# Authors: Martin Luessi <mluessi@nmr.mgh.harvard.edu>
# Denis A. Engemann <denis.engemann@gmail.com>
# Adam Li <adam2392@gmail.com>
# Thomas S. Binns <t.s.binns@outlook.com>
#
# License: BSD (3-clause)
import inspect
from functools import partial
import numpy as np
from mne.epochs import BaseEpochs
from mne.parallel import parallel_func
from mne.source_estimate import _BaseSourceEstimate
from mne.time_frequency.multitaper import (
_compute_mt_params,
_csd_from_mt,
_mt_spectra,
_psd_from_mt,
_psd_from_mt_adaptive,
)
from mne.time_frequency.tfr import cwt, morlet
from mne.utils import _arange_div, _check_option, _time_mask, logger, verbose, warn
from ..base import SpectralConnectivity, SpectroTemporalConnectivity
from ..utils import _check_multivariate_indices, check_indices, fill_doc
from .epochs_bivariate import _CON_METHOD_MAP_BIVARIATE
from .epochs_multivariate import (
_CON_METHOD_MAP_MULTIVARIATE,
_check_rank_input,
_gc_methods,
_multivariate_methods,
)
def _compute_freqs(n_times, sfreq, cwt_freqs, mode):
from scipy.fft import rfftfreq
# get frequencies of interest for the different modes
if mode in ("multitaper", "fourier"):
# fmin fmax etc is only supported for these modes
# decide which frequencies to keep
freqs_all = rfftfreq(n_times, 1.0 / sfreq)
elif mode == "cwt_morlet":
# cwt_morlet mode
if cwt_freqs is None:
raise ValueError("define frequencies of interest using " "cwt_freqs")
else:
cwt_freqs = cwt_freqs.astype(np.float64)
if any(cwt_freqs > (sfreq / 2.0)):
raise ValueError(
"entries in cwt_freqs cannot be " "larger than Nyquist (sfreq / 2)"
)
freqs_all = cwt_freqs
else:
raise ValueError("mode has an invalid value")
return freqs_all
def _compute_freq_mask(freqs_all, fmin, fmax, fskip):
# create a frequency mask for all bands
freq_mask = np.zeros(len(freqs_all), dtype=bool)
for f_lower, f_upper in zip(fmin, fmax):
freq_mask |= (freqs_all >= f_lower) & (freqs_all <= f_upper)
# possibly skip frequency points
for pos in range(fskip):
freq_mask[pos + 1 :: fskip + 1] = False
return freq_mask
def _prepare_connectivity(
epoch_block,
times_in,
tmin,
tmax,
fmin,
fmax,
sfreq,
indices,
method,
mode,
fskip,
n_bands,
cwt_freqs,
faverage,
):
"""Check and precompute dimensions of results data."""
first_epoch = epoch_block[0]
# get the data size and time scale
n_signals, n_times_in, times_in, warn_times = _get_and_verify_data_sizes(
first_epoch, sfreq, times=times_in
)
n_times_in = len(times_in)
if tmin is not None and tmin < times_in[0]:
warn(
"start time tmin=%0.2f s outside of the time scope of the data "
"[%0.2f s, %0.2f s]" % (tmin, times_in[0], times_in[-1])
)
if tmax is not None and tmax > times_in[-1]:
warn(
"stop time tmax=%0.2f s outside of the time scope of the data "
"[%0.2f s, %0.2f s]" % (tmax, times_in[0], times_in[-1])
)
mask = _time_mask(times_in, tmin, tmax, sfreq=sfreq)
tmin_idx, tmax_idx = np.where(mask)[0][[0, -1]]
tmax_idx += 1
tmin_true = times_in[tmin_idx]
tmax_true = times_in[tmax_idx - 1] # time of last point used
times = times_in[tmin_idx:tmax_idx]
n_times = len(times)
if any(this_method in _multivariate_methods for this_method in method):
multivariate_con = True
else:
multivariate_con = False
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"
)
else:
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")
# only compute r for lower-triangular region
indices_use = np.tril_indices(n_signals, -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"
)
else:
indices_use = check_indices(indices)
# number of connectivities to compute
n_cons = len(indices_use[0])
logger.info(" computing connectivity for %d connections" % n_cons)
logger.info(
" using t=%0.3fs..%0.3fs for estimation (%d points)"
% (tmin_true, tmax_true, n_times)
)
# check that fmin corresponds to at least 5 cycles
dur = float(n_times) / sfreq
five_cycle_freq = 5.0 / dur
if len(fmin) == 1 and fmin[0] == -np.inf:
# we use the 5 cycle freq. as default
fmin = np.array([five_cycle_freq])
else:
if np.any(fmin < five_cycle_freq):
warn(
"fmin=%0.3f Hz corresponds to %0.3f < 5 cycles "
"based on the epoch length %0.3f sec, need at least %0.3f "
"sec epochs or fmin=%0.3f. Spectrum estimate will be "
"unreliable."
