Phase-I analysis for

manufacturing process control-

a combination of T2 and

multivariate-CUSUM method

Authors: Sandeep Nemmani

Vanshaj Handoo

EXECUTIVE SUMMARY

The existing dependency between the multiple variables of most physical processes has led to the need for monitoring each

parameter closely, so that total quality control can be achieved by making the necessary detection of the data samples which

are contributing to the process going out of the acceptable control limits. Some of the most widely-used control charts for

monitoring multivariate data are the T2 control chart, m-EWMA control chart, and the m-CUSUM control chart, to name a few.

In this project, we started our analysis procedure by determining the mean of the variables in each sample and representing this

graphically. The profile of this graph led us to believe that the data was indicative of a process which was following a trend of a

unimodal function, where the values keep increasing up to a particular value, and then keep curving downward again. We

studied the individual profiles for the different samples present in the data and have expectedly observed a similar trend. Also,

as this is a manufacturing physical process with a single unit value for each variable in a sample dataset, we chose to use

covariance matrix instead of the correlation matrix, as we were interested in maintaining the original relationship between the

these variables throughout the process.

An important assumption made by us is that the data provided here follows a normal distribution. This is a reasonable

assumption as any process with sufficiently large continuous data will eventually approximate to a normal distribution. For the

purpose of data reduction, we have done the principal component analysis (PCA) by using the minimum description length

(MDL), scree plot, and the pareto plot in conjunction.

To identify the in-control samples values, we have used both T2 chart and the m-CUSUM chart. The idea here is that although

both T2 and m-CUSUM charts are used for a similar purpose, i.e. detecting the out-of-control samples from a given dataset, they

have different operational purposes. While the T2 control chart is effective in detecting a spike-type of change, the m-CUSUM

chart detects the mean shift experienced in the process. Using both the charts together is a conclusive way to find the in control

parameters of a process. This further enhanced our understanding about the multivariate natured detection and application of

the aforementioned process control methods.

Initially, we began the analysis by performing the required number of iterations on the T2 control chart. This eliminated the

spike-type changes observed for the samples. Then we used the m-CUSUM control chart to detect the mean shift for the in-

control samples. Once we were certain that there was no sustained mean shift (with a statistical distance of 3) in the process,

we proceeded to perform the necessary iterations on the new set of in-control samples by using the T2 chart again to check if

the new data had any more spike type changes . Following this, the m-CUSUM chart iteration was done for the second time, and

this resulted in no sample being out of control. We thus concluded that there were no samples which had any spike type

changes, nor were experiencing a sustained mean shift.

In retrospection, although we did achieve the desired result, we could have avoided extra computations by performing the m-

CUSUM control chart iterations in the first instance instead of the T2 chart calculations. This is because some of the spike-type

changes that were eliminated in the T2 chart iterations could have been eliminated in the first iteration of the m-CUSUM itself.

The average run length (ARL0) value (which is 200 for a corresponding UCL of 6), is quite less than that for the Shewhart chart

(which is 370 for the 3-sigma control limits). Additionally, for a mean shift of 2.984, the ARL1 value for the T2 control chart was

found to be 2.67. But in the case of an m-CUSUM control chart, for a mean shift of 11.2, the ARL1 value obtained was 2.

Therefore, the mean shift required for a similar ARL1 value in the case of the m-CUSUM control chart iterations was significantly

large when compared with the T2 chart. Our understanding for this phenomenon is that it is the offset value (k ni) which varies

(increases) on summation, thus preventing a sample to go out of control and in turn resulting in an increase in the ARL1 value.

However, we are not certain if this is the only reason for having a large ARL1 in m-CUSUM when compared with the T2 case.

Another observation was that the final in-control sample values covariance matrix obtained from doing both the m-CUSUM and

the T2 control charts iterations was a correlated one (i.e it had non-zero values in the non-principal diagonal elements). This is a

counter intuitive observation as we had employed the principal component analysis to the variables initially, which transforms a

correlated matrix to an uncorrelated one.

MULTIVARIATE STATISTICAL DETECTION AND ITS BENEFITS:

The primary advantage of a multivariate data analysis over the univariate one is that it eliminates the need to prepare the

univariate charts for each monitoring characteristic. As there are 209 variables available with us in this dataset, it can be

particularly taxing to prepare 209 control charts and then determine which of them are contributing for the process signalling

the out of control condition. Therefore, multivariate statistical analysis is preferred over the univariate detection. This also

eliminates the need to adjust the α and β errors, as they will need to be adjusted owing to the inflation in their values caused by

the use of the multiple univariate charts.

