# Authors: Adam Li <adam2392@gmail.com>
# Santeri Ruuskanen <santeriruuskanen@gmail.com>
# Thomas S. Binns <t.s.binns@outlook.com>
#
# License: BSD (3-clause)
import inspect
import numpy as np
import xarray as xr
from mne.epochs import BaseEpochs
from mne.parallel import parallel_func
from mne.time_frequency import dpss_windows, tfr_array_morlet, tfr_array_multitaper
from mne.utils import logger, verbose
from ..base import EpochSpectralConnectivity, SpectralConnectivity
from ..utils import _check_multivariate_indices, check_indices, fill_doc
from .epochs import _compute_freq_mask
from .epochs_multivariate import (
_CON_METHOD_MAP_MULTIVARIATE,
_check_rank_input,
_gc_methods,
_multivariate_methods,
)
from .smooth import _create_kernel, _smooth_spectra
[docs]
@verbose
@fill_doc
def spectral_connectivity_time(
data,
freqs,
method="coh",
average=False,
indices=None,
sfreq=None,
fmin=None,
fmax=None,
fskip=0,
faverage=False,
sm_times=0,
sm_freqs=1,
sm_kernel="hanning",
padding=0,
mode="cwt_morlet",
mt_bandwidth=None,
n_cycles=7,
gc_n_lags=40,
rank=None,
decim=1,
n_jobs=1,
verbose=None,
):
r"""Compute time-frequency-domain connectivity measures.
This function computes spectral connectivity over time from epoched data.
The data may consist of a single epoch.
The connectivity method(s) are specified using the ``method`` parameter.
All methods are based on time-resolved estimates of the cross- and
power spectral densities (CSD/PSD) Sxy and Sxx, Syy.
Parameters
----------
data : array_like, shape (n_epochs, n_signals, n_times) | Epochs
The data from which to compute connectivity.
freqs : array_like
Array of frequencies of interest for time-frequency decomposition.
Only the frequencies within the range specified by ``fmin`` and
``fmax`` are used.
method : str | list of str
Connectivity measure(s) to compute. These can be
``['coh', 'mic', 'mim', 'plv', 'ciplv', 'pli', 'wpli', 'gc',
'gc_tr']``. These are:
* 'coh' : Coherence
* 'mic' : Maximised Imaginary part of Coherency (MIC)
* 'mim' : Multivariate Interaction Measure (MIM)
* 'plv' : Phase-Locking Value (PLV)
* 'ciplv' : Corrected imaginary Phase-Locking Value
* 'pli' : Phase-Lag Index
* 'wpli' : Weighted Phase-Lag Index
* 'gc' : State-space Granger Causality (GC)
* 'gc_tr' : State-space GC on time-reversed signals
Multivariate methods (``['mic', 'mim', 'gc', 'gc_tr]``) cannot be
called with the other methods.
average : bool
Average connectivity scores over epochs. If ``True``, output will be
an instance of :class:`SpectralConnectivity`, otherwise
:class:`EpochSpectralConnectivity`.
indices : tuple of array_like | None
Two arrays with indices of connections for which to compute
connectivity. If a bivariate method is called, each array for the seeds
and targets should contain the channel indices for the each bivariate
connection. If a multivariate method is called, each array for the
seeds and targets should consist of nested arrays containing
the channel indices for each multivariate connection. If None,
connections between all channels are computed, unless a Granger
causality method is called, in which case an error is raised.
sfreq : float
The sampling frequency. Required if data is not
:class:`Epochs <mne.Epochs>`.
fmin : float | tuple of float | None
The lower frequency of interest. Multiple bands are defined using
a tuple, e.g., ``(8., 20.)`` for two bands with 8 Hz and 20 Hz lower
bounds. If `None`, the lowest frequency in ``freqs`` is used.
fmax : float | tuple of float | None
The upper frequency of interest. Multiple bands are defined using
a tuple, e.g. ``(13., 30.)`` for two band with 13 Hz and 30 Hz upper
bounds. If `None`, the highest frequency in ``freqs`` is used.
fskip : int
Omit every ``(fskip + 1)``-th frequency bin to decimate in frequency
domain.
faverage : bool
Average connectivity scores for each frequency band. If `True`,
the output ``freqs`` will be an array of the median frequencies of each
band.
sm_times : float
Amount of time to consider for the temporal smoothing in seconds.
If zero, no temporal smoothing is applied.
sm_freqs : int
Number of points for frequency smoothing. By default, 1 is used which
is equivalent to no smoothing.
sm_kernel : {'square', 'hanning'}
Smoothing kernel type. Choose either 'square' or 'hanning'.
padding : float
Amount of time to consider as padding at the beginning and end of each
epoch in seconds. See Notes for more information.
mode : str
Time-frequency decomposition method. Can be either: 'multitaper', or
'cwt_morlet'. See :func:`mne.time_frequency.tfr_array_multitaper` and
:func:`mne.time_frequency.tfr_array_morlet` for reference.
mt_bandwidth : float | None
Product between the temporal window length (in seconds) and the full
frequency bandwidth (in Hz). This product can be seen as the surface
of the window on the time/frequency plane and controls the frequency
bandwidth (thus the frequency resolution) and the number of good
tapers. See :func:`mne.time_frequency.tfr_array_multitaper`
documentation.