% (
np.min(fmin),
dur * np.min(fmin),
dur,
5.0 / np.min(fmin),
five_cycle_freq,
)
)
# compute frequencies to analyze based on number of samples,
# sampling rate, specified wavelet frequencies and mode
freqs = _compute_freqs(n_times, sfreq, cwt_freqs, mode)
# compute the 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]
n_freqs = len(freqs)
# get the freq. indices and points for each band
freq_idx_bands = [
np.where((freqs >= fl) & (freqs <= fu))[0] for fl, fu in zip(fmin, fmax)
]
freqs_bands = [freqs[freq_idx] for freq_idx in freq_idx_bands]
# make sure we don't have empty bands
for i, n_f_band in enumerate([len(f) for f in freqs_bands]):
if n_f_band == 0:
raise ValueError(
"There are no frequency points between "
"%0.1fHz and %0.1fHz. Change the band "
"specification (fmin, fmax) or the "
"frequency resolution." % (fmin[i], fmax[i])
)
if n_bands == 1:
logger.info(
" frequencies: %0.1fHz..%0.1fHz (%d points)"
% (freqs_bands[0][0], freqs_bands[0][-1], n_freqs)
)
else:
logger.info(" computing connectivity for the bands:")
for i, bfreqs in enumerate(freqs_bands):
logger.info(
" band %d: %0.1fHz..%0.1fHz "
"(%d points)" % (i + 1, bfreqs[0], bfreqs[-1], len(bfreqs))
)
if faverage:
logger.info(" connectivity scores will be averaged for " "each band")
return (
n_cons,
times,
n_times,
times_in,
n_times_in,
tmin_idx,
tmax_idx,
n_freqs,
freq_mask,
freqs,
freqs_bands,
freq_idx_bands,
n_signals,
indices_use,
warn_times,
)
def _assemble_spectral_params(
mode,
n_times,
mt_adaptive,
mt_bandwidth,
sfreq,
mt_low_bias,
cwt_n_cycles,
cwt_freqs,
freqs,
freq_mask,
):
"""Prepare time-frequency decomposition."""
spectral_params = dict(eigvals=None, window_fun=None, wavelets=None)
n_tapers = None
n_times_spectrum = 0
if mode == "multitaper":
window_fun, eigvals, mt_adaptive = _compute_mt_params(
n_times, sfreq, mt_bandwidth, mt_low_bias, mt_adaptive
)
spectral_params.update(window_fun=window_fun, eigvals=eigvals)
elif mode == "fourier":
logger.info(" using FFT with a Hanning window to estimate " "spectra")
spectral_params.update(window_fun=np.hanning(n_times), eigvals=1.0)
elif mode == "cwt_morlet":
logger.info(" using CWT with Morlet wavelets to estimate " "spectra")
# reformat cwt_n_cycles if we have removed some frequencies
# using fmin, fmax, fskip
cwt_n_cycles = np.array((cwt_n_cycles,), dtype=float).ravel()
if len(cwt_n_cycles) > 1:
if len(cwt_n_cycles) != len(cwt_freqs):
raise ValueError(
"cwt_n_cycles must be float or an "
"array with the same size as cwt_freqs"
)
cwt_n_cycles = cwt_n_cycles[freq_mask]
# get the Morlet wavelets
spectral_params.update(
wavelets=morlet(sfreq, freqs, n_cycles=cwt_n_cycles, zero_mean=True)
)
n_times_spectrum = n_times
else:
raise ValueError("mode has an invalid value")
return spectral_params, mt_adaptive, n_times_spectrum, n_tapers
########################################################################
# Connectivity estimators base class
class _AbstractConEstBase(object):
"""ABC for connectivity estimators."""
