6.3. Univariate and Multivariate Control Charts
6.3.4. |
What are Multivariate Control Charts? |
collected. The collection is displayed and analyzed, either in "real time", which means that a very fast program works on the data as soon as they are generated, or later after perhaps some cleaning up. It is a fact of life, however, that most data are naturally multivariate.
The classical Shewhart approach, dating back to 1924, tracked one variable at a time by detecting shifts in the mean or variance with the assistance of control limits. Later the Western Electric Company (WECO) introduced a set of rules to interpret the manner by which out-of-control situations occurred. Over time a variety of additional charts have been developed and used as auxiliary tools in the arena of the SPC, SQC and TQM efforts in industry. These include CUSUM and EWMA. These charts were previously described.
Hotelling in 1947 introduced a statistic which uniquely lends itself to plotting of multiple observations. This statistic, appropriately named "Hotelling T2 is a scalar that combines information from the dispersion and mean of several variables. Due to the fact that computations are plentiful and fairly complex and require some knowledge of matrix algebra, acceptance of multivariate control charts by industry was slow and hesitant.
Nowadays, modern computers in general and the PC in particular have made complex calculations accessible and during the last decade, multivariate control charts have started to appear. In fact the multivariate charts which display the Hotelling T2 statistic became so popular that they sometimes are called Shewhart charts as well, (e.g. Crosier, 1988), although Shewhart had nothing to do with them.
As in the univariate case, when data are grouped, the T2 chart can be paired with a chart that displays a measure of variability within the subgroups for all the analyzed characteristics. The combined T2 and T2d (dispersion) charts are thus a multivariate counterpart of the univariate Xbar and S (or Xbar - R) charts.
Each chart represent 14 consecutive measurements on the means of
four variables. The T2 chart for means indicates an out-of-control
state for groups 1,2 and 9-11. The T2d
chart for dispersions indicate that groups 10, 13 and 14 are out-of-control.
The interpretation is that the p-variate system is suspect. To find an
assignable cause, one has to resort to the individual univariate control
charts that should accompany this multivariate chart.
For more details and examples see the next page and also Tutorials, section 5, subsections 4.3, 4.3.1 and 4.3.2. Click here to link to section 5.4.3. An introduction to Elements of multivariate analysis is also given in the Tutorials.
