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mccube._regions

Defines the integration regions (measure spaces) against which AbstractCubatures can be defined.

AbstractRegion ¤

Bases: Module

Abstract base class for all (weighted) integration regions.

Integration regions are measure spaces \((\Omega, \mathcal{F}, \mu)\), where \(\mathcal{F}\) is the Borel \(\sigma\)-algebra on the region \(\Omega\), and \(\mu\) is some suitable positive Borel (probability) measure that 'weights' the region.

Attributes:

  • dimension (int) –

    dimension \(d\) of the integration region \(\Omega\).

volume ¤

volume() -> float

Measure \(\mu\) of the entirety of \(\Omega\), denoted by \(V\).

Source code in mccube/_regions.py 
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@abc.abstractproperty
def volume(self) -> float:
    r"""Measure $\mu$ of the entirety of $\Omega$, denoted by $V$."""
    ...

GaussianRegion ¤

Bases: AbstractRegion

Euclidean space \(\mathbb{R}^d\) with Gaussian probability measure.

The probability space \((\mathbb{R}^d, \mathcal{F}, \mu)\), where \(\mu\) is the standard d-dimensional Gaussian measure, with mean zero and identity covariance. I.E. \(\mu(x_1, \dots, x_d) = (2\pi)^{-d/2}\exp(-\frac{x_1^2}{2} - \dots - \frac{x_d^2}{2})\). This is the measure against which the "probabilist's" Hermite polynomials are orthogonal.

Note: if the covariance is scaled by one half, then \(\mu\) is the measure for which the "physicist's" Hermite polynomials are orthogonal. I.E. \(\mu(x_1, \dots, x_d) = \pi^{-d/2}\exp(-x_1^2 - \dots - x_d^2)\).

Attributes:

  • dimension (int) –

    dimension \(d\) of the Euclidean space \(\mathbb{R}^d\).

volume property ¤

volume: float

Measure \(\mu\) of the entirety of \(\Omega\), given by \(V=1\).