Application of Principal Components Analysisin Quality Control Problem

Application of Principal
Components Analysis
in Quality Control Problem
Date: March 4, 2014

Introduction to Principal Components Analysis (PCA)
 Why we need to bring principal components analysis into quality control field?
When the number of variables is relatively large, the conventional
multivariate control charts have a poor performance(large ARL1) on
detecting a specific mean shift.
Therefore, it is necessary to find out some new methods to monitor
processes with large number of variables. A group of methods, called
latent structure methods, are particularly efficient when the variability is
unequally distributed. Principal components is one of them.

 What is principal components?
Principal components are set of linear combinations of the original data.
z1 = c11x1 + c12x2 + …… + c1pxp
z2 = c21x1 + c22x2 + …… + c2pxp
:
zp = cp1x1 + cp2x2 + …… + cppxp
The z’s could form a new coordinate system which is rotated from the
original x system. The new axes have the largest variability.

 What is principal components?
The idea of principal components is to find that new coordinate system
which has fewer variables(principal components) than in the original
system.
 How to calculate the principal components?
To calculate the principal components, we first need to find out the
coefficients of the original variables. This could be computed by the
following formula.
C’ ∑C = Λ
cij is the element of the jth eigenvector associated with the eigenvalue λi.

 How to monitor the process with the principal components?
First determine how many principal components are needed to keep into
the monitoring by checking the proportion of variability explained by the
first r principal components.
Then plot the principal components scores of the preliminary dataset onto
a scatter plot with confidence contours of certain levels.
Finally plot the principal components scores of future values to check if
they are outside the certain confidence contours or not. The outside point
is a signal of process out of control.

 Final comments
This plot is called the principal component trajectory plots.
Monitoring more than two principal components, the pairwise scatter plots
is needed.
The PCA method is not efficient to detect a shift which is orthogonal to the
retained principal component directions.

Statistical Process
Monitoring with
Principal Components
By Mastrangelo, C. M., G. C. Runger, and D. C.
Paper Review 1 about PCA

Review of the first paper about PCA
 Introduction
In this paper, the authors are trying to discuss the performance of the
conventional statistical process monitoring techniques with the methods
based on latent structure, particularly the principal components under the
circumstance that there is dynamic behavior(autocorrelation) in the
processing data.
The paper also presents an example of using PCA method and gives the
discussion on the efficiency of this method.

 The performance of the Classical Multivariate SPC
The univariate control charts have a main disadvantage that they may need
to monitor a huge number of variables.
The multivariate quality methods, like Hotelling T2, on the other hand, only
to monitor one criteria.
The authors also discuss that when there is autocorrelation is the
multivariate data the monitoring techniques based on T2 would be stable
than the univariate control charts.

 Monitoring with Latent Variables
In the paper, the authors review two advantages of PCA that the principal
components variables can be found to be some kinds of physical characters
of the process and that the sensitivity to assignable causes is increased by
using PCA.
One disadvantage of PCA the author argued in the paper is that when the
shift is orthogonal to the axes of our selected principal components the
signal will be missed from monitoring process.

 Example of an autocorrelated data
A dataset of 7 variables with
autocorrelation is selected on purpose.
The authors choose the first two principal components to build the in control
status of the process. The dots circled on the lower left is the in control area.
A shift of 1σ also plotted on upper right.
The dots 51 to 60 represent the shift increasing from 0 to 1σ. As we can see
from the plot, the PCA successfully detects the shift when there is
autocorrelation in the data.
Figure on this slice is from Mastrangelo, Runger, and Montgomery (1996) . See reference 1.

 Conclusion
The authors draw conclusions that:
1. The T2 control chart is less sensitive to autocorrelation comparing to the
univariate control charts;
2. The PCA could reduce the dimensionality of monitoring process and is a
good method when there is autocorrelation in the data.

Efficient Shift Detection Using
Multivariate Exponentially-
weighted Moving Average
Control Charts And Principal
Components
By Scranton, R., G. C. Runger, J. B. Keats, and D.C.
Montgomery
Paper Review 2 about PCA

Review of the second paper about PCA
 Introduction
In this paper, the authors combine the principal components with
multivariate exponentially-weighted moving average (MEWMA) to monitor
the process.
Both the number of variables and the ARL1 of monitoring process could be
reduced by PCA.
The advantages of the principal components MEWMA is also presented in
this paper.
To illustrate these, an example is discussed at the end of their paper.

 What are MEWMA and principal components?
The MEWMA is a direct extension of the exponentially weighted moving
average (EWMA). And the MEWMA is a better method to chi-square
control chart when analyzing the ARL of the chart.
By using the principal components method, the analyst could build a
dimensional space with fewer degrees. Therefore, with smaller space to
monitor, the process control becomes more efficient.

 Combination of MEWMA and principal components
By the authors, it could be shown that the results by using the full principal
components to MEWMA is equivalent to results generated by T2.
However, MEWMA could be enhanced by using the principal components.
By the authors, the ARL performance of MEWMA chart is shortened by
applying principal components to the data.
Therefore, the new method is superior to the chi-square chart applied to
all the variables.

 A typical Example
Figure Source: Scranton, Runger, Keats, and Montgomery (1996). See reference 2.
This dataset has 10
variables.
If, for example , we use the
first three principal
components to monitor this
process, then the average
length of ARL to detect the
shift is 6.69. In contrast, the
average length is 9.16 when
using the original MEWMA.

 Conclusion
The principal components MEWMA chart is very useful in the circumstance
that there are many variables in the dataset.
This combination maintains the advantage of principal components and
gains sensitivity for MEWMA to detect the shifts.

Conclusion
 The principal components analysis method is a powerful method to
monitor processes with a large number of variables.
 The principal components analysis method is also an appropriate method
when there is autocorrelation in the data.
 The MEWMA chart combining with principal components could reduce
the average length of detecting a shift in the process comparing with the
original MEWMA.

References
 Mastrangelo, C. M., G. C. Runger, and D. C. Montgomery (1996). “Statistical
Process Monitoring with Principal Components,” Quality and Reliability
Engineering International, Vol.12(2), pp. 79-89
 Scranton, R., G. C. Runger, J. B. Keats, and D.C. Montgomery (1996). “Efficient
Shift Detection Using Exponentially Weighted Moving Average Control Charts and
Principal Components,” Quality and Reliability Engineering International, Vol.
12(3), pp. 158-163
 Montgomery, D. C. (1999). “Introduction to Statistical Quality Control. 6th edition.
Wiley, New York.

Application of Principal Components Analysisin Quality Control Problem

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