n_cycles : float | array_like of float
Number of cycles in the wavelet, either a fixed number or one per
frequency. The number of cycles ``n_cycles`` and the frequencies of
interest ``cwt_freqs`` define the temporal window length. For details,
see :func:`mne.time_frequency.tfr_array_morlet` documentation.
gc_n_lags : int
Number of lags to use for the vector autoregressive model when
computing Granger causality. Higher values increase computational cost,
but reduce the degree of spectral smoothing in the results. Only used
if ``method`` contains any of ``['gc', 'gc_tr']``.
rank : tuple of array | None
Two arrays with the rank to project the seed and target data to,
respectively, using singular value decomposition. If `None`, the rank
of the data is computed and projected to. Only used if ``method``
contains any of ``['mic', 'mim', 'gc', 'gc_tr']``.
decim : int
To reduce memory usage, decimation factor after time-frequency
decomposition. Returns ``tfr[…, ::decim]``.
n_jobs : int
Number of connections to compute in parallel. Memory mapping must be
activated. Please see the Notes section for details.
%(verbose)s
Returns
-------
con : instance of Connectivity | list
Computed connectivity measure(s). An instance of
:class:`EpochSpectralConnectivity`, :class:`SpectralConnectivity`
or a list of instances corresponding to connectivity measures if
several connectivity measures are specified.
The shape of each connectivity dataset is (n_epochs, n_cons, n_freqs).
When "indices" is None and a bivariate method is called,
"n_cons = n_signals ** 2", or if a multivariate method is called
"n_cons = 1". When "indices" is specified, "n_con = len(indices[0])"
for bivariate and multivariate methods.
See Also
--------
mne_connectivity.spectral_connectivity_epochs
mne_connectivity.SpectralConnectivity
mne_connectivity.EpochSpectralConnectivity
Notes
-----
Please note that the interpretation of the measures in this function
depends on the data and underlying assumptions and does not necessarily
reflect a causal relationship between brain regions.
The connectivity measures are computed over time within each epoch and
optionally averaged over epochs. High connectivity values indicate that
the phase coupling (interpreted as estimated connectivity) differences
between signals stay consistent over time.
The spectral densities can be estimated using a multitaper method with
digital prolate spheroidal sequence (DPSS) windows, or a continuous wavelet
transform using Morlet wavelets. The spectral estimation mode is specified
using the ``mode`` parameter.
When using the multitaper spectral estimation method, the
cross-spectral density is computed separately for each taper and aggregated
using a weighted average, where the weights correspond to the concentration
ratios between the DPSS windows.
Spectral estimation using multitaper or Morlet wavelets introduces edge
effects that depend on the length of the wavelet. To remove edge effects,
the parameter ``padding`` can be used to prune the edges of the signal.
Please see the documentation of
:func:`mne.time_frequency.tfr_array_multitaper` and
:func:`mne.time_frequency.tfr_array_morlet` for details on wavelet length
(i.e., time window length).
By default, the connectivity between all signals is computed (only
connections corresponding to the lower-triangular part of the connectivity
matrix). If one is only interested in the connectivity between some
signals, the "indices" parameter can be used. For example, to compute the
connectivity between the signal with index 0 and signals "2, 3, 4" (a total
of 3 connections) one can use the following::
indices = (np.array([0, 0, 0]), # row indices
np.array([2, 3, 4])) # col indices
con = spectral_connectivity_time(data, method='coh',
indices=indices, ...)
In this case ``con.get_data().shape = (3, n_freqs)``. The connectivity
scores are in the same order as defined indices.
For multivariate methods, this is handled differently. If "indices" is
None, connectivity between all signals will be computed and a single
connectivity spectrum will be returned (this is not possible if a Granger
causality method is called). If "indices" is specified, seed and target
indices for each connection should be specified as nested array-likes. For
example, to compute the connectivity between signals (0, 1) -> (2, 3) and
(0, 1) -> (4, 5), indices should be specified as::
indices = (np.array([[0, 1], [0, 1]]), # seeds
np.array([[2, 3], [4, 5]])) # targets
More information on working with multivariate indices and handling
connections where the number of seeds and targets are not equal can be
found in the :doc:`../auto_examples/handling_ragged_arrays` example.
**Supported Connectivity Measures**
The connectivity method(s) is specified using the ``method`` parameter. The
following methods are supported (note: ``E[]`` denotes average over
epochs). Multiple measures can be computed at once by using a list/tuple,
e.g., ``['coh', 'pli']`` to compute coherence and PLI.
'coh' : Coherence given by::
| E[Sxy] |
C = ---------------------
sqrt(E[Sxx] * E[Syy])
'mic' : Maximised Imaginary part of Coherency (MIC)
:footcite:`EwaldEtAl2012` given by:
:math:`MIC=\Large{\frac{\boldsymbol{\alpha}^T \boldsymbol{E \beta}}
{\parallel\boldsymbol{\alpha}\parallel \parallel\boldsymbol{\beta}
\parallel}}`
where: :math:`\boldsymbol{E}` is the imaginary part of the
transformed cross-spectral density between seeds and targets; and
:math:`\boldsymbol{\alpha}` and :math:`\boldsymbol{\beta}` are
eigenvectors for the seeds and targets, such that
:math:`\boldsymbol{\alpha}^T \boldsymbol{E \beta}` maximises
connectivity between the seeds and targets.
