Nonrandom or systematic patterns occurring in a univariate Shewhart control chart have often been used as indicators of extraneous sources of process variation. Proper diagnosis of these patterns can lead to process improvement by reducing the overall system variation. Similarly, in multivariate statistical process control, many di.erent systematic patterns may occur in the control charts used to monitor the process. The purpose of this paper is to examine the process conditions that lead to the occurrence of certain nonrandom patterns in a T^{2}control chart. Examples resulting from cycles, mixtures, trends, process shifts, and autocorrelated data are identi.ed and presented. Results are applicable to a Phase I operation or a Phase II operation where the T^{2}statistic is based on the most common covariance matrix estimator.

Key Words: Autocorrelation, Multivariate Statistical Process Control, Trends.

*By* **ROBERT L. MASON, Southwest Research Institute,
San Antonio, TX 78228-0510
YOUN-MIN CHOU, The University of Texas at San Antonio, San
Antonio, TX 78249-0664
JOE H. SULLIVAN, Mississippi State University, MS 39762-9582
ZACHARY G. STOUMBOS, Rutgers, The State University of New
Jersey, Piscataway, NJ 08854-8054
JOHN C. YOUNG, McNeese State University, Lake Charles, LA
70609-2340**

**Introduction**

One of the most popular control chart statistics for monitoring
a multivariate normal process is Hotelling's *T*^{2}.
This recognition, in large part, is due to the fact that,
among similar statistics, it is the easiest to interpret (e.g.,
see Mason, Tracy, and Young (1997) or Ryan (2000)). The *T*^{2}
chart is sometimes referred to as a multivariate Shewhart
chart, and several of its properties, such as its performance
under varying process conditions, have been examined recently.
For example, Stoumbos and Sullivan (2002) show that, for some
multivariate non-normal distributions, the *T*^{2}
chart based on known in-control parameters has an excessive
false-alarm rate as well as a reduced probability for detecting
shifts in the mean vector. Also, Linna, Woodall, and Busby
(2001) study the effect of measurement error on the performance
of the *T*^{2} chart.

Control charts indicate out-of-control
conditions when the value of the corresponding control chart
statistic is not contained in the control region. For univariate
Shewhart charts, signals occur when the value of the statistic
exceeds the upper control limit (UCL), or when the plotted
statistic falls below the lower control limit (LCL). Lack
of statistical control also may be indicated by a nonrandom
pattern in the plotted statistic. These patterns often are
detected using supplemental run rules, such as those specified
in *The Western Electric Handbook* (1956), or in many
control chart texts (e.g., see Montgomery (2001)). However,
application of run rules is known to cause a considerable
reduction in the in-control average run length (e.g., see
Champ and Woodall (1987)).

The *T*^{2} statistic can be expressed as a
univariate measure of the squared statistical distance that
the p-dimensional multivariate normal observation vector x
is from the process center represented by the population mean
vector _{}.
One form of the *T*^{2} statistic is given by

In applications, we estimate the unknown values of _{}
and
using the common unbiased estimators,
and ,
respectively. These estimates are obtained using the historical
data set (HDS), x1, x2, ..., x_{n}, under baseline
conditions that are indicative of good operational conditions.
The form of the *T*^{2} statistic under these
conditions is given as *T*^{2}

In-control data taken on a multivariate
process reflect the process conditions. In turn, the conditions
are reflected in the *T*^{2} control chart. For
example, data obtained from a fixed-parts industry often exhibit
only random variation when the process is in control. A *T*^{2}
statistic for such data should produce a random signature
in the corresponding *T*^{2} control chart. Any
pattern other than a random signature would indicate lack
of statistical control for the process. For such a situation,
some form of run rules might be useful for generating signals
based on seemingly non-random *T*^{2} patterns.
However, because the *T*^{2} statistic has a
non-normal distribution, it is not appropriate to apply the
Western Electric run rules to the corresponding *T*^{2}
control chart. Further, only limited run rules have been studied
for *T*^{2} control charts. For example, Bozzello
(1989) discusses the use of run rules with a *T*^{2}
control chart in the case where the observation vectors are
taken from a bivariate normal distribution.

In contrast to the above example,
in-control data for some non-stationary processes, such as
occur in the chemical processing industry, may actually contain
inherent variation due to ramp changes, step changes, and
even weak levels of autocorrelation. A *T*^{2}
control chart for these data will often contain nonrandom
patterns corresponding to the data characteristics. Application
of run rules would be inappropriate for this type of process
data, however, as the nonrandom patterns occurring in the
*T*^{2} chart are not indicative of lack of statistical
process control, but due to the inherent variation in the
data. Since multivariate statistical process control is achieved
when the mean vector and covariance matrix remain stable over
rational subgroups of observations (e.g., see Wierda (1994)),
it becomes most important to recognize the source of the patterns
in a *T*^{2} chart.

Given this situation, the major purpose
of this paper is to aid the *T*^{2} control-chart
user by identifying process conditions that can lead to nonrandom
patterns in a *T*^{2} chart. In doing so, we
extend the use of the *T*^{2} statistic in a
Phase I operation and show that certain nonrandom patterns
that occur in the *T*^{2} chart of a preliminary
data set can be used as a diagnostic data tool. This provides
the user with an additional means of locating cycles, mixtures,
trends, process shifts, and autocorrelation within the data
set. These procedures also have applications in a Phase II
operation.

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