Gaussian Model

univariate data

• Formula:
  $p(x;\mu ,{\sigma }^2)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right)$
• Steps:
  • Estimate $\mu$ and ${\sigma }^2$ from the data.
  • Compute the probability density for each data point.
  • Points with low probabilities (below a threshold $ϵ$) are considered anomalies.

Multivariate Gaussian Distribution

Steps: - Extend the Gaussian model to include covariance across features. - Fit the multivariate Gaussian model:
$p(x;\mu ,\Sigma )=\frac{1}{(2\pi {)}^{n/2}|\Sigma {|}^{1/2}}\exp \left(-\frac{1}{2}(x-\mu {)}^T{\Sigma }^{-1}(x-\mu )\right)$ - $\mu$: Mean vector - $\Sigma$: Covariance matrix - Threshold low-probability examples to identify anomalies.