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6.
Process or Product Monitoring and Control
6.5. Tutorials 6.5.4. Elements of Multivariate Analysis
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| 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. |
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| 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 \(\sigma^2\)
$$ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} \mbox{exp}
\left[ -\frac{(x - m)^2}{2 \sigma^2} \right] \,\,\,\,\,\, -\infty < x < \infty \, . $$
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| 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 $$ \rho = \frac{\sigma_{21}}{\sigma_1 \sigma_2} \, . $$ | ||