IMPLEMENTATION OF PRINCIPAL COMPONENT ANALYSIS (PCA) IN THE MULTIVARIATE STATISTICAL DETECTION:

Although multivariate detection does help in effectively representing the procedural characteristics of a physical process

consisting of multiple number of variables on a single control chart, it will still be difficult to monitor the process efficiently when

there are high number of variables present, such as in this instance. Instead of monitoring all the 209 variables, we have used

the concept of Principal Component Analysis (PCA) to determine which variables contribute the maximum towards the process

readings hovering outside the specified control limits. The rule of thumb which we have implemented here was that the

cumulative variance of the variables chosen by PCA had to account for at least 80% of the total sum of the variance of all the

parameters of the process.

DETERMINATION OF THE PRINCIPAL COMPONENTS:

To achieve this dimensional data reduction, the first step was to determine the minimum description length (MDL), as MDL

facilitates the identification of the few variables which can represent the pattern of the entire data. The MDL was calculated as

36, which implied that there were 36 variables which represented the regularity present in the data. Further scrutiny proved

that the number of principal components could still be reduced, and therefore a scree plot was plotted between the variable

number and the corresponding eigen value of these 36 variables, where the eigen value are plotted in the descending order.

Upon observing the scree plot, it was noticed that the elbow bend was after the 9th variable, and therefore, it entailed that the

number of principal components could further be narrowed to four.

To ensure that the reading was accurate, a pareto plot was also constructed, which also arranges the eigen values of the

corresponding variables in the descending order. As the eigen values were comparatively low and uniform after the fourth

variable, the number of principal components to be analysed were conclusively selected as four. More importantly, the

proportion of the cumulative variance of these principal components to the total variance exhibited by all the variables in the

dataset was found to be 0.8011 or 80.11%, thereby fulfilling our aforementioned condition of the proportion being at least 80%.

Illustration of the MDL plot:

Illustration of the scree plot:

Illustration of the pareto plot:

USAGE OF THE T2 HOTELLING STATISTIC AND CONTROL CHART:

After the selection of the principal components, the T2 chart was obtained by plotting the sample number on the X-axis and the

corresponding T2 statistic on the Y-axis. Here the objective is to conduct a Phase-I analysis with the value of n equal to 1. This

value is considered because every variable from 1.….209 has a unique observational value in each sample data set. Therefore,

with n=1 and a Phase-I analysis, the Upper Control Limit (UCL) value is determined as the value of χ2 1-α(p).

Here p=number of principal components selected=4.

α=0.0027 (comparable with 3 sigma control limits of Shewhart charts)

The formula of the T2 statistic is given by:

T2= (xj-x̄ ) s-1 x̄ T

Where, xj = Individual data sample values matrix

x̄ = Average values of individual xj’s matrix

s = Sample covariance matrix

After the UCL was calculated, we plotted the available data values and noticed that 11 of these values were beyond the UCL. As

they were responsible for the process being detected as an out of control one, we remove these points. However, after the

analysis we observe that there were 7 values which were still out of the control limits. As a result, the Phase-I analysis was

performed two more times until all the points obtained were within the upper control limit. The details and results of each

iteration are provided below:

Phase-I Analysis Number of Out-of-Control samples observed

After the 1st iteration 11

After the 2nd iteration 7

After the 3rd iteration 2

After the 4th iteration 0

(All the T2 chart readings are provided in the appendix+ provided at the end of the report)

Total in-control samples after completing the Phase-I analysis using a T2 control chart = 552-(11+7+2) = 532

Therefore, by performing the Phase-I iterations on a T2 control chart using the existing data and eliminating the out-of-control

samples, we had successfully managed to identify the in-control sample values. Although the T2 chart is useful in detecting the

spike-type change fluctuation in the existing uncorrelated data, it cannot present a clear picture about the discrepancy caused

by the sustained mean shift undergone in the process. Using other forms of multivariate control charts, such as multivariate-

CUSUM (m-CUSUM) or multivariate EWMA (m-EWMA), which are susceptible to the mean shift which takes place in a process

can help overcome this limitation. In this instance, we consider the m-CUSUM control chart to detect the mean shift.

USAGE OF MULTIVARIATE CUSUM CONTROL CHART:

The logic employed in using an m-CUSUM chart is that we identify the samples which have a mean shift equivalent to the

statistical distance of 3. These points are deemed out-of-control and are removed by repeating the Phase-I analysis the required

number of times, until there are no more points which have undergone a mean shift of 3. The desired ARL0 is expected to be in

the vicinity of the value of 200. From the available UCL and ARL0 values for p=2,3 and 10, the corresponding UCL and ARL0 values

for p=4 are 6.00 and 201, respectively.

