def test_get_spectra_bi():
"""
Test the bi-variate get_spectra function
"""
methods = (None,
{"this_method": 'welch', "NFFT": 256, "Fs": 2 * np.pi},
{"this_method": 'welch', "NFFT": 1024, "Fs": 2 * np.pi})
for method in methods:
arsig1, _, _ = utils.ar_generator(N=2 ** 16)
arsig2, _, _ = utils.ar_generator(N=2 ** 16)
avg_pwr1 = (arsig1 ** 2).mean()
avg_pwr2 = (arsig2 ** 2).mean()
avg_xpwr = (arsig1 * arsig2.conjugate()).mean()
tseries = np.vstack([arsig1, arsig2])
f, fxx, fyy, fxy = tsa.get_spectra_bi(arsig1, arsig2, method=method)
# \sum_{\omega} PSD(\omega) d\omega:
est_pwr1 = np.sum(fxx * (f[1] - f[0]))
est_pwr2 = np.sum(fyy * (f[1] - f[0]))
est_xpwr = np.sum(fxy * (f[1] - f[0])).real
# Test that we have the right order of magnitude:
npt.assert_array_almost_equal(est_pwr1, avg_pwr1, decimal=-1)
npt.assert_array_almost_equal(est_pwr2, avg_pwr2, decimal=-1)
npt.assert_array_almost_equal(np.mean(est_xpwr),
np.mean(avg_xpwr),
decimal=-1)
def test_get_spectra():
"""
Testing spectral estimation
"""
methods = (None,
{"this_method": 'welch', "NFFT": 256, "Fs": 2 * np.pi},
{"this_method": 'welch', "NFFT": 1024, "Fs": 2 * np.pi})
for method in methods:
avg_pwr1 = []
avg_pwr2 = []
est_pwr1 = []
est_pwr2 = []
arsig1, _, _ = utils.ar_generator(N=2 ** 16) # It needs to be that long for
# the answers to converge
arsig2, _, _ = utils.ar_generator(N=2 ** 16)
avg_pwr1.append((arsig1 ** 2).mean())
avg_pwr2.append((arsig2 ** 2).mean())
tseries = np.vstack([arsig1,arsig2])
f, c = tsa.get_spectra(tseries,method=method)
# \sum_{\omega} psd d\omega:
est_pwr1.append(np.sum(c[0, 0]) * (f[1] - f[0]))
est_pwr2.append(np.sum(c[1, 1]) * (f[1] - f[0]))
# Get it right within the order of magnitude:
npt.assert_array_almost_equal(est_pwr1, avg_pwr1, decimal=-1)
npt.assert_array_almost_equal(est_pwr2, avg_pwr2, decimal=-1)
def test_crosscov():
N = 128
ar_seq1, _, _ = utils.ar_generator(N=N)
ar_seq2, _, _ = utils.ar_generator(N=N)
for all_lags in (True, False):
sxy = utils.crosscov(ar_seq1, ar_seq2, all_lags=all_lags)
sxy_ref = ref_crosscov(ar_seq1, ar_seq2, all_lags=all_lags)
err = sxy_ref - sxy
mse = np.dot(err, err) / N
npt.assert_(mse < 1e-12, "Large mean square error w.r.t. reference cross covariance")
def test_crosscov():
N = 128
ar_seq1, _, _ = utils.ar_generator(N=N)
ar_seq2, _, _ = utils.ar_generator(N=N)
for all_lags in (True, False):
sxy = utils.crosscov(ar_seq1, ar_seq2, all_lags=all_lags)
sxy_ref = ref_crosscov(ar_seq1, ar_seq2, all_lags=all_lags)
err = sxy_ref - sxy
mse = np.dot(err, err) / N
nt.assert_true(mse < 1e-12, \
'Large mean square error w.r.t. reference cross covariance')
def test_periodogram():
arsig, _, _ = ut.ar_generator(N=512)
avg_pwr = (arsig * arsig.conjugate()).mean()
f, psd = tsa.periodogram(arsig, N=2048)
df = 2. * np.pi / 2048
avg_pwr_est = np.trapz(psd, dx=df)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=1)
def test_AR_LD():