'mim' : Multivariate Interaction Measure (MIM)
:footcite:`EwaldEtAl2012` given by:
:math:`MIM=tr(\boldsymbol{EE}^T)`
'plv' : Phase-Locking Value (PLV) :footcite:`LachauxEtAl1999` given
by::
PLV = |E[Sxy/|Sxy|]|
'ciplv' : Corrected imaginary PLV (ciPLV) :footcite:`BrunaEtAl2018`
given by::
|E[Im(Sxy/|Sxy|)]|
ciPLV = ------------------------------------
sqrt(1 - |E[real(Sxy/|Sxy|)]| ** 2)
'pli' : Phase Lag Index (PLI) :footcite:`StamEtAl2007` given by::
PLI = |E[sign(Im(Sxy))]|
'wpli' : Weighted Phase Lag Index (WPLI) :footcite:`VinckEtAl2011`
given by::
|E[Im(Sxy)]|
WPLI = ------------------
E[|Im(Sxy)|]
'gc' : State-space Granger Causality (GC) :footcite:`BarnettSeth2015`
given by:
:math:`GC = ln\Large{(\frac{\lvert\boldsymbol{S}_{tt}\rvert}{\lvert
\boldsymbol{S}_{tt}-\boldsymbol{H}_{ts}\boldsymbol{\Sigma}_{ss
\lvert t}\boldsymbol{H}_{ts}^*\rvert}})`,
where: :math:`s` and :math:`t` represent the seeds and targets,
respectively; :math:`\boldsymbol{H}` is the spectral transfer
function; :math:`\boldsymbol{\Sigma}` is the residuals matrix of
the autoregressive model; and :math:`\boldsymbol{S}` is
:math:`\boldsymbol{\Sigma}` transformed by :math:`\boldsymbol{H}`.
'gc_tr' : State-space GC on time-reversed signals
:footcite:`BarnettSeth2015,WinklerEtAl2016` given by the same equation
as for 'gc', but where the autocovariance sequence from which the
autoregressive model is produced is transposed to mimic the reversal of
the original signal in time.
Parallel computation can be activated by setting the ``n_jobs`` parameter.
Under the hood, this utilizes the ``joblib`` library. For effective
parallelization, you should activate memory mapping in MNE-Python by
setting ``MNE_MEMMAP_MIN_SIZE`` and ``MNE_CACHE_DIR``. Activating memory
mapping will make ``joblib`` store arrays greater than the minimum size on
disc, and forego direct RAM access for more efficient processing.
For example, in your code, run
mne.set_config('MNE_MEMMAP_MIN_SIZE', '10M')
mne.set_config('MNE_CACHE_DIR', '/dev/shm')
When ``MNE_MEMMAP_MIN_SIZE=None``, the underlying joblib implementation
results in pickling and unpickling the whole array each time a pair of
indices is accessed, which is slow, compared to memory mapping the array.
This function is based on the ``frites.conn.conn_spec`` implementation in
Frites.
.. versionadded:: 0.3
References
----------
.. footbibliography::
"""
events = None
event_id = None
# extract data from Epochs object
if isinstance(data, BaseEpochs):
names = data.ch_names
sfreq = data.info["sfreq"]
events = data.events
event_id = data.event_id
# Extract metadata from the Epochs data structure.
# Make Annotations persist through by adding them to the metadata.
metadata = data.metadata
if metadata is None:
annots_in_metadata = False
else:
annots_in_metadata = all(
name not in metadata.columns
for name in ["annot_onset", "annot_duration", "annot_description"]
)
if hasattr(data, "annotations") and not annots_in_metadata:
data.add_annotations_to_metadata(overwrite=True)
metadata = data.metadata
# XXX: remove logic once support for mne<1.6 is dropped
kwargs = dict()
if "copy" in inspect.getfullargspec(data.get_data).kwonlyargs:
kwargs["copy"] = False
data = data.get_data(**kwargs)
n_epochs, n_signals, n_times = data.shape
else:
data = np.asarray(data)
n_epochs, n_signals, n_times = data.shape
names = np.arange(0, n_signals)
metadata = None
if sfreq is None:
raise ValueError(
"Sampling frequency (sfreq) is required with " "array input."
)
# check that method is a list
if isinstance(method, str):
method = [method]
# defaults for fmin and fmax
if fmin is None:
fmin = np.min(freqs)
logger.info("Fmin was not specified. Using fmin=min(freqs)")
if fmax is None:
fmax = np.max(freqs)
logger.info("Fmax was not specified. Using fmax=max(freqs).")
