Cumulative Sum charts - Guidance on quality control and data analysis using CUSUM techniques
Some standard content:
1CS 03. 120. 30
National Standardization Guidance Technical Document of the People's Republic of China GB/Z 4887-2006/ISO/TR 7871: 1997 replaces GB/T4887-1985
Cumulative sum charts-
Guidance on quality control
and data analysis using CUSUM techniques(ISO/TR 7871,1997,IDT)
Published on 2006-05-24
General Administration of Quality Supervision, Inspection and Quarantine of the People's Republic of China Administration of Standardization of the People's Republic of China
GB/Z 4887-2006/IS0/TR 7871:1997 Foreword
0 Introduction
Models and general principles
Preparation of cumulative sum diagrams
Graph presentation
4 Decision rules for monitoring
Decision rules for retrospective analysis
Cumulative sum and techniques of calculation
Application examples
Appendix A (normative)
Appendix 3 (normative)
Appendix C (normative)
Appendix D (normative)
Appendix E (Wei Model)
Appendix F (normative)
Appendix G Normative Appendix H (Normative Appendix)
Measures of variation
Scale of graph
Calculation of local mean value
Redrawing of cumulative sum graph
Full V-shaped template
Local decision line
Non-parametric test of cumulative sum
Alternative standard scheme
GB/Z4887—2006/IS0/TR7871:1997 This guidance technical document is equivalent to the ISU technical report IS0/TR7871:1997 "Guide to quality control and data analysis using accumulation and technology for cumulative sum graphs".
Compared with ISO/TR7871:1997, the changes in the technical content of this guidance technical document are mainly the corrections of errors in the text of ISO/TR7871:1997.
This technical guidance document makes the following corrections to the errors in ISO/TR7871:1997: 1) In 5.2.3, Vm, = C, C, -(C, -C), m = -, the denominator 1 in the original text is corrected to 21-)(2 -(-1)) —25 -
2) In 5.2.a, the original calculation Vmx — -26 — (—1) - No 14
X(-1) = —24. 42
31---7
21(—2-(— 1)) -— 25 —
is corrected to Vex = — 26 —(- 1) - 314 × ( 1) - 24. 42;
3) Corrected the coordinates of Figure 17:
4) Table 10 was changed to be arranged in row order; 5) The. in Table 12 was changed to u-;
6) The section "from 1=7 to 30" in Figure 11 was changed to "from i7 to 31"; 7) 2.41 in 7.3.1b) was changed to 2×41; 8) 12 in A.4 was changed to 1±1.96/2.
This guidance technical document replaces CB/T4887-1985 from the date of implementation. The appendixes A, B, C, D, E, F, G and H of this guidance technical document are all normative appendices.
This guidance technical document is proposed and managed by the National Technical Committee for the Application of Statistical Forces and Standardization. The main drafting units of this guidance technical document are: China National Institute of Standardization, Shenzhen Quality Inspection Institute, Institute of Mathematics and Systems Science, Chinese Academy of Sciences, Ordnance Engineering College, Liaoning Exit-Entry Inspection and Quarantine Bureau, Beijing Hongfu Group Co., Ltd. The main drafters of this guidance technical document are: Yu Zhenfan, Jiang Jian, Tu Zhujuan, Yang Jun, Li Mingzhao, Zhang Yuzhu, Liu Wen, Zhu Douwen, Ma Yilin, Xiao Li, Fang Xi, Liu Yang.
0 Introduction
0. 1 Basics of Cumulative Sum Charts
CB/Z4887—2006/IS0/TR7871:1997 Cumulative Sum Charts
Guidelines for Quality Control and Data Analysis Using Cumulative Sum Techniques
Cumulative Sum Charts (CUISUM Charts) are a graphical representation of a set of data information with a reasonable (or natural) order, which usually corresponds to the chronological order of observations.
Subtract the reference value T from each observation. The reference value T is usually a constant, but it may also be a predicted value or a variable target value obtained from a forecast model. The deviations from T are accumulated and summed, and the sum C is plotted in the order of observations, which is a cumulative sum chart.
In a cumulative sum chart that detects the deviation of a process from a mean reference value, the reference value is also called a monthly target value or target. Before further discussing the accumulation and technique, the following two concepts must be clarified: target value and reference value. The former refers to the actual or expected process average, while the latter refers to the reference value used in the accumulation and technique. Because the "target value" is very intuitive, in most cases in Chapters 0 to 6 of this guidance technical document, when the target value and the reference value are the same value, it is generally called the target value without causing confusion. In Chapter 6, because the upper reference value and the lower reference value are introduced, they must be distinguished from the target value. The cumulative sum method of plotting the cumulative results uses the local slope of the chart to represent the mean. When the local mean is consistent with the target value, the cumulative sum line is roughly parallel to the series axis. When the local mean of the series is greater than the monthly target value, the cumulative sum line slopes upward; conversely, when the local mean is less than the target value, the cumulative sum line slopes downward. The greater the difference between the local mean and the target value, the steeper the cumulative sum line. By drawing a cumulative sum chart, the changes in the mean level of different parts of the observation series can be clearly represented by the changes in the slope of the chart. The local mean of each part can be easily estimated by the following methods: either from the data of the drawn cumulative sum chart or directly from the cumulative sum table.