"""
Test the Levinson Durbin estimate of the AR coefficients against the
expercted PSD
"""
arsig, _, _ = utils.ar_generator(N=512)
avg_pwr = (arsig * arsig.conjugate()).real.mean()
order = 8
ak, sigma_v = tsa.AR_est_LD(arsig, order)
w, psd = tsa.AR_psd(ak, sigma_v)
# the psd is a one-sided power spectral density, which has been
# multiplied by 2 to preserve the property that
# 1/2pi int_{-pi}^{pi} Sxx(w) dw = Rxx(0)
# evaluate this integral numerically from 0 to pi
dw = np.pi / len(psd)
avg_pwr_est = np.trapz(psd, dx=dw) / (2 * np.pi)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=0)
# Test for providing the autocovariance as an input:
ak, sigma_v = tsa.AR_est_LD(arsig, order, utils.autocov(arsig))
w, psd = tsa.AR_psd(ak, sigma_v)
avg_pwr_est = np.trapz(psd, dx=dw) / (2 * np.pi)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=0)
def armdl_func (sys,h,w):
s = (h,w)
x_c = np.zeros(s)
x_c[0] = sys;
npts1 = w
for i in np.arange(9):
xi=x_c[i]
c_e, sigma_e = alg.AR_est_YW(xi, 15)
xo, _, _ = utils.ar_generator(N=npts1, sigma=sigma_e, coefs=c_e, drop_transients=45)
x_c[i+1][:]=xo
plt.figure
plt.plot(x_c[h-1], label='estimated process')
plt.plot(sys, 'k', label='original process')
plt.legend()
err = x_c[h-1] - sys
mse = np.dot(err, err) / w
plt.title('MSE = %1.3e' % mse)
plt.grid(True)
plt.show()
f_sys, Pxx_sys = signal.periodogram(x_c[h-1], fs)
f_syswel, Pxx_syswel = signal.welch(sys,fs)
plt.figure
plt.semilogy(f_sys[1:],Pxx_sys[1:], label='estimated process')
plt.semilogy(f_syswel[1:],Pxx_syswel[1:], 'k', label='original process')
plt.legend()
plt.title('PSD')
plt.grid(True)
plt.show()
def test_autocorr():
N = 128
ar_seq, _, _ = utils.ar_generator(N=N)
rxx = utils.autocorr(ar_seq)
npt.assert_(rxx[0] == rxx.max(), "Zero lag autocorrelation is not maximum autocorrelation")
rxx = utils.autocorr(ar_seq, all_lags=True)
npt.assert_(rxx[127] == rxx.max(), "Zero lag autocorrelation is not maximum autocorrelation")
def test_autocorr():
N = 128
ar_seq, _, _ = utils.ar_generator(N=N)
rxx = utils.autocorr(ar_seq)
yield nt.assert_true, rxx[0] == 1, "Zero lag autocorrelation is not equal to 1"
rxx = utils.autocorr(ar_seq, all_lags=True)
yield nt.assert_true, rxx[127] == 1, "Zero lag autocorrelation is not equal to 1"
def test_AR_LD():
"""
Test the Levinson Durbin estimate of the AR coefficients against the
expercted PSD
"""
arsig,_,_ = utils.ar_generator(N=512)
avg_pwr = (arsig*arsig.conjugate()).real.mean()
order = 8
ak, sigma_v = tsa.AR_est_LD(arsig, order)
w, psd = tsa.AR_psd(ak, sigma_v)
# the psd is a one-sided power spectral density, which has been
# multiplied by 2 to preserve the property that
# 1/2pi int_{-pi}^{pi} Sxx(w) dw = Rxx(0)
# evaluate this integral numerically from 0 to pi
dw = np.pi/len(psd)
avg_pwr_est = np.trapz(psd, dx=dw) / (2*np.pi)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=0)
