In gbonte/gbcode: Code from the handbook "Statistical foundations of machine learning"

Shiny dashboard "Statistical foundations of machine learning"

Gaussian visualization

Univariate standard

Goal: computation of the probability mass associated to the interval (inner or outer).

Bivariate

Goal: computation of the probability mass associated to the interval (inner or outer).

• Slider $\lambda_1$: 1st eigenvalue of covariance matrix (variance along first axis)
• Slider $\lambda_2$: 2nd eigenvalue of covariance matrix (variance along second axis)
• Slider rotation : rotation angle $\theta$ of covariance matrix

Note that the diagonal matrix corresponds to $\lambda_1=\lambda_2$ and $\theta=0$.

A generic bivariate covariance matrix may be written as $$ \Sigma= R S R^T $$ where $S$ and $R$ denote the scaling and rotation of a diagonal covariance matrix. The scaling matrix $$ S=\begin{bmatrix} \lambda_1 & 0 \ 0 & \lambda_2 \end{bmatrix}$$ stretches the x-axis and the y-axis by the factors $\lambda_1>0$ and $\lambda_1>0$ respectively.

The matrix $$ R=\begin{bmatrix} \cos(\theta) & -\sin(\theta) \ sin(\theta) &\cos(\theta) \end{bmatrix}$$ performs a rotation of angle $\theta$.

Suggested manipulations:

• by acting on the sliders, you can change the axis size and the rotation of the bivariate gaussian distribution. The associate covariance matrix and the eigenvalues change accordingly.

gbonte/gbcode documentation built on Feb. 27, 2024, 7:38 a.m.