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.