# Test for providing the autocovariance as an input:
ak,sigma_v = tsa.AR_est_LD(arsig, order, utils.autocov(arsig))
w, psd = tsa.AR_psd(ak, sigma_v)
avg_pwr_est = np.trapz(psd, dx=dw) / (2*np.pi)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=0)
def test_periodogram():
arsig, _, _ = ut.ar_generator(N=512)
avg_pwr = (arsig * arsig.conjugate()).mean()
f, psd = tsa.periodogram(arsig, N=2048)
df = 2. * np.pi / 2048
avg_pwr_est = np.trapz(psd, dx=df)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=1)
def test_get_spectra_complex():
"""
Testing spectral estimation
"""
methods = (None, {
"this_method": 'welch',
"NFFT": 256,
"Fs": 2 * np.pi
}, {
"this_method": 'welch',
"NFFT": 1024,
"Fs": 2 * np.pi
})
for method in methods:
avg_pwr1 = []
avg_pwr2 = []
est_pwr1 = []
est_pwr2 = []
# Make complex signals:
r, _, _ = utils.ar_generator(N=2**16) # It needs to be that long for
# the answers to converge
c, _, _ = utils.ar_generator(N=2**16)
arsig1 = r + c * scipy.sqrt(-1)
r, _, _ = utils.ar_generator(N=2**16)
c, _, _ = utils.ar_generator(N=2**16)
arsig2 = r + c * scipy.sqrt(-1)
avg_pwr1.append((arsig1 * arsig1.conjugate()).mean())
avg_pwr2.append((arsig2 * arsig2.conjugate()).mean())
tseries = np.vstack([arsig1, arsig2])
f, c = tsa.get_spectra(tseries, method=method)
# \sum_{\omega} psd d\omega:
est_pwr1.append(np.sum(c[0, 0]) * (f[1] - f[0]))
est_pwr2.append(np.sum(c[1, 1]) * (f[1] - f[0]))
# Get it right within the order of magnitude:
npt.assert_array_almost_equal(est_pwr1, avg_pwr1, decimal=-1)
npt.assert_array_almost_equal(est_pwr2, avg_pwr2, decimal=-1)
def test_AR_est_consistency():
order = 10 # some even number
ak = _random_poles(order/2)
x, v, _ = utils.ar_generator(N=512, coefs=-ak[1:], drop_transients=100)
ak_yw, ssq_yw = tsa.AR_est_YW(x, order)
ak_ld, ssq_ld = tsa.AR_est_LD(x, order)
npt.assert_almost_equal(ak_yw, ak_ld)
npt.assert_almost_equal(ssq_yw, ssq_ld)
def test_AR_est_consistency():
order = 10 # some even number
ak = _random_poles(order // 2)
x, v, _ = utils.ar_generator(N=512, coefs=-ak[1:], drop_transients=100)
ak_yw, ssq_yw = tsa.AR_est_YW(x, order)
ak_ld, ssq_ld = tsa.AR_est_LD(x, order)
npt.assert_almost_equal(ak_yw, ak_ld)
npt.assert_almost_equal(ssq_yw, ssq_ld)
def test_LD_AR():
arsig,_,_ = ut.ar_generator(N=512)
avg_pwr = (arsig*arsig.conjugate()).mean()
w, psd = tsa.LD_AR_est(arsig, 8, 1024)
# for efficiency, let's leave out the 2PI in the numerator and denominator
# for the following integral
dw = 1./1024
avg_pwr_est = np.trapz(psd, dx=dw)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=0)
def test_autocorr():
N = 128
ar_seq, _, _ = utils.ar_generator(N=N)
rxx = utils.autocorr(ar_seq)
nt.assert_true(rxx[0] == rxx.max(), \
'Zero lag autocorrelation is not maximum autocorrelation')
rxx = utils.autocorr(ar_seq, all_lags=True)
nt.assert_true(rxx[127] == rxx.max(), \
'Zero lag autocorrelation is not maximum autocorrelation')
def test_periodogram():
