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"""
.. _multi-taper-coh:
================================
Multi-taper coherence estimation
================================
Coherence estimation can be done using windowed-spectra. This is the method
used in the example :ref:`resting-state`. In addition, multi-taper spectral
estimation can be used in order to calculate coherence and also confidence
intervals for the coherence values that result (see :ref:`multi-taper-psd`)
The data analyzed here is an fMRI data-set contributed by Beth Mormino. The
data is taken from a single subject in a"resting-state" scan, in which subjects
are fixating on a cross and maintaining alert wakefulness, but not performing
any other behavioral task.
We start by importing modules/functions we will use in this example and define
variables which will be used as the sampling interval of the TimeSeries
objects and as upper and lower bounds on the frequency range analyzed:
"""
import os
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.mlab import csv2rec
import scipy.stats.distributions as dist
from scipy import fftpack
import nitime
from nitime.timeseries import TimeSeries
from nitime import utils
import nitime.algorithms as alg
import nitime.viz
from nitime.viz import drawmatrix_channels
from nitime.analysis import CoherenceAnalyzer, MTCoherenceAnalyzer
TR = 1.89
f_ub = 0.15
f_lb = 0.02
"""
We read in the data into a recarray from a csv file:
"""
data_path = os.path.join(nitime.__path__[0], 'data')
data_rec = csv2rec(os.path.join(data_path, 'fmri_timeseries.csv'))
"""
The first line in the file contains the names of the different brain regions
(or ROI = regions of interest) from which the time-series were derived. We
extract the data into a regular array, while keeping the names to be used later:
"""
roi_names = np.array(data_rec.dtype.names)
nseq = len(roi_names)
n_samples = data_rec.shape[0]
data = np.zeros((nseq, n_samples))
for n_idx, roi in enumerate(roi_names):
data[n_idx] = data_rec[roi]
"""
We normalize the data in each of the ROIs to be in units of % change:
"""
pdata = utils.percent_change(data)
"""
We start by performing the detailed analysis, but note that a significant
short-cut is presented below, so if you just want to know how to do this
(without needing to understand the details), skip on down.
We start by defining how many tapers will be used and calculate the values of
the tapers and the associated eigenvalues of each taper:
"""
NW = 4
K = 2 * NW - 1
tapers, eigs = alg.dpss_windows(n_samples, NW, K)
"""
We multiply the data by the tapers and derive the fourier transform and the
magnitude of the squared spectra (the power) for each tapered time-series:
"""
tdata = tapers[None, :, :] * pdata[:, None, :]
tspectra = fftpack.fft(tdata)
## mag_sqr_spectra = np.abs(tspectra)
## np.power(mag_sqr_spectra, 2, mag_sqr_spectra)
"""
Coherence for real sequences is symmetric, so we calculate this for only half
the spectrum (the other half is equal):
"""
L = n_samples / 2 + 1
sides = 'onesided'
"""
We estimate adaptive weighting of the tapers, based on the data (see
:ref:`multi-taper-psd` for an explanation and references):
"""
w = np.empty((nseq, K, L))
for i in xrange(nseq):
w[i], _ = utils.adaptive_weights(tspectra[i], eigs, sides=sides)
"""
We proceed to calculate the coherence. We initialize empty data containers:
"""
csd_mat = np.zeros((nseq, nseq, L), 'D')
psd_mat = np.zeros((2, nseq, nseq, L), 'd')
coh_mat = np.zeros((nseq, nseq, L), 'd')
coh_var = np.zeros_like(coh_mat)
"""
Looping over the ROIs:
"""
for i in xrange(nseq):
for j in xrange(i):
"""
We calculate the multi-tapered cross spectrum between each two