fmin = np.array((fmin,), dtype=float).ravel()
fmax = np.array((fmax,), dtype=float).ravel()
if len(fmin) != len(fmax):
raise ValueError("fmin and fmax must have the same length")
if np.any(fmin > fmax):
raise ValueError("fmax must be larger than fmin")
if len(fmin) != 1 and any(this_method in _gc_methods for this_method in method):
raise ValueError(
"computing Granger causality on multiple frequency "
"bands is not yet supported"
)
if any(this_method in _multivariate_methods for this_method in method):
if not all(this_method in _multivariate_methods for this_method in method):
raise ValueError(
"bivariate and multivariate connectivity methods cannot be "
"used in the same function call"
)
multivariate_con = True
else:
multivariate_con = False
# convert kernel width in time to samples
if isinstance(sm_times, (int, float)):
sm_times = int(np.round(sm_times * sfreq))
# convert frequency smoothing from hz to samples
if isinstance(sm_freqs, (int, float)):
sm_freqs = int(np.round(max(sm_freqs, 1)))
# temporal decimation
if isinstance(decim, int):
sm_times = int(np.round(sm_times / decim))
sm_times = max(sm_times, 1)
# Create smoothing kernel
kernel = _create_kernel(sm_times, sm_freqs, kernel=sm_kernel)
# get indices of pairs of (group) regions
if indices is None:
if multivariate_con:
if any(this_method in _gc_methods for this_method in method):
raise ValueError(
"indices must be specified when computing Granger "
"causality, as all-to-all connectivity is not supported"
)
logger.info("using all indices for multivariate connectivity")
# indices expected to be a masked array, even if not ragged
indices_use = (
np.arange(n_signals, dtype=int)[np.newaxis, :],
np.arange(n_signals, dtype=int)[np.newaxis, :],
)
indices_use = np.ma.masked_array(indices_use, mask=False, fill_value=-1)
else:
logger.info("only using indices for lower-triangular matrix")
indices_use = np.tril_indices(n_signals, k=-1)
else:
if multivariate_con:
# pad ragged indices and mask the invalid entries
indices_use = _check_multivariate_indices(indices, n_signals)
if any(this_method in _gc_methods for this_method in method):
for seed, target in zip(indices_use[0], indices_use[1]):
intersection = np.intersect1d(
seed.compressed(), target.compressed()
)
if intersection.size > 0:
raise ValueError(
"seed and target indices must not intersect when "
"computing Granger causality"
)
# make sure padded indices are stored in the connectivity object
# create a copy so that `indices_use` can be modified
indices = (indices_use[0].copy(), indices_use[1].copy())
else:
indices_use = check_indices(indices)
n_cons = len(indices_use[0])
# unique signals for which we actually need to compute the CSD of
if multivariate_con:
signals_use = np.unique(indices_use.compressed())
remapping = {ch_i: sig_i for sig_i, ch_i in enumerate(signals_use)}
remapped_inds = indices_use.copy()
# multivariate functions expect seed/target remapping
for idx in signals_use:
remapped_inds[indices_use == idx] = remapping[idx]
source_idx = remapped_inds[0]
target_idx = remapped_inds[1]
max_n_channels = len(indices_use[0][0])
else:
# no indices remapping required for bivariate functions
signals_use = np.unique(np.r_[indices_use[0], indices_use[1]])
source_idx = indices_use[0].copy()
target_idx = indices_use[1].copy()
max_n_channels = len(indices_use[0])
# check rank input and compute data ranks if necessary
if multivariate_con:
rank = _check_rank_input(rank, data, indices_use)
else:
rank = None
gc_n_lags = None
# check freqs
if isinstance(freqs, (int, float)):
freqs = [freqs]
# array conversion
freqs = np.asarray(freqs)
# check order for multiple frequencies
if len(freqs) >= 2:
delta_f = np.diff(freqs)
increase = np.all(delta_f > 0)
assert increase, "Frequencies should be in increasing order"
# check that freqs corresponds to at least n_cycles cycles
dur = float(n_times) / sfreq
cycle_freq = n_cycles / dur
if np.any(freqs < cycle_freq):
raise ValueError(
"At least one value in n_cycles corresponds to a"
"wavelet longer than the signal. Use less cycles, "
"higher frequencies, or longer epochs."
)
# check for Nyquist
if np.any(freqs > sfreq / 2):
raise ValueError(
f"Frequencies {freqs[freqs > sfreq / 2]} Hz are "
f"larger than Nyquist = {sfreq / 2:.2f} Hz"
)
# compute frequency mask based on specified min/max and decimation factor
freq_mask = _compute_freq_mask(freqs, fmin, fmax, fskip)
# the frequency points where we compute connectivity
freqs = freqs[freq_mask]
# compute central frequencies
_f = xr.DataArray(np.arange(len(freqs)), dims=("freqs",), coords=(freqs,))
foi_s = _f.sel(freqs=fmin, method="nearest").data
foi_e = _f.sel(freqs=fmax, method="nearest").data
foi_idx = np.c_[foi_s, foi_e]
f_vec = freqs[foi_idx].mean(1)
if faverage:
n_freqs = len(fmin)
out_freqs = f_vec
else:
n_freqs = len(freqs)
out_freqs = freqs
conn = dict()
conn_patterns = dict()
for m in method:
conn[m] = np.zeros((n_epochs, n_cons, n_freqs))
# patterns shape of [epochs x seeds/targets x cons x channels x freqs]
conn_patterns[m] = np.full(
(n_epochs, 2, n_cons, max_n_channels, n_freqs), np.nan
)
logger.info("Connectivity computation...")