arsig,_,_ = ut.ar_generator(N=512)
avg_pwr = (arsig*arsig.conjugate()).mean()
f, psd = tsa.periodogram(arsig, N=2048)
# for efficiency, let's leave out the 2PI in the numerator and denominator
# for the following integral
dw = 1./2048
avg_pwr_est = np.trapz(psd, dx=dw)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=1)
def test_periodogram():
arsig, _, _ = ut.ar_generator(N=512)
avg_pwr = (arsig * arsig.conjugate()).mean()
f, psd = tsa.periodogram(arsig, N=2048)
# for efficiency, let's leave out the 2PI in the numerator and denominator
# for the following integral
dw = 1. / 2048
avg_pwr_est = np.trapz(psd, dx=dw)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=1)
def test_periodogram():
"""Test some of the inputs to periodogram """
arsig, _, _ = utils.ar_generator(N=1024)
Sk = np.fft.fft(arsig)
f1, c1 = tsa.periodogram(arsig)
f2, c2 = tsa.periodogram(arsig, Sk=Sk)
npt.assert_equal(c1, c2)
# Check that providing a complex signal does the right thing
# (i.e. two-sided spectrum):
N = 1024
r, _, _ = utils.ar_generator(N=N)
c, _, _ = utils.ar_generator(N=N)
arsig = r + c * scipy.sqrt(-1)
f, c = tsa.periodogram(arsig)
npt.assert_equal(f.shape[0], N) # Should be N, not the one-sided N/2 + 1
def test_periodogram():
"""Test some of the inputs to periodogram """
arsig, _, _ = utils.ar_generator(N=1024)
Sk = fftpack.fft(arsig)
f1, c1 = tsa.periodogram(arsig)
f2, c2 = tsa.periodogram(arsig, Sk=Sk)
npt.assert_equal(c1, c2)
# Check that providing a complex signal does the right thing
# (i.e. two-sided spectrum):
N = 1024
r, _, _ = utils.ar_generator(N=N)
c, _, _ = utils.ar_generator(N=N)
arsig = r + c * scipy.sqrt(-1)
f, c = tsa.periodogram(arsig)
npt.assert_equal(f.shape[0], N) # Should be N, not the one-sided N/2 + 1
def test_periodogram_csd():
"""Test corner cases of periodogram_csd"""
arsig1, _, _ = utils.ar_generator(N=1024)
arsig2, _, _ = utils.ar_generator(N=1024)
tseries = np.vstack([arsig1, arsig2])
Sk = fftpack.fft(tseries)
f1, c1 = tsa.periodogram_csd(tseries)
f2, c2 = tsa.periodogram_csd(tseries, Sk=Sk)
npt.assert_equal(c1, c2)
# Check that providing a complex signal does the right thing
# (i.e. two-sided spectrum):
N = 1024
r, _, _ = utils.ar_generator(N=N)
c, _, _ = utils.ar_generator(N=N)
arsig1 = r + c * scipy.sqrt(-1)
r, _, _ = utils.ar_generator(N=N)
c, _, _ = utils.ar_generator(N=N)
arsig2 = r + c * scipy.sqrt(-1)
tseries = np.vstack([arsig1, arsig2])
f, c = tsa.periodogram_csd(tseries)
npt.assert_equal(f.shape[0], N) # Should be N, not the one-sided N/2 + 1
def test_get_spectra_bi():
"""
Test the bi-variate get_spectra function
"""
methods = (None, {
"this_method": 'welch',
"NFFT": 256,
"Fs": 2 * np.pi
}, {
"this_method": 'welch',
"NFFT": 1024,
"Fs": 2 * np.pi
})
for method in methods:
arsig1, _, _ = utils.ar_generator(N=2**16)
arsig2, _, _ = utils.ar_generator(N=2**16)
avg_pwr1 = (arsig1**2).mean()
avg_pwr2 = (arsig2**2).mean()