Another result of the cumulative sum method is that there is an inherent serial dependence between successive cumulative sums. The determination of acceptable shifts in the series axis requires the method of random processes. 0. 2 Simple example of a cumulative sum chart
The above principle can be illustrated by a simple example. Here, the calculation and plotting process can be expressed without mathematical formulas. Assume that the following sample observations have been obtained in chronological order and the given reference value is assumed to be 15. Table 1 Data used to draw the cumulative sum chart
Observation number
Observation value
Deviation from reference (7=15)
Cumulative sum of deviations
GB/Z 4887—2006/1S0/TR 7871:1997Observation number
Observation value
Table 1 (continued)
Deviation from reference (—15)
Cumulative sum of deviations
For a conventional control chart (such as Figure 1), the observations are plotted according to their observation number. The last 12 observations and the first 20 or so observations seem to be clustered around two different mean levels. Compared with the conventional control chart, the cumulative sum chart is clearer. The method of the cumulative sum diagram is to plot the cumulative sum (the 4th column in Table 1) according to the observed sequence number, with the Y axis (vertical axis) representing the cumulative sum value and the X axis (horizontal axis) representing the observed sequence number, as shown in Figure 2. Observation
and marking
1618202224
12628303234
Present sequencing
Figure 1 Conventional control chart drawn according to the data in Table 1 10
Oxygen cumulative sum
Reference value [5
CB/Z 4B87—2006/IS0/TR 7871:19971012
1416182022242628303234
Sequence number
Figure 2 is a cumulative sum graph drawn according to the data in Table 1
The cumulative sum graph is clearly divided into three sections. From the 1st observation to the 21st observation (inclusive), the cumulative sum line is generally parallel to the observation sequence number axis, that is, this section of the line is roughly horizontal: from the 8th observation to the 21st observation (inclusive), the line slopes downward (although there are some irregular changes in the 14th and 20th observations); from the 22nd to the 33rd observation, the line slopes upward (there are also some irregular changes).
Therefore, we can preliminarily conclude that:
α) Observations 1 to 7 constitute a sample from a population with a mean of 15 or close to it; h) Observations 8 to 21 come from a population with a mean less than 15; c) Starting from observation 22, the observation sequence seems to come from a population with a mean greater than 15. The following questions arise: Given the variability in the
data (as indicated by the irregularities in the cumulative sum line), can we assert that changes in slope represent a) real changes in the
average value, rather than just differences in chains of "good" and "bad" samples from a stable population? If the changes are real, how should the data be used to estimate local means? b) How much influence does the choice of
reference value and the cumulative sum and scaling coefficient have on the inference or estimate? Figures 3 and 4 show dot plots of the same series: the cumulative sum scaling coefficient in Figure 3 is alternating with that in Figure 1, but the target value is 12; Figure 4 still uses 15. Figure 3 Cumulative sum of data in Table 1 when reference value T is 12. GB/Z4887—2006/IS0/TR7871:1997 Male Cumulative sum is compressed.
4681012
20222426283032
View serial number
Figure 4 Cumulative sum graph of the data in Table 1 after scaling, with the test value T being 15. In Figures 3 and 4, the change in slope around the 8th observation is not obvious. In Figure 3, the change around the 21st observation is still visible, but it is not easy to identify in Figure 4. Thus, in order to avoid ignoring useful information or exaggerating false components, special attention should be paid to the selection of reference values and scaling factors. It is also obvious from Figure 3 that using an inappropriate scaling value may cause the graph to exceed the upper and lower limits of the drawing, although this problem can be reduced by redrawing with any point in the series as a new zero point. 1 Scope and general rules
1.1 Overview
This technical guidance document proposes the principles of cumulative sum diagrams, including guidance on the drawing and interpretation of diagrams. This technical guidance document is applicable to the quality control of continuous production processes when the quality indicators of the inspected products are the number of defectives, the number of non-conforming products or the mean of a certain quality characteristic. 1.2 Basic requirements
When using cumulative sum diagrams, the basic requirements for observations are as follows: a) The observations should at least be measurements within a certain scale range; b) The observations should be ordered, which is natural in process control. The above two requirements are explained below. First, the difference between any two given sets of observations has a consistent interpretation over the entire range of variation. For example, a .1mm difference in length between two objects has the same meaning, regardless of whether the objects are 10.1mm in length. mm and 10.0 mm, or steel beams with lengths of 10,000.1 mm and 10,000.0 mm, although the difference in the latter is negligible, the difference between them has the same meaning. Many kitchen properties do not have this property. Take "score" as an example. If each serious non-conformity score is recorded as 10, the general non-conformity score is recorded as 5, and the minor non-conformity score is recorded as 1, then we cannot say that the following products have the same meaning, although the difference in their scores is 0: - Product A has 1 serious non-conformity
- Product B has 2 general non-conformities - Product B has 1 general non-conformity and 5 minor non-conformities - Product D has 10 minor non-conformities
score - 10 yuan
score = 10,
score = 10:
score - 10 yuan
score = ...