# parameters to pass to the connectivity function
call_params = dict(
method=method,
kernel=kernel,
foi_idx=foi_idx,
source_idx=source_idx,
target_idx=target_idx,
signals_use=signals_use,
mode=mode,
sfreq=sfreq,
freqs=freqs,
faverage=faverage,
n_cycles=n_cycles,
mt_bandwidth=mt_bandwidth,
gc_n_lags=gc_n_lags,
rank=rank,
decim=decim,
padding=padding,
kw_cwt={},
kw_mt={},
n_jobs=n_jobs,
verbose=verbose,
multivariate_con=multivariate_con,
)
for epoch_idx in np.arange(n_epochs):
logger.info(f" Processing epoch {epoch_idx+1} / {n_epochs} ...")
scores, patterns = _spectral_connectivity(data[epoch_idx], **call_params)
for m in method:
conn[m][epoch_idx] = np.stack(scores[m], axis=0)
if multivariate_con and patterns[m] is not None:
conn_patterns[m][epoch_idx] = np.stack(patterns[m], axis=0)
for m in method:
if np.isnan(conn_patterns[m]).all():
conn_patterns[m] = None
else:
# transpose to [seeds/targets x epochs x cons x channels x freqs]
conn_patterns[m] = conn_patterns[m].transpose((1, 0, 2, 3, 4))
if indices is None and not multivariate_con:
conn_flat = conn
conn = dict()
for m in method:
this_conn = np.zeros(
(n_epochs, n_signals, n_signals) + conn_flat[m].shape[2:],
dtype=conn_flat[m].dtype,
)
this_conn[:, source_idx, target_idx] = conn_flat[m]
this_conn = this_conn.reshape(
(
n_epochs,
n_signals**2,
)
+ conn_flat[m].shape[2:]
)
conn[m] = this_conn
# create the connectivity containers
out = []
for m in method:
store_params = {
"data": conn[m],
"patterns": conn_patterns[m],
"freqs": out_freqs,
"n_nodes": n_signals,
"names": names,
"indices": indices,
"method": method,
"spec_method": mode,
"events": events,
"event_id": event_id,
"metadata": metadata,
"rank": rank,
"n_lags": gc_n_lags if m in _gc_methods else None,
}
if average:
store_params["data"] = np.mean(store_params["data"], axis=0)
if conn_patterns[m] is not None:
store_params["patterns"] = np.mean(store_params["patterns"], axis=1)
out.append(SpectralConnectivity(**store_params))
else:
out.append(EpochSpectralConnectivity(**store_params))
logger.info("[Connectivity computation done]")
# return the object instead of list of length one
if len(out) == 1:
return out[0]
else:
return out
def _spectral_connectivity(
data,
method,
kernel,
foi_idx,
source_idx,
target_idx,
signals_use,
mode,
sfreq,
freqs,
faverage,
n_cycles,
mt_bandwidth,
gc_n_lags,
rank,
decim,
padding,
kw_cwt,
kw_mt,
n_jobs,
verbose,
multivariate_con,
):
"""Estimate time-resolved connectivity for one epoch.
Parameters
----------
data : array_like, shape (n_channels, n_times)
Time-series data.
method : list of str
List of connectivity metrics to compute.
kernel : array_like, shape (n_sm_fres, n_sm_times)
Smoothing kernel.
foi_idx : array_like, shape (n_foi, 2)
Upper and lower bound indices of frequency bands.
source_idx : array_like, shape (n_cons,) or (n_cons, n_channels)
Defines the signal pairs of interest together with ``target_idx``.
target_idx : array_like, shape (n_cons,) or (n_cons, n_channels)
Defines the signal pairs of interest together with ``source_idx``.
signals_use : list of int
The unique signals on which connectivity is to be computed.
mode : str
Time-frequency transformation method.
sfreq : float
Sampling frequency.
freqs : array_like
Array of frequencies of interest for time-frequency decomposition.
Only the frequencies within the range specified by ``fmin`` and
``fmax`` are used.
faverage : bool
Average over frequency bands.
n_cycles : float | array_like of float
Number of cycles in the wavelet, either a fixed number or one per
frequency.
mt_bandwidth : float | None
Multitaper time-bandwidth.
gc_n_lags : int
Number of lags to use for the vector autoregressive model when
computing Granger causality.
rank : tuple of array
Ranks to project the seed and target data to.
decim : int
Decimation factor after time-frequency
decomposition.
padding : float
Amount of time to consider as padding at the beginning and end of each
epoch in seconds.
multivariate_con : bool
Whether or not multivariate connectivity is to be computed.
Returns
-------
scores : dict
Dictionary containing the connectivity estimates corresponding to the
metrics in ``method``. Each element is an array of shape (n_cons,
n_freqs) or (n_cons, n_fbands) if ``faverage`` is `True`.
patterns : dict
Dictionary containing the connectivity patterns (for reconstructing the
connectivity components in source-space) corresponding to the metrics
in ``method``, if multivariate methods are called, else an empty
dictionary. Each element is an array of shape (2, n_channels, n_freqs)
or (2, n_channels, 1) if ``faverage`` is `True`, where 2 corresponds to
the seed and target signals (respectively).