avg_xpwr = (arsig1 * arsig2.conjugate()).mean()
tseries = np.vstack([arsig1, arsig2])
f, fxx, fyy, fxy = tsa.get_spectra_bi(arsig1, arsig2, method=method)
# \sum_{\omega} PSD(\omega) d\omega:
est_pwr1 = np.sum(fxx * (f[1] - f[0]))
est_pwr2 = np.sum(fyy * (f[1] - f[0]))
est_xpwr = np.sum(fxy * (f[1] - f[0])).real
# Test that we have the right order of magnitude:
npt.assert_array_almost_equal(est_pwr1, avg_pwr1, decimal=-1)
npt.assert_array_almost_equal(est_pwr2, avg_pwr2, decimal=-1)
npt.assert_array_almost_equal(np.mean(est_xpwr),
np.mean(avg_xpwr),
decimal=-1)
def test_periodogram_csd():
"""Test corner cases of periodogram_csd"""
arsig1, _, _ = utils.ar_generator(N=1024)
arsig2, _, _ = utils.ar_generator(N=1024)
tseries = np.vstack([arsig1, arsig2])
Sk = np.fft.fft(tseries)
f1, c1 = tsa.periodogram_csd(tseries)
f2, c2 = tsa.periodogram_csd(tseries, Sk=Sk)
npt.assert_equal(c1, c2)
# Check that providing a complex signal does the right thing
# (i.e. two-sided spectrum):
N = 1024
r, _, _ = utils.ar_generator(N=N)
c, _, _ = utils.ar_generator(N=N)
arsig1 = r + c * scipy.sqrt(-1)
r, _, _ = utils.ar_generator(N=N)
c, _, _ = utils.ar_generator(N=N)
arsig2 = r + c * scipy.sqrt(-1)
tseries = np.vstack([arsig1, arsig2])
f, c = tsa.periodogram_csd(tseries)
npt.assert_equal(f.shape[0], N) # Should be N, not the one-sided N/2 + 1
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 xrange(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 = np.fft.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:
npt.assert_raises(ValueError,
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:
npt.assert_raises(ValueError, tsa.mtm_cross_spectrum, tspectra,
np.r_[tspectra, tspectra], (w, w))
def test_multi_taper_psd_csd():
"""
Test the multi taper psd and csd estimation functions.
Based on the example in
doc/examples/multi_taper_spectral_estimation.py
"""
N = 2**10
n_reps = 10
psd = []
est_psd = []
est_csd = []
for jk in [True, False]:
for k in range(n_reps):
for adaptive in [True, False]:
ar_seq, nz, alpha = utils.ar_generator(N=N, drop_transients=10)
ar_seq -= ar_seq.mean()
fgrid, hz = tsa.freq_response(1.0,
a=np.r_[1, -alpha],
n_freqs=N)
psd.append(2 * (hz * hz.conj()).real)
f, psd_mt, nu = tsa.multi_taper_psd(ar_seq,
adaptive=adaptive,
jackknife=jk)
est_psd.append(psd_mt)
f, csd_mt = tsa.multi_taper_csd(np.vstack([ar_seq, ar_seq]),
adaptive=adaptive)
# Symmetrical in this case, so take one element out:
est_csd.append(csd_mt[0][1])
fxx = np.mean(psd, axis=0)
fxx_est1 = np.mean(est_psd, axis=0)
fxx_est2 = np.mean(est_csd, axis=0)
# Tests the psd:
psd_ratio1 = np.mean(fxx_est1 / fxx)
npt.assert_array_almost_equal(psd_ratio1, 1, decimal=-1)
# Tests the csd:
psd_ratio2 = np.mean(fxx_est2 / fxx)
npt.assert_array_almost_equal(psd_ratio2, 1, decimal=-1)
def test_multi_taper_psd_csd():
"""
Test the multi taper psd and csd estimation functions.