The mere use of the "average" score, without listing the type of nonconformance and the frequency of its occurrence, can be misleading. Reasonable sequentiality can be obtained in many ways. Observations may occur in a sequence measured in time or length, thus forming a natural sequence. Quality monitoring and process control provide many such examples. Observations may also be ordered according to the measured value of some auxiliary variation of the product. Thus, the CUSUM chart also provides a means of describing or examining the relationship between variations and can be used for regression or correlation analysis. Any classification or grouping that exploits the structural characteristics of the observations or the patterns that they produce provides the basis for a CUSUM sequence. 1.3 Types of data required for CUSUM charts Many data types meet the basic requirements a) and b) of 1.2. Perhaps the most common application of CUSUM charts is in quality control. In a process, observations (such as sample means or ranges) are plotted in sequence to assess the state of the process. When the CUSUM chart is used as an effective data display tool, it is not necessary to specify its probability distribution or to require that successive observations are independent. These conditions are important for decision rules, but not for data presentation. In fact, CUSUM plots can also help to identify distributional characteristics, such as serial correlation or periodicity.
Sample range and sample standard deviation can be plotted against sample mean: ... This situation is similar to the significance test between groups of observations, but the difference is that the grouping is based on the preliminary examination results of the cumulative sum chart. Statistical tests for retrospective analysis are described in Chapter 5.
2 Preparation of cumulative sum charts
2.1 Notation
Let T be the reference value. For each observation, sum the difference J-T until it occurs, and the cumulative sum is: C, =
++++++
In the cumulative sum chart, the cumulative sum C is used as the ordinate (vertical axis) and the points are plotted against the abscissa (horizontal axis). Assume that i takes a series of consecutive integer values 0 (origin), 1, 2, etc. The scale factor on the ordinate is denoted by A. This means that the unit on the ordinate corresponding to one unit on the abscissa is A. The scale factor A is often expressed as a multiple of the standard error (.) of the value of the plotted point. The standardized scale factor is denoted by α (i.e. A-α.). The meaning and estimation of . See Appendix A for details. Usually, the lower mean of the fan of the series of plot points from 1 to , or from t to -1, or from t to ! r is calculated first. These values are expressed as: 3.jh, +...
Other symbols will be defined accordingly when introduced. 2.2 Selection of reference value T
In the process of derivation, the selection of an appropriate target value T is one of the two most important steps. An inappropriate target value T will cause the cumulative sum chart to always tilt upward or downward, making it difficult to detect changes, and when the broken line exceeds the top or bottom of the chart, it must be redrawn frequently (see Appendix)
2.3 In most cases, I is a measure of the specified target level of quality. It is best to make the process ensure this quality level, otherwise not only the instrument and the cumulative sum chart will continue to send out process abnormality signals, but also any control system will be ineffective in this case. And the target value T does not always exist, there is no time to use the average level of quality characteristic measurements under recent stable conditions as a reference value. 2.4When a CUSUM chart is used to examine a set of historical data or the residuals of an experiment retrospectively, a natural choice of the reference value is the arithmetic mean of the data. This choice of reference value will keep the ordinate of the CUSUM chart constant, except for small differences due to rounding errors about the mean. The CUSUM chart can be completed on ordinary coordinate paper by using the appropriate scaling factor (see Appendix 1).
2.5 For binary data, that is, data that only take the values "0\ and \1\, what needs to be estimated is the proportion of data that take the value \1\. This proportion can be used as the target value. In quality control applications, this value is the failure rate or the acceptance quality level (AQ[.) specified in the contract, or an appropriate value provided by the feasibility study. In such experiments involving binary responses, the target value can be the total ratio of the complete experiment, or in the case of continuous experiments, the ratio observed in the initial stage of the experiment is set as the target value and then revised if it is proved to be inappropriate. 2.6 Types of variation
In order to more effectively determine the scale of the cumulative sum diagram and use it as the basis for the significance test, a measure of the potential short-term variation of the series is required. In engineering terms, in order to systematically determine the degree of detection signal, the noise must be measured. 5
GB/Z4887—2006/ISO/TR 7871:1997 The basic statistical measure of variation is the standard deviation. From a sample of n values, the standard deviation can be estimated by the following formula: (z(3. )
where the sum is calculated over the II sample observations. ---{ 2)
Usually, the value to be plotted is a function of a set of observations, such as the mean, range, or defective rate. The well-posed measure of variation is the standard error of these sample statistics (when it is unknown, use an estimate of it). In a sample containing 100 units of product, the simplest statistic used is the sample mean, or the defective rate. Note that the reference here refers to the sample mean of each sample, that is, the number of observations of each sample, not the number of samples to be plotted. Assuming that the process is in a state of statistical control in these two simple cases, the standard error of 0, =/ is often replaced by its estimated value.