"""
n_cons = len(source_idx)
data = np.expand_dims(data, axis=0)
if mode == "cwt_morlet":
out = tfr_array_morlet(
data,
sfreq,
freqs,
n_cycles=n_cycles,
output="complex",
decim=decim,
n_jobs=n_jobs,
**kw_cwt,
)
out = np.expand_dims(out, axis=2) # same dims with multitaper
weights = None
elif mode == "multitaper":
out = tfr_array_multitaper(
data,
sfreq,
freqs,
n_cycles=n_cycles,
time_bandwidth=mt_bandwidth,
output="complex",
decim=decim,
n_jobs=n_jobs,
**kw_mt,
)
if isinstance(n_cycles, (int, float)):
n_cycles = [n_cycles] * len(freqs)
mt_bandwidth = mt_bandwidth if mt_bandwidth else 4
n_tapers = int(np.floor(mt_bandwidth - 1))
weights = np.zeros((n_tapers, len(freqs), out.shape[-1]))
for i, (f, n_c) in enumerate(zip(freqs, n_cycles)):
window_length = np.arange(0.0, n_c / float(f), 1.0 / sfreq).shape[0]
half_nbw = mt_bandwidth / 2.0
n_tapers = int(np.floor(mt_bandwidth - 1))
_, eigvals = dpss_windows(window_length, half_nbw, n_tapers, sym=False)
weights[:, i, :] = np.sqrt(eigvals[:, np.newaxis])
# weights have shape (n_tapers, n_freqs, n_times)
else:
raise ValueError("Mode must be 'cwt_morlet' or 'multitaper'.")
out = np.squeeze(out, axis=0)
if padding:
if padding < 0:
raise ValueError(f"Padding cannot be negative, got {padding}.")
if padding >= data.shape[-1] / sfreq / 2:
raise ValueError(
f"Padding cannot be larger than half of data " f"length, got {padding}."
)
pad_idx = int(np.floor(padding * sfreq / decim))
out = out[..., pad_idx:-pad_idx]
weights = weights[..., pad_idx:-pad_idx] if weights is not None else None
# compute for each connectivity method
scores = {}
patterns = {}
conn = _parallel_con(
out,
method,
kernel,
foi_idx,
source_idx,
target_idx,
signals_use,
gc_n_lags,
rank,
n_jobs,
verbose,
n_cons,
faverage,
weights,
multivariate_con,
)
for i, m in enumerate(method):
if multivariate_con:
scores[m] = conn[0][i]
patterns[m] = conn[1][i] if conn[1][i] is not None else None
else:
scores[m] = [out[i] for out in conn]
patterns[m] = None
return scores, patterns
###############################################################################
###############################################################################
# TIME-RESOLVED CORE FUNCTIONS
###############################################################################
###############################################################################
def _parallel_con(
w,
method,
kernel,
foi_idx,
source_idx,
target_idx,
signals_use,
gc_n_lags,
rank,
n_jobs,
verbose,
total,
faverage,
weights,
multivariate_con,
):
"""Compute spectral connectivity in parallel.
Parameters
----------
w : array_like, shape (n_chans, n_tapers, n_freqs, n_times)
Time-frequency data (complex signal).
method : list of str
List of connectivity metrics to compute.
kernel : array_like, shape (n_sm_fres, n_sm_times)
Smoothing kernel.
foi_idx : array_like, shape (n_foi, 2)
Upper and lower bound indices of frequency bands.
source_idx : array_like, shape (n_cons,) or (n_cons, n_channels)
Defines the signal pairs of interest together with ``target_idx``.
target_idx : array_like, shape (n_cons,) or (n_cons, n_channels)
Defines the signal pairs of interest together with ``source_idx``.
signals_use : list of int
The unique signals on which connectivity is to be computed.
gc_n_lags : int
Number of lags to use for the vector autoregressive model when
computing Granger causality.
rank : tuple of array of int
Ranks to project the seed and target data to.
n_jobs : int
Number of parallel jobs.
total : int
Number of pairs of signals.
faverage : bool
Average over frequency bands.
weights : array_like, shape (n_tapers, n_freqs, n_times)
Multitaper weights.
multivariate_con : bool
Whether or not multivariate connectivity is being computed.
Returns
-------
out : tuple of list of array
Connectivity estimates for each signal pair, method, and frequency or
frequency band. If bivariate methods are called, the output is a tuple
of a list of arrays containing the connectivity scores. If multivariate
methods are called, the output is a tuple of lists containing arrays
for the connectivity scores and patterns, respectively.
"""
if "coh" in method:
# psd
if weights is not None:
psd = weights * w
psd = psd * np.conj(psd)
psd = psd.real.sum(axis=1)
psd = psd * 2 / (weights * weights.conj()).real.sum(axis=0)
else:
psd = w.real**2 + w.imag**2
psd = np.squeeze(psd, axis=1)
# smooth
psd = _smooth_spectra(psd, kernel)
else:
psd = None
if not multivariate_con:
# only show progress if verbosity level is DEBUG
if verbose != "DEBUG" and verbose != "debug" and verbose != 10:
total = None
# define the function to compute in parallel
parallel, my_pairwise_con, n_jobs = parallel_func(
_pairwise_con, n_jobs=n_jobs, verbose=verbose, total=total
)
return tuple(
parallel(
my_pairwise_con(
w, psd, s, t, method, kernel, foi_idx, faverage, weights
)
for s, t in zip(source_idx, target_idx)
)
)
return _multivariate_con(
w,
source_idx,
target_idx,
signals_use,
method,
kernel,
foi_idx,
faverage,
weights,
gc_n_lags,
rank,
n_jobs,
)
def _pairwise_con(w, psd, x, y, method, kernel, foi_idx, faverage, weights):
"""Compute spectral connectivity metrics between two signals.
