CUSUM chart

CUSUM chart (cumulative sum control chart) - a type of control chart that's using E. Page's sequential analysis technique . That technique is an alternative to traditional individual charts, as it uses the cumulative sum of deviations from a target. It is typically used for detecting and monitoring small shifts in the process mean.

Due to chart's superior sensitivity, the primary purpose of using a CUSUM control chart is to detect small changes from the process target, as it improved the ability to detect shifts less than 1.5σ by charting a statistic that incorporates process' current and previous data values .

Cumulative sum

Cumulative sum let us collect m samples, each of size of n, and compute the mean of each sample. Then the CUSUM control chart is formed by plotting one of these quantities:

$S_{m}=\sum _{i=1}^{m}{\bar {x}}_{i}-{\hat {\mu }}_{0}\quad or\quad S'_{m}={\frac {1}{\sigma _{\bar {x}}}}\ \sum _{i=1}^{m}({\bar {x}}_{i}-{\hat {\mu }}_{0})$

where ${\hat {\mu }}_{0}$ is an estimation of the in-control mean and $\sigma _{\bar {x}}$ is the known (or estimated) standard deviation of the sample means. The choice of which of these two quantities are plotted is standardly determined by the statistical software package. As long as the process remains in control centered at ${\hat {\mu }}_{0}$, the CUSUM plot shows variation in a random pattern centered around zero. The charted CUSUM points will eventually drift upwards in case the process mean shifts upward and vice versa if the process mean decreases .

CSUM charts usage and creation

Tabular CUSUM chart is useful in situations where sub-grouping is not desired or feasible, yet great sensitivity is necessary. CUSUM charts are also a good fit where the process average is expected to naturally shift or trend from the target and there is a need to make process adjustments in a timely manner to bring the process back on target .

• Upper control limit $S_{m}^{+}=max[0,(x_{m}-({\hat {\mu }}_{0}+K)+S_{(}m-1)^{+}]$
• Lower control limit $S_{m}^{-}=max[0,({\hat {\mu }}_{0}-K)-x_{m}+S_{(}m-1)^{-}]$
• Plotted statistic $S_{m}=\sum _{i=1}^{m}{\bar {x}}_{i}-{\hat {\mu }}_{0}$

CSUM charts and Shewhart charts

A CUSUM chart plots the cumulative sums of the aberration of the sample values from a target value. The incorporation of using several samples in the cumulative sum results in greater sensitivity and speed for detecting shifts or trends over the traditional Shewhart charts .

Examples of CUSUM chart

• A CUSUM chart can be used to monitor production output over time. For example, a manufacturing facility may use a CUSUM chart to monitor the number of units produced over a period of time. The chart can be used to identify any small shifts in the process mean, and alert when the mean number of units produced deviates from the pre-determined target.
• CUSUM charts can also be used to monitor the quality of a product over time. For example, a company may use a CUSUM chart to monitor the number of defects per unit produced. This chart can be used to identify any small changes in the process mean, and alert when the mean number of defects per unit deviates from the pre-determined target.
• CUSUM charts can also be used to monitor performance over time. For example, a company may use a CUSUM chart to monitor the number of sales made over a period of time. The chart can be used to identify any small shifts in the process mean, and alert when the mean number of sales deviates from the pre-determined target.

Advantages of CUSUM chart

A CUSUM chart is an effective tool for identifying small changes in process mean and monitoring its performance over time. Its main advantages are:

• Its ability to detect small shifts quickly allowing for quick corrective action.
• Its flexibility as it can be adjusted to monitor different types of processes.
• Its ability to track trends over time, making it easier to identify long-term process changes.
• Its ability to combine with other control charts to provide a comprehensive view of process performance.
• Its ease of use as it requires minimal data manipulation and calculation.

Limitations of CUSUM chart

CUSUM charts have several limitations that should be taken into account when considering their use. These include:

• CUSUM charts are sensitive to small process shifts, making them unsuitable for detecting large changes in the process.
• CUSUM charts require reliable data and can be difficult to interpret if the data is noisy.
• CUSUM charts can be difficult to set up and require careful interpretation to identify true changes in the process.
• CUSUM charts are sensitive to the selection of parameters, including the target value, decision interval and reference change value.
• CUSUM charts are not able to identify the source of a shift in the process.

Other approaches related to CUSUM chart

• Shewhart Control Chart - this chart is used to monitor the process performance and detect any changes in the process. It consists of a series of points plotted on a graph, and the points are calculated from the process data.
• Stratification Control Chart - this chart is used to detect changes in the process by analyzing the data in terms of its subgroups. The subgroups are defined according to the data’s characteristics and the chart is used to compare the subgroups.
• EWMA Control Chart - this chart is used to detect small changes in the process mean. It uses exponentially weighted moving averages to identify changes in process behavior.
• Adaptive Control Chart - this chart is used to adjust the control limits as the process changes. It uses a series of algorithms to identify the changes in process behavior and adjust the control limits accordingly.

In summary, CUSUM chart is one of several approaches used to detect and monitor small shifts in the process mean. Other approaches include Shewhart Control Chart, Stratification Control Chart, EWMA Control Chart and Adaptive Control Chart. Each of these approaches has its own advantages and disadvantages, so the choice of which to use depends on the specific application.

Footnotes

CUSUM chart — recommended articles

Control chart — P chart — Np chart — Attribute control chart — Interval scale — Run chart — Parametric analysis — Sequential sampling — Support vector machine

References

• Basseville M., Nikiforov I. (1993), Detection of Abrupt Changes: Theory and Application Englewood Cliffs, New Jersey
• Mishra S., Vanli O. A., Park C. (2015), A Multivariate Cumulative Sum Method for Continuous Damage Monitoring with Lamb-wave Sensors International Journal of Prognostics and Health Management, Tallahassee, Florida
• NIST/SEMATECH, (2012), e-Handbook of Statistical Methods, United States
• Page E. S., (1954) Continuous inspection schemes, Biometrika
• Ryan T. P., (2011), Statistical Methods for Quality Improvement, John Wiley & Sons
• Wachs S. (2010), What is a CUSUM Chart and When Should I Use One? , Integral Concepts

Author: Anna Strzelecka