Discussion about Discriminant Functions for the Multivariate Normal Density
(partially based on Prof. Mireille Boutin's ECE 662 lecture)
Multivariate Normal Density
Because of the mathematical tractability as well as because of the central limit theorem, Multivariate Normal Density, as known as Gaussian Density, received more attention than other various density functions that have been investigated in pattern recognition.
The general multivariate normal density in d dimensions is:
$ p(\mathbf{x})=\frac{1}{(2\pi)^{d/2}| \mathbf{\Sigma} |^{1/2} }exp\begin{bmatrix} -\frac{1}{2} (\mathbf{x}-\mathbf{\mu})^t \mathbf{\Sigma} ^{-1}(\mathbf{x}-\mathbf{\mu}) \end{bmatrix}~~~~(1) $
where
$ \mathbf{x}=[x_1,x_2,\cdots ,x_d]^t $ is a d-component column vector,
$ \mathbf{\mu}=[\mu_1,\mu_2,\cdots ,\mu_d]^t $ is the d-component mean vector,
$ \mathbf{\Sigma}=\begin{bmatrix} \sigma_1^2 & \cdots & \sigma_{1d}^2\\ \vdots & \ddots & \\ \vdots \sigma_{d1}^2 & \cdots & \sigma_d^2 \end{bmatrix} $ is the d-by-d covariance matrix,
$ |\mathbf{\Sigma|} $ is its determinant,
$ \mathbf{\Sigma}^{-1} $ is its inverse.
Each $ p(x_i)\sim N(\mu_i,\sigma_i^2) $, so $ \Sigma $ plays role similar to the role that $ \sigma^2 $ plays in one dimension. Thus, we can find out the relationship among different features (such as $ x_i $ and $ x_j $)through $ \Sigma $:
- $ \sigma_{ij}=0 $, $ x_i $ and $ x_j $ are independent;
- $ \sigma_{ij}>0 $, $ x_i $ and $ x_j $ are positive correlated;
- $ \sigma_{ij}<0 $, $ x_i $ and $ x_j $ are negative correlated.
