spey.backends.default_pdf.third_moment.compute_third_moments

spey.backends.default_pdf.third_moment.compute_third_moments(absolute_upper_uncertainties: ndarray, absolute_lower_uncertainties: ndarray, return_integration_error: bool = False) → Tuple[ndarray, ndarray] | ndarray

Estimate the diagonal third central moments \(\{m^{(3)}_i\}\) from asymmetric uncertainty envelopes using a bifurcated Gaussian model.

When only the upper and lower uncertainty envelopes \((\sigma^+_i, \sigma^-_i)\) are available (rather than the full background distribution), the background fluctuation in bin \(i\) is modelled as a bifurcated Gaussian:

\[\begin{split}p_i(\theta) = \frac{2}{\sigma^+_i + \sigma^-_i} \begin{cases} \mathcal{N}(\theta \mid 0,\, \sigma^-_i), & \theta < 0, \\ \mathcal{N}(\theta \mid 0,\, \sigma^+_i), & \theta \geq 0. \end{cases}\end{split}\]

The third central moment of this distribution is

\[m^{(3)}_i = \mathbb{E}_i[\theta^3] = \frac{2}{\sigma^+_i + \sigma^-_i} \left[ \sigma^-_i \int_{-\infty}^{0} \theta^3\, \mathcal{N}(\theta \mid 0, \sigma^-_i)\,d\theta + \sigma^+_i \int_{0}^{\infty} \theta^3\, \mathcal{N}(\theta \mid 0, \sigma^+_i)\,d\theta \right],\]

which is evaluated numerically via scipy.integrate.quad().

Note

The \(k\)-th central moment of a distribution \(p\) is defined as

\[m^{(k)} = \mathbb{E}[(X - c)^k] = \int_{-\infty}^{\infty} (x - c)^k\, p(x)\, dx,\]

with \(c = 0\) for the bifurcated Gaussian (which is already centred).

Attention

third_moment_expansion() requires \(8\,\Sigma_{ii}^3 \geq (m^{(3)}_i)^2\). Because this function derives \(m^{(3)}_i\) independently from the covariance matrix, the condition is not guaranteed to hold. Whether it is satisfied depends on the interplay between the envelopes and the diagonal of \(\Sigma\).

Parameters:

• absolute_upper_uncertainties (np.ndarray) – Per-bin upper absolute uncertainties \(\{\sigma^+_i\}\) (positive values).

• absolute_lower_uncertainties (np.ndarray) – Per-bin lower absolute uncertainties \(\{\sigma^-_i\}\) (positive values; signs are taken as absolute internally).

• return_integration_error (bool, default False) – If True, also return the per-bin numerical integration errors from scipy.integrate.quad().

Returns:

Array of diagonal third moments \(\{m^{(3)}_i\}\) (shape \(N\)). When return_integration_error=True, a 2-tuple (third_moments, errors) is returned.

Return type:

np.ndarray or Tuple[np.ndarray, np.ndarray]