ellipsoid level-set confidence sampling

For a standard normal distribution the P’th quantile confidence ellispoid is given: $x^T x \leq {}^P\chi^2_n$ where $x \in \mathbb{R}^n$ and ${}^P\chi^2_n$ is the P’th quantile of the chi-squared distribution with n degrees of freedom.

For an arbitrary zero mean normal distribution with covariance matrix $\Sigma$, we use the z score transformation $z = \Sigma^{-1/2} x$ to transform the distribution to a standard normal distribution. Then the P’th quantile confidence ellipsoid is given by:

$$z^T z \leq {}^P\chi^2_n \implies x^T \Sigma^{-1} x \leq {}^P\chi^ 2_n$$

Then, if we want a normal distribution with the P’th percentile confidence ellipsoid to be given by $x^T S x = \rho$, we can choose $\Sigma ^{-1} = \frac{\chi^2_n}{\rho} S$, thus:

$$x^T \Sigma^{-1} x \leq {}^P\chi^2_n \implies x^T \frac{\chi^2_n}{\rho} S x \leq {}^P\chi^2_n \implies x^T S x = \rho$$

This is useful if we want to sample from a normal distribution with a confidence ellipsoid cooresponding to the level set of the LQR cost-to-go.