The Monitoring of Linear Profiles Keun Pyo Kim Mahmoud A. Mahmoud William H. Woodall Virginia Tech Blacksburg, VA 24061-0439 (Send request for paper, submitted to JQT , to firstname.lastname@example.org).
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The Monitoring of Linear ProfilesKeun Pyo Kim Mahmoud A. MahmoudWilliam H. WoodallVirginia Tech Blacksburg, VA 24061-0439(Send request for paper, submitted to JQT, to email@example.com)
It is assumed that when the process is in statistical control, the underlying model isi = 1, 2, …, n, where the ’s are independent, identically distributed (i.i.d.) N(0, ).
The least squares estimators and have have a bivariate normal distribution with the mean vector and the variance-covariance matrix
First we consider the Phase II case involving process monitoring with in-control values of the parameters assumed to be known.
Since now the least squares estimators are independent, we recommend three EWMA charts in Phase II to detect sustained shifts in the parameters. There is a chart for each regression coefficient and one for the variation about the line.
We use the in-control model
with error terms i.i.d. N(0, 1). The values for X are 2, 4, 6, 8.
Phase I In Phase I, one has k sets of bivariate observations. One checks for stability of the linear profiles over time and estimates parameters.
Relationship to Regression-adjusted Control Charts
Monitoring linear profiles is a generalization of regression-adjusted methods studied by Mandel (1969), Zhang (1992), Wade and Woodall (1993), Hawkins (1991, 1993), and Hauck et. al (1999).
Suppose X is an input quality variable and Y is the output quality variable with k = 1 and n = 1. Then we have the simplest regression-adjusted chart, sometimes referred to as the cause-selecting chart. (Note X-values are random.)
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