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:

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

2. The inverse of the covariance matrix, S^-1,

3. The vector of deviations, (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:

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 X₁,...X_n be n p-dimensional vectors of observations that are sampled independently from N_p(m, ) with p < n-1, with the covariance matrix of X. The observed mean vector and the sample dispersion matrix

are the unbiased estimators of m and , 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.