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


The first step in analyzing multivariate data is computing the mean vector and the variancecovariance matrix.  
Sample data matrix  Consider the following matrix: $$ {\bf X} = \left[ \begin{array}{ccc} 4.0 & 2.0 & 0.60 \\ 4.2 & 2.1 & 0.59 \\ 3.9 & 2.0 & 0.58 \\ 4.3 & 2.1 & 0.62 \\ 4.1 & 2.2 & 0.63 \end{array} \right] $$ The set of 5 observations, measuring 3 variables, can be described by its mean vector and variancecovariance matrix. The three variables, from left to right are length, width, and height of a certain object, for example. Each row vector \({\bf X}_i\) is another observation of the three variables (or components).  
Definition of mean vector and variance covariance matrix 
The mean vector consists of the
means of each
variable and the variancecovariance matrix consists of the
variances of the
variables along the main diagonal and the covariances between each
pair of variables in the other matrix positions.
The formula for computing the covariance of the variables \(X\) and \(Y\) is $$ \mbox{COV} = \frac{\sum_{i=1}^n (X_i  \bar{x})(Y_i  \bar{y})}{n1} \, , $$ with \(\bar{x}\) and \(\bar{y}\) denoting the means of \(X\) and \(Y\), respectively. 

Mean vector and variance covariance matrix for sample data matrix 
The results are:
$$ {\bf \bar{x}} = \left[ \begin{array}{ccc}
4.10 & 2.08 & 0.604
\end{array} \right] $$
$$ {\bf S} = \left[ \begin{array}{ccc}
0.025 & 0.0075 & 0.00175 \\
0.0075 & 0.0070 & 0.00135 \\
0.00175 & 0.00135 & 0.00043
\end{array} \right] $$
where the mean vector contains the arithmetic averages of the three
variables and the (unbiased) variancecovariance matrix \({\bf S}\)
is calculated by
$$ S = \frac{1}{n1} \sum_{i=1}^n (X_i  \bar{X})(X_i  \bar{X})' \, , $$
where \(n=5\)
for this example.
Thus, 0.025 is the variance of the length variable, 0.0075 is the covariance between the length and the width variables, 0.00175 is the covariance between the length and the height variables, 0.007 is the variance of the width variable, 0.00135 is the covariance between the width and height variables and 0.00043 is the variance of the height variable. 

Centroid, dispersion matix 
The mean vector is often referred to as the centroid and the
variancecovariance matrix as the dispersion or dispersion matrix.
Also, the terms variancecovariance matrix and covariance matrix are used
interchangeably.
