesh(times, rf_lap.feature_names, rf_lap.coef_[0], **kwargs)
    ax.set(xticks=[], yticks=[], xlim=xlim)
    if ii == 0:
        ax.set(ylabel='Laplacian')
    ax = plt.subplot2grid([3, len(alphas)], [1, ii], 1, 1)
    ax.pcolormesh(times, rf.feature_names, rf.coef_[0], **kwargs)
    ax.set(xticks=[], yticks=[], xlim=xlim)
    if ii == 0:
        ax.set(ylabel='Ridge')
fig.suptitle('Model coefficients / scores for laplacian regularization', y=1)
mne.viz.tight_layout()
Explanation: Compare model performance
Below we visualize the model performance of each regularization method
(ridge vs. Laplacian) for different levels of alpha. As you can see, the
Laplacian method performs better in general, because it imposes a smoothness
constraint along the time and feature dimensions of the coefficients.
This matches the "true" receptive field structure and results in a better
model fit.
End of explanation
rf = models[ix_best_alpha]
rf_lap = models_lap[ix_best_alpha_lap]
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(9, 3),
                                    sharey=True, sharex=True)
ax1.pcolormesh(delays_sec, freqs, weights, **kwargs)
ax2.pcolormesh(times, rf.feature_names, rf.coef_[0], **kwargs)
ax3.pcolormesh(times, rf_lap.feature_names, rf_lap.coef_[0], **kwargs)
ax1.set_title('Original STRF')
ax2.set_title('Best Ridge STRF')
ax3.set_title('Best Laplacian STRF')
plt.setp([iax.get_xticklabels() for iax in [ax1, ax2, ax3]], rotation=45)
plt.autoscale(tight=True)
mne.viz.tight_layout()
plt.show()
Explanation: Plot the original STRF, and the one that we recovered with modeling.
End of explanation
<END_TASK>