def start_epoch(self):
raise NotImplementedError("start_epoch method not implemented")
def accumulate(self, con_idx, csd_xy):
raise NotImplementedError("accumulate method not implemented")
def combine(self, other):
raise NotImplementedError("combine method not implemented")
def compute_con(self, con_idx, n_epochs):
raise NotImplementedError("compute_con method not implemented")
########################################################################
def _epoch_spectral_connectivity(
data,
sig_idx,
tmin_idx,
tmax_idx,
sfreq,
method,
mode,
window_fun,
eigvals,
wavelets,
freq_mask,
mt_adaptive,
idx_map,
n_cons,
block_size,
psd,
accumulate_psd,
con_method_types,
con_methods,
n_signals,
n_signals_use,
n_times,
gc_n_lags,
accumulate_inplace=True,
):
"""Estimate connectivity for one epoch (see spectral_connectivity)."""
if any(this_method in _multivariate_methods for this_method in method):
n_con_signals = n_signals_use**2
else:
n_con_signals = n_cons
if wavelets is not None:
n_times_spectrum = n_times
n_freqs = len(wavelets)
else:
n_times_spectrum = 0
n_freqs = np.sum(freq_mask)
if not accumulate_inplace:
# instantiate methods only for this epoch (used in parallel mode)
con_methods = []
for mtype in con_method_types:
method_params = list(inspect.signature(mtype).parameters)
if "n_signals" in method_params:
# if it's a multivariate connectivity method
if "n_lags" in method_params:
# if it's a Granger causality method
con_methods.append(
mtype(
n_signals_use, n_cons, n_freqs, n_times_spectrum, gc_n_lags
)
)
else:
# if it's a coherence method
con_methods.append(
mtype(n_signals_use, n_cons, n_freqs, n_times_spectrum)
)
else:
con_methods.append(mtype(n_cons, n_freqs, n_times_spectrum))
_check_option("mode", mode, ("cwt_morlet", "multitaper", "fourier"))
if len(sig_idx) == n_signals:
# we use all signals: use a slice for faster indexing
sig_idx = slice(None, None)
# compute tapered spectra
x_t = list()
this_psd = list()
for this_data in data:
if mode in ("multitaper", "fourier"):
if isinstance(this_data, _BaseSourceEstimate):
_mt_spectra_partial = partial(_mt_spectra, dpss=window_fun, sfreq=sfreq)
this_x_t = this_data.transform_data(
_mt_spectra_partial,
idx=sig_idx,
tmin_idx=tmin_idx,
tmax_idx=tmax_idx,
)
else:
this_x_t, _ = _mt_spectra(
this_data[sig_idx, tmin_idx:tmax_idx], window_fun, sfreq
)
if mt_adaptive:
# compute PSD and adaptive weights
_this_psd, weights = _psd_from_mt_adaptive(
this_x_t, eigvals, freq_mask, return_weights=True
)
# only keep freqs of interest
this_x_t = this_x_t[:, :, freq_mask]
else:
# do not use adaptive weights
this_x_t = this_x_t[:, :, freq_mask]
if mode == "multitaper":
weights = np.sqrt(eigvals)[np.newaxis, :, np.newaxis]
else:
# hack to so we can sum over axis=-2
weights = np.array([1.0])[:, None, None]
if accumulate_psd:
_this_psd = _psd_from_mt(this_x_t, weights)
else: # mode == 'cwt_morlet'
if isinstance(this_data, _BaseSourceEstimate):
cwt_partial = partial(cwt, Ws=wavelets, use_fft=True, mode="same")
this_x_t = this_data.transform_data(
cwt_partial, idx=sig_idx, tmin_idx=tmin_idx, tmax_idx=tmax_idx
)
else:
this_x_t = cwt(
this_data[sig_idx, tmin_idx:tmax_idx],
wavelets,
use_fft=True,
mode="same",
)
_this_psd = (this_x_t * this_x_t.conj()).real
x_t.append(this_x_t)
if accumulate_psd:
this_psd.append(_this_psd)
x_t = np.concatenate(x_t, axis=0)
if accumulate_psd:
this_psd = np.concatenate(this_psd, axis=0)