In a sample of sample loss, when the observation is the number or ratio of a certain attribute, if the process is in a state of statistical control, the probability that the unit product has this attribute is, then according to the binomial distribution, it can be given as: the standard error of force
is often recorded as
(1=p)
pede,ga - The standard deviation of the number of units with this attribute in a sample is: pa
= Vnpa
. (4)
Combined with the graph of the original data, this is a case where the standard deviation itself is helpful for diagnosis. Another case is the number of unqualified products, points, defects or other incidental events that occur in a certain product or material or in a certain period of observation, such as the number of points in 1m (or 1m) of cloth, the number of accidents that occur in a factory every week, etc. If the assumptions of Poisson distribution are met, then; o=m
where ㎡ is the average level of incidental events in each sample. In particular, if the probability of a non-conformity in a very small product is very small and proportional to the product quantity, and if the occurrence of non-conformities is independent of each other, then the number of non-conformities follows a Poisson distribution. The requirement of statistical control means that in any time period, when no identifiable changes (causes of variation) occur, all sampled unit products can be regarded as simple random samples from the entire process (or population, or time period, etc.). In this case, the observed short-term variation in the units of the sample provides an appropriate basis for estimating the expected variation of the entire series (using the standard error of the statistic chosen at the time). Any variation greater than the normal variance is considered to be due to an identifiable cause: it indicates that the mean of the series has been skewed or that the nature or magnitude of the variation has changed. In many cases, simply using the overall estimate of the standard deviation is inappropriate. Some cases may lead to errors, such as the following: In the case of observing a continuous process, many small and insignificant variations in the average level may occur; these variations are completely different from the short-term variations in the process (systematic or continuous changes that should be determined). For example, an industrial process may be controlled by a thermostat or other automatic control device; although there may be some small changes in the quality of the input raw materials that will not violate the specification. In monitoring a patient's response to treatment, there may be small metabolic changes related to food, hospital or home habits, etc., and the judgment of any treatment effect should be based on the typical variation of the body. b) The sampling method itself may also lead to effects such as those mentioned in a) above. Samples are often taken from units that are close together on the production line because it is difficult to draw a truly random sample. These units then form a "clustered" sample that is too similar to be used as a basis for assessing overall variation. d) The sample may be observations from different sources (machines, operators, management areas). There may also be too much local variation to provide a practical basis for assessing whether meaningful changes have occurred. For this reason, data from a mixture of these sources should be treated with caution because any local characteristics within each contributing source may be ignored and changes between these sources over time may mask changes that occur in the whole system. d) There may be serial correlation among the observations, that is, one observation is correlated with other observations that are related to it. For example, if a moving average is used, the weight between the data used for one half of the average and the next average will cause a positive correlation. For example, when assessing the quality of a batch of bulk material, one overestimation may lead to the next underestimation due to the difference between successive gauges or readings, which is a negative correlation. We need to recognize this possible interaction. In some industrial processes, positive correlation is very likely to exist, when one batch of material is partially mixed with the previous and subsequent batches, sometimes producing what is called a "heel" effect. For example, refueling a vehicle's tank is an everyday example, because new fuel is generally added before the tank is exhausted.
Therefore, in a series or data series, other measures of variation must be considered and where they are appropriate. These measures of variation, including the differences between successive sample values (8, -3, -1), are considered appropriate representations of variation, with all sample observations treated as if they were drawn from a single population. These measures are discussed in Appendix A. 2.7 Measures of variation See Appendix A. 2.8 Scaling of graphs See Appendix B. 2.9 Preparation of CUSUM graphs. Checklist As a reminder (but not a substitute) of the detailed steps in this chapter to prepare CUSUM graphs, the following checklist may be useful. Choose an appropriate standard value, which may be: a) some fixed value b) a satisfactory level that is likely to be achieved (for a process); c) the average performance level over a recent typical period or time; d) the average level of all observations (including those used for retrospective analysis). Consider the points in 2.9 and choose an appropriate measure of variation. Decide on the scaling convention to be used. In general, the B.1 method is the simplest for the preparation and interpretation of graphs, but for some special purposes, one of the other two conventions may be preferred, or some special forms may be prepared for routine use. It is important to ensure that all personnel involved in the preparation and analysis of cumulative sum graphs are familiar with the method. 3. Presentation of Graphs
3.1 The preparation of cumulative sum graphs was described in detail in Chapter 2, including the choice of target values, definitions, calculation of measures of variation, and selection of cumulative sum graphs and coefficients of variation.
Some practical points marked with a dot should also be noted. The following information should be included in the CUSUM chart: a) target value (a brief explanation of the reason for choosing the target value (e.g., agreed mean, mean of historical data) can be added); standard error of the observed value (the method used to estimate the standard error can be noted); b) characteristics of the observed values (initial value, sample mean, number of failures, etc.); d) title indicating the purpose of the CUSUM chart (e.g., "CUSUM chart for control" or "CUSUM chart for inter-correlation analysis of data"); the sample interval of the horizontal axis (i.e., coordinate axis) and the scale of the CUSUM coordinate axis should be clearly marked. 3.2 To explain changes in the pattern of the CUSUM graph when known conditions change, the corresponding points on the abscissa (2) may be marked. For example, the introduction of new raw materials into the production process, changes in operators or operating methods. When examining residuals from experiments, the points where the experiment changes due to the level of the experiment should be marked. For data collected in chronological order, it is advisable to indicate the specific period on the abscissa axis. The possible connotations of this information will affect the choice of sample interval scale. If considerable annotation is expected, it is advisable to use a more general scale rather than a scale that is only applicable to the current situation, so as to avoid excessive cluttering of the graph with auxiliary information. 7
CB/Z4887—2006/IS0/TR7871:19973.3 Choice of the origin of the graph
In most cases, considering that positive and negative CUSUMs are to be plotted, the origin of the CUSUM coordinates is taken as zero. However, the visual impression of the graph is not affected by the actual origin used. If nonnegative values are preferred, the CUSUM plot may be started at an appropriate end value. For example, when the CUSUM plot is a continuation of some historical points, or when the line of the CUSUM plot extends beyond the top or bottom of the plot, it may be useful to start a new plot (or replot one) at a new value that corresponds to a general level close to the conclusion of the previous plot (or segment). 3.4 Calculation of the average value
See Appendix C.
