This post introduces multivariate fault detection with multivariate statistical process monitoring (MSPM) and discusses its benefits over univariate methods.

What is MSPM?

Multivariate statistical process monitoring (MSPM) is a commonly used data-driven statistical approach to detect faults (anomalies/abnormalities) in a process when more than one variable is measured at a time. It is common in many modern processes for many process variables to be measured simultaneously. Often, regular sensor or analyzer readings are automatically recorded and uploaded via a Supervisory Control and Data Acquisition (SCADA) system. Common examples of areas where SCADA systems are used include:

• public utilities, such as water/wastewater treatment or electric utilities
• manufacturing processes
• environmental monitoring, such as air quality monitoring
• building management systems that help monitor and control things such as heating/cooling

MSPM is also commonly referred to as multivariate statistical process control (MSPC). Personally, I prefer the terminology monitoring over control because these methods are not used to directly intervene and control a process, but to monitor the process and alert others in the event of any unexpected abnormalities.

Why is MSPM important?

As explained in this post, quick and accurate detection of faults is essential; by quickly identifying the presence of abnormalities in a process, the source of those abnormalities can be promptly investigated and corrected, helping minimize:

• damage to process equipment
• poor production quality
• process downtime (often due to maintenance for damaged equipment)
• overall operational costs (less damage and downtime = lower cost)
• negative impacts on environmental or human health

Although it is possible to simply create a control chart for each variable in the process that we want to monitor, this approach is very inefficient.

MSPM allows us to monitor all variables of interest simultaneously without having to make individual control charts for each variable. MSPM also allows us to monitor not only changes in the mean values of each variable, but also in the relationships (covariances) between the variables, which is impossible to do if only univariate control charts are used.

When we want to monitor many variables in a process simultaneously, MSPM provides a much simpler and faster framework over univariate methods. This becomes especially useful as the number of variables we want to monitor increases. Some MSPM methods are even designed to monitor high-dimensional data where the number of variables of interest is large (in the dozens or hundreds).

How does MSPM work?

There are many different methods to perform MSPM, two of which are described in detail here). Recall that univariate SPM (or SPC) methods entail working with two phases of data:

• Phase I, where we obtain and use in-control training data to estimate parameters for our method
• Phase II, where we compare new observations to the in-control behavior observed in Phase I

In univariate SPM, we typically estimate the mean and standard deviation (or variance) of the variable of interest during Phase I.

In multivariate SPM, we estimate the mean vector and covariance matrix during Phase I. This requires us to compute the mean of each variable, the variance of each variable, and the covariances between each pair of variables.

The beauty of the multivariate approach is that we can mathematically condense the information from each of the process variables into a single statistic, called the monitoring statistic (or charting statistic). The monitoring statistic is then monitored over time for any abnormalities, and it represents the aggregated information across all the variables in the process.

Example: Monitoring a process with 5 variables

An example of MSPM method applied to a process with 5 variables can be seen below. The individual process variables are shown on the left, with gray lines showing the Phase I data, black lines showing the Phase II data, and a red vertical line on each plot showing when Phase I ends and Phase II begins.

On the plot on the right, a multivariate control chart is shown, where the information from the 5 process variables is condensed into a single monitoring statistic. Observations where the monitoring statistic exceeds the dashed horizontal purple line are flagged and labeled “out-of-control”.

Example of applying a MSPM method to monitor a process with 5 variables. Phase I data is shown in gray, and Phase II data is shown in black.

We see that the MSPM control chart shows threshold exceedances shortly after Phase II begins. In fact, the consistent exceedances coincide with the negative shift in Variable 3 on the left. The monitoring statistics also continue to increase throughout Phase II, with an upward shift around the same time as the upward shift in Variable 1 and consistently increasing values that correspond to the increasing upward drift in Variable 5.

The MSPM control chart was clearly able to detect changes in the process without making separate control charts for each of the variables.

Conclusion

Using MSPM is especially beneficial when the number of variables in a process we want to monitor is large or when we are interested in detecting changes in the relationships (covariance) between variables in the process. Above, a basic example showed the simplicity afforded by a multivariate approach that still provides useful and actionable information.

For more detailed information on popular MSPM control charts and their specifics, see this post).