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Add chi-squared confidence level attribute to PyROS EllipsoidalSet #3434

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@shermanjasonaf shermanjasonaf commented Nov 23, 2024

Summary/Motivation:

Motivated by recent user feedback, this PR adds a new PyROS EllipsoidalSet attribute and constructor argument gaussian_conf_lvl that specifies, as an alternative to the the squared semi-axis scaling factor attribute and constructor argument scale, the level of the (ellipsoidal) confidence region of the multivariate Gaussian distribution with mean center and covariance shape_matrix. Mathematically, the PyROS ellipsoidal set with center $q^0 \in \mathbb{R}^n$, positive definite shape matrix $P \in \mathbb{S}_{++}^n$, and squared semi-axis scale factor $s \geq 0$ is defined by

$$ \{ q \in \mathbb{R}^n \mid (q - q^0)^\top P^{-1} (q - q^0) \leq s \}. $$

For confidence level $p \in [0, 1)$, the $100p$% multivariate normal confidence region is obtained by setting $s = \chi_{n,p}^2$, where $\chi_{n,\cdot}^2$ is the (invertible) lower-tail quantile function of the chi-squared distribution with $n$ degrees of freedom. Thus, we ensure that setting gaussian_conf_lvl triggers, through the invertible mapping between the squared scaling factor $s$ and confidence level $p$, an update of scale, and vice-versa.

Changes proposed in this PR:

  • Add new settable attribute and constructor argument gaussian_conf_lvl to PyROS EllipsoidalSet.

Legal Acknowledgement

By contributing to this software project, I have read the contribution guide and agree to the following terms and conditions for my contribution:

  1. I agree my contributions are submitted under the BSD license.
  2. I represent I am authorized to make the contributions and grant the license. If my employer has rights to intellectual property that includes these contributions, I represent that I have received permission to make contributions and grant the required license on behalf of that employer.

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