3.5 Replotting of the CUSUM plot
See Appendix D.
4 Decision rules for monitoring
4.1 Introduction
4.1.1 This chapter and Chapter 5 give simple decision rules for determining whether apparent changes in the slope of the CUSUM plot (the average level of the variable used to plot the points) are genuine or should be considered part of the inherent variation of the population. This chapter is intended for monitoring and control: Plot the data as you obtain the observations in a sequence that allows any shift from the target level to be detected as quickly as possible. Chapter 5 gives decision rules for retrospective analysis of historical data to determine whether individual segments differ from preceding or succeeding segments in terms of mean level.
4.1.2 The use of decision rules is not always necessary, especially when the CUSUM plot is used only as an efficient representation of sequence data. However, when decisions need to be made and possible actions taken, the CUSUM method often makes more effective use of the data than other traditional techniques. Effective here means that the decision of level change can be made more quickly, or the number of false alarms (false decisions that mean shifts have occurred when there is actually no change) is reduced, or both. 4.1.3 When used for monitoring and control, the CUSUM method has a dual purpose: to detect significant changes and the point at which such changes occur. To measure the performance of the method, this is related to the average number of samples taken until the decision rule signals that a significant change has occurred. When no actual change has occurred, the average chain length will be very long - perhaps containing hundreds of samples - and any signal will be a false alarm. When there are large deviations from the standard, a shorter average chain length is required to quickly detect deviations from the standard. The response of the decision rule to different degrees of deviation from the standard can be represented by the average chain length curve (AL curve). This curve gives a means of comparing decision rules, similar to the sampling characteristic curve (AL curve) used to compare acceptance sampling procedures or the power line of comparing statistical hypothesis tests. It should be noted that the chain length itself will also have statistical variation. Sometimes you may be lucky and no false alarms will be issued for a long time, or changes will be detected very quickly; sometimes you may be unlucky and the chain passing through the sample may issue a false alarm, or mask a real change so that no signal is issued. The average chain length is used as a summary measure for comparison between solutions and decision rules, but other characteristics of the chain length (the mode or median of the chain length - or the distribution of chain lengths) are sometimes worth noting. A typical chain length distribution is shown in Figure 5. g) Average chain length = 5
Median - 3, 5
Mode - 2
2.5% of chain lengths ≥ 16
1% of chain lengths ≥ 19. etc.
b) Average chain length = 20
Median =) 4
Mode = 3
12. 5% chain length ≥ 4 l
7.5% chain length ≥ 50, etc.
Figure 5 Typical chain length distribution
203040
c) Average chain length = 96.5
Median - 67
Mode - 4
12.5% chain length ≥ 200
7.5% chain length ≥ 250, etc.
10203040
Figure 5 (continued)
GB/7 4887-—2006/1S0/TR7871:19971n0
4.1.4 The basic characteristics of the CUSUM chart used to detect potential mean level changes are the steepness of the CUSUM slope and the number of sample months that the slope trend persists. For monitoring and control, the purpose of the CUSUM chart is to detect deviations from the target level. When the process is running near the target level, it is inevitable that the cumulative sum line may deviate from the horizontal direction due to sampling. In this case, the decision rule should not issue a false alarm that the sequence is seriously deviated from the target at that moment. On the other hand, when the process runs into an unsatisfactory situation, the decision rule should react as soon as possible. This is urgent:
The actual process condition
is at or close to the monthly standard
actual deviation from the daily standard
The product and sum chart responds
Long average chain length (few false alarms) Short average chain length (quick detection of abnormalities) Detailed instructions on how to give a specified average chain length for the cumulative sum chart at a specific quality level are given in other standards. Only simple rules that are widely used are given here. Their chain length characteristics are related to the quality level, which is usually defined as the deviation from the target, measured by the standard error of the variable used to plot the points. The standard error can be obtained based on preliminary estimates and assumed to be known, as described in Appendix A of this guidance technical document.
4.2 Cumulative sum V-form templates
4.2.1 Overview
The simplest decision rule used in conjunction with the cumulative sum diagram is embodied in the V-form template. There are three slightly different forms, but they are equivalent in principle and effect. Each form can be applied to any of the four conventional degrees, see B, 1 in Appendix H of this technical guidance document.