# accumulate or return psd
if accumulate_psd:
if accumulate_inplace:
psd += this_psd
else:
psd = this_psd
else:
psd = None
# tell the methods that a new epoch starts
for method in con_methods:
method.start_epoch()
# accumulate connectivity scores
if mode in ["multitaper", "fourier"]:
for i in range(0, n_con_signals, block_size):
n_extra = max(0, i + block_size - n_con_signals)
con_idx = slice(i, i + block_size - n_extra)
if mt_adaptive:
csd = _csd_from_mt(
x_t[idx_map[0][con_idx]],
x_t[idx_map[1][con_idx]],
weights[idx_map[0][con_idx]],
weights[idx_map[1][con_idx]],
)
else:
csd = _csd_from_mt(
x_t[idx_map[0][con_idx]], x_t[idx_map[1][con_idx]], weights, weights
)
for method in con_methods:
method.accumulate(con_idx, csd)
else: # mode == 'cwt_morlet' # reminder to add alternative TFR methods
for i in range(0, n_con_signals, block_size):
n_extra = max(0, i + block_size - n_con_signals)
con_idx = slice(i, i + block_size - n_extra)
# this codes can be very slow
csd = x_t[idx_map[0][con_idx]] * x_t[idx_map[1][con_idx]].conjugate()
for method in con_methods:
method.accumulate(con_idx, csd)
# future estimator types need to be explicitly handled here
return con_methods, psd
def _get_n_epochs(epochs, n):
"""Generate lists with at most n epochs."""
epochs_out = list()
for epoch in epochs:
if not isinstance(epoch, (list, tuple)):
epoch = (epoch,)
epochs_out.append(epoch)
if len(epochs_out) >= n:
yield epochs_out
epochs_out = list()
if 0 < len(epochs_out) < n:
yield epochs_out
def _check_method(method):
"""Test if a method implements the required interface."""
interface_members = [
m[0]
for m in inspect.getmembers(_AbstractConEstBase)
if not m[0].startswith("_")
]
method_members = [
m[0] for m in inspect.getmembers(method) if not m[0].startswith("_")
]
for member in interface_members:
if member not in method_members:
return False, member
return True, None
def _get_and_verify_data_sizes(
data, sfreq, n_signals=None, n_times=None, times=None, warn_times=True
):
"""Get and/or verify the data sizes and time scales."""
if not isinstance(data, (list, tuple)):
raise ValueError("data has to be a list or tuple")
n_signals_tot = 0
# Sometimes data can be (ndarray, SourceEstimate) groups so in the case
# where ndarray comes first, don't use it for times
times_inferred = False
for this_data in data:
this_n_signals, this_n_times = this_data.shape
if n_times is not None:
if this_n_times != n_times:
raise ValueError(
"all input time series must have the same " "number of time points"
)
else:
n_times = this_n_times
n_signals_tot += this_n_signals
if hasattr(this_data, "times"):
assert isinstance(this_data, _BaseSourceEstimate)
this_times = this_data.times
if times is not None and not times_inferred:
if warn_times and not np.allclose(times, this_times):
with np.printoptions(threshold=4, linewidth=120):
warn(
"time scales of input time series do not match:\n"
f"{this_times}\n{times}"
)
warn_times = False
else:
times = this_times
elif times is None:
times_inferred = True
times = _arange_div(n_times, sfreq)
if n_signals is not None:
if n_signals != n_signals_tot:
raise ValueError(
"the number of time series has to be the same in " "each epoch"
)
n_signals = n_signals_tot
return n_signals, n_times, times, warn_times
# map names to estimator types
_CON_METHOD_MAP = {**_CON_METHOD_MAP_BIVARIATE, **_CON_METHOD_MAP_MULTIVARIATE}
def _check_estimators(method):
"""Check construction of connectivity estimators."""