Parameters
----------
w : array_like, shape (n_chans, n_tapers, n_freqs, n_times)
Time-frequency data.
psd : array_like, shape (n_chans, n_freqs, n_times)
Power spectrum between signals ``x`` and ``y``.
x : int
Channel index.
y : int
Channel index.
method : str
Connectivity method.
kernel : array_like, shape (n_sm_fres, n_sm_times)
Smoothing kernel.
foi_idx : array_like, shape (n_foi, 2)
Upper and lower bound indices of frequency bands.
faverage : bool
Average over frequency bands.
weights : array_like, shape (n_tapers, n_freqs, n_times) | None
Multitaper weights.
Returns
-------
out : list
List of connectivity estimates between signals ``x`` and ``y``
corresponding to the methods in ``method``. Each element is an array
with shape (n_freqs,) or (n_fbands) depending on ``faverage``.
"""
w_x, w_y = w[x], w[y]
if weights is not None:
s_xy = np.sum(weights * w_x * np.conj(weights * w_y), axis=0)
s_xy = s_xy * 2 / (weights * np.conj(weights)).real.sum(axis=0)
else:
s_xy = w_x * np.conj(w_y)
s_xy = np.squeeze(s_xy, axis=0)
s_xy = _smooth_spectra(s_xy, kernel)
out = []
conn_func = {"plv": _plv, "ciplv": _ciplv, "pli": _pli, "wpli": _wpli, "coh": _coh}
for m in method:
if m == "coh":
s_xx = psd[x]
s_yy = psd[y]
out.append(conn_func[m](s_xx, s_yy, s_xy))
else:
out.append(conn_func[m](s_xy))
for i, _ in enumerate(out):
# mean inside frequency sliding window (if needed)
if isinstance(foi_idx, np.ndarray) and faverage:
out[i] = _foi_average(out[i], foi_idx)
# squeeze time dimension
out[i] = out[i].squeeze(axis=-1)
return out
def _multivariate_con(
w,
seeds,
targets,
signals_use,
method,
kernel,
foi_idx,
faverage,
weights,
gc_n_lags,
rank,
n_jobs,
):
"""Compute spectral connectivity metrics between multiple signals.
Parameters
----------
w : array_like, shape (n_chans, n_tapers, n_freqs, n_times)
Time-frequency data.
seeds : array, shape of (n_cons, n_channels)
Seed channel indices. ``n_channels`` is the largest number of channels
across all connections, with missing entries padded with ``-1``.
targets : array, shape of (n_cons, n_channels)
Target channel indices. ``n_channels`` is the largest number of
channels across all connections, with missing entries padded with
``-1``.
signals_use : list of int
The unique signals on which connectivity is to be computed.
method : str
Connectivity method.
kernel : array_like, shape (n_sm_fres, n_sm_times)
Smoothing kernel.
foi_idx : array_like, shape (n_foi, 2)
Upper and lower bound indices of frequency bands.
faverage : bool
Average over frequency bands.
weights : array_like, shape (n_tapers, n_freqs, n_times) | None
Multitaper weights.
gc_n_lags : int
Number of lags to use for the vector autoregressive model when
computing Granger causality.
rank : tuple of array, shape of (2, n_cons)
Ranks to project the seed and target data to.
n_jobs : int
Number of jobs to run in parallel.
Returns
-------
scores : list
List of connectivity scores between seed and target signals for each
connectivity method. Each element is an array with shape (n_freqs,) or
(n_fbands) depending on ``faverage``.
patterns : list
List of connectivity patterns between seed and target signals for each
connectivity method. Each element is an array of length 2 corresponding
to the seed and target patterns, respectively, each with shape
(n_channels, n_freqs) or (n_channels, n_fbands)
depending on ``faverage``. ``n_channels`` is the largest number of
channels across all connections, with missing entries padded with
``np.nan``.