If the product sum is realized by calculation: the decision rule is also equivalent in principle and effect, but the form is different - for a detailed explanation, see Chapter 6
4.2.2 Truncated V-form templates
Although full V-form templates can be used, truncated V-form templates are quite convenient in practice. The geometry of the template introduces a point, namely point A in Figure 6, which is located on the vertical line segment IC and is marked by a small notch. One arm of the template runs from B to D and the other arm runs from C to E. If necessary, BD and CE can be extended indefinitely. To be consistent with the rest of this technical guidance document, the length of the vertical line segment, i.e., the length of AB and AC, is called the judgment interval, and the straight lines BJ and CE are called judgment lines. The slopes of these judgment lines will correspond to the reference values introduced in 6.1.3, and can be measured by the number of scale units corresponding to the intervals where each sample is plotted on the graph. The construction of the template is shown in Figure 6. Note that if necessary, the lengths 5α, 10g can be obtained from the cumulative scale conversion, or calculated using the scale factor.
The method of using the template is to place the geometric reference point A of the template on the most recently obtained point, or the last 9 of a segment of particular interest.1 This chapter and Chapter 5 give concise decision rules for determining whether apparent changes in the slope of a CUSUM plot (the mean level of the variable used to plot the points) are real or should be considered part of the inherent variation in the data. This chapter is used for monitoring and control: the observations are plotted in order so that any shifts from a target level can be detected as quickly as possible. Chapter 5 gives corresponding decision rules for retrospective analysis of historical data to determine whether individual segments differ from previous or subsequent segments in terms of mean level.
4.1.2 The use of decision rules is not always necessary, especially when the CUSUM plot is used only as an effective representation of the sequence data. However, when decisions need to be made and possible actions taken, the CUSUM method often makes more effective use of the data than other traditional techniques. Effectiveness here means that the decision of level change is made more quickly, or fewer false alarms are issued (false decisions of mean shift when there is actually no change), or both. 4.1.3 When used for monitoring and control, the CUSUM method has a dual purpose: to detect significant changes and the point at which such changes occur. To determine the performance of the method, the average number of samples taken until the decision rule signals that a significant change has occurred is related. When no actual change has occurred, the average chain length will be very long - perhaps containing hundreds of samples - and any signal will be a false alarm. When there are large excursions from the standard, shorter average chain lengths are needed to quickly detect deviations from the standard. The response of the decision rule to varying degrees of deviation from the standard can be represented by an average chain length curve (AL curve). This curve gives a means of comparing decision rules, similar to the sampling characteristic curve (AL curve) used to compare acceptance sampling procedures or the power line for comparing statistical hypothesis tests. It should be noted that the chain length itself will also have statistical variation. Sometimes you may be lucky and no false alarms will be issued for a long time, or a change will be detected very quickly; sometimes you may be unlucky and the chain of samples may issue a false alarm, or mask a real change so that no signal is issued. The average chain length is used as a summary measure for comparison between solutions and decision rules, but other characteristics of the chain length (the mode or median of the chain length - or the distribution of chain lengths) are sometimes worth noting. A typical chain length distribution is shown in Figure 5. g) Average chain length = 5
Median - 3, 5
Mode - 2
2.5% of chain lengths ≥ 16
1% of chain lengths ≥ 19. etc.
b) Average chain length = 20
Median =) 4
Mode = 3
12. 5% chain length ≥ 4 l
7.5% chain length ≥ 50, etc.
Figure 5 Typical chain length distribution
203040
c) Average chain length = 96.5
Median - 67
Mode - 4
12.5% chain length ≥ 200
7.5% chain length ≥ 250, etc.
10203040
Figure 5 (continued)
GB/7 4887-—2006/1S0/TR7871:19971n0
4.1.4 The basic characteristics of the CUSUM chart used to detect potential mean level changes are the steepness of the CUSUM slope and the number of sample months that the slope trend persists. For monitoring and control, the purpose of the CUSUM chart is to detect deviations from the target level. When the process is running near the target level, it is inevitable that the cumulative sum line may deviate from the horizontal direction due to sampling. In this case, the decision rule should not issue a false alarm that the sequence is seriously deviated from the target at that moment. On the other hand, when the process runs into an unsatisfactory situation, the decision rule should react as soon as possible. This is urgent: bZxz.net
The actual process condition
is at or close to the monthly standard
actual deviation from the daily standard
The product and sum chart responds
Long average chain length (few false alarms) Short average chain length (quick detection of abnormalities) Detailed instructions on how to give a specified average chain length for the cumulative sum chart at a specific quality level are given in other standards. Only simple rules that are widely used are given here. Their chain length characteristics are related to the quality level, which is usually defined as the deviation from the target, measured by the standard error of the variable used to plot the points. The standard error can be obtained based on preliminary estimates and assumed to be known, as described in Appendix A of this guidance technical document.
4.2 Cumulative sum V-form templates
4.2.1 Overview
The simplest decision rule used in conjunction with the cumulative sum diagram is embodied in the V-form template. There are three slightly different forms, but they are equivalent in principle and effect. Each form can be applied to any of the four conventional degrees, see B, 1 in Appendix H of this technical guidance document.
