SpectralDensityResult.partial_coherence#

SpectralDensityResult.partial_coherence(first, second)[source]#

Squared partial coherence, conditioning on all other variables.

Read off the inverse density matrix \(g = f^{-1}\) as

\[\tilde C_{ij}(\omega) = \frac{|g_{ij}(\omega)|^2}{g_{ii}(\omega)\, g_{jj}(\omega)},\]

the shared power between the two variables at each frequency once every other variable’s contribution is removed – the frequency-domain partial correlation, squared. For a bivariate system there is nothing to condition on and it equals coherence().

Parameters:

  • first (str) – An endogenous variable.

  • second (str) – Another endogenous variable.

Return type:

NDArray[double]

Returns:

A real (n,) array in [0, 1].

Raises:

SpecificationError – If either label is unknown.

Example

>>> import numpy as np
>>> from cultivars.multivariate.reduced_form.vector_autoregression import VAR
>>> rng = np.random.default_rng(0)
>>> z = np.zeros((800, 3))
>>> for t in range(1, 800):
...     z[t, 2] = 0.6 * z[t - 1, 2] + rng.standard_normal()
...     z[t, 0] = 0.3 * z[t - 1, 0] + 0.8 * z[t - 1, 2] + rng.standard_normal()
...     z[t, 1] = 0.3 * z[t - 1, 1] + 0.8 * z[t - 1, 2] + rng.standard_normal()
>>> density = SpectralDensity(VAR(z, order=1).fit(), n_frequencies=64).compute()
>>> plain = density.coherence("y1", "y2")[1]
>>> given = density.partial_coherence("y1", "y2")[1]
>>> bool(plain > 0.3), bool(given < 0.1)
(True, True)