Quality and Reliability Methods - SAS

114 Multivariate Control Charts Chapter 7
Change Point Detection
Method
Example
Suppose there are m independent observations from a multivariate normal distribution of dimensionality p
such that
x i
∼ N p
( μ i
, Σ i
), i = 1 , …,
m.
where x i is an individual observation, and N p (μ i ,Σ i ) represents a multivariate normally distributed mean
vector and covariance matrix, respectively.
If a distinct change occurs in the mean vector, the covariance matrix, or both, after m 1 observations, all
observations through m 1 have the same mean vector and the same covariance matrix (μ a ,Σ a ). Similarly, all
ensuing observations, beginning with m 1 + 1, have the same mean vector and covariance matrix (μ b ,Σ b ). If
the data are from an in-control process, then μ a = μ b and Σ a = Σ b for all values of m, and the parameters of
the in-control process can be estimated directly from the data.
A likelihood ratio test approach is used to determine changes or a combination of changes in the mean
vector and covariance matrix. The likelihood ratio statistic is plotted for all possible m 1 values, and an
appropriate Upper Control Limit (UCL) is chosen. The location (observation or row number) of the
maximum test statistic value corresponds to the maximum likelihood location of only one shift, assuming
that exactly one change (or shift) occurred. For technical details of this method, refer to the Statistical
Details in the section “Change Point Detection” on page 118.
As an example of determining a possible change or shift in the data, open Gravel.jmp from the Quality
Control subfolder in the Sample Data directory. This data set can be found in Sullivan and Woodall (2000)
and contains 56 observations from a European gravel production plant. The two columns of the data set
show the percent of the particles (by weight) that are large and medium in size. Select Analyze > Quality
And Process > Control Chart > Multivariate Control Chart. Select Large and Medium as Y, Columns,
and click OK. Select Change Point Detection from the Multivariate Control Chart platform menu in the
report. The resulting Change Point Detection Plot is shown in Figure 7.8.
Figure 7.8 Gravel.jmp Change Point Detection Plot
Control chart statistics for the Change Point Detection plot are obtained by dividing the likelihood ratio
statistic of interest (either a mean vector or a covariance matrix) by a normalizing factor. Plotted values
above 1.0 indicate a possible shift in the data. The change point of the data occurs for the observation