n_methods = len(method)
con_method_types = list()
for this_method in method:
if this_method in _CON_METHOD_MAP:
con_method_types.append(_CON_METHOD_MAP[this_method])
elif isinstance(this_method, str):
raise ValueError("%s is not a valid connectivity method" % this_method)
else:
# support for custom class
method_valid, msg = _check_method(this_method)
if not method_valid:
raise ValueError(
"The supplied connectivity method does "
"not have the method %s" % msg
)
con_method_types.append(this_method)
# if none of the comp_con functions needs the PSD, we don't estimate it
accumulate_psd = any(this_method.accumulate_psd for this_method in con_method_types)
return con_method_types, n_methods, accumulate_psd
[docs]
@verbose
@fill_doc
def spectral_connectivity_epochs(
data,
names=None,
method="coh",
indices=None,
sfreq=None,
mode="multitaper",
fmin=None,
fmax=np.inf,
fskip=0,
faverage=False,
tmin=None,
tmax=None,
mt_bandwidth=None,
mt_adaptive=False,
mt_low_bias=True,
cwt_freqs=None,
cwt_n_cycles=7,
gc_n_lags=40,
rank=None,
block_size=1000,
n_jobs=1,
verbose=None,
):
r"""Compute frequency- and time-frequency-domain connectivity measures.
The connectivity method(s) are specified using the "method" parameter.
All methods are based on 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. Note that it is also
possible to combine multiple signals by providing a list of tuples,
e.g., data = [(arr_0, stc_0), (arr_1, stc_1), (arr_2, stc_2)],
corresponds to 3 epochs, and arr_* could be an array with the same
number of time points as stc_*. The array-like object can also
be a list/generator of array, shape =(n_signals, n_times),
or a list/generator of SourceEstimate or VolSourceEstimate objects.
%(names)s
method : str | list of str
Connectivity measure(s) to compute. These can be ``['coh', 'cohy',
'imcoh', 'mic', 'mim', 'plv', 'ciplv', 'ppc', 'pli', 'dpli', 'wpli',
'wpli2_debiased', 'gc', 'gc_tr']``. Multivariate methods (``['mic',
'mim', 'gc', 'gc_tr]``) cannot be called with the other methods.
indices : tuple of array | 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 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>`.
mode : str
Spectrum estimation mode can be either: 'multitaper', 'fourier', or
'cwt_morlet'.
fmin : float | tuple of float
The lower frequency of interest. Multiple bands are defined using
a tuple, e.g., (8., 20.) for two bands with 8Hz and 20Hz lower freq.
fmax : float | tuple of float
The upper frequency of interest. Multiple bands are dedined using
a tuple, e.g. (13., 30.) for two band with 13Hz and 30Hz upper freq.
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 a list with arrays of the frequencies
that were averaged.
tmin : float | None
Time to start connectivity estimation. Note: when "data" is an array,
the first sample is assumed to be at time 0. For other types
(Epochs, etc.), the time information contained in the object is used
to compute the time indices.
tmax : float | None
Time to end connectivity estimation. Note: when "data" is an array,
the first sample is assumed to be at time 0. For other types
(Epochs, etc.), the time information contained in the object is used
to compute the time indices.
mt_bandwidth : float | None
The bandwidth of the multitaper windowing function in Hz.
Only used in 'multitaper' mode.
mt_adaptive : bool
Use adaptive weights to combine the tapered spectra into PSD.
Only used in 'multitaper' mode.
mt_low_bias : bool
Only use tapers with more than 90 percent spectral concentration
within bandwidth. Only used in 'multitaper' mode.
cwt_freqs : array
Array of frequencies of interest. Only used in 'cwt_morlet' mode.
cwt_n_cycles : float | array of float
Number of cycles. Fixed number or one per frequency. Only used in
'cwt_morlet' mode.
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']``.
block_size : int
How many connections to compute at once (higher numbers are faster
but require more memory).
n_jobs : int
How many samples to process in parallel.
%(verbose)s
Returns
-------
con : array | list of array
Computed connectivity measure(s). Either an instance of
``SpectralConnectivity`` or ``SpectroTemporalConnectivity``.
The shape of the connectivity result will be:
- ``(n_cons, n_freqs)`` for multitaper or fourier modes
- ``(n_cons, n_freqs, n_times)`` for cwt_morlet mode
- ``n_cons = n_signals ** 2`` for bivariate methods with
``indices=None``
- ``n_cons = 1`` for multivariate methods with ``indices=None``
- ``n_cons = len(indices[0])`` for bivariate and multivariate methods
when indices is supplied.
