def coherence(self):
nrows = self.input.data.shape[0]
psd_mat = np.zeros((2, nrows, nrows, self._L), 'd')
coh_mat = np.zeros((nrows, nrows, self._L), 'd')
for i in range(self.input.data.shape[0]):
for j in range(i):
sxy = tsa.mtm_cross_spectrum(
self.spectra[i],
self.spectra[j], (self.weights[i], self.weights[j]),
sides='onesided')
sxx = tsa.mtm_cross_spectrum(self.spectra[i],
self.spectra[i],
self.weights[i],
sides='onesided')
syy = tsa.mtm_cross_spectrum(self.spectra[j],
self.spectra[j],
self.weights[i],
sides='onesided')
psd_mat[0, i, j] = sxx
psd_mat[1, i, j] = syy
coh_mat[i, j] = np.abs(sxy)**2
coh_mat[i, j] /= (sxx * syy)
idx = triu_indices(self.input.data.shape[0], 1)
coh_mat[idx[0], idx[1], ...] = coh_mat[idx[1], idx[0], ...].conj()
return coh_mat
def test_mtm_lin_combo():
"Test the functionality of cross and autospectrum MTM combinations"
spec1 = np.random.randn(5, 100) + 1j * np.random.randn(5, 100)
spec2 = np.random.randn(5, 100) + 1j * np.random.randn(5, 100)
# test on both broadcasted weights and per-point weights
for wshape in ((2, 5, 1), (2, 5, 100)):
weights = np.random.randn(*wshape)
sides = 'onesided'
mtm_cross = tsa.mtm_cross_spectrum(spec1,
spec2, (weights[0], weights[1]),
sides=sides)
npt.assert_(mtm_cross.dtype in np.sctypes['complex'],
'Wrong dtype for crossspectrum')
npt.assert_(len(mtm_cross) == 51, 'Wrong length for halfband spectrum')
sides = 'twosided'
mtm_cross = tsa.mtm_cross_spectrum(spec1,
spec2, (weights[0], weights[1]),
sides=sides)
npt.assert_(
len(mtm_cross) == 100, 'Wrong length for fullband spectrum')
sides = 'onesided'
mtm_auto = tsa.mtm_cross_spectrum(spec1,
spec1,
weights[0],
sides=sides)
npt.assert_(mtm_auto.dtype in np.sctypes['float'],
'Wrong dtype for autospectrum')
npt.assert_(len(mtm_auto) == 51, 'Wrong length for halfband spectrum')
sides = 'twosided'
mtm_auto = tsa.mtm_cross_spectrum(spec1,
spec2,
weights[0],
sides=sides)
npt.assert_(len(mtm_auto) == 100, 'Wrong length for fullband spectrum')
def coherence(self):
nrows = self.input.data.shape[0]
psd_mat = np.zeros((2, nrows, nrows, self._L), 'd')
coh_mat = np.zeros((nrows, nrows, self._L), 'd')
for i in range(self.input.data.shape[0]):
for j in range(i):
sxy = tsa.mtm_cross_spectrum(self.spectra[i], self.spectra[j],
(self.weights[i], self.weights[j]),
sides='onesided')
sxx = tsa.mtm_cross_spectrum(self.spectra[i], self.spectra[i],
self.weights[i],
sides='onesided')
syy = tsa.mtm_cross_spectrum(self.spectra[j], self.spectra[j],
self.weights[i],
sides='onesided')
psd_mat[0, i, j] = sxx
psd_mat[1, i, j] = syy
coh_mat[i, j] = np.abs(sxy) ** 2
coh_mat[i, j] /= (sxx * syy)
idx = triu_indices(self.input.data.shape[0], 1)
coh_mat[idx[0], idx[1], ...] = coh_mat[idx[1], idx[0], ...].conj()
return coh_mat
def test_mtm_lin_combo():
"Test the functionality of cross and autospectrum MTM combinations"
spec1 = np.random.randn(5, 100) + 1j*np.random.randn(5, 100)
spec2 = np.random.randn(5, 100) + 1j*np.random.randn(5, 100)
# test on both broadcasted weights and per-point weights
for wshape in ( (2,5,1), (2,5,100) ):
weights = np.random.randn(*wshape)
sides = 'onesided'
mtm_cross = tsa.mtm_cross_spectrum(
spec1, spec2, (weights[0], weights[1]), sides=sides
)
nt.assert_true(mtm_cross.dtype in np.sctypes['complex'],
'Wrong dtype for crossspectrum')
nt.assert_true(len(mtm_cross) == 51,
'Wrong length for halfband spectrum')
sides = 'twosided'
mtm_cross = tsa.mtm_cross_spectrum(
spec1, spec2, (weights[0], weights[1]), sides=sides
)
nt.assert_true (len(mtm_cross) == 100,
'Wrong length for fullband spectrum')
sides = 'onesided'
mtm_auto = tsa.mtm_cross_spectrum(
spec1, spec1, weights[0], sides=sides
)
nt.assert_true(mtm_auto.dtype in np.sctypes['float'],
'Wrong dtype for autospectrum')
nt.assert_true(len(mtm_auto) == 51,
'Wrong length for halfband spectrum')
sides = 'twosided'
mtm_auto = tsa.mtm_cross_spectrum(
spec1, spec2, weights[0], sides=sides
)
nt.assert_true(len(mtm_auto) == 100,
'Wrong length for fullband spectrum')
def test_mtm_cross_spectrum():
"""
Test the multi-taper cross-spectral estimation. Based on the example in
doc/examples/multi_taper_coh.py
"""
NW = 4
K = 2 * NW - 1
N = 2 ** 10
n_reps = 10
n_freqs = N
tapers, eigs = tsa.dpss_windows(N, NW, 2 * NW - 1)
est_psd = []
for k in range(n_reps):
data, nz, alpha = utils.ar_generator(N=N)
fgrid, hz = tsa.freq_response(1.0, a=np.r_[1, -alpha], n_freqs=n_freqs)
# 'one-sided', so multiply by 2:
psd = 2 * (hz * hz.conj()).real
tdata = tapers * data
tspectra = fftpack.fft(tdata)
L = N / 2 + 1