If the product sum is realized by calculation: the decision rule is also equivalent in principle and effect, but the form is different - for a detailed explanation, see Chapter 6
4.2.2 Truncated V-form templates
Although full V-form templates can be used, truncated V-form templates are quite convenient in practice. The geometry of the template introduces a point, namely point A in Figure 6, which is located on the vertical line segment IC and is marked by a small notch. One arm of the template runs from B to D and the other arm runs from C to E. If necessary, BD and CE can be extended indefinitely. To be consistent with the rest of this technical guidance document, the length of the vertical line segment, i.e., the length of AB and AC, is called the judgment interval, and the straight lines BJ and CE are called judgment lines. The slopes of these judgment lines will correspond to the reference values introduced in 6.1.3, and can be measured by the number of scale units corresponding to the intervals where each sample is plotted on the graph. The construction of the template is shown in Figure 6. Note that if necessary, the lengths 5α, 10g can be obtained from the cumulative scale conversion, or calculated using the scale factor.
The method of using the template is to place the geometric reference point A of the template on the most recently obtained point, or the last 9 of a segment of particular interest.1 This chapter and Chapter 5 give concise decision rules for determining whether apparent changes in the slope of a CUSUM plot (the mean level of the variable used to plot the points) are real or should be considered part of the inherent variation in the data. This chapter is used for monitoring and control: the observations are plotted in order so that any shifts from a target level can be detected as quickly as possible. Chapter 5 gives corresponding decision rules for retrospective analysis of historical data to determine whether individual segments differ from previous or subsequent segments in terms of mean level.
4.1.2 The use of decision rules is not always necessary, especially when the CUSUM plot is used only as an effective representation of the sequence data. However, when decisions need to be made and possible actions taken, the CUSUM method often makes more effective use of the data than other traditional techniques. Effectiveness here means that the decision of level change is made more quickly, or fewer false alarms are issued (false decisions of mean shift when there is actually no change), or both. 4.1.3 When used for monitoring and control, the CUSUM method has a dual purpose: to detect significant changes and the point at which such changes occur. To determine the performance of the method, the average number of samples taken until the decision rule signals that a significant change has occurred is related. When no actual change has occurred, the average chain length will be very long - perhaps containing hundreds of samples - and any signal will be a false alarm. When there are large excursions from the standard, shorter average chain lengths are needed to quickly detect deviations from the standard. The response of the decision rule to varying degrees of deviation from the standard can be represented by an average chain length curve (AL curve). This curve gives a means of comparing decision rules, similar to the sampling characteristic curve (AL curve) used to compare acceptance sampling procedures or the power line for comparing statistical hypothesis tests. It should be noted that the chain length itself will also have statistical variation. Sometimes you may be lucky and no false alarms will be issued for a long time, or a change will be detected very quickly; sometimes you may be unlucky and the chain of samples may issue a false alarm, or mask a real change so that no signal is issued. The average chain length is used as a summary measure for comparison between solutions and decision rules, but other characteristics of the chain length (the mode or median of the chain length - or the distribution of chain lengths) are sometimes worth noting. A typical chain length distribution is shown in Figure 5. g) Average chain length = 5
Median - 3, 5
Mode - 2
2.5% of chain lengths ≥ 16
1% of chain lengths ≥ 19. etc.
b) Average chain length = 20
Median =) 4
Mode = 3
12. 5% chain length ≥ 4 l
7.5% chain length ≥ 50, etc.
Figure 5 Typical chain length distribution
203040
c) Average chain length = 96.5
Median - 67
Mode - 4
12.5% chain length ≥ 200
7.5% chain length ≥ 250, etc.
10203040
Figure 5 (continued)
GB/7 4887-—2006/1S0/TR7871:19971n0
4.1.4 The basic characteristics of the CUSUM chart used to detect potential mean level changes are the steepness of the CUSUM slope and the number of sample months that the slope trend persists. For monitoring and control, the purpose of the CUSUM chart is to detect deviations from the target level. When the process is running near the target level, it is inevitable that the cumulative sum line may deviate from the horizontal direction due to sampling. In this case, the decision rule should not issue a false alarm that the sequence is seriously deviated from the target at that moment. On the other hand, when the process runs into an unsatisfactory situation, the decision rule should react as soon as possible. This is urgent:
The actual process condition
is at or close to the monthly standard
actual deviation from the daily standard
The product and sum chart responds
Long average chain length (few false alarms) Short average chain length (quick detection of abnormalities) Detailed instructions on how to give a specified average chain length for the cumulative sum chart at a specific quality level are given in other standards. Only simple rules that are widely used are given here. Their chain length characteristics are related to the quality level, which is usually defined as the deviation from the target, measured by the standard error of the variable used to plot the points. The standard error can be obtained based on preliminary estimates and assumed to be known, as described in Appendix A of this guidance technical document.
4.2 Cumulative sum V-form templates
4.2.1 Overview
The simplest decision rule used in conjunction with the cumulative sum diagram is embodied in the V-form template. There are three slightly different forms, but they are equivalent in principle and effect. Each form can be applied to any of the four conventional degrees, see B, 1 in Appendix H of this technical guidance document.
