6.
Process or Product Monitoring and Control
6.3. Univariate and Multivariate Control Charts 6.3.4. What are Multivariate Control Charts?


Definition of Hotelling's \(T^2\) "distance" statistic 
The Hotelling \(T^2\)
distance is a measure that
accounts for the covariance structure of a multivariate normal
distribution. It was proposed by Harold Hotelling in 1947
and is called Hotelling \(T^2\).
It may be thought
of as the multivariate counterpart of the Student'st
statistic.
The \(T^2\) distance is a constant multiplied by a quadratic form. This quadratic form is obtained by multiplying the following three quantities. It should be mentioned that for independent variables, the covariance matrix is a diagonal matrix and \(T^2\) becomes proportional to the sum of squared standardized variables. In general, the higher the \(T^2\) value, the more distant is the observation from the mean. The formula for computing the \(T^2\) is: $$ T^2 = c( \bf{Xm'} ) \bf{S}^{1} ( \bf{Xm} ) \, . $$ The constant \(c\) is the sample size from which the covariance matrix was estimated. 

\(T^2\) readily graphable  The \(T^2\) distances lend themselves readily to graphical displays and as a result the \(T^2\) chart is the most popular among the multivariate control charts.  
Estimation of the Mean and Covariance Matrix  
Mean and Covariance matrices  Let \( \bf{X}_1, \, \bf{X}_2, \, \ldots, \, \bf{X}_n \) be \(n\) \(p\)dimensional vectors of observations that are sampled independently from \( N_p(\bf{m}, \bf{\Sigma}) \), where \(p < n1\), and \(\bf{\Sigma}\) is the covariance matrix of \(\bf{X}\). The observed mean vector \(\bf{\bar{X}}\) and the sample dispersion matrix $$ \bf{S} = \frac{1}{n1} \sum_{i=1}^n (\bf{X}_i  \bf{\bar{X}})(\bf{X}_i  \bf{\bar{X}})' $$ are the unbiased estimators of \(\bf{m}\) and \(\bf{\Sigma}\), respectively.  
Additional discussion  See Tutorials (section 5), subsections 4.3, 4.3.1 and 4.3.2 for more details and examples. An introduction to Elements of multivariate analysis is also given in the Tutorials. 