||Multivariate Statistical Process Control Applications for Autocorrelated Data
||Kulahci, Murat (Mathematical Statistics, Department of Informatics and Mathematical Modeling, Technical University of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark)
||Technical University of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark
||This thesis deals with applications of auto-correlated data in principal component analysis
and statistical process control. The main purpose of the thesis is to search for and explore
effects and patterns of not only correlation but also auto-correlation in PCA and SPC.
Since nowadays we work with a different kind of data sets auto-correlation has grown
to be more and more applied form of relation in time series analysis. The series used
here is autoregressive model AR(1). It is a particular model of autoregressive integrated
moving average (ARIMA) models. The data, which has been simulated from standard
normal distribution, are a mix of correlated or not correlated variables. Some of which
are also auto-correlated. The effects of auto-correlation are seen in the eigenvalues and
the eigenvectors and visualized on the plots.
Standardization method is used to provide data in the same scale. Thereafter, the data are
used in PCA. However, non-standardization is briefly compared with standardization to
show why the second way is mostly used in the analysis. Some examples of auto-correlated
variables are given and presented in the figures to see the differences in variability of these
Afterwards, the principal components are taken to the statistical control. Shewhart control
chart is used to control the scores. The crucial point is to find Average Run Length for
each principal component assuming the process to be in statistical control. Three different
approaches have been considered in this case. Mainly, three covariance estimators have
been calculated for constructing the control limits to Shewhart control chart. Based on
these estimators eigenvalues are extracted and used as standard deviations in the control
limits. The way of estimating and analyzing is presented in the theoretical and practical
parts. The results of the approaches is to find which covariance estimate is appropriate
when working with auto-correlated data.
The analysis done in the thesis is divided into two practical parts. Each of them consists
of cases and scenarios of the auto-correlation and correlation of the variables. The point
is to present, step by step, the influence of different kinds of correlation in the data and
show how it effects PCA and SPC.
In the thesis I distinguish between correlation and cross-correlation. Correlation of the
variables means that the variables are correlated with each other but not correlated in
time. The second term means that the variables are correlated with each other AND
correlated in time, meaning they are auto-correlated and correlated.
||Technical University of Denmark (DTU) : Kgs. Lyngby, Denmark
Creation date: 2009-12-15
Update date: 2010-08-25