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Interpreting
the Multivariate Chart Statistics
When the process is stable,
it has a stable set of Principal Components. Each Principal Component (PC)
is a linear combination of all the process variables. Unlike the process
variables, which may be correlated, the PCs are constructed to be orthogonal
(independent) of one-another. The PCs may be used to estimate the data and
thereby provide a basis for an estimate of the prediction error. The number
of PCs may never exceed the number of process variables and is often constrained
to be fewer.
- Begin by constructing
the MVA model of the process. Ideally, this model will use data from
a period (a range of initial subgroups) of known process stability.
The T2 control chart will then detect when the process covariance has
shifted relative to this model (or Reference) data. Generally, all Principal
Components (or an arbitrary realistic maximum of 10 PC's) are used for
a first pass, which usually does not provide a realistic model, but
allows calculation of a suggested number of PC's for a realistic model.
- Review the Principal
Component Statistics table , which indicates (with an asterisk)
the suggested number of PC's, using either the Cumulative Percentage
or the Leverage Correction criteria. Re-generate the model with the
suggested number of PC's.
- Evaluate the SPE
(Squared Prediction Error) chart. Groups out of control indicate when
there is significant error between the data (at that group) and the
fitted model. If a few groups are out of control, consider removing
them from the model (Reference data). If many groups are out of control,
then reconsider the number of PC's in the model, the range of groups
in the model, or both (i.e. repeat steps 1 and 2, above).
- If the T2 control
chart is in control, then the process has a stable set of PC's, and
is said to be in control. Consider using this model for evaluating the
future status of the process.
- If the T2 control
chart is out of control, consider removing the out of control points
from the Model (Reference) data range. This provides a model that excludes
known deviations from the remaining model set. In order to determine
which PC has the strongest effect on the T2 statistic, consider the
following:
a) In the Principal Component
t-Scores table, compare the scores from each Principal Component for the
out of control group. PC's with a higher absolute score influence the group
more than other groups.
b) Another approach which
is sometimes useful is to copy the Principal Component t-Scores table into
a blank Data Editor (.qdb file). Construct an Individual-X chart for each
of the PC's. The PC's are assumed to be independent and Normally distributed.
If any of the PC's are out of control for the group in question, it is likely
to be influencing the T2 control chart.
- For any PC which
seems to be influencing the out of control group on the T2 control chart,
determine which underlying process variables have the strongest effect
on the PC.
a) Use the Loading
Chart to determine which Process Variables have higher influence on
the PC in question. (A PC is a linear combination of the Process Variables,
and the Loading is the weight applied to each Process Variable in the stated
PC).
b) Process Variables having
approximately equal Loading are likely to be highly correlated to each other.
This can be verified by review of the Process Variable Correlation Matrix
. When correlation is high between variables (close to one), it is recommended
to remove the extraneous variables from the T2 control chart (i.e. all but
one of the highly correlated variables).
c) Use the Score Contribution
Chart to determine which Process Variables have the highest influence
on the PC in question at the selected group.
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