sides = 'onesided'
w, _ = utils.adaptive_weights(tspectra, eigs, sides=sides)
sxx = tsa.mtm_cross_spectrum(tspectra, tspectra, w, sides=sides)
est_psd.append(sxx)
fxx = np.mean(est_psd, 0)
psd_ratio = np.mean(fxx / psd)
# This is a rather lenient test, making sure that the average ratio is 1 to
# within an order of magnitude. That is, that they are equal on average:
npt.assert_array_almost_equal(psd_ratio, 1, decimal=1)
# Test raising of error in case the inputs don't make sense:
with pytest.raises(ValueError) as e_info:
tsa.mtm_cross_spectrum(tspectra, np.r_[tspectra, tspectra], (w, w))
def test_mtm_cross_spectrum():
"""
Test the multi-taper cross-spectral estimation. Based on the example in
doc/examples/multi_taper_coh.py
"""
NW = 4
K = 2 * NW - 1
N = 2**10
n_reps = 10
n_freqs = N
tapers, eigs = tsa.dpss_windows(N, NW, 2 * NW - 1)
est_psd = []
for k in range(n_reps):
data, nz, alpha = utils.ar_generator(N=N)
fgrid, hz = tsa.freq_response(1.0, a=np.r_[1, -alpha], n_freqs=n_freqs)
# 'one-sided', so multiply by 2:
psd = 2 * (hz * hz.conj()).real
tdata = tapers * data
tspectra = fftpack.fft(tdata)
L = N / 2 + 1
sides = 'onesided'
w, _ = utils.adaptive_weights(tspectra, eigs, sides=sides)
sxx = tsa.mtm_cross_spectrum(tspectra, tspectra, w, sides=sides)
est_psd.append(sxx)
fxx = np.mean(est_psd, 0)
psd_ratio = np.mean(fxx / psd)
# This is a rather lenient test, making sure that the average ratio is 1 to
# within an order of magnitude. That is, that they are equal on average:
npt.assert_array_almost_equal(psd_ratio, 1, decimal=1)
# Test raising of error in case the inputs don't make sense:
with pytest.raises(ValueError) as e_info:
tsa.mtm_cross_spectrum(tspectra, np.r_[tspectra, tspectra], (w, w))
def jackknifed_variances(tx, ty, eigvals, adaptive=True, deg=True):
"""
Returns the variance of the admittance (real-part),
gain (modulus) and phase of the transfer function and
gamma^2 (modulus-squared coherence) between x and y,
estimated through jack-knifing the tapered samples in {tx, ty}.
Parameters
----------
tx : ndarray, (K, L)
The K complex spectra of tapered timeseries x
ty : ndarray, (K, L)
The K complex spectra of tapered timeseries y
eigvals : ndarray (K,)
The eigenvalues associated with the K DPSS tapers
Returns
-------
jk_var : dictionary of ndarrays
(entries are 'admittance', 'gain', 'phase',
'magnitude_squared_coherence')
The variance computed in the transformed domain
"""
K = tx.shape[0]
# calculate leave-one-out estimates of the admittance
jk_admittance = []
jk_gain = []
jk_phase = []
jk_magnitude_squared_coherence = []
sides = 'onesided'
all_orders = set(range(K))
import nitime.algorithms as alg
# get the leave-one-out estimates
for i in range(K):
items = list(all_orders.difference([i]))
tx_i = np.take(tx, items, axis=0)
ty_i = np.take(ty, items, axis=0)
eigs_i = np.take(eigvals, items)
if adaptive:
wx, _ = utils.adaptive_weights(tx_i, eigs_i, sides=sides)
wy, _ = utils.adaptive_weights(ty_i, eigs_i, sides=sides)
else:
wx = wy = eigs_i[:, None]
# The CSD
sxy_i = alg.mtm_cross_spectrum(tx_i, ty_i, (wx, wy), sides=sides)
# The PSDs
sxx_i = alg.mtm_cross_spectrum(tx_i, tx_i, wx, sides=sides)
syy_i = alg.mtm_cross_spectrum(ty_i, ty_i, wy, sides=sides)
# these are the Zr_i samples
Z = sxy_i / syy_i
jk_admittance.append(np.real(Z))
jk_gain.append(np.absolute(Z))
jk_phase.append(np.angle(Z, deg=deg))
jk_magnitude_squared_coherence.append(
np.abs(sxy_i)**2 / (sxx_i * syy_i))
# The jackknifed variance is equal to
# (K-1)/K * sum_i ( (x_i - mean(x_i))^2 )
jk_var = {}
for (name, jk_variance) in [('admittance', np.array(jk_admittance)),
('gain', np.array(jk_gain)),
('phase', np.array(jk_phase)),
('magnitude_squared_coherence',
np.array(jk_magnitude_squared_coherence))]:
jk_avg = np.mean(jk_variance, axis=0)
jk_var[name] = (float(K - 1.) / K) * (np.power(
(jk_variance - jk_avg), 2.)).sum(axis=0)
return jk_var
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')
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'
)
def compute_mtcoherence(s1, s2, sample_rate, window_size, bandwidth=15.0, chunk_len_percentage_tolerance=0.30,
frequency_cutoff=None, tanh_transform=False, debug=False):
"""
Computing the multi-taper coherence between signals s1 and s2. To do so, the signals are broken up into segments of length
specified by window_size. Then the multi-taper coherence is computed between each segment. The mean coherence
is computed across segments, and an estimate of the coherence variance is computed across segments.