See Also
--------
mne_connectivity.spectral_connectivity_time
mne_connectivity.SpectralConnectivity
mne_connectivity.SpectroTemporalConnectivity
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.
These measures are not to be interpreted over time. Each Epoch passed into
the dataset is interpreted as an independent sample of the same
connectivity structure. Within each Epoch, it is assumed that the spectral
measure is stationary. The spectral measures implemented in this function
are computed across Epochs. **Thus, spectral measures computed with only
one Epoch will result in errorful values and spectral measures computed
with few Epochs will be unreliable.** Please see
``spectral_connectivity_time`` for time-resolved connectivity estimation.
The spectral densities can be estimated using a multitaper method with
digital prolate spheroidal sequence (DPSS) windows, a discrete Fourier
transform with Hanning windows, or a continuous wavelet transform using
Morlet wavelets. The spectral estimation mode is specified using the
"mode" parameter.
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_epochs(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])
'cohy' : Coherency given by::
E[Sxy]
C = ---------------------
sqrt(E[Sxx] * E[Syy])
'imcoh' : Imaginary coherence :footcite:`NolteEtAl2004` given by::
Im(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)
'ppc' : Pairwise Phase Consistency (PPC), an unbiased estimator
of squared PLV :footcite:`VinckEtAl2010`.
'pli' : Phase Lag Index (PLI) :footcite:`StamEtAl2007` given by::
PLI = |E[sign(Im(Sxy))]|
'pli2_unbiased' : Unbiased estimator of squared PLI
:footcite:`VinckEtAl2011`.
'dpli' : Directed Phase Lag Index (DPLI) :footcite:`StamEtAl2012`
given by (where H is the Heaviside function)::
DPLI = E[H(Im(Sxy))]
'wpli' : Weighted Phase Lag Index (WPLI) :footcite:`VinckEtAl2011`
given by::
|E[Im(Sxy)]|
WPLI = ------------------
E[|Im(Sxy)|]
'wpli2_debiased' : Debiased estimator of squared WPLI
:footcite:`VinckEtAl2011`.
'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.
References
----------
.. footbibliography::
"""
if n_jobs != 1:
parallel, my_epoch_spectral_connectivity, _ = parallel_func(
_epoch_spectral_connectivity, n_jobs, verbose=verbose
)
# format fmin and fmax and check inputs
if fmin is None:
fmin = -np.inf # set it to -inf, so we can adjust it later
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")
n_bands = len(fmin)
# assign names to connectivity methods
if not isinstance(method, (list, tuple)):
method = [method] # make it a list so we can iterate over it
if n_bands != 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
# handle connectivity estimators
(con_method_types, n_methods, accumulate_psd) = _check_estimators(method)
events = None
event_id = None
if isinstance(data, BaseEpochs):
names = data.ch_names
times_in = data.times # input times for Epochs input type
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
else:
times_in = None
metadata = None
if sfreq is None:
raise ValueError(
"Sampling frequency (sfreq) is required with " "array input."
)
# loop over data; it could be a generator that returns
# (n_signals x n_times) arrays or SourceEstimates
epoch_idx = 0
logger.info("Connectivity computation...")
warn_times = True
for epoch_block in _get_n_epochs(data, n_jobs):
if epoch_idx == 0:
# initialize everything times and frequencies
(
n_cons,
times,
n_times,
times_in,
n_times_in,
tmin_idx,
tmax_idx,
n_freqs,
freq_mask,
freqs,
freqs_bands,
freq_idx_bands,
n_signals,
indices_use,
warn_times,
) = _prepare_connectivity(
epoch_block=epoch_block,
times_in=times_in,
tmin=tmin,
tmax=tmax,
fmin=fmin,
fmax=fmax,
sfreq=sfreq,
indices=indices,
method=method,
mode=mode,
fskip=fskip,
n_bands=n_bands,
cwt_freqs=cwt_freqs,
faverage=faverage,
)
# 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
# make sure padded indices are stored in the connectivity object
if multivariate_con and indices is not None:
# create a copy so that `indices_use` can be modified
indices = (indices_use[0].copy(), indices_use[1].copy())
# get the window function, wavelets, etc for different modes
(
spectral_params,
mt_adaptive,
n_times_spectrum,
n_tapers,
) = _assemble_spectral_params(
mode=mode,
n_times=n_times,
mt_adaptive=mt_adaptive,
mt_bandwidth=mt_bandwidth,
sfreq=sfreq,
mt_low_bias=mt_low_bias,
cwt_n_cycles=cwt_n_cycles,
cwt_freqs=cwt_freqs,
freqs=freqs,
freq_mask=freq_mask,
)
