Cumulative sum

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... 

Figure 2.19. Intersection of two linear regression lines (schematic). In the intersection zone (gray area), at a given c-value two PD-curves of equal area exist that at a specific y-value yield the densities zi and Z2 depicted by the dashed and the full lines. The product zi Z2 is added over the whole y-range, giving the probability-of-intersection value for that x. The cumulative sum of such probabilities is displayed as a sigmoidal curve the r-values at which 5, respectively 95% of Z2) s reached are indicated by vertical arrows. These can be...

Purpose A technique to detect deviations from random scatter in the residuals (symmetrical about 0, frequent change of sign) Cumulative sum of residuals detects changes in trend or average. Here, an average is subtracted to yield residuals these residuals are then summed over points 1. .. k. .. N, with the sum being plotted at every point x k). Two uses are possible ... 

Marshall, R. A. G., Cumulative Sum Charts for Monitoring of Radioactivity Background Count Rates, Anal. Chem. 49, 1977, 2193-2196. 

Marshall AG (1977) Cumulative sum charts for monitoring of radioactivity background count rates. Anal Chem 49 2193... 

Woodward RH, Goldsmith PL (1964) Cumulative sum techniques. Oliver Boyd, London... 

Figure 4.5. Cumulative sums of extractable elements in soil J3 (CaC03 33.3%) in consecutive steps of the selective sequential dissolution procedure as a function of the cumulative steps. 1 CARB (carbonate) 2 CARB (carbonate) + ERO (easily reducible oxide) 3 CARB + ERO + OM (organic matter) 4 CARB + ERO + OM + RO (reducible oxide) (after Han and Banin, 1995. Reprinted from Cornmun Soil Sci Plant Anal, 26, Han and Banin A., Selective sequential dissolution techniques for trace metals in arid-zone soils The carbonate dissolution step, p 573, Copyright (1995), with permission from Taylor Francis US)...

The cumulative sums of selected major and trace metals extracted by the two SSD procedures from representative arid-zone soils are shown in Fig. 4.6. As can be seen from the figure, the Rehovot procedure is stronger in attacking desired fractions, such as the carbonate bound, Mn oxide bound and organically bound fractions. Extraction of certain major elements, indicating selectivity, specificity and completeness of extraction of given soil components, was found to differ between the two procedures. Calcium and Mg were more completely extracted from the carbonate fraction in arid zone soils by the Rehovot procedure. Calcium and relevant trace elements bound in the carbonate fraction, which were not completely dissolved by the Bonn procedure at this step, were released at the following steps, such as the ERO, OM or RO fractions. 

Figure 4.6. Cumulative sums of extractable Ca, Mg, and Mn in arid soil J3, BIO, and SI in consecutive steps of two sequential dissolution procedures.

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,... 

Contaminated Land Proficiency Testing Scheme Control of Substances Hazardous to Health Certified Reference Material cumulative sum... 

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. 

Cusum = cumulative sum = sum of all differences from one target value... 

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 ... 

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. 

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. 

Next, you divide the lake into zones which are bounded by planes located halfway between the depths for which temperature data exist (see Fig. 1). Since A(z) is linear, the volume for each zone is simply the arithmetic mean of the cross-sections above and below the layer multiplied by the thickness of the layer. These volumes are multiplied by the corresponding temporal temperature changes. The results are summarized in the following table. The last column, the cumulative sum of the weighted temporal temperature gradient summed from below, is an approximation of the numerator of Eq. 22-36. 

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. 

The procedure consists of plotting the cumulative sum of the observations, taken with ii(. as the origin, on a chart such as Fig. 1.14 on which both sets of boundary lines are shown. 

In one experiment the following values were recorded for the differences in breaking load between synthetic fibers prepared in two different ways 7 5 8 -ll 10 8 -9 6 -7. The cumulative sums of the observations are 7 12 20 9 19 27 18 24 17, and the points are plotted in Fig. 1.15. The line crosses the limit T0 at the ninth observation, so the hypothesis that no important change in breaking load has occurred is accepted. Exactly the same procedure can be done analytically. 

If the procedure is reversed the so-called cumulative sum (CUSUM) integrates the deviations from the mean of the series [DOERFFEL et al., 1990 1991] ... 

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. 

Because analytical results can be accumulated over long periods on easily accessible disks — a big improvement on laboratory notebooks — it is easier to evaluate the performance of a method or process over a period of time. It is easy to determine the frequency of defects in performance, both by inspection and by more formal methods. One of these which is being used more widely is the cusum procedure. A reference value, k, is selected typically it is the mean value of the expected result. Then sequentially the cumulative sum, Si is calculated... 

The Phytoplankton Colour (PCI) cumulative sum plot showed a (single) shift (Fig. 5) in the mid 1980s. This was very different from the two shifts in the Phaeocystis plot (mid 1960s and early 1990s, see Fig. 5), which shows again that Phaeocystis does not reflect the variation observed in phytoplankton biomass. Between 1970 and 1985, low phytoplankton abundance (PCI) in the North Sea coincided with a late spring bloom. This... 

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