EWMA Control Charts and Shift Detection: A PCA Approach

EFFICIENT SHIFT_DETECTION USING EXPONENTIALLY WEIGHTED MOVING AVERAGE CONTROL CHARTS AND PRINCIPAL COMPONENTS
Jayam Dharani Krishna, Rongali Harish
Abstract- The control limits for exponentially weighted moving average (EWMA) varies with time and approaches_asymptotic limit as the time passes. The shift detection is measured by how much the process goes out of the control limits. The shift is auto corrected by using the variable chart for subgroups. The main assumption behind the Principal Component Analysis (PCA) is discussed and comparisons are made between the multivariate EWMA used in PCA to other methods of statistical control processes.
I. INTRODUCTION
The EWMA#is one of the ways by which the control charts are plotted for the variables.

a) STEPS TO BE FOLLOWED TO PERFORM SHIFT DETECTION USING PRINCIPAL COMPONENT ANALYSIS WITH EWMA
1. Plot the data set into a matrix that has dimension time X variable.
2. Normalize the data.
3. Obtain the mean per column.
4. Calculate the variance – covariance matrix of X(S).
5. Obtain the Eigen values and Eigen vectors of S.
6. Plot the Eigen vectors using a chart. Each explains λ*100% of the total variance.
7. Analyze the results obtained.
b) A TYPICAL EXAMPLE
The following example shows the application of PCA applied to the numerous variables. The principal components are calculated to give the estimate of how the number of variables can be reduced and the correlation between the variables is also calculated. [4]
S.No Income Education Age Residence Employ Savings Debt Credit cards Variable PC1 PC2 vPC1
1 50000 16 28 2 2 5000 1200 2 Income 0.313901 0.144645

The score plot for the first two components shows the distribution of the Eigen values thus obtained and helps in the creation of correlation matrix required for calculating the reduced number of variables. This helps in plotting the multivariate control charts. The ARL is also reduced from 200 to 50 with the help of correlation matrix. It has been found out that the upper control limit for the MEWMA chart is much above the required process and thus it is concluded that the process is in control.
d) GENERAL RESULTS OBTAINED BY APPLYING PRINCIPAL COMPONENTS TO MEWMA CHARTS
1. When the complete set of principal component variables Y is given, it is found that a MEWMA chart applied to Y generates the same value of T^2 as applying MEWMA to original variables, X. [6]
2. For shifts in the subspace of the important principal components the non-centrality parameters of MEWMA charts applied to the chief_principal component variables and the original variables are identical. [2]
3. Implementation of the MEWMA chart can be improved by reducing the dimension of the matrix.