# unique signals for which we actually need to compute PSD etc.
if multivariate_con:
sig_idx = np.unique(indices_use.compressed())
remapping = {ch_i: sig_i for sig_i, ch_i in enumerate(sig_idx)}
remapped_inds = indices_use.copy()
for idx in sig_idx:
remapped_inds[indices_use == idx] = remapping[idx]
remapped_sig = np.unique(remapped_inds.compressed())
else:
sig_idx = np.unique(np.r_[indices_use[0], indices_use[1]])
n_signals_use = len(sig_idx)
# map indices to unique indices
if multivariate_con:
indices_use = remapped_inds # use remapped seeds & targets
idx_map = [
np.sort(np.repeat(remapped_sig, len(sig_idx))),
np.tile(remapped_sig, len(sig_idx)),
]
else:
idx_map = [np.searchsorted(sig_idx, ind) for ind in indices_use]
# allocate space to accumulate PSD
if accumulate_psd:
if n_times_spectrum == 0:
psd_shape = (n_signals_use, n_freqs)
else:
psd_shape = (n_signals_use, n_freqs, n_times_spectrum)
psd = np.zeros(psd_shape)
else:
psd = None
# create instances of the connectivity estimators
con_methods = []
for mtype_i, mtype in enumerate(con_method_types):
method_params = dict(
n_cons=n_cons, n_freqs=n_freqs, n_times=n_times_spectrum
)
if method[mtype_i] in _multivariate_methods:
method_params.update(dict(n_signals=n_signals_use))
if method[mtype_i] in _gc_methods:
method_params.update(dict(n_lags=gc_n_lags))
con_methods.append(mtype(**method_params))
sep = ", "
metrics_str = sep.join([meth.name for meth in con_methods])
logger.info(" the following metrics will be computed: %s" % metrics_str)
# check dimensions and time scale
for this_epoch in epoch_block:
_, _, _, warn_times = _get_and_verify_data_sizes(
this_epoch,
sfreq,
n_signals,
n_times_in,
times_in,
warn_times=warn_times,
)
call_params = dict(
sig_idx=sig_idx,
tmin_idx=tmin_idx,
tmax_idx=tmax_idx,
sfreq=sfreq,
method=method,
mode=mode,
freq_mask=freq_mask,
idx_map=idx_map,
n_cons=n_cons,
block_size=block_size,
psd=psd,
accumulate_psd=accumulate_psd,
mt_adaptive=mt_adaptive,
con_method_types=con_method_types,
con_methods=con_methods if n_jobs == 1 else None,
n_signals=n_signals,
n_signals_use=n_signals_use,
n_times=n_times,
gc_n_lags=gc_n_lags,
accumulate_inplace=True if n_jobs == 1 else False,
)
call_params.update(**spectral_params)
if n_jobs == 1:
# no parallel processing
for this_epoch in epoch_block:
logger.info(
" computing cross-spectral density for epoch %d"
% (epoch_idx + 1)
)
# con methods and psd are updated inplace
_epoch_spectral_connectivity(data=this_epoch, **call_params)
epoch_idx += 1
else:
# process epochs in parallel
logger.info(
" computing cross-spectral density for epochs %d..%d"
% (epoch_idx + 1, epoch_idx + len(epoch_block))
)
out = parallel(
my_epoch_spectral_connectivity(data=this_epoch, **call_params)
for this_epoch in epoch_block
)
# do the accumulation
for this_out in out:
for _method, parallel_method in zip(con_methods, this_out[0]):
_method.combine(parallel_method)
if accumulate_psd:
psd += this_out[1]
epoch_idx += len(epoch_block)
# normalize
n_epochs = epoch_idx
if accumulate_psd:
psd /= n_epochs
# compute final connectivity scores
con = list()
patterns = list()
for method_i, conn_method in enumerate(con_methods):
# future estimators will need to be handled here
if conn_method.accumulate_psd:
# compute scores block-wise to save memory
for i in range(0, n_cons, block_size):
con_idx = slice(i, i + block_size)
psd_xx = psd[idx_map[0][con_idx]]