If the product sum is realized by calculation: the decision rule is also equivalent in principle and effect, but the form is different - for a detailed explanation, see Chapter 6
4.2.2 Truncated V-form templates
Although full V-form templates can be used, truncated V-form templates are quite convenient in practice. The geometry of the template introduces a point, namely point A in Figure 6, which is located on the vertical line segment IC and is marked by a small notch. One arm of the template runs from B to D and the other arm runs from C to E. If necessary, BD and CE can be extended indefinitely. To be consistent with the rest of this technical guidance document, the length of the vertical line segment, i.e., the length of AB and AC, is called the judgment interval, and the straight lines BJ and CE are called judgment lines. The slopes of these judgment lines will correspond to the reference values introduced in 6.1.3, and can be measured by the number of scale units corresponding to the intervals where each sample is plotted on the graph. The construction of the template is shown in Figure 6. Note that if necessary, the lengths 5α, 10g can be obtained from the cumulative scale conversion, or calculated using the scale factor.
The method of using the template is to place the geometric reference point A of the template on the most recently obtained point, or the last 9 of a segment of particular interest.4 The basic characteristics of the CUSUM chart used to detect potential mean level changes are the steepness of the CUSUM slope and the number of samples that follow the slope. For monitoring and control, the purpose of the CUSUM chart is to detect deviations from the target level. When the process is operating near the target level, there is an inevitable possibility that the CUSUM curve deviates from the horizontal direction due to sampling. In this case, the decision rule should not issue a false alarm that the process is seriously deviating from the target at that moment. On the other hand, when the process is running into an unsatisfactory situation, the decision rule should react as quickly as possible. The following are urgent requirements:
The actual process conditions
are at or near the monthly target
Actually deviate from the daily target
The CUSUM chart responds
Long average chain length (few false alarms) Short average chain length (quick detection of anomalies) Detailed instructions on how to give a specified average chain length for a CUSUM chart at a specific quality level are given in other standards. Only simple rules that are widely used are given here. Their chain length characteristics are related to the quality level, which is usually defined as the deviation from the target, measured by the standard error of the variable used to plot the points. The standard error can be obtained based on preliminary estimates and assumed to be known, as described in Appendix A of this technical guidance document.
4.2 Cumulative sum V-type template
4.2.1 Overview
The simplest decision rule used in conjunction with the cumulative sum diagram is embodied in the V-type template. There are three slightly different forms, but they are equivalent in principle and function. Each form can apply any of the four conventional degrees, as detailed in B, 1 in Appendix H of this technical guidance document.
If the product and the sum are realized by calculation: the judgment rule is also equivalent in principle and effect, but the form is different - see Chapter 6 for a detailed explanation
4.2.2 Truncated V-shaped template
Although a full V-shaped template can be used, the truncated V-shaped template is quite convenient to use in practice. The geometry of the template introduces a point, namely point A in Figure 6, which is located on the vertical line segment IC and is marked by a small notch. The arm of the template runs from B to D and the other arm runs from C to E. If necessary, BD and CE can be extended indefinitely. In order to be consistent with the following text of this guiding technical document, the length of the vertical line segment, that is, AB and AC, is called the judgment interval, and the straight lines BJ and CE are called judgment lines. The slopes of these judgment lines will correspond to the reference values introduced in 6.1.3 and can be measured by the number of scale units corresponding to the intervals in which each sample is plotted on the graph. The construction of the template is shown in Figure 6. Note that the lengths 5α, 10g, if necessary, can be obtained from the cumulative scale conversion or calculated using the scale factor.
The template is used by placing the geometric reference point A of the template at the most recently obtained point, or at the last 9 of a segment of particular interest.4 The basic characteristics of the CUSUM chart used to detect potential mean level changes are the steepness of the CUSUM slope and the number of samples that follow the slope. For monitoring and control, the purpose of the CUSUM chart is to detect deviations from the target level. When the process is operating near the target level, there is an inevitable possibility that the CUSUM curve deviates from the horizontal direction due to sampling. In this case, the decision rule should not issue a false alarm that the process is seriously deviating from the target at that moment. On the other hand, when the process is running into an unsatisfactory situation, the decision rule should react as quickly as possible. The following are urgent requirements:
The actual process conditions
are at or near the monthly target
Actually deviate from the daily target
The CUSUM chart responds
Long average chain length (few false alarms) Short average chain length (quick detection of anomalies) Detailed instructions on how to give a specified average chain length for a CUSUM chart at a specific quality level are given in other standards. Only simple rules that are widely used are given here. Their chain length characteristics are related to the quality level, which is usually defined as the deviation from the target, measured by the standard error of the variable used to plot the points. The standard error can be obtained based on preliminary estimates and assumed to be known, as described in Appendix A of this technical guidance document.
4.2 Cumulative sum V-type template
4.2.1 Overview
The simplest decision rule used in conjunction with the cumulative sum diagram is embodied in the V-type template. There are three slightly different forms, but they are equivalent in principle and function. Each form can apply any of the four conventional degrees, as detailed in B, 1 in Appendix H of this technical guidance document.
If the product and the sum are realized by calculation: the judgment rule is also equivalent in principle and effect, but the form is different - see Chapter 6 for a detailed explanation
4.2.2 Truncated V-shaped template
Although a full V-shaped template can be used, the truncated V-shaped template is quite convenient to use in practice. The geometry of the template introduces a point, namely point A in Figure 6, which is located on the vertical line segment IC and is marked by a small notch. The arm of the template runs from B to D and the other arm runs from C to E. If necessary, BD and CE can be extended indefinitely. In order to be consistent with the following text of this guiding technