CUSUM

When data of a single type accumulate, new forms of statistical analysis become possible. In the following, conventional control and Cusum charts will be presented. In the authors opinion, newer developments in the form of tight (multiple) specifications and the proliferation of PCs have increased the value of control charts especially in the case of on-line in-process controlling, monitors depicting several stacked charts allow floor supervi-... 

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... 

The Cusum technique (Section 1.8.5, program CUSUM) can also be regarded as a special type of filtering function, one that detects changes in the... 

The quality control unit in a cosmetics company supervised the processing of the weekly batch of shampoo by determining, among other parameters, the viscosity and the dry residue. Control charts showed nothing spectacular. (See Fig. 4.10, top.) The cusum charts were just as uneventful, except for that displaying the dry residue (Fig. 4.10, middle and bottom) The change in trend in the middle of the chart was unmistakable. Since the analytical method was very simple and well-proven, no change in laboratory personnel had taken place in the period, and the calibration of the balances was done on a weekly basis, suspicions turned elsewhere. A first hypothesis,... 

Figure 4.10. At the top the raw data for dry residue for 63 successive batches is shown in a standard control chart format. The fact that as of batch 34 (arrow ) a different composition was manufactured can barely be discerned, see the horizontals that indicate the means DRi 33 resp. DR34 g3- A hypothesis that a change occurred as of batch 37 would find support, though. Cusum charts for base period 1. .. 63 resp. base period 1. .. 37 make the change fairly obvious, but the causative event cannot be pinpointed without further information. Starting with batch 55 (second arrow ), production switched back to the old composition.

No averaging has taken place (option 5 in the menu) the individual average is equal to the over-all mean y ,ean which is displayed as a horizontal line this corresponds to the classical use of the Cusum technique. By this means, slight shifts in the average (e.g., when plotting process parameters on control charts) can be detected even when the shift is much smaller than the process dispersion, because the Cusum trace changes slope. 

If a data set was first appropriately treated in program SMOOTH and the smoothed coordinates were saved, the difference between raw and smoothed values (use subtract function in DATA) can be analyzed essentially, Cusum now detects how well the smoothed trace represents the measurements. For example, if peak shapes are to be filtered (see data file SIMl.dat) and too wide a filter is used, the smoothed trace might cut comers as a result, the Cusum trace will change slope twice, the Cusum trace can be shifted vertically, and an expansion factor can be chosen. Ordinate rescaling is done automatically. 

PKG ClASS.dat One hundred items weights are given in Vectors 2 and 3 the items are classified as either Hi or Lo, to be used with SMOOTH, MSD, HISTO, HUBER, and CUSUM. 

The reactor pressure is reduced to 0 psig to flash off any remaining water after a desired temperature is reached. Simultaneous ramp up of the heat source to a new setpoint is also carried out. The duration spent at this second setpoint is monitored using CUSUM plots to ensure the batch reaches a desired final reactor temperature within the prescribed batch time. The heat source subsequently is removed and the material is allowed to continue reacting until the final desired temperature is reached. The last stage involves the removal of the finished polymer as evidenced by the rise in the reactor pressure. Each reactor is equipped with sensors that measure the relevant temperature, pressure, and the heat source variable values. These sensors are interfaced to a distributed control system that monitors and controls the processing steps. 

In addition, the GPC trace, an example of which is shown in Fig. 42, reflects the composition signature of a given product and reflects the spectrum of molecular chains that are present. Analysis of the area, height, and location of each peak provides valuable quantitative information that is used as input to a CUSUM analysis. Numeric input data from the GPC is mapped into high, normal, and low, based on variance from established normal operating experience. Both the sensor and GPC interpretations are accomplished by individual numeric-symbolic interpreters using limit checking for each individual measurement. 

Figure 50. CUSUM test, stability and convergence of the estimates of ground thermal conductivity as a function of starting time and amount of data included, obtained in the reference...

During the many tests we performed we found that that ground water flow will significantly affect results of the In Situ Geothermal Response Test. This is especially clear in the CUSUM graphs, that show an increasing estimate of ground thermal conductivity with time. The reason for this is that the effect... 

Figure 51. Conductivity convergence (CUSUM graph) curves for the reference (no groundwater) and ground water flow experiment...

Appropriately designed, Cusum control charts give sensitive and instructive impressions on process changes. Cumulative stuns S = YT,= ( — 0) also contain information on actual as well as on previously obtained values. Therefore their display enables one to perceive earlier changes leading to OCS than by means of the chart of original values (see Woodward and Goldsmith [1964] Marshall [1977] Doerffel [1990]). 

The validation process begun in Phase I is extended during Phase II. In this phase, selectivity is investigated using various batches of drugs, available impurities, excipients, and samples from stability studies. Accuracy should be determined using at least three levels of concentration, and the intermediate precision and the quantitation limit should be tested. For quality assurance evaluation of the analysis results, control charts can be used, such as the Shewart-charts, the R-charts, or the Cusum-charts. In this phase, the analytical method is refined for routine use. 

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). 

Figure 6.8 CUSUM charts showing (a) a step-change and (b) drift.

Figure 6.9 (a) The V-mask - for interpretation of CUSUM charts and (b) a CUSUM chart illustrating the use of a V-mask. 

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

Guide to Data Analysis and Quality Control Using CUSUM Techniques. Uses and Value of CUSUM Charts in Business, Industry, Commerce and Public Service , BS 5703-1 2003, British Standards Institute (BSI), London, UK, 2003. 

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