At a recent event held by the JMP Japan division, one of the presenters introduced multivariate control charts as one method of identifying root causes.
Furthermore, in a survey of those who attended the event, we received many comments stating that they found the "Multivariate Control Chart" feature useful.
Multivariate control charts themselves have long been known as a type of control chart used in quality control, but the Model Driven Multivariate Control Chart (MDMVCC) included in JMP not only allows you to monitor multivariate processes, but also allows you to interactively drill down and investigate the contribution of each individual process to the overall change, allowing you to diagnose the process.
「モデルに基づく多変量管理図」のレポートModel-Based Multivariate Control Chart Report
This article explains the overview and role of multivariate control charts and introduces an example of process control using JMP's "model-based multivariate control chart."
What is a multivariate control chart?
A multivariate control chart is a control chart that monitors multiple quality characteristics simultaneously. When there are multiple process parameters, it is common to create a univariate control chart for each parameter ( univariate control chart ).
At the same time, when dealing with multiple correlated process parameters, multivariate control charts are used to set control limits taking that correlation into account.
The chart below compares univariate and multivariate control charts.
Looking at the red points, we can see that the data that was within the control limits in the univariate control chart is outside the probability ellipse that takes correlation into account in multivariate (bivariate in this example) process control, that is, the data is outlying.
The multivariate control chart created using the two variables above is shown below.
T², the vertical axis of the multivariate control chart, is calculated as the squared Mahalanobis distance. Data above the red control limit line is outside the control limits, and you can see that all red points have a large T².
Furthermore, if you select all the red points on the control chart and return to the data table, you will see that these points are from 10/15/2019 to 10/19/2019. There may have been factors affecting the process during this period.
"Model-based multivariate control charts" and examples of their use
JMP's Model Driven Multivariate Control Chart (MDMVCC) is a feature that creates multivariate control charts based on principal component analysis or PLS models.
Here we will introduce an example of creating a control chart based on principal component analysis.
The data used is the "WaterTreatment_rev.jmp" file attached to this blog. The sensor data was collected daily from January 2018 to October 2019, and data on 47 items was obtained. We will consider these as process data and plot a multivariate control chart.
*This data has been slightly improved by adding more variables to the JMP sample data "WaterTreatment.jmp."
Basic Operation
Select [Analyze] > [Quality and Process] > [Model-Based Multivariate Control Chart] from the JMP menu bar and specify the 47 process variables as [Process]. The following report of "Multivariate Control Chart Based on PCA (Principal Component Analysis) Model" will be displayed.
The report shows the T² for 13 principal components extracted from 47 variables and plots multivariate control charts of the principal component scores.
The red line on the control chart is the control limit for T² (UCL = 22.16). Any point beyond this value is considered to be outside the control limits in the multivariate space. In this example, there are as many as 45 points outside the control limits.
Interactively explore your process
Looking at the multivariate control chart, there are some points at the beginning where T² exceeds 200., I'd like to know the following:
1) When did this occur?
②What caused it to fall outside the control limits?
In this case, when you move your mouse over the data on the graph, a hover label appears, and the date at the top of the label tells you that this is data for 3/13/2018.
Clicking on the hover label will display a "T² Contribution Plot" on the right side of the report, showing the individual contributions to all variables in a bar graph.
In the Ratio Plot, the color of the bar graph indicates whether the selected data row is univariately out of control (red bars) or in control (green bars).
Additionally, when you move the mouse cursor over a particular bar in the Contribution Plot, a univariate control chart is displayed. In this example, the univariate control chart shows that the point for 2018/3/13 for the variable "RD-SED-G" is well below the lower control limit.
In summary, using JMP's Model-Based Multivariate Control Charts you can gain detailed understanding of your process in the following steps:
Check for points outside the control limits on a multivariate control chart⇒ Select the relevant point on the graph
⇒ Identify the factors in the contribution ratio plot (check the contribution ratio value and univariate control chart)
Displaying the Principal Component Scores Plot
In this example, we have created a multivariate control chart based on principal component analysis, so we can also display the score plot from the principal component analysis. The figure below plots the score of the first principal component on the horizontal axis and the score of the second principal component on the vertical axis, and draws the 95% confidence ellipse.
If we check the points outside the confidence ellipse, we can see that the data from 2018/3/13 to 2018/3/16 is on the left side of the first principal component axis, and the data from 2018/12/10 to 2018/12/13 is on the above side of the second principal component axis.
If a large number of outliers occur in a specific period like this, you should investigate whether there were any process problems during that period.
Reference: Multivariate control charts using historical data
The "Model-Based Multivariate Control Chart" also allows you to apply a control chart to current data based on past data (historical data).
For example, let's say you want to use the data above from 2018 as historical data to manage the process for 2019.
In this case, if you enter the last row of 2018 (2018/12/30) in "Row number of the last row of historical data", the control chart will be divided into historical data (2018) and current data (2019) as shown below, and control limits will be set for each.
Reference : There is an explanatory video for "Model-Based Multivariate Control Charts"!
Creating a control chart using JMP (Japanese)
by Naohiro Masukawa (JMP Japan)
Naohiro Masukawa - JMP User Community
