# Understanding Multivariate Gaussian, Gaussian Properties and Gaussian Mixture Model

## (1) Multivariate Gaussian

We already discuss Gaussian distribution function with one variable (univariate) here. In this post, we will discuss about Gaussian distribution function with multi variables (multivariate), which is the general form of Gaussian distribution. For $k-$dimentional vector $\textbf{x}$, multivariate Gaussian distribution is defined as follows.

$\boxed{ \mathcal{N}(x|\boldsymbol{\mu,\Sigma})=\frac{1}{\sqrt{(2\pi)^k}|\boldsymbol{ \Sigma}|^{\frac{1}{2}}}e^{-\frac{1}{2}(\boldsymbol{x-\mu})^T\boldsymbol{\Sigma^{-1}}(\boldsymbol{x-\mu})}}$

$\boldsymbol{ \Sigma}=\begin{bmatrix} &\sigma_{_{X_1}}&COV(X_1,X_2)&COV(X_1,X_3)&...&COV(X_1,X_k)\\\\ &COV(X_2,X_1)&\sigma_{_{X_2}}&COV(X_2,X_3)&...&COV(X_2,X_k)\\\\ &COV(X_3,X_1)&COV(X_3,X_2)&\sigma_{_{X_3}}&...&COV(X_3,X_k)\\\\ &\vdots& & &\ddots&\vdots\\\\ &COV(X_k,X_1)&COV(X_k,X_2)&COV(X_k,X_3)&...&\sigma_{_{X_k}} \end{bmatrix},\, \boldmath{x=\begin{bmatrix}x_1\\\\x_2\\\\x_3\\\\\vdots\\\\x_k\end{bmatrix}},\, \boldmath{\mu=\begin{bmatrix}\mu_1\\\\\mu_2\\\\\mu_3\\\\\vdots\\\\\mu_k\end{bmatrix}}$

where $\boldsymbol{\mu}$ is a $k-$dimensional mean vector, $\boldsymbol{\Sigma}$ is $k \times k$ covariance matrix, and $|\boldsymbol{\Sigma}|$ is the determinant of matrix $\boldsymbol{\Sigma}$. Continue reading “Understanding Multivariate Gaussian, Gaussian Properties and Gaussian Mixture Model”