sample_rate: the sample rate in Hz of s1 and s2
window_size: size of the segments in seconds
bandwidth: related to the # of tapers used to compute the spectral density. The higher the bandwidth, the more tapers.
chunk_len_percentage_tolerance: If there are leftover segments whose lengths are less than window_size, use them
if they comprise at least the fraction of window_size specified by chunk_len_percentage_tolerance
frequency_cutoff: the frequency at which to cut off the coherence when computing the normal mutual information
tanh_transform: whether to transform the coherences when computing the upper and lower bounds, supposedly
improves the estimate of variance.
"""
minlen = min(len(s1), len(s2))
if s1.shape != s2.shape:
s1 = s1[:minlen]
s2 = s2[:minlen]
sample_length_bins = min(len(s1), int(window_size * sample_rate))
#compute DPSS tapers for signals
NW = int(window_size*bandwidth)
K = 2*NW - 1
#print 'compute_coherence: NW=%d, K=%d' % (NW, K)
tapers,eigs = ntalg.dpss_windows(sample_length_bins, NW, K)
if debug:
print '[compute_coherence] bandwidth=%0.1f, # of tapers: %d' % (bandwidth, len(eigs))
#break signal into chunks and estimate coherence for each chunk
nchunks = int(np.floor(len(s1) / float(sample_length_bins)))
nleft = len(s1) % sample_length_bins
if nleft > 0:
nchunks += 1
#print 'sample_length_bins=%d, # of chunks:%d, # samples in last chunk: %d' % (sample_length_bins, nchunks, nleft)
coherence_estimates = list()
for k in range(nchunks):
s = k*sample_length_bins
e = min(len(s1), s + sample_length_bins)
chunk_len = e - s
chunk_percentage = chunk_len / float(sample_length_bins)
if chunk_percentage < chunk_len_percentage_tolerance:
#don't compute coherence for a chunk whose length is less than a certain percentage of sample_length_bins
continue
s1_chunk = np.zeros([sample_length_bins])
s2_chunk = np.zeros([sample_length_bins])
s1_chunk[:chunk_len] = s1[s:e]
s2_chunk[:chunk_len] = s2[s:e]
#taper the signals
s1_tap = tapers * s1_chunk
s2_tap = tapers * s2_chunk
#compute fft of tapered signals
s1_fft = fftpack.fft(s1_tap, axis=1)
s2_fft = fftpack.fft(s2_tap, axis=1)
#compute adaptive weights for each taper
w1,nu1 = ntutils.adaptive_weights(s1_fft, eigs, sides='onesided')
w2,nu2 = ntutils.adaptive_weights(s2_fft, eigs, sides='onesided')
#compute cross spectral density
sxy = ntalg.mtm_cross_spectrum(s1_fft, s2_fft, (w1, w2), sides='onesided')
#compute individual power spectrums
sxx = ntalg.mtm_cross_spectrum(s1_fft, s1_fft, w1, sides='onesided')
syy = ntalg.mtm_cross_spectrum(s2_fft, s2_fft, w2, sides='onesided')
#compute coherence
coherence = np.abs(sxy)**2 / (sxx * syy)
coherence_estimates.append(coherence)
#compute variance
coherence_estimates = np.array(coherence_estimates)
if tanh_transform:
coherence_estimates = np.arctanh(coherence_estimates)
coherence_variance = np.zeros([coherence_estimates.shape[1]])
coherence_mean = coherence_estimates.mean(axis=0)
#mean subtract and square
cv = np.sum((coherence_estimates - coherence_mean)**2, axis=0)
coherence_variance[:] = (1.0 - 1.0/nchunks) * cv
if tanh_transform:
coherence_variance = np.tanh(coherence_variance)
coherence_mean = np.tanh(coherence_mean)
#compute frequencies
sampint = 1.0 / sample_rate
L = sample_length_bins / 2 + 1
freq = np.linspace(0, 1 / (2 * sampint), L)
#compute upper and lower bounds
coherence_lower = coherence_mean - 2*np.sqrt(coherence_variance)
coherence_upper = coherence_mean + 2*np.sqrt(coherence_variance)
cdata = CoherenceData(frequency_cutoff=frequency_cutoff)
cdata.coherence = coherence_mean
cdata.coherence_lower = coherence_lower
cdata.coherence_upper = coherence_upper
cdata.frequency = freq
cdata.sample_rate = sample_rate
return cdata
def compute_coherence_original(s1, s2, sample_rate, bandwidth, jackknife=False, tanh_transform=False):
"""
An implementation of computing the coherence. Don't use this.
