Quadratic Discriminant Analysis

Quadratic discriminant analysis (QDA) assumes observations from each class come from a Gaussian distribution and assumes each class has it's own covariance matrix.

Assume that an observation from the $k^{th}$ class is of the form $X\sim N({\mu }_k,{\mathrm{\Sigma }}_k)$, where ${\mathrm{\Sigma }}_k$ is a covariance matrix for the $k^{th}$ class. Under this assumption, the Bayes classifier assigns an observation $X=x$ to the class for which the following is largest:

QDA uses estimates of ${\mathrm{\Sigma }}_k$, ${\mu }_k$, and ${\pi }_k$ for the Bayes classifier and assigns an observation $X=x$ to the class for which the quantity is largest. The quantity $x$ appears as a quadratic function.

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