psd_yy = psd[idx_map[1][con_idx]]
conn_method.compute_con(con_idx, n_epochs, psd_xx, psd_yy)
else:
# compute all scores at once
if method[method_i] in _multivariate_methods:
conn_method.compute_con(indices_use, rank, n_epochs)
else:
conn_method.compute_con(slice(0, n_cons), n_epochs)
# get the connectivity scores
this_con = conn_method.con_scores
this_patterns = conn_method.patterns
if this_con.shape[0] != n_cons:
raise RuntimeError(
"first dimension of connectivity scores does not match the "
"number of connections; please contact the mne-connectivity "
"developers"
)
if faverage:
if this_con.shape[1] != n_freqs:
raise RuntimeError(
"second dimension of connectivity scores does not match "
"the number of frequencies; please contact the "
"mne-connectivity developers"
)
con_shape = (n_cons, n_bands) + this_con.shape[2:]
this_con_bands = np.empty(con_shape, dtype=this_con.dtype)
for band_idx in range(n_bands):
this_con_bands[:, band_idx] = np.mean(
this_con[:, freq_idx_bands[band_idx]], axis=1
)
this_con = this_con_bands
if this_patterns is not None:
patterns_shape = list(this_patterns.shape)
patterns_shape[3] = n_bands
this_patterns_bands = np.empty(
patterns_shape, dtype=this_patterns.dtype
)
for band_idx in range(n_bands):
this_patterns_bands[:, :, :, band_idx] = np.mean(
this_patterns[:, :, :, freq_idx_bands[band_idx]], axis=3
)
this_patterns = this_patterns_bands
con.append(this_con)
patterns.append(this_patterns)
freqs_used = freqs
if faverage:
# for each band we return the frequencies that were averaged
freqs = [np.mean(x) for x in freqs_bands]
# make sure freq_bands is a list of equal-length lists
# XXX: we lose information on which frequency points went into the
# computation. If h5netcdf supports numpy objects in the future, then
# we can change the min/max to just make it a list of lists.
freqs_used = freqs_bands
freqs_used = [[np.min(band), np.max(band)] for band in freqs_used]
if indices is None and not multivariate_con:
# return all-to-all connectivity matrices
# raveled into a 1D array
logger.info(" assembling connectivity matrix")
con_flat = con
con = list()
for this_con_flat in con_flat:
this_con = np.zeros(
(n_signals, n_signals) + this_con_flat.shape[1:],
dtype=this_con_flat.dtype,
)
this_con[indices_use] = this_con_flat
# ravel 2D connectivity into a 1D array
# while keeping other dimensions
this_con = this_con.reshape((n_signals**2,) + this_con_flat.shape[1:])
con.append(this_con)
# number of nodes in the original data
n_nodes = n_signals
# create a list of connectivity containers
conn_list = []
for _con, _patterns, _method in zip(con, patterns, method):
kwargs = dict(
data=_con,
patterns=_patterns,
names=names,
freqs=freqs,
method=_method,
n_nodes=n_nodes,
spec_method=mode,
indices=indices,
n_epochs_used=n_epochs,
freqs_used=freqs_used,
times_used=times,
n_tapers=n_tapers,
metadata=metadata,
events=events,
event_id=event_id,
rank=rank,
n_lags=gc_n_lags if _method in _gc_methods else None,
)
# create the connectivity container
if mode in ["multitaper", "fourier"]:
klass = SpectralConnectivity
else:
assert mode == "cwt_morlet"
klass = SpectroTemporalConnectivity
kwargs.update(times=times)
conn_list.append(klass(**kwargs))
logger.info("[Connectivity computation done]")
if n_methods == 1:
# for a single method return connectivity directly
conn_list = conn_list[0]
return conn_list