time-series:
"""
sxy = alg.mtm_cross_spectrum(
tspectra[i], tspectra[j], (w[i], w[j]), sides='onesided'
)
"""
And the individual PSD for each:
"""
sxx = alg.mtm_cross_spectrum(
tspectra[i], tspectra[i], w[i], sides='onesided'
)
syy = alg.mtm_cross_spectrum(
tspectra[j], tspectra[j], w[j], sides='onesided'
)
psd_mat[0, i, j] = sxx
psd_mat[1, i, j] = syy
"""
Coherence is : $Coh_{xy}(\lambda) = \frac{|{f_{xy}(\lambda)}|^2}{f_{xx}(\lambda) \cdot f_{yy}(\lambda)}$
"""
coh_mat[i, j] = np.abs(sxy) ** 2
coh_mat[i, j] /= (sxx * syy)
csd_mat[i, j] = sxy
"""
The variance from the different samples is calculated using a jack-knife
approach:
"""
if i != j:
coh_var[i, j] = utils.jackknifed_coh_variance(
tspectra[i], tspectra[j], eigs, adaptive=True,
)
"""
This measure is normalized, based on the number of tapers:
"""
coh_mat_xform = utils.normalize_coherence(coh_mat, 2 * K - 2)
"""
We calculate 95% confidence intervals based on the jack-knife variance
calculation:
"""
t025_limit = coh_mat_xform + dist.t.ppf(.025, K - 1) * np.sqrt(coh_var)
t975_limit = coh_mat_xform + dist.t.ppf(.975, K - 1) * np.sqrt(coh_var)
utils.normal_coherence_to_unit(t025_limit, 2 * K - 2, t025_limit)
utils.normal_coherence_to_unit(t975_limit, 2 * K - 2, t975_limit)
if L < n_samples:
freqs = np.linspace(0, 1 / (2 * TR), L)
else:
freqs = np.linspace(0, 1 / TR, L, endpoint=False)
"""
We look only at frequencies between 0.02 and 0.15 (the physiologically
relevant band, see http://imaging.mrc-cbu.cam.ac.uk/imaging/DesignEfficiency:
"""
freq_idx = np.where((freqs > f_lb) * (freqs < f_ub))[0]
"""
We extract the coherence and average over all these frequency bands:
"""
coh = np.mean(coh_mat[:, :, freq_idx], -1) # Averaging on the last dimension
"""
The next line calls the visualization routine which displays the data
"""
fig01 = drawmatrix_channels(coh,
roi_names,
size=[10., 10.],
color_anchor=0,
title='MTM Coherence')
"""
.. image:: fig/multi_taper_coh_01.png
Next we perform the same analysis, using the nitime object oriented interface.
We start by initializing a TimeSeries object with this data and with the
sampling_interval provided above. We set the metadata 'roi' field with the ROI
names.
"""
T = TimeSeries(pdata, sampling_interval=TR)
T.metadata['roi'] = roi_names
"""
We initialize an MTCoherenceAnalyzer object with the TimeSeries object
"""
C2 = MTCoherenceAnalyzer(T)
"""
The relevant indices in the Analyzer object are derived:
"""
freq_idx = np.where((C2.frequencies > 0.02) * (C2.frequencies < 0.15))[0]
"""
The call to C2.coherence triggers the computation and this is averaged over the
frequency range of interest in the same line and then displayed:
"""
coh = np.mean(C2.coherence[:, :, freq_idx], -1) # Averaging on the last dimension
fig02 = drawmatrix_channels(coh,
roi_names,
size=[10., 10.],
color_anchor=0,
title='MTCoherenceAnalyzer')
"""
.. image:: fig/multi_taper_coh_02.png
For comparison, we also perform the analysis using the standard
CoherenceAnalyzer object, which does the analysis using Welch's windowed
periodogram, instead of the multi-taper spectral estimation method (see
:ref:`resting_state` for a more thorough analysis of this data using this
method):
"""
C3 = CoherenceAnalyzer(T)
freq_idx = np.where((C3.frequencies > f_lb) * (C3.frequencies < f_ub))[0]
#Extract the coherence and average across these frequency bands:
coh = np.mean(C3.coherence[:, :, freq_idx], -1) # Averaging on the last dimension
fig03 = drawmatrix_channels(coh,
roi_names,
size=[10., 10.],
color_anchor=0,
title='CoherenceAnalyzer')
"""
.. image:: fig/multi_taper_coh_03.png
plt.show() is called in order to display the figures:
"""
plt.show()
|