"""
minlen = min(len(s1), len(s2))
if s1.shape != s2.shape:
s1 = s1[:minlen]
s2 = s2[:minlen]
window_length = len(s1) / sample_rate
window_length_bins = int(window_length * sample_rate)
#compute DPSS tapers for signals
NW = int(window_length*bandwidth)
K = 2*NW - 1
print 'compute_coherence: NW=%d, K=%d' % (NW, K)
tapers,eigs = ntalg.dpss_windows(window_length_bins, NW, K)
njn = len(eigs)
jn_indices = [range(njn)]
#compute jackknife indices
if jackknife:
jn_indices = list()
for i in range(len(eigs)):
jn = range(len(eigs))
jn.remove(i)
jn_indices.append(jn)
#taper the signals
s1_tap = tapers * s1
s2_tap = tapers * s2
#compute fft of tapered signals
s1_fft = fftpack.fft(s1_tap, axis=1)
s2_fft = fftpack.fft(s2_tap, axis=1)
#compute adaptive weights for each taper
w1,nu1 = ntutils.adaptive_weights(s1_fft, eigs, sides='onesided')
w2,nu2 = ntutils.adaptive_weights(s2_fft, eigs, sides='onesided')
coherence_estimates = list()
for jn in jn_indices:
#compute cross spectral density
sxy = ntalg.mtm_cross_spectrum(s1_fft[jn, :], s2_fft[jn, :], (w1[jn], w2[jn]), sides='onesided')
#compute individual power spectrums
sxx = ntalg.mtm_cross_spectrum(s1_fft[jn, :], s1_fft[jn, :], w1[jn], sides='onesided')
syy = ntalg.mtm_cross_spectrum(s2_fft[jn, :], s2_fft[jn, :], w2[jn], sides='onesided')
#compute coherence
coherence = np.abs(sxy)**2 / (sxx * syy)
coherence_estimates.append(coherence)
#compute variance
coherence_estimates = np.array(coherence_estimates)
coherence_variance = np.zeros([coherence_estimates.shape[1]])
coherence_mean = coherence_estimates[0]
if jackknife:
coherence_mean = coherence_estimates.mean(axis=0)
#mean subtract and square
cv = np.sum((coherence_estimates - coherence_mean)**2, axis=0)
coherence_variance[:] = (1.0 - 1.0/njn) * cv
#compute frequencies
sampint = 1.0 / sample_rate
L = minlen / 2 + 1
freq = np.linspace(0, 1 / (2 * sampint), L)
#compute upper and lower bounds
cmean = coherence_mean
coherence_lower = cmean - 2*np.sqrt(coherence_variance)
coherence_upper = cmean + 2*np.sqrt(coherence_variance)
cdata = CoherenceData()
cdata.coherence = coherence_mean
cdata.coherence_lower = coherence_lower
cdata.coherence_upper = coherence_upper
cdata.frequency = freq
cdata.sample_rate = sample_rate
return cdata
def multitaper_cross_spectral_estimates(traces,
delta,
NW,
compute_confidence_intervals=True,
confidence_interval=0.95):
# Define the number of tapers, their values and associated eigenvalues:
npts = len(traces[0])
K = 2 * NW - 1
tapers, eigs = alg.dpss_windows(npts, NW, K)
# Multiply the data by the tapers, calculate the Fourier transform
# 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, :, :] * traces[:, None, :]
tspectra = fftpack.fft(tdata)
# The coherency for real sequences is symmetric so only half
# the spectrum if required
L = npts // 2 + 1
if L < npts:
freqs = np.linspace(0, 1. / (2. * delta), L)
else:
freqs = np.linspace(0, 1. / delta, L, endpoint=False)
# Estimate adaptive weighting of the tapers, based on the data
# (see Thomsen, 2007; 10.1109/MSP.2007.4286561)
w = np.empty((2, K, L))
for i in range(2):
w[i], _ = utils.adaptive_weights(tspectra[i], eigs, sides='onesided')
# Calculate the multi-tapered cross spectrum
# and the PSDs for the two time-series:
sxy = alg.mtm_cross_spectrum(tspectra[0],
tspectra[1], (w[0], w[1]),
sides='onesided')
sxx = alg.mtm_cross_spectrum(tspectra[0],
tspectra[0],
w[0],
sides='onesided')
syy = alg.mtm_cross_spectrum(tspectra[1],
tspectra[1],
w[1],
sides='onesided')
Z = sxy / syy
spectral_estimates = {}
spectral_estimates['frequencies'] = freqs
spectral_estimates['magnitude_squared_coherence'] = np.abs(sxy)**2 / (sxx *
syy)
spectral_estimates['transfer_function'] = Z # Transfer function
spectral_estimates['admittance'] = np.real(Z)
spectral_estimates['gain'] = np.absolute(Z)
spectral_estimates['phase'] = np.angle(Z, deg=True)
# Estimate confidence intervals
if compute_confidence_intervals:
spectral_estimates['confidence_bounds'] = {}
c_bnds = [
0.5 - confidence_interval / 2., 0.5 + confidence_interval / 2.