"""
csd = []
for x in signals_use:
for y in signals_use:
w_x, w_y = w[x], w[y]
if weights is not None:
s_xy = np.sum(weights * w_x * np.conj(weights * w_y), axis=0)
s_xy = s_xy * 2 / (weights * np.conj(weights)).real.sum(axis=0)
else:
s_xy = w_x * np.conj(w_y)
s_xy = np.squeeze(s_xy, axis=0)
csd.append(_smooth_spectra(s_xy, kernel).mean(axis=-1))
csd = np.array(csd)
# initialise connectivity estimators and add CSD information
conn = []
for m in method:
call_params = {
"n_signals": len(signals_use),
"n_cons": len(seeds),
"n_freqs": csd.shape[1],
"n_times": 0,
"n_jobs": n_jobs,
}
if m in _gc_methods:
call_params["n_lags"] = gc_n_lags
con_est = _CON_METHOD_MAP_MULTIVARIATE[m](**call_params)
for con_i, con_csd in enumerate(csd):
con_est.accumulate(con_i, con_csd)
conn.append(con_est)
# compute connectivity
scores = []
patterns = []
for con_est in conn:
con_est.compute_con((seeds, targets), rank)
scores.append(con_est.con_scores[..., np.newaxis])
patterns.append(con_est.patterns)
if patterns[-1] is not None:
patterns[-1] = patterns[-1][..., np.newaxis]
for i, _ in enumerate(scores):
# mean inside frequency sliding window (if needed)
if isinstance(foi_idx, np.ndarray) and faverage:
scores[i] = _foi_average(scores[i], foi_idx)
if patterns[i] is not None:
patterns[i] = _foi_average(patterns[i], foi_idx)
# squeeze time dimension
scores[i] = scores[i].squeeze(axis=-1)
if patterns[i] is not None:
patterns[i] = patterns[i].squeeze(axis=-1)
return scores, patterns
def _plv(s_xy):
"""Compute phase-locking value given the cross power spectral density.
Parameters
----------
s_xy : array-like, shape (n_freqs, n_times)
The cross PSD between channel 'x' and channel 'y' across
frequency and time points.
Returns
-------
plv : array-like, shape (n_freqs, n_times)
The estimated PLV.
"""
s_xy = s_xy / np.abs(s_xy)
plv = np.abs(s_xy.mean(axis=-1, keepdims=True))
return plv
def _ciplv(s_xy):
"""Compute corrected imaginary phase-locking value.
Parameters
----------
s_xy : array-like, shape (n_freqs, n_times)
The cross PSD between channel 'x' and channel 'y' across
frequency and time points.
Returns
-------
ciplv : array-like, shape (n_freqs, n_times)
The estimated ciPLV.
"""
s_xy = s_xy / np.abs(s_xy)
rplv = np.abs(np.mean(np.real(s_xy), axis=-1, keepdims=True))
iplv = np.abs(np.mean(np.imag(s_xy), axis=-1, keepdims=True))
ciplv = iplv / (np.sqrt(1 - rplv**2))
return ciplv
def _pli(s_xy):
"""Compute phase-lag index given the cross power spectral density.
Parameters
----------
s_xy : array-like, shape (n_freqs, n_times)
The cross PSD between channel 'x' and channel 'y' across
frequency and time points.
Returns
-------
pli : array-like, shape (n_freqs, n_times)
The estimated PLI.
"""
pli = np.abs(np.mean(np.sign(np.imag(s_xy)), axis=-1, keepdims=True))
return pli
def _wpli(s_xy):
"""Compute weighted phase-lag index given the cross power spectral density.
Parameters
----------
s_xy : array-like, shape (n_freqs, n_times)
The cross PSD between channel 'x' and channel 'y' across
frequency and time points.
Returns
-------
wpli : array-like, shape (n_freqs, n_times)
The estimated wPLI.
"""
con_num = np.abs(s_xy.imag.mean(axis=-1, keepdims=True))
con_den = np.mean(np.abs(s_xy.imag), axis=-1, keepdims=True)
wpli = con_num / con_den
return wpli
def _coh(s_xx, s_yy, s_xy):
"""Compute coherence given the cross spectral density and PSD.
Parameters
----------
s_xx : array-like, shape (n_freqs, n_times)
The PSD of channel 'x'.
s_yy : array-like, shape (n_freqs, n_times)
The PSD of channel 'y'.
s_xy : array-like, shape (n_freqs, n_times)
The cross PSD between channel 'x' and channel 'y' across
frequency and time points.
Returns
-------
coh : array-like, shape (n_freqs, n_times)
The estimated COH.
"""
con_num = np.abs(s_xy.mean(axis=-1, keepdims=True))
con_den = np.sqrt(
s_xx.mean(axis=-1, keepdims=True) * s_yy.mean(axis=-1, keepdims=True)
)
coh = con_num / con_den
return coh
def _compute_csd(x, y, weights):
"""Compute cross spectral density between signals x and y."""
if weights is not None:
s_xy = np.sum(weights * x * np.conj(weights * y), axis=-3)
s_xy = s_xy * 2 / (weights * np.conj(weights)).real.sum(axis=-3)
else:
s_xy = x * np.conj(y)
s_xy = np.squeeze(s_xy, axis=-3)
return s_xy
def _foi_average(conn, foi_idx):
"""Average inside frequency bands.
The frequency dimension should be located at -2.
Parameters
----------
conn : array_like, shape (..., n_freqs, n_times)
Connectivity estimate array.
foi_idx : array_like, shape (n_foi, 2)
Upper and lower frequency bounds of each frequency band.
Returns
-------
conn_f : np.ndarray, shape (..., n_fbands, n_times)
Connectivity estimate array, averaged within frequency bands.
"""
# get the number of foi
n_foi = foi_idx.shape[0]
# get input shape and replace n_freqs with the number of foi
sh = list(conn.shape)
sh[-2] = n_foi
# compute average
conn_f = np.zeros(sh, dtype=conn.dtype)
for n_f, (f_s, f_e) in enumerate(foi_idx):
f_e += 1 if f_s == f_e else f_e
conn_f[..., n_f, :] = conn[..., f_s:f_e, :].mean(-2)
return conn_f