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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:
 ==============================================
Compute effect-matched-spatial filtering (EMS)
==============================================
This example computes the EMS to reconstruct the time course of the
experimental effect as described in [1]_.
This technique is used to create spatial filters based on the difference
between two conditions. By projecting the trial onto the corresponding spatial
filters, surrogate single trials are created in which multi-sensor activity is
reduced to one time series which exposes experimental effects, if present.
We will first plot a trials x times image of the single trials and order the
trials by condition. A second plot shows the average time series for each
condition. Finally a topographic plot is created which exhibits the temporal
evolution of the spatial filters.
References
.. [1] Aaron Schurger, Sebastien Marti, and Stanislas Dehaene, "Reducing
       multi-sensor data to a single time course that reveals experimental
       effects", BMC Neuroscience 2013, 14
Step1:  Note that a similar transformation can be applied with compute_ems
However, this function replicates Schurger et al's original paper, and thus
applies the normalization outside a leave-one-out cross-validation, which we
recommend not to do.
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Python Code:
# Author: Denis Engemann <denis.engemann@gmail.com>
#         Jean-Remi King <jeanremi.king@gmail.com>
#
# License: BSD (3-clause)
import numpy as np
import matplotlib.pyplot as plt
import mne
from mne import io, EvokedArray
from mne.datasets import sample
from mne.decoding import EMS, compute_ems
from sklearn.cross_validation import StratifiedKFold
print(__doc__)
data_path = sample.data_path()
# Preprocess the data
raw_fname = data_path + '/MEG/sample/sample_audvis_filt-0-40_raw.fif'
event_fname = data_path + '/MEG/sample/sample_audvis_filt-0-40_raw-eve.fif'
event_ids = {'AudL': 1, 'VisL': 3}
# Read data and create epochs
raw = io.read_raw_fif(raw_fname, preload=True)
raw.filter(0.5, 45, l_trans_bandwidth='auto', h_trans_bandwidth='auto',
           filter_length='auto', phase='zero')
events = mne.read_events(event_fname)
picks = mne.pick_types(raw.info, meg='grad', eeg=False, stim=False, eog=True,
                       exclude='bads')
epochs = mne.Epochs(raw, events, event_ids, tmin=-0.2, tmax=0.5, picks=picks,
                    baseline=None, reject=dict(grad=4000e-13, eog=150e-6),
                    preload=True)
epochs.drop_bad()
epochs.pick_types(meg='grad')
# Setup the data to use it a scikit-learn way:
X = epochs.get_data()  # The MEG data
y = epochs.events[:, 2]  # The conditions indices
n_epochs, n_channels, n_times = X.shape
# Initialize EMS transformer
ems = EMS()
# Initialize the variables of interest
X_transform = np.zeros((n_epochs, n_times))  # Data after EMS transformation
filters = list()  # Spatial filters at each time point
# In the original paper, the cross-validation is a leave-one-out. However,
# we recommend using a Stratified KFold, because leave-one-out tends
# to overfit and cannot be used to estimate the variance of the
# prediction within a given fold.
for train, test in StratifiedKFold(y):
    # In the original paper, the z-scoring is applied outside the CV.
    # However, we recommend to apply this preprocessing inside the CV.
    # Note that such scaling should be done separately for each channels if the
    # data contains multiple channel types.
    X_scaled = X / np.std(X[train])
    # Fit and store the spatial filters
    ems.fit(X_scaled[train], y[train])
    # Store filters for future plotting
    filters.append(ems.filters_)
    # Generate the transformed data
    X_transform[test] = ems.transform(X_scaled[test])
# Average the spatial filters across folds
filters = np.mean(filters, axis=0)
# Plot individual trials
plt.figure()
plt.title('single trial surrogates')
plt.imshow(X_transform[y.argsort()], origin='lower', aspect='auto',
           extent=[epochs.times[0], epochs.times[-1], 1, len(X_transform)],
           cmap='RdBu_r')
plt.xlabel('Time (ms)')
plt.ylabel('Trials (reordered by condition)')
# Plot average response
plt.figure()
plt.title('Average EMS signal')
mappings = [(key, value) for key, value in event_ids.items()]
for key, value in mappings:
    ems_ave = X_transform[y == value]
    plt.plot(epochs.times, ems_ave.mean(0), label=key)
plt.xlabel('Time (ms)')
plt.ylabel('a.u.')
plt.legend(loc='best')
plt.show()
# Visualize spatial filters across time
evoked = EvokedArray(filters, epochs.info, tmin=epochs.tmin)
evoked.plot_topomap()
Explanation: ==============================================
Compute effect-matched-spatial filtering (EMS)
==============================================
This example computes the EMS to reconstruct the time course of the
experimental effect as described in [1]_.
This technique is used to create spatial filters based on the difference
between two conditions. By projecting the trial onto the corresponding spatial
filters, surrogate single trials are created in which multi-sensor activity is
reduced to one time series which exposes experimental effects, if present.
We will first plot a trials x times image of the single trials and order the
trials by condition. A second plot shows the average time series for each
condition. Finally a topographic plot is created which exhibits the temporal
evolution of the spatial filters.
References
.. [1] Aaron Schurger, Sebastien Marti, and Stanislas Dehaene, "Reducing
       multi-sensor data to a single time course that reveals experimental
       effects", BMC Neuroscience 2013, 14:122.
End of explanation
epochs.equalize_event_counts(event_ids)
X_transform, filters, classes = compute_ems(epochs)
Explanation: Note that a similar transformation can be applied with compute_ems
However, this function replicates Schurger et al's original paper, and thus
applies the normalization outside a leave-one-out cross-validation, which we
recommend not to do.
End of explanation
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