]
variances = jackknifed_variances(tspectra[0],
tspectra[1],
eigs,
adaptive=True)
spectral_estimates['confidence_bounds']['admittance'] = [
spectral_estimates['admittance'] +
dist.t.ppf(c_bnds[0], K - 1) * np.sqrt(variances['admittance']),
spectral_estimates['admittance'] +
dist.t.ppf(c_bnds[1], K - 1) * np.sqrt(variances['admittance'])
]
spectral_estimates['confidence_bounds']['gain'] = [
spectral_estimates['gain'] +
dist.t.ppf(c_bnds[0], K - 1) * np.sqrt(variances['gain']),
spectral_estimates['gain'] +
dist.t.ppf(c_bnds[1], K - 1) * np.sqrt(variances['gain'])
]
spectral_estimates['confidence_bounds']['phase'] = [
spectral_estimates['phase'] +
dist.t.ppf(c_bnds[0], K - 1) * np.sqrt(variances['phase']),
spectral_estimates['phase'] +
dist.t.ppf(c_bnds[1], K - 1) * np.sqrt(variances['phase'])
]
spectral_estimates['confidence_bounds'][
'magnitude_squared_coherence'] = [
spectral_estimates['magnitude_squared_coherence'] +
dist.t.ppf(c_bnds[0], K - 1) *
np.sqrt(variances['magnitude_squared_coherence']),
spectral_estimates['magnitude_squared_coherence'] +
dist.t.ppf(c_bnds[1], K - 1) *
np.sqrt(variances['magnitude_squared_coherence'])
]
return spectral_estimates
def jackknifed_coh_variance(tx, ty, weights=None, last_freq=None):
"""
Returns the variance of the coherency between x and y, estimated
through jack-knifing the tapered samples in {tx, ty}.
Parameters
----------
tx: ndarray, (K, L)
The K complex spectra of tapered timeseries x
ty: ndarray, (K, L)
The K complex spectra of tapered timeseries y
weights: ndarray, or sequence-of-ndarrays 2 x (K, [N]), optional
The weights to use for combining the K spectra in tx and ty
last_freq: int, optional
The last frequency for which to compute variance (e.g., if only
computing half of the coherence spectrum)
Returns
-------
jk_var: ndarray
The variance computed in the transformed domain (see normalize_coherence)
"""
K = tx.shape[0]
L = tx.shape[1] if last_freq is None else last_freq
tx = tx[:,:L]
ty = ty[:,:L]
# prepare weights
if weights is None:
weights = ( np.ones(K), np.ones(K) )
if len(weights) != 2:
raise ValueError('Must provide 2 sets of weights')
weights_x, weights_y = weights
if len(weights_x.shape) < 2:
weights_x = weights_x.reshape(K, 1)
weights_y = weights_y.reshape(K, 1)
if weights_x.shape[1] > L:
weights_x = weights_x[:,:L]
weights_y = weights_y[:,:L]
# calculate leave-one-out estimates of MSC (magnitude squared coherence)
jk_coh = np.empty((K, L), 'd')
all_orders = set(range(K))
import nitime.algorithms as alg
# get the leave-one-out estimates
for i in xrange(K):
items = list(all_orders.difference([i]))
tx_i = np.take(tx, items, axis=0)
ty_i = np.take(ty, items, axis=0)
wx = np.take(weights_x, items, axis=0)
wy = np.take(weights_y, items, axis=0)
weights = (wx, wy)
# The CSD
sxy_i = alg.mtm_cross_spectrum(tx_i, ty_i, weights)
# The PSDs
sxx_i = alg.mtm_cross_spectrum(tx_i, tx_i, weights).real
syy_i = alg.mtm_cross_spectrum(ty_i, ty_i, weights).real
# these are the | c_i | samples
jk_coh[i] = np.abs(sxy_i)
jk_coh[i] /= np.sqrt(sxx_i * syy_i)
jk_avg = np.mean(jk_coh, axis=0)
# now normalize the coherence estimates and the avg
normalize_coherence(jk_coh, 2*K-2, jk_coh)
normalize_coherence(jk_avg, 2*K-2, jk_avg)
jk_var = (jk_coh - jk_avg)
np.power(jk_var, 2, jk_var)
jk_var = jk_var.sum(axis=0)
# Do/Don't use the alternative scaling here??
f = float(K-1)/K
jk_var *= f
return jk_var
def compute_mtcoherence(s1,
s2,
sample_rate,
window_size,
bandwidth=15.0,
chunk_len_percentage_tolerance=0.30,
frequency_cutoff=None,
tanh_transform=False,
debug=False):
"""
Computing the multi-taper coherence between signals s1 and s2. To do so, the signals are broken up into segments of length
specified by window_size. Then the multi-taper coherence is computed between each segment. The mean coherence
is computed across segments, and an estimate of the coherence variance is computed across segments.
sample_rate: the sample rate in Hz of s1 and s2
window_size: size of the segments in seconds
bandwidth: related to the # of tapers used to compute the spectral density. The higher the bandwidth, the more tapers.
chunk_len_percentage_tolerance: If there are leftover segments whose lengths are less than window_size, use them
if they comprise at least the fraction of window_size specified by chunk_len_percentage_tolerance
frequency_cutoff: the frequency at which to cut off the coherence when computing the normal mutual information
tanh_transform: whether to transform the coherences when computing the upper and lower bounds, supposedly
improves the estimate of variance.
"""
minlen = min(len(s1), len(s2))
if s1.shape != s2.shape:
s1 = s1[:minlen]
s2 = s2[:minlen]
sample_length_bins = min(len(s1), int(window_size * sample_rate))
#compute DPSS tapers for signals
NW = int(window_size * bandwidth)
K = 2 * NW - 1
#print 'compute_coherence: NW=%d, K=%d' % (NW, K)
tapers, eigs = ntalg.dpss_windows(sample_length_bins, NW, K)
if debug:
print '[compute_coherence] bandwidth=%0.1f, # of tapers: %d' % (
bandwidth, len(eigs))
#break signal into chunks and estimate coherence for each chunk
nchunks = int(np.floor(len(s1) / float(sample_length_bins)))
nleft = len(s1) % sample_length_bins
if nleft > 0:
nchunks += 1
#print 'sample_length_bins=%d, # of chunks:%d, # samples in last chunk: %d' % (sample_length_bins, nchunks, nleft)
coherence_estimates = list()
for k in range(nchunks):
s = k * sample_length_bins
e = min(len(s1), s + sample_length_bins)
chunk_len = e - s
chunk_percentage = chunk_len / float(sample_length_bins)
if chunk_percentage < chunk_len_percentage_tolerance:
#don't compute coherence for a chunk whose length is less than a certain percentage of sample_length_bins
continue
s1_chunk = np.zeros([sample_length_bins])
s2_chunk = np.zeros([sample_length_bins])
s1_chunk[:chunk_len] = s1[s:e]
s2_chunk[:chunk_len] = s2[s:e]
#taper the signals
s1_tap = tapers * s1_chunk
s2_tap = tapers * s2_chunk
#compute fft of tapered signals
s1_fft = fftpack.fft(s1_tap, axis=1)
s2_fft = fftpack.fft(s2_tap, axis=1)
#compute adaptive weights for each taper
w1, nu1 = ntutils.adaptive_weights(s1_fft, eigs, sides='onesided')
w2, nu2 = ntutils.adaptive_weights(s2_fft, eigs, sides='onesided')
#compute cross spectral density
sxy = ntalg.mtm_cross_spectrum(s1_fft,
s2_fft, (w1, w2),
sides='onesided')
#compute individual power spectrums
sxx = ntalg.mtm_cross_spectrum(s1_fft, s1_fft, w1, sides='onesided')
syy = ntalg.mtm_cross_spectrum(s2_fft, s2_fft, w2, sides='onesided')
#compute coherence
coherence = np.abs(sxy)**2 / (sxx * syy)
coherence_estimates.append(coherence)
#compute variance
coherence_estimates = np.array(coherence_estimates)
if tanh_transform:
coherence_estimates = np.arctanh(coherence_estimates)
coherence_variance = np.zeros([coherence_estimates.shape[1]])
coherence_mean = coherence_estimates.mean(axis=0)
#mean subtract and square
cv = np.sum((coherence_estimates - coherence_mean)**2, axis=0)
coherence_variance[:] = (1.0 - 1.0 / nchunks) * cv
if tanh_transform:
coherence_variance = np.tanh(coherence_variance)
coherence_mean = np.tanh(coherence_mean)
#compute frequencies
sampint = 1.0 / sample_rate
L = sample_length_bins / 2 + 1
freq = np.linspace(0, 1 / (2 * sampint), L)
#compute upper and lower bounds
coherence_lower = coherence_mean - 2 * np.sqrt(coherence_variance)
coherence_upper = coherence_mean + 2 * np.sqrt(coherence_variance)
cdata = CoherenceData(frequency_cutoff=frequency_cutoff)
cdata.coherence = coherence_mean
cdata.coherence_lower = coherence_lower
cdata.coherence_upper = coherence_upper
cdata.frequency = freq
cdata.sample_rate = sample_rate
return cdata
def compute_coherence_original(s1,
s2,
sample_rate,
bandwidth,
jackknife=False,
tanh_transform=False):
"""
An implementation of computing the coherence. Don't use this.
"""
minlen = min(len(s1), len(s2))
if s1.shape != s2.shape:
s1 = s1[:minlen]
s2 = s2[:minlen]
window_length = len(s1) / sample_rate
window_length_bins = int(window_length * sample_rate)
#compute DPSS tapers for signals
NW = int(window_length * bandwidth)
K = 2 * NW - 1
print 'compute_coherence: NW=%d, K=%d' % (NW, K)
tapers, eigs = ntalg.dpss_windows(window_length_bins, NW, K)
njn = len(eigs)
jn_indices = [range(njn)]
#compute jackknife indices
if jackknife:
jn_indices = list()
for i in range(len(eigs)):
jn = range(len(eigs))
jn.remove(i)
jn_indices.append(jn)
#taper the signals
s1_tap = tapers * s1
s2_tap = tapers * s2
#compute fft of tapered signals
s1_fft = fftpack.fft(s1_tap, axis=1)
s2_fft = fftpack.fft(s2_tap, axis=1)
#compute adaptive weights for each taper
w1, nu1 = ntutils.adaptive_weights(s1_fft, eigs, sides='onesided')
w2, nu2 = ntutils.adaptive_weights(s2_fft, eigs, sides='onesided')
coherence_estimates = list()
for jn in jn_indices:
#compute cross spectral density
sxy = ntalg.mtm_cross_spectrum(s1_fft[jn, :],
s2_fft[jn, :], (w1[jn], w2[jn]),
sides='onesided')
#compute individual power spectrums
sxx = ntalg.mtm_cross_spectrum(s1_fft[jn, :],
s1_fft[jn, :],
w1[jn],
sides='onesided')
syy = ntalg.mtm_cross_spectrum(s2_fft[jn, :],
s2_fft[jn, :],
w2[jn],
sides='onesided')
#compute coherence
coherence = np.abs(sxy)**2 / (sxx * syy)
coherence_estimates.append(coherence)
#compute variance
coherence_estimates = np.array(coherence_estimates)
coherence_variance = np.zeros([coherence_estimates.shape[1]])
coherence_mean = coherence_estimates[0]
if jackknife:
coherence_mean = coherence_estimates.mean(axis=0)
#mean subtract and square
cv = np.sum((coherence_estimates - coherence_mean)**2, axis=0)
coherence_variance[:] = (1.0 - 1.0 / njn) * cv
#compute frequencies
sampint = 1.0 / sample_rate
L = minlen / 2 + 1
freq = np.linspace(0, 1 / (2 * sampint), L)
#compute upper and lower bounds
cmean = coherence_mean
coherence_lower = cmean - 2 * np.sqrt(coherence_variance)
coherence_upper = cmean + 2 * np.sqrt(coherence_variance)
cdata = CoherenceData()
cdata.coherence = coherence_mean
cdata.coherence_lower = coherence_lower
cdata.coherence_upper = coherence_upper
cdata.frequency = freq
cdata.sample_rate = sample_rate
return cdata
def __vespa_az(self, st):
def find_nearest(array, value):
idx, val = min(enumerate(array), key=lambda x: abs(x[1] - value))
return idx, val
sides = 'onesided'
pi = math.pi
st.sort()
n = len(st)
for i in range(n):
coords = self.inv.get_coordinates(st[i].id)
st[i].stats.coordinates = AttribDict({
'latitude':
coords['latitude'],
'elevation':
coords['elevation'],
'longitude':
coords['longitude']
})
coord = get_geometry(st, coordsys='lonlat', return_center=True)
tr = st[0]
win = len(tr.data)
if (win % 2) == 0:
nfft = win / 2 + 1
else:
nfft = (win + 1) / 2
nr = st.count() # number of stations
delta = st[0].stats.delta
fs = 1 / delta
fn = fs / 2
freq = np.arange(0, fn, fn / nfft)
value1, freq1 = find_nearest(freq, self.linf)
value2, freq2 = find_nearest(freq, self.lsup)
df = value2 - value1
m = np.zeros((win, nr))
WW = np.hamming(int(win))
WW = np.transpose(WW)
for i in range(nr):
tr = st[i]
if self.method == "FK":
m[:, i] = (tr.data - np.mean(tr.data)) * WW
else:
m[:, i] = (tr.data - np.mean(tr.data))
pdata = np.transpose(m)
#####Coherence######
NW = 2 # the time-bandwidth product##Buena seleccion de 2-3
K = 2 * NW - 1
tapers, eigs = alg.dpss_windows(win, NW, K)
tdata = tapers[None, :, :] * pdata[:, None, :]
tspectra = fftpack.fft(tdata)
w = np.empty((nr, int(K), int(nfft)))
for i in range(nr):
w[i], _ = utils.adaptive_weights(tspectra[i], eigs, sides=sides)
Cx = np.ones((nr, nr, df), dtype=np.complex128)
if self.method == "MTP.COHERENCE":
for i in range(nr):
for j in range(nr):
sxy = alg.mtm_cross_spectrum(tspectra[i], (tspectra[j]),
(w[i], w[j]),
sides='onesided')
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')
s = sxy / np.sqrt((sxx * syy))
cxcohe = s[value1:value2]
Cx[i, j, :] = cxcohe
####Calculates Conventional FK-power ##without normalization
if self.method == "FK":
for i in range(nr):
for j in range(nr):
A = np.fft.rfft(m[:, i])
B = np.fft.rfft(m[:, j])
#Power
#out = A * np.conjugate(B)
#Relative Power
den = np.absolute(A) * np.absolute(np.conjugate(B))
out = (A * np.conjugate(B)) / den
cxcohe = out[value1:value2]
Cx[i, j, :] = cxcohe
r = np.zeros((nr, 2))
S = np.zeros((1, 2))
Pow = np.zeros((360, df))
for n in range(nr):
r[n, :] = coord[n][0:2]
freq = freq[value1:value2]
rad = np.pi / 180
slow_range = np.linspace(0, self.slow, 360)
for j in range(360):
ang = self.azimuth2mathangle(self.baz)
S[0, 0] = slow_range[j] * np.cos(rad * ang)
S[0, 1] = slow_range[j] * np.sin(rad * ang)
k = (S * r)
K = np.sum(k, axis=1)
n = 0
for f in freq:
A = np.exp(-1j * 2 * pi * f * K)
B = np.conjugate(np.transpose(A))
D = np.matmul(B, Cx[:, :, n]) / nr
P = np.matmul(D, A) / nr
Pow[j, n] = np.abs(P)
n = n + 1
Pow = np.mean(Pow, axis=1)
return Pow
def FKCoherence(self, st, inv, DT, linf, lsup, slim, win_len, sinc,
method):
def find_nearest(array, value):
idx, val = min(enumerate(array), key=lambda x: abs(x[1] - value))
return idx, val
sides = 'onesided'
pi = math.pi
smax = slim
smin = -1 * smax
Sx = np.arange(smin, smax, sinc)[np.newaxis]
Sy = np.arange(smin, smax, sinc)[np.newaxis]
nx = ny = len(Sx[0])
Sy = np.fliplr(Sy)
#####Convert start from Greogorian to actual date###############
Time = DT
Time = Time - int(Time)
d = date.fromordinal(int(DT))
date1 = d.isoformat()
H = (Time * 24)
H1 = int(H) # Horas
minutes = (H - int(H)) * 60
minutes1 = int(minutes)
seconds = (minutes - int(minutes)) * 60
H1 = str(H1).zfill(2)
minutes1 = str(minutes1).zfill(2)
seconds = "%.2f" % seconds
seconds = str(seconds).zfill(2)
DATE = date1 + "T" + str(H1) + minutes1 + seconds
t1 = UTCDateTime(DATE)
########End conversion###############################
st.trim(starttime=t1, endtime=t1 + win_len)
st.sort()
n = len(st)
for i in range(n):
coords = inv.get_coordinates(st[i].id)
st[i].stats.coordinates = AttribDict({
'latitude':
coords['latitude'],
'elevation':
coords['elevation'],
'longitude':
coords['longitude']
})
coord = get_geometry(st, coordsys='lonlat', return_center=True)
tr = st[0]
win = len(tr.data)
if (win % 2) == 0:
nfft = win / 2 + 1
else:
nfft = (win + 1) / 2
nr = st.count() # number of stations
delta = st[0].stats.delta
fs = 1 / delta
fn = fs / 2
freq = np.arange(0, fn, fn / nfft)
value1, freq1 = find_nearest(freq, linf)
value2, freq2 = find_nearest(freq, lsup)
df = value2 - value1
m = np.zeros((win, nr))
WW = np.hamming(int(win))
WW = np.transpose(WW)
for i in range(nr):
tr = st[i]
if method == "FK":
m[:, i] = (tr.data - np.mean(tr.data)) * WW
else:
m[:, i] = (tr.data - np.mean(tr.data))
pdata = np.transpose(m)
#####Coherence######
NW = 2 # the time-bandwidth product##Buena seleccion de 2-3
K = 2 * NW - 1
tapers, eigs = alg.dpss_windows(win, NW, K)
tdata = tapers[None, :, :] * pdata[:, None, :]
tspectra = fftpack.fft(tdata)
w = np.empty((nr, int(K), int(nfft)))
for i in range(nr):
w[i], _ = utils.adaptive_weights(tspectra[i], eigs, sides=sides)
nseq = nr
L = int(nfft)
#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)
Cx = np.ones((nr, nr, df), dtype=np.complex128)
if method == "MTP.COHERENCE":
for i in range(nr):
for j in range(nr):
sxy = alg.mtm_cross_spectrum(tspectra[i], (tspectra[j]),
(w[i], w[j]),
sides='onesided')
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')
s = sxy / np.sqrt((sxx * syy))
cxcohe = s[value1:value2]
Cx[i, j, :] = cxcohe
# Calculates Conventional FK-power
if method == "FK":
for i in range(nr):
for j in range(nr):
A = np.fft.rfft(m[:, i])
B = np.fft.rfft(m[:, j])
#Relative Power
den = np.absolute(A) * np.absolute(np.conjugate(B))
out = (A * np.conjugate(B)) / den
cxcohe = out[value1:value2]
Cx[i, j, :] = cxcohe
r = np.zeros((nr, 2), dtype=np.complex128)
S = np.zeros((1, 2), dtype=np.complex128)
Pow = np.zeros((len(Sx[0]), len(Sy[0]), df))
for n in range(nr):
r[n, :] = coord[n][0:2]
freq = freq[value1:value2]
for i in range(ny):
for j in range(nx):
S[0, 0] = Sx[0][j]
S[0, 1] = Sy[0][i]
k = (S * r)
K = np.sum(k, axis=1)
n = 0
for f in freq:
A = np.exp(-1j * 2 * pi * f * K)
B = np.conjugate(np.transpose(A))
D = np.matmul(B, Cx[:, :, n]) / nr
P = np.matmul(D, A) / nr
Pow[i, j, n] = np.abs(P)
n = n + 1
Pow = np.mean(Pow, axis=2)
#Pow = Pow / len(freq)
Pow = np.fliplr(Pow)
x = y = np.linspace(smin, smax, nx)
nn = len(x)
maximum_power = np.where(Pow == np.amax(Pow))
Sxpow = (maximum_power[1] - nn / 2) * sinc
Sypow = (maximum_power[0] - nn / 2) * sinc
return Pow, Sxpow, Sypow, coord