def test_coherence_linear_dependence():
"""
Tests that the coherence between two linearly dependent time-series
behaves as expected.
From William Wei's book, according to eq. 14.5.34, if two time-series are
linearly related through:
y(t) = alpha*x(t+time_shift)
then the coherence between them should be equal to:
.. :math:
C(\nu) = \frac{1}{1+\frac{fft_{noise}(\nu)}{fft_{x}(\nu) \cdot \alpha^2}}
"""
t = np.linspace(0,16*np.pi,2**14)
x = np.sin(t) + np.sin(2*t) + np.sin(3*t) + 0.1 *np.random.rand(t.shape[-1])
N = x.shape[-1]
alpha = 10
m = 3
noise = 0.1 * np.random.randn(t.shape[-1])
y = alpha*(np.roll(x,m)) + noise
f_noise = np.fft.fft(noise)[0:N/2]
f_x = np.fft.fft(x)[0:N/2]
c_t = ( 1/( 1 + ( f_noise/( f_x*(alpha**2)) ) ) )
f,c = tsa.coherence(np.vstack([x,y]))
c_t = np.abs(signaltools.resample(c_t,c.shape[-1]))
np.testing.assert_array_almost_equal(c[0,1],c_t,2)
def test_coherence_multi_taper():
"""Tests that the code runs and that the resulting matrix is symmetric """
t = np.linspace(0,16*np.pi,1024)
x = np.sin(t) + np.sin(2*t) + np.sin(3*t) + np.random.rand(t.shape[-1])
y = x + np.random.rand(t.shape[-1])
method = {"this_method":'multi_taper_csd',
"Fs":2*np.pi}
f,c = tsa.coherence(np.vstack([x,y]),csd_method=method)
np.testing.assert_array_almost_equal(c[0,1],c[1,0])
def test_coherence_mlab():
"""Tests that the code runs and that the resulting matrix is symmetric """
t = np.linspace(0,16*np.pi,1024)
x = np.sin(t) + np.sin(2*t) + np.sin(3*t) + np.random.rand(t.shape[-1])
y = x + np.random.rand(t.shape[-1])
method = {"this_method":'mlab',
"NFFT":256,
"Fs":2*np.pi}
f,c = tsa.coherence(np.vstack([x,y]),csd_method=method)
np.testing.assert_array_almost_equal(c[0,1],c[1,0])
f_theoretical = ut.get_freqs(method['Fs'],method['NFFT'])
np.testing.assert_array_almost_equal(f,f_theoretical)