The formula for the m-CUSUM statistic (MCi) is:

MCi = max {0, (CTi ϵ-1 Ci)1/2- k . ni}

Where, Ci = Cumulative sum of previous ‘ni’ number of xi’s

ϵ = Population covariance matrix

k = offset constant, typically chosen as half of the statistical mean shift which we intend to detect, i.e, in this case, k = 1.5

To begin with, the 532 in-control samples (obtained by the Phase-I analysis performed on the T2 charts) are represented on the

X-axis and the corresponding m-CUSUM statistics are represented on the Y-axis. Again, we observed that 61 of the samples were

showing an m-CUSUM statistic value higher than the UCL (6.00). Consequently, we proceeded to conduct the Phase-I analysis on

these charts again, similar to how we did for the T2 control charts. After six Phase-I analysis iterations, we obtained all the

sample number m-CUSUM statistical points in control, i.e. within the specified UCL. The results of the initial m-CUSUM and the

subsequent Phase-I iterations are tabulated below:

Phase-I Analysis Number of Out-Of-Control samples observed

After the 1st iteration 61

After the 2nd iteration 19

After the 3rd iteration 2

After the 4th iteration 7

After the 5th iteration 1

After the 6th iteration 0

(All the m-CUSUM chart readings are provided in the appendix+ at the end of the report)

In-control samples after completing the Phase-I analysis using the m-CUSUM control chart = 532-(61+19+7+2+1) = 442

After the 442 in-control data samples are obtained, we deduce that there is no undetected mean shift (mean shift of 3) in the

process. But it still remains to be seen if there is still a spike-type change for these 442 samples. To confirm that there is indeed

no such change still left in the process, we revert back to the T2 control chart to confirm our supposition.

REVERSION TO T2 CONTROL CHART:

The corresponding T2 statistics for all the 442 samples obtained from the Phase-I analysis of the m-CUSUM control chart were

plotted again. As suspected earlier, there were 6 more points which were out-of-control on the chart. Hence, we conducted the

Phase-I analysis again to ensure that there were no more out-of-control points. The results for the same are available below:

Phase-I Analysis Number of Out-Of-Control samples observed

After the 1st iteration 6

After the 2nd iteration 2

After the 3rd iteration 0

(All the T2 control chart iterations are provided in the appendix+ at the end of the report)

Total in-control samples after completing the Phase-I analysis using the m-CUSUM control chart = 442-(6+2) = 434.

Therefore, there are now 434 in-control samples which do not exhibit any spike-type changes. But there nevertheless might be

samples whose mean shift value is equivalent to 3. To confirm if this is indeed the case, we repeat the procedure done by us

earlier, i.e. we use the m-CUSUM chart again.

FINAL CONFIRMATION USING THE M-CUSUM CONTROL CHART:

On plotting the m-CUSUM statistic values on the control chart, we could see that there are now no points which were exceeding

the UCL. This implies that we had obtained a set of samples which were not experiencing any spike-type changes in the process,

as well as did not have a mean shift of an equivalent statistical distance equal to 3. In conclusion, we can state that there are 434

data samples from the initial 552 samples which are in control.

CHART PERFORMANCE PARAMETERS:

By performing the Monte Carlo simulation for the T2 control chart with a specified mean shift of 2.984, the ARL1 value was found

to be 2.67. Whereas in the case of an m-CUSUM control chart, for a mean shift of 9.8, the ARL1 obtained was 5. Also, to obtain

the required ARL1 of 2, the corresponding mean shift required was 11.2. Some other ARL1 values for specified mean shifts for

both the processes are detailed in the following page:

For T2 control chart:

Specified mean shift ARL1 value observed

0 371

0.915 115.44

1.66 27.44

2.99 2.67

4.35 0.36

For m-CUSUM control chart:

Specified mean shift ARL1 value observed

0 200

5.72 568

8.45 18

9.8 5

11.2 2

Therefore, the specified mean shift values required for obtaining almost identical ARL1 values differs significantly in both the

processes. As mentioned earlier, our understanding for this phenomenon is that it is the variation in the offset value (k ni) which

results in an increase in the ARL1 value. However, we are not certain if this is the sole reason for having a large ARL1 in m-CUSUM

compared to the T2 case.

Therefore, from the ARL0 and ARL1 values obtained by us for both the control charts, we can specify the run lengths for

conducting the Phase-II analysis for any further data samples in the future. The in-control samples mean is found as:

[

7.98102718.241214.875598−0.25935

]

And the in-control samples covariance matrix is determined as:

[

6386.49 505.4985 −362.29 −126.755505.4985 1175.924 −383.666 208.188−362.29 −383.666 1122.933 118.3388−126.755 208.188 118.3388 409.0725

]

Ideally, the covariance matrix obtained from the iterations of the data sample values for either of the control charts must consist

of uncorrelated variables only. But we have noticed here that the covariance matrix values are still correlated, as the non-

diagonal elements of the above matrix still consists of non-zero elements. We had obtained this result even though we had

employed the PCA as described initially.

Appendix+:

T2 control chart (1st round of iterations):

In-Control samples T2 chart:

m-CUSUM control chart (1st round of iterations):

Reversion to T2 control chart (2nd round of iterations):

Final confirmation through m-CUSUM (2nd round of iterations):