CUSUM control chart
The CUSUM control chart is a measure of descriptive statistics . It indicates the number of feature carriers in an empirical study for which the feature value is greater than a certain limit. The CUSUM control chart is calculated as the cumulative sum of the differences between the data values and the reference values. CUSUM is the cumulative sum.
description
The reference value of a CUSUM control chart is zero. The reference value is equal to the content of a certified standard sample. The reference value can, however, also be equated with the sample mean value of the measurement series of a preliminary analysis. In this case, the mean must be carefully determined so that the sums spread evenly around the reference line zero.
The CUSUM values are plotted in a diagram. From this diagram you can then easily see whether the process and the mean value of the process remain constant. If you normalize the cumulative sums by dividing them by the standard deviation (normalized cumulative sums), CUSUM control charts can be more easily compared with one another.
The choice of the scaling of the Y-axis is important, since changes in the slope are more difficult to detect with compressed or elongated CUSUM curves. The scaling factor of the y-axis is given in units of the standard deviation:
W = q ⋅ s
1 ≤ q ≤ 2
Legend:
W - scaling factor
s - standard deviation
q - factor
V mask
The warning and control areas of a CUSUM control chart are given by the so-called V mask. The V mask is a graphic construction that is entered in the CUSUM control chart.
The V-mask depends on two parameters, which in turn depend on the probability of error and the smallest deviation. These parameters are connected to the last data value by a straight line. This creates a V-mask. If the data values are contained within this V-mask, one speaks of an "in-control situation". One speaks of an “out of control situation” when a data value intersects a straight line of the V mask.
Another, older approach assumes that you define a limit value from which you have to intervene in the process.
CUSUM control charts are particularly suitable for recognizing changes in mean values. Small changes in the slope of the CUSUM curve can often be visually determined without a V-mask. Measurements with a strong spread of the measured values can be monitored by CUSUM control cards
Examples
example 1
| Time unit | Data value | Reference value | Data value - reference value | CUSUM sum |
|---|---|---|---|---|
| 0 | 0 | |||
| 1 | 2 | 5 | −3 | −3 |
| 2 | 4th | 5 | −1 | −4 |
| 3 | 7th | 5 | 2 | −2 |
| 4th | 3 | 5 | −2 | −4 |
| 5 | 9 | 5 | 4th | 0 |
Procedure: The difference between the data value and the measured value is calculated for each time unit. These differences are added up for each time unit. This creates a CUSUM curve.
Example 2
Situation: The data values move evenly around the value 10. The reference value is 10.
Evaluation: Because the difference between the data value and the reference value is formed and these differences are added up for each time unit, the CUSUM remains at 0 due to the even distribution of the data values around the value. The positive outliers and the negative outliers offset each other. It is therefore an in-control situation. There is no CUSUM increase. This means that the mean value of the process does not change and the process remains continuous in the long term over the observation period.
Situation: The data values slowly increase from the value 10. The reference value is 10.
Evaluation: As the difference between the data value and the reference value is formed and these differences are added up for each time unit, the CUSUM value increases slowly due to the slow increase in the data values. Because these data values are not evenly around the value 10, the positive outliers and negative outliers do not even out. It is therefore an out of control situation. This slowly increases the mean value of the process and the process does not remain continuous in the long term over the observation period.
Situation: Up to the time unit 60, the values remain evenly distributed around the value 10. Then the data value rises continuously to 11. The reference value is 10.
Evaluation: Because the difference between the data value and the reference value is formed and these differences are added up for each time unit, the CUSUM value remains constant up to time unit 60, from time unit 60 the CUSUM value increases slowly due to the higher data values compared to the reference values. The CUSUM value therefore increases by an average of 1 per unit of time. It is therefore an out of control situation. This slowly increases the mean value of the process and the process does not remain continuous in the long term over the observation period.
literature
- Michèle Basseville, Igor V. Nikiforov: Detection of Abrupt Changes: Theory and Application. Prentice-Hall, Englewood Cliffs, NJ 1993, ISBN 0-13-126780-9
- Douglas M. Hawkins, David H. Olwell: Cumulative sum charts and charting for quality improvement Springer Verlag, 1998, ISBN 0-387-98365-1