Cumulative Sum CUSUM Charts

The cumulative sum (CUSUM) chart incorporates all the information in a data sequence to highlight changes in the process average level. The values to be plotted on the chart are computed by subtracting the overall mean /io from the data and then accumulating the differences. The quantity 

A two-sided CUSUM chart can be generated by running two one-sided CUSUM charts simultaneously with the upper and lower reference values. The recursive formulae for high and low side shifts that include resetting to 

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The control charts discussed earlier are very useful in the diagnostic aspects of quality process improvement. They can be used to stabilize a process by identifying out-of-control situations. After the process is stabilized and brought in control, further improvement of the process can be achieved by using some special control charts such as the cumulative sum (CUSUM) control chart and the exponentially weighted moving average (EWMA) control chart. These control charts can be used when small shifts in a process are of interest. 

Another way to present control data is the cumulative sum control chart. In this method, the x-axis is either time or number of control observations and the y-axis is the cumulative sum (cusum), or the difference between the measured control value and the predicted mean... 

Figure 1.1 Examples of (a) Levey-Jennings and (b) cumulative sum (cusum) control charts using inter-assay quality control data from an alprazolam GC-MS assay. Since the cusum chart persents the cumulative sum of deviations from the mean, it is more sensitive to small biases that develop over time, whereas Levey-Jennings charts are most useful for detecting changes in the precision of the assay...

It should be noted that the cumulative sum (CUSUM) and exponential weighted moving average (EWMA) control charts have also been proposed for the use in high-quality processes (Yeh et al. 2008 Szarka and Woodall 2012). However, in some cases, they are very complex and complicated to understand for the practitioners. 

A disadvantage of the conventional control charts is that a small or gradual shift in the observed process parameter is only confirmed long after it has occurred, because the shift is swamped in statistical (analytical) noise. A simple way out is the Cusum chart (cumulated sum of residuals, see program CUSUM.exe), because changes in a parameter s average quickly show up, see Fig. 1.32. The... 

Because it uses all of the data, the CUSUM chart is the best way of detecting small changes in the mean. Consider a process for which there is a known target value, T. For each new measurement, the difference between the measurement and T is calculated and added to a running total. This running total is plotted against successive measurements (CUSUM is short for cumulative sum). 

Cumulative Sum Charts - Guidance on Quality Control and Data Analysis Using CUSUM Techniques , ISO/TR 7871 1997, International Organization for Standardization (ISO), Geneva,... 

The Cusum Control Chart is a very special chart from which a lot of information can be drawn. Cusum is the abbreviation for cumulative sum and means the sum of all differences from the target value. Every day the difference of the control analysis from the target value is added to the sum of all the previous ddferences. 

This shde gives an example of the use of a Cusum Chart. The upper chart is a conventional X-chart, the lower one a Cusum Chart. Starting from the 11th value, all values are below the target value originating from slow between-batch-drift in the analyses. This can be seen in the Cusum Chart by a descending cumulative sum. 

Shewhart charts are adept at detecting mean value shifts on the order of 3(7 or higher. To detect more subtle shifts in the mean value, the CUSUM chart has been developed [10,11]. The cumulative sum is defined as ... 

CUSUM Control Chart A CUSUM chart provides an efficient way of detecting small shifts in the mean of a process (l/2 a), the chart is usually used.The CUSUM chart incorporates information contained in a sequence of sample points. It keeps track of the cumulative sum of the deviations between each sample point (a sample mean) and a target value. Unlike the x chart, which often bases its out-of-control decision on just the most recently collected sample, the CUSUM calculated for a sample point carries the history prior to that sample. For example, a sequence of sample points above the centerline can trigger an out-of-control signal although all of them stayed well below the UCLs of the x chart. 

A cumulative sum (or cusum) chart is a type of control chart that can detect changes in process average more powerfully than an x chart. A reference value K is chosen. K can be the process target value, historical average, or any convenient value. As new values xh. .., xn are observed, the cumulative sums... 

Example Table 1 gives an example of using a cusum chart for manufacturing data. The slope of cumulative sums changes for the sums formed from batch 103, suggesting that the process operated at a lower mean level. 

Apart from the standard Shewart charts, the analyst can also apply X-charts, on which the mean of several replicate measurements is plotted, or R-charts, where the difference between two replicate measurements is plotted. X- and R-charts give an indication of the reproducibility of the method. Drift in analytical procedure, for example, slows changes in the system caused by the aging of parts of instruments, decalibration in wavelength, or the aging of calibration stock solutions, can be detected early when a Cusum chart (cumulative sum) is applied. In Cusum charts, the analyst reports the cumulative sum of the differences between delivered and reference values. If this reference value is certified (CRM), the Cusum chart allows the accuracy of the determination to be monitored. 

Two alternatives to the Shewhart control chart, which are more complicated to calculate but generally more effective to detect small shifts, are the Cumulative Sum (or Cusum) control chart and the Exponentially Weighted Moving Average (EWMA) control chart. These control charts will not be discussed here, but are described in standard references. ... 

The single-value control charts discussed in the previous sections provide a display of the differences between the observed values and the expected mean. Control rules, such as 22s, 4i5, and 10 provide one way of determining when these successive differences no longer appear to be random (too many in a row on one side of a limit). A more exact and quantitative method is the cumulative sum control procedure or cusum chart. 

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