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

Hotelling 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's-t statistic.

The \(T^2\) distance is a constant multiplied by a quadratic form. This quadratic form is obtained by multiplying the following three quantities.

  1. The vector of deviations between the observations and the mean, \( \bf{m} \), which is expressed by \((\bf{X-m})'\),

  2. the inverse of the covariance matrix, \(\bf{S}^{-1}\),

  3. and the vector of deviations, \((\bf{X-m})\).
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: $$ T^2 = c( \bf{X-m'} ) \bf{S}^{-1} ( \bf{X-m} ) \, . $$ 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 \(p < n-1\), and \(\bf{\Sigma}\) is the covariance matrix of \(\bf{X}\). The observed mean vector \(\bf{\bar{X}}\) and the sample dispersion matrix $$ \bf{S} = \frac{1}{n-1} \sum_{i=1}^n (\bf{X}_i - \bf{\bar{X}})(\bf{X}_i - \bf{\bar{X}})' $$ 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.
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