6.3.4.1. Hotelling Control Charts

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

6.3.4.1. 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.

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

the inverse of the covariance matrix, $\bf{S}^{-1}$,

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.