6. Process or Product Monitoring and Control
6.5. Tutorials
6.5.4. Elements of Multivariate Analysis

## The Multivariate Normal Distribution

Multivariate normal model When multivariate data are analyzed, the multivariate normal model is the most commonly used model.

The multivariate normal distribution model extends the univariate normal distribution model to fit vector observations.

Definition of multivariate normal distribution A $$p$$-dimensional vector of random variables, $${\bf X} = X_1, \, X_2, \, \ldots, \, X_p \,\,\,\,\,\, -\infty < X_i < \infty, \,\, i = 1, \, \ldots, \, p \, ,$$ is said to have a multivariate normal distribution if its density function $$f({\bf X})$$ is of the form $$\begin{eqnarray} f({\bf X}) & = & f(X_1, \, X_2, \, \ldots, \, X_p) \\ & = & \left( \frac{1}{2 \pi} \right)^{p / 2} |{\bf \Sigma}|^{-1/2} \mbox{exp} \left[ -\frac{1}{2} ({\bf X} - {\bf m})' {\bf \Sigma}^{-1} ({\bf X} - {\bf m}) \right] \, , \end{eqnarray}$$ where $${\bf m} = (m_1, \, \ldots, \, m_p)$$ is the vector of means and $${\bf \Sigma}$$ is the variance-covariance matrix of the multivariate normal distribution. The shortcut notation for this density is $${\bf X} = \mbox{N}_p ({\bf m}, {\bf \Sigma}) \, .$$
Univariate normal distribution When $$p=1$$, the one-dimensional vector $${\bf X} = X_1$$ has the normal distribution with mean $$m$$ and variance $$\sigma^2$$ $$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} \mbox{exp} \left[ -\frac{(x - m)^2}{2 \sigma^2} \right] \,\,\,\,\,\, -\infty < x < \infty \, .$$

Bivariate normal distribution When $$p=2$$, $${\bf X} = (X_1, \, X_2)$$ has the bivariate normal distribution with a two-dimensional vector of means, $${\bf m} = (m_1, \, m_2)$$ and covariance matrix $${\bf \Sigma} = \left[ \begin{array}{cc} \sigma_1^2 & \sigma_{12} \\ \sigma_{21} & \sigma_2^2 \end{array} \right] \, .$$ The correlation between the two random variables is given by $$\rho = \frac{\sigma_{21}}{\sigma_1 \sigma_2} \, .$$