Home | Business News | Browse by Publication | G | Georgia Journal of Science

Principal components for diagnosing dispersion in multivariate statistical process control.

Article Excerpt
ABSTRACT

We provide an easily implemented procedure to help data analysts systematically diagnose which quality characteristics may be driving the dispersion of a multivariate process out of control. Multivariate statistical process control (MSPC) commonly uses Hotelling's [T.sup.2] statistic to indicate when a multivariate observation goes out-of-control. Several techniques currently exist that accurately diagnose which specific variables are driving the [T.sup.2] statistic out-of-control. For subgroups of independently and identically distributed multivariate normal observations, we advocate decomposing the overall [T.sup.2] into independent [T.sup.2] statistics for separate monitoring of location and dispersion. We propose a procedure based on principal components to diagnose the specific variables responsible for driving subgroup dispersion out-of-control. The procedure is demonstrated on a publicly available data-set.

Keywords: multivariate, statistical process control, principal components, diagnosis of dispersion

INTRODUCTION

It is increasingly clear that new methods of diagnosing the dispersion of multivariate processes are needed. (1) The purpose of this article is to present a principal component based procedure for diagnosing which specific variable(s) in multivariate statistical process control (MSPC) are driving the process dispersion out of control. Well regarded sources in the literature that summarize the state of MSPC (2, 3, 4, 5) indicate that one of the test statistics most commonly used to monitor a multivariate process is Hotelling's [T.sup.2] statistic. Although principal components have been used in the past to diagnose which variables are driving the [T.sup.2] statistic out of control (2, 3, 4, 5), the efficacy of principal component based (PC-based) diagnosis has been contingent upon how well these components approximate a physically interpretable latent factor. Kourti and MacGregor (6) show that under multivariate normality the normalized scores of the principal components can accurately diagnose the causal variables regardless of their physical meaning. In contrast, Mason and Young (5) diagnose the responsible variables by an orthogonal decomposition of the [T.sup.2] statistic not based on principal components.

Data analysts are increasingly monitoring multivariate processes that are subject to shifts in scale as well as location. When monitoring rational subgroups of multivariate observations, the techniques of Mason and Young (5) and Kourti and MacGregor (6) have limited diagnostic capability because they can't discern between these two different kinds of shifts, i.e. scale and location. Mason and Young (5) state that their decomposition procedure does not discretely identify the specific variables responsible for shifting location versus those responsible for driving dispersion out-of-control. Kourti and MacGregor (6) did not demonstrate how their procedure might be applied to diagnosis of multivariate dispersion.

Our contribution is to show how the procedure of Kourti and MacGregor (6) can be extended to systematically identify variables driving the [T.sup.2] statistic for subgroup dispersion out-of-control. We demonstrate the diagnostic potential of our procedure on a data-set from Fuchs and Kenett (4) containing subgroups of data whose significant [T.sup.2] values are directly attributable to out-of-control subgroup dispersion. To our knowledge there is no such procedure based on a simply implemented decomposition of Hotelling's [T.sup.2] statistic. We hope this procedure will be useful to data analysts who monitor multivariate processes by providing a systematic method of investigating which variables may be driving process variance out of control.

The paper is organized as follows. In Materials and Methods we show how the PC-based procedure of Kourti and MacGregor (6) can be directly applied to diagnose shifts in subgroup location and extended to diagnose shifts in subgroup dispersion. In Results we demonstrate the diagnostic procedure for subgroup dispersion on data from Fuchs and Kenett (4). In Discussion we present concluding remarks.

MATERIALS AND METHODS

Diagnosis of Shifts in Hotelling's [T.sup.2]

In the practice of statistical process control, it is preferable to collect rational subgroups of multivariate observations whenever possible. Subgroups yield more reliable process information than individual vectors as well as an estimate of the correlation structure within the subgroup. Research including that of Hawkins (7), Kourti and MacGregor (6) and Mason and Young (5) can be used to diagnose the variables which drive the subgroup's overall [T.sup.2] out of control. However these do not typically differentiate between the variables shifting location versus those driving higher dispersion. In order to distinguish which variables cause which type of...

View this article FREE - Now for a Limited Time, try Goliath Business News
Free for 3 Days!