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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Demonstrate impact of whitening on source estimates
This example demonstrates the relationship between the noise covariance
estimate and the MNE / dSPM source amplitudes. It computes source estimates for
the SPM faces data and compares proper regularization with insufficient
regularization based on the methods described in [1]. The example demonstrates
that improper regularization can lead to overestimation of source amplitudes.
This example makes use of the previous, non-optimized code path that was used
before implementing the suggestions presented in [1].
<div class="alert alert-danger"><h4>Warning</h4><p>Please do not copy the patterns presented here for your own
             analysis, this is example is purely illustrative.</p></div>
<div class="alert alert-info"><h4>Note</h4><p>This example does quite a bit of processing, so even on a
          fast machine it can take a couple of minutes to complete.</p></div>
References
.. [1] Engemann D. and Gramfort A. (2015) Automated model selection in
       covariance estimation and spatial whitening of MEG and EEG signals,
       vol. 108, 328-342, NeuroImage.
Step1:  Get data
Step2:  Estimate covariances
Step3:  Show the resulting source estimates
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Python Code:
# Author: Denis A. Engemann <denis.engemann@gmail.com>
#
# License: BSD (3-clause)
import numpy as np
import matplotlib.pyplot as plt
import mne
from mne import io
from mne.datasets import spm_face
from mne.minimum_norm import apply_inverse, make_inverse_operator
from mne.cov import compute_covariance
print(__doc__)
Explanation: Demonstrate impact of whitening on source estimates
This example demonstrates the relationship between the noise covariance
estimate and the MNE / dSPM source amplitudes. It computes source estimates for
the SPM faces data and compares proper regularization with insufficient
regularization based on the methods described in [1]. The example demonstrates
that improper regularization can lead to overestimation of source amplitudes.
This example makes use of the previous, non-optimized code path that was used
before implementing the suggestions presented in [1].
<div class="alert alert-danger"><h4>Warning</h4><p>Please do not copy the patterns presented here for your own
             analysis, this is example is purely illustrative.</p></div>
<div class="alert alert-info"><h4>Note</h4><p>This example does quite a bit of processing, so even on a
          fast machine it can take a couple of minutes to complete.</p></div>
References
.. [1] Engemann D. and Gramfort A. (2015) Automated model selection in
       covariance estimation and spatial whitening of MEG and EEG signals,
       vol. 108, 328-342, NeuroImage.
End of explanation
data_path = spm_face.data_path()
subjects_dir = data_path + '/subjects'
raw_fname = data_path + '/MEG/spm/SPM_CTF_MEG_example_faces%d_3D.ds'
raw = io.read_raw_ctf(raw_fname % 1)  # Take first run
# To save time and memory for this demo, we'll just use the first
# 2.5 minutes (all we need to get 30 total events) and heavily
# resample 480->60 Hz (usually you wouldn't do either of these!)
raw = raw.crop(0, 150.).load_data()
picks = mne.pick_types(raw.info, meg=True, exclude='bads')
raw.filter(1, 20., n_jobs=1, fir_design='firwin')
events = mne.find_events(raw, stim_channel='UPPT001')
event_ids = {"faces": 1, "scrambled": 2}
tmin, tmax = -0.2, 0.5
baseline = None  # no baseline as high-pass is applied
reject = dict(mag=3e-12)
# Make source space
trans = data_path + '/MEG/spm/SPM_CTF_MEG_example_faces1_3D_raw-trans.fif'
src = mne.setup_source_space('spm', spacing='oct6', subjects_dir=subjects_dir,
                             add_dist=False)
bem = data_path + '/subjects/spm/bem/spm-5120-5120-5120-bem-sol.fif'
forward = mne.make_forward_solution(raw.info, trans, src, bem)
del src
# inverse parameters
conditions = 'faces', 'scrambled'
snr = 3.0
lambda2 = 1.0 / snr ** 2
method = 'dSPM'
clim = dict(kind='value', lims=[0, 2.5, 5])
Explanation: Get data
End of explanation
samples_epochs = 5, 15,
method = 'empirical', 'shrunk'
colors = 'steelblue', 'red'
evokeds = list()
stcs = list()
methods_ordered = list()
for n_train in samples_epochs:
    # estimate covs based on a subset of samples
    # make sure we have the same number of conditions.
    events_ = np.concatenate([events[events[:, 2] == id_][:n_train]
                              for id_ in [event_ids[k] for k in conditions]])
    events_ = events_[np.argsort(events_[:, 0])]
    epochs_train = mne.Epochs(raw, events_, event_ids, tmin, tmax, picks=picks,
                              baseline=baseline, preload=True, reject=reject,
                              decim=8)
    epochs_train.equalize_event_counts(event_ids)
    assert len(epochs_train) == 2 * n_train
    # We know some of these have too few samples, so suppress warning
    # with verbose='error'
    noise_covs = compute_covariance(
        epochs_train, method=method, tmin=None, tmax=0,  # baseline only
        return_estimators=True, verbose='error')  # returns list
    # prepare contrast
    evokeds = [epochs_train[k].average() for k in conditions]
    del epochs_train, events_
    # do contrast
    # We skip empirical rank estimation that we introduced in response to
    # the findings in reference [1] to use the naive code path that
    # triggered the behavior described in [1]. The expected true rank is
    # 274 for this dataset. Please do not do this with your data but
    # rely on the default rank estimator that helps regularizing the
    # covariance.
    stcs.append(list())
    methods_ordered.append(list())
    for cov in noise_covs:
        inverse_operator = make_inverse_operator(evokeds[0].info, forward,
                                                 cov, loose=0.2, depth=0.8,
                                                 rank=274)
        stc_a, stc_b = (apply_inverse(e, inverse_operator, lambda2, "dSPM",
                                      pick_ori=None) for e in evokeds)
        stc = stc_a - stc_b
        methods_ordered[-1].append(cov['method'])
        stcs[-1].append(stc)
    del inverse_operator, evokeds, cov, noise_covs, stc, stc_a, stc_b
del raw, forward  # save some memory
Explanation: Estimate covariances
End of explanation
fig, (axes1, axes2) = plt.subplots(2, 3, figsize=(9.5, 5))
for ni, (n_train, axes) in enumerate(zip(samples_epochs, (axes1, axes2))):
    # compute stc based on worst and best
    ax_dynamics = axes[1]
    for stc, ax, method, kind, color in zip(stcs[ni],
                                            axes[::2],
                                            methods_ordered[ni],
                                            ['best', 'worst'],
                                            colors):
        brain = stc.plot(subjects_dir=subjects_dir, hemi='both', clim=clim,
                         initial_time=0.175, background='w', foreground='k')
        brain.show_view('ven')
        im = brain.screenshot()
        brain.close()
        ax.axis('off')
        ax.get_xaxis().set_visible(False)
        ax.get_yaxis().set_visible(False)
        ax.imshow(im)
        ax.set_title('{0} ({1} epochs)'.format(kind, n_train * 2))
        # plot spatial mean
        stc_mean = stc.data.mean(0)
        ax_dynamics.plot(stc.times * 1e3, stc_mean,
                         label='{0} ({1})'.format(method, kind),
                         color=color)
        # plot spatial std
        stc_var = stc.data.std(0)
        ax_dynamics.fill_between(stc.times * 1e3, stc_mean - stc_var,
                                 stc_mean + stc_var, alpha=0.2, color=color)
    # signal dynamics worst and best
    ax_dynamics.set(title='{0} epochs'.format(n_train * 2),
                    xlabel='Time (ms)', ylabel='Source Activation (dSPM)',
                    xlim=(tmin * 1e3, tmax * 1e3), ylim=(-3, 3))
    ax_dynamics.legend(loc='upper left', fontsize=10)
fig.subplots_adjust(hspace=0.2, left=0.01, right=0.99, wspace=0.03)
mne.viz.utils.plt_show()
Explanation: Show the resulting source estimates
End of explanation
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