This example demonstrates the different time-frequency estimation methods on simulated data. It shows the time-frequency resolution trade-off and the problem of estimation variance. In addition it highlights alternative functions for generating TFRs without averaging across trials, or by operating on numpy arrays.

```
# Authors: Hari Bharadwaj <hari@nmr.mgh.harvard.edu>
# Denis Engemann <denis.engemann@gmail.com>
# Chris Holdgraf <choldgraf@berkeley.edu>
#
# License: BSD (3-clause)
import numpy as np
from matplotlib import pyplot as plt
from mne import create_info, EpochsArray
from mne.baseline import rescale
from mne.time_frequency import (tfr_multitaper, tfr_stockwell, tfr_morlet,
tfr_array_morlet)
print(__doc__)
```

We’ll simulate data with a known spectro-temporal structure.

```
sfreq = 1000.0
ch_names = ['SIM0001', 'SIM0002']
ch_types = ['grad', 'grad']
info = create_info(ch_names=ch_names, sfreq=sfreq, ch_types=ch_types)
n_times = int(sfreq) # 1 second long epochs
n_epochs = 40
seed = 42
rng = np.random.RandomState(seed)
noise = rng.randn(n_epochs, len(ch_names), n_times)
# Add a 50 Hz sinusoidal burst to the noise and ramp it.
t = np.arange(n_times, dtype=np.float) / sfreq
signal = np.sin(np.pi * 2. * 50. * t) # 50 Hz sinusoid signal
signal[np.logical_or(t < 0.45, t > 0.55)] = 0. # Hard windowing
on_time = np.logical_and(t >= 0.45, t <= 0.55)
signal[on_time] *= np.hanning(on_time.sum()) # Ramping
data = noise + signal
reject = dict(grad=4000)
events = np.empty((n_epochs, 3), dtype=int)
first_event_sample = 100
event_id = dict(sin50hz=1)
for k in range(n_epochs):
events[k, :] = first_event_sample + k * n_times, 0, event_id['sin50hz']
epochs = EpochsArray(data=data, info=info, events=events, event_id=event_id,
reject=reject)
```

Out:

```
40 matching events found
0 projection items activated
0 bad epochs dropped
```

Below we’ll demonstrate the output of several TFR functions in MNE:

`mne.time_frequency.tfr_multitaper()`

`mne.time_frequency.tfr_stockwell()`

`mne.time_frequency.tfr_morlet()`

First we’ll use the multitaper method for calculating the TFR.
This creates several orthogonal tapering windows in the TFR estimation,
which reduces variance. We’ll also show some of the parameters that can be
tweaked (e.g., `time_bandwidth`

) that will result in different multitaper
properties, and thus a different TFR. You can trade time resolution or
frequency resolution or both in order to get a reduction in variance.

```
freqs = np.arange(5., 100., 3.)
vmin, vmax = -3., 3. # Define our color limits.
```

**(1) Least smoothing (most variance/background fluctuations).**

```
n_cycles = freqs / 2.
time_bandwidth = 2.0 # Least possible frequency-smoothing (1 taper)
power = tfr_multitaper(epochs, freqs=freqs, n_cycles=n_cycles,
time_bandwidth=time_bandwidth, return_itc=False)
# Plot results. Baseline correct based on first 100 ms.
power.plot([0], baseline=(0., 0.1), mode='mean', vmin=vmin, vmax=vmax,
title='Sim: Least smoothing, most variance')
```

Out:

```
Applying baseline correction (mode: mean)
```

**(2) Less frequency smoothing, more time smoothing.**

```
n_cycles = freqs # Increase time-window length to 1 second.
time_bandwidth = 4.0 # Same frequency-smoothing as (1) 3 tapers.
power = tfr_multitaper(epochs, freqs=freqs, n_cycles=n_cycles,
time_bandwidth=time_bandwidth, return_itc=False)
# Plot results. Baseline correct based on first 100 ms.
power.plot([0], baseline=(0., 0.1), mode='mean', vmin=vmin, vmax=vmax,
title='Sim: Less frequency smoothing, more time smoothing')
```

Out:

```
Applying baseline correction (mode: mean)
```

**(3) Less time smoothing, more frequency smoothing.**

```
n_cycles = freqs / 2.
time_bandwidth = 8.0 # Same time-smoothing as (1), 7 tapers.
power = tfr_multitaper(epochs, freqs=freqs, n_cycles=n_cycles,
time_bandwidth=time_bandwidth, return_itc=False)
# Plot results. Baseline correct based on first 100 ms.
power.plot([0], baseline=(0., 0.1), mode='mean', vmin=vmin, vmax=vmax,
title='Sim: Less time smoothing, more frequency smoothing')
```

Out:

```
Applying baseline correction (mode: mean)
```

Stockwell uses a Gaussian window to balance temporal and spectral resolution.
Importantly, frequency bands are phase-normalized, hence strictly comparable
with regard to timing, and, the input signal can be recoverd from the
transform in a lossless way if we disregard numerical errors. In this case,
we control the spectral / temporal resolution by specifying different widths
of the gaussian window using the `width`

parameter.

```
fig, axs = plt.subplots(1, 3, figsize=(15, 5), sharey=True)
fmin, fmax = freqs[[0, -1]]
for width, ax in zip((0.2, .7, 3.0), axs):
power = tfr_stockwell(epochs, fmin=fmin, fmax=fmax, width=width)
power.plot([0], baseline=(0., 0.1), mode='mean', axes=ax, show=False,
colorbar=False)
ax.set_title('Sim: Using S transform, width = {:0.1f}'.format(width))
plt.tight_layout()
```

Out:

```
The input signal is shorter (1000) than "n_fft" (1024). Applying zero padding.
Applying baseline correction (mode: mean)
The input signal is shorter (1000) than "n_fft" (1024). Applying zero padding.
Applying baseline correction (mode: mean)
The input signal is shorter (1000) than "n_fft" (1024). Applying zero padding.
Applying baseline correction (mode: mean)
```

Finally, show the TFR using morlet wavelets, which are a sinusoidal wave
with a gaussian envelope. We can control the balance between spectral and
temporal resolution with the `n_cycles`

parameter, which defines the
number of cycles to include in the window.

```
fig, axs = plt.subplots(1, 3, figsize=(15, 5), sharey=True)
all_n_cycles = [1, 3, freqs / 2.]
for n_cycles, ax in zip(all_n_cycles, axs):
power = tfr_morlet(epochs, freqs=freqs,
n_cycles=n_cycles, return_itc=False)
power.plot([0], baseline=(0., 0.1), mode='mean', vmin=vmin, vmax=vmax,
axes=ax, show=False, colorbar=False)
n_cycles = 'scaled by freqs' if not isinstance(n_cycles, int) else n_cycles
ax.set_title('Sim: Using Morlet wavelet, n_cycles = %s' % n_cycles)
plt.tight_layout()
```

Out:

```
Applying baseline correction (mode: mean)
Applying baseline correction (mode: mean)
Applying baseline correction (mode: mean)
```

It is also possible to calculate a TFR without averaging across trials.
We can do this by using `average=False`

. In this case, an instance of
`mne.time_frequency.EpochsTFR`

is returned.

```
n_cycles = freqs / 2.
power = tfr_morlet(epochs, freqs=freqs,
n_cycles=n_cycles, return_itc=False, average=False)
print(type(power))
avgpower = power.average()
avgpower.plot([0], baseline=(0., 0.1), mode='mean', vmin=vmin, vmax=vmax,
title='Using Morlet wavelets and EpochsTFR', show=False)
```

Out:

```
<class 'mne.time_frequency.tfr.EpochsTFR'>
Applying baseline correction (mode: mean)
```

MNE also has versions of the functions above which operate on numpy arrays
instead of MNE objects. They expect inputs of the shape
`(n_epochs, n_channels, n_times)`

. They will also return a numpy array
of shape `(n_epochs, n_channels, n_freqs, n_times)`

.

```
power = tfr_array_morlet(epochs.get_data(), sfreq=epochs.info['sfreq'],
freqs=freqs, n_cycles=n_cycles,
output='avg_power')
# Baseline the output
rescale(power, epochs.times, (0., 0.1), mode='mean', copy=False)
fig, ax = plt.subplots()
mesh = ax.pcolormesh(epochs.times * 1000, freqs, power[0],
cmap='RdBu_r', vmin=vmin, vmax=vmax)
ax.set_title('TFR calculated on a numpy array')
ax.set(ylim=freqs[[0, -1]], xlabel='Time (ms)')
fig.colorbar(mesh)
plt.tight_layout()
plt.show()
```

Out:

```
Applying baseline correction (mode: mean)
```

**Total running time of the script:** ( 0 minutes 12.330 seconds)