Based on the example in
doc/examples/multi_taper_spectral_estimation.py
"""
N = 2 ** 10
n_reps = 10
psd = []
est_psd = []
est_csd = []
for jk in [True, False]:
for k in range(n_reps):
for adaptive in [True, False]:
ar_seq, nz, alpha = utils.ar_generator(N=N, drop_transients=10)
ar_seq -= ar_seq.mean()
fgrid, hz = tsa.freq_response(1.0, a=np.r_[1, -alpha],
n_freqs=N)
psd.append(2 * (hz * hz.conj()).real)
f, psd_mt, nu = tsa.multi_taper_psd(ar_seq, adaptive=adaptive,
jackknife=jk)
est_psd.append(psd_mt)
f, csd_mt = tsa.multi_taper_csd(np.vstack([ar_seq, ar_seq]),
adaptive=adaptive)
# Symmetrical in this case, so take one element out:
est_csd.append(csd_mt[0][1])
fxx = np.mean(psd, axis=0)
fxx_est1 = np.mean(est_psd, axis=0)
fxx_est2 = np.mean(est_csd, axis=0)
# Tests the psd:
psd_ratio1 = np.mean(fxx_est1 / fxx)
npt.assert_array_almost_equal(psd_ratio1, 1, decimal=-1)
# Tests the csd:
psd_ratio2 = np.mean(fxx_est2 / fxx)
npt.assert_array_almost_equal(psd_ratio2, 1, decimal=-1)
def test_AR_YW():
arsig,_,_ = utils.ar_generator(N=512)
avg_pwr = (arsig*arsig.conjugate()).mean()
order = 8
ak,sigma_v = tsa.AR_est_YW(arsig, order)
w, psd = tsa.AR_psd(ak, sigma_v)
# the psd is a one-sided power spectral density, which has been
# multiplied by 2 to preserve the property that
# 1/2pi int_{-pi}^{pi} Sxx(w) dw = Rxx(0)
# evaluate this integral numerically from 0 to pi
dw = np.pi / len(psd)
avg_pwr_est = np.trapz(psd, dx=dw) / (2*np.pi)
# consistency on the order of 10**0 is pretty good for this test
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=0)
# Test for providing the autocovariance as an input:
ak,sigma_v = tsa.AR_est_YW(arsig, order, utils.autocov(arsig))
w, psd = tsa.AR_psd(ak, sigma_v)
avg_pwr_est = np.trapz(psd, dx=dw) / (2*np.pi)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=0)
def test_AR_YW():
arsig, _, _ = utils.ar_generator(N=512)
avg_pwr = (arsig * arsig.conjugate()).mean()
order = 8
ak, sigma_v = tsa.AR_est_YW(arsig, order)
w, psd = tsa.AR_psd(ak, sigma_v)
# the psd is a one-sided power spectral density, which has been
# multiplied by 2 to preserve the property that
# 1/2pi int_{-pi}^{pi} Sxx(w) dw = Rxx(0)
# evaluate this integral numerically from 0 to pi
dw = np.pi / len(psd)
avg_pwr_est = np.trapz(psd, dx=dw) / (2 * np.pi)
# consistency on the order of 10**0 is pretty good for this test
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=0)
# Test for providing the autocovariance as an input:
ak, sigma_v = tsa.AR_est_YW(arsig, order, utils.autocov(arsig))
w, psd = tsa.AR_psd(ak, sigma_v)
avg_pwr_est = np.trapz(psd, dx=dw) / (2 * np.pi)
npt.assert_almost_equal(avg_pwr, avg_pwr_est, decimal=0)
sig[1] = sxx_m[0] - phi[1, 1] * sxx_m[1]
for k in range(2, order + 1):
phi[k, k] = (sxx_m[k] -
np.dot(phi[1:k, k - 1], sxx_m[1:k][::-1])) / sig[k - 1]
for j in range(1, k):
phi[j, k] = phi[j, k - 1] - phi[k, k] * phi[k - j, k - 1]
sig[k] = sig[k - 1] * (1 - phi[k, k]**2)
sigma_v = sig[-1]
arcoefs = phi[1:, -1]
return sigma_v, arcoefs, pacf, phi #return everything
import nitime.utils as ut
sxx = None
order = 10
npts = 2048 * 10
sigma = 1
drop_transients = 1024
coefs = np.array([0.9, -0.5])
# Generate AR(2) time series
X, v, _ = ut.ar_generator(npts, sigma, coefs, drop_transients)
s = X
import statsmodels.api as sm
sm.tsa.stattools.pacf(X)
tspectra = np.fft.fft(tdata)
mag_sqr_spectra = np.abs(tspectra)
np.power(mag_sqr_spectra, 2, mag_sqr_spectra)
rms_err_fmri, _, _ = generate_subset_weights_and_err(
mag_sqr_spectra[10], l
)
w_err_fmri = compare_weight_methods(mag_sqr_spectra[10], l)
plot_err(rms_err_fmri, 'FMRI re-weight err')
plot_err(w_err_fmri, 'FMRI weights err')
pp.show()
# ---- AUTOREGRESSIVE SEQUENCE
ar_seq, nz, alpha = utils.ar_generator(N=n_samples, drop_transients=10)
ar_seq -= ar_seq.mean()
ar_spectra = np.abs(np.fft.fft(tapers * ar_seq))
np.power(ar_spectra, 2, ar_spectra)
rms_err_arseq, _, _ = generate_subset_weights_and_err(
ar_spectra, l
)
w_err_arseq = compare_weight_methods(ar_spectra, l)
plot_err(rms_err_arseq, 'AR re-weight err')
plot_err(w_err_arseq, 'AR weights err')
pp.show()
""" import numpy as np import matplotlib.pyplot as plt import nitime.utils as utils import nitime.timeseries as ts import nitime.viz as viz """ For this example, we generate an auto-regressive sequence to be the signal: """ ar_seq, nz, alpha = utils.ar_generator(N=128, drop_transients=10) ar_seq -= ar_seq.mean() """ The signal will be repeated several times, adding noise to the signal in each repetition: """ n_trials = 12 fig_snr = [] sample = [] fig_tseries = [] """
""" import numpy as np import matplotlib.pyplot as plt import nitime.utils as utils import nitime.timeseries as ts import nitime.viz as viz """ For this example, we generate an auto-regressive sequence to be the signal: """ ar_seq, nz, alpha = utils.ar_generator(N=128, drop_transients=10) ar_seq -= ar_seq.mean() """ The signal will be repeated several times, adding noise to the signal in each repetition: """ n_trials = 12 fig_snr = [] sample = [] fig_tseries = []
Fs = 1000 """ In this case, we generate an order 2 AR process, with the following coefficients: """ coefs = np.array([0.9, -0.5]) """ This generates the AR(2) time series: """ X, noise, _ = utils.ar_generator(npts, sigma, coefs, drop_transients) ts_x = TimeSeries(X, sampling_rate=Fs, time_unit='s') ts_noise = TimeSeries(noise, sampling_rate=1000, time_unit='s') """ We use the plot_tseries function in order to visualize the process: """ fig01 = plot_tseries(ts_x, label='AR signal') fig01 = plot_tseries(ts_noise, fig=fig01, label='Noise') fig01.axes[0].legend() """ .. image:: fig/ar_est_1var_01.png
""" In this case, we generate an order 2 AR process, with the following coefficients: """ coefs = np.array([0.9, -0.5]) """ This generates the AR(2) time series: """ X, noise, _ = utils.ar_generator(npts, sigma, coefs, drop_transients) ts_x = TimeSeries(X, sampling_rate=Fs, time_unit='s') ts_noise = TimeSeries(noise, sampling_rate=1000, time_unit='s') """ We use the plot_tseries function in order to visualize the process: """ fig01 = plot_tseries(ts_x,label='AR signal') fig01 = plot_tseries(ts_noise,fig=fig01,label='Noise') fig01.axes[0].legend() """
"""Simple example AR(p) fitting."""
import numpy as np
from matplotlib import pyplot as plt
from nitime import utils
from nitime import algorithms as alg
reload(utils)
npts = 2048*10
sigma = 1
drop_transients = 1024
coefs = np.array([0.9, -0.5])
# Generate AR(2) time series
X, v, _ = utils.ar_generator(npts, sigma, coefs, drop_transients)
# Visualize
plt.figure()
plt.plot(v)
plt.title('noise')
plt.figure()
plt.plot(X)
plt.title('AR signal')
# Estimate the model parameters
sigma_est, coefs_est = alg.yule_AR_est(X, 2, 2*npts, system=True)
print 'coefs :', coefs
print 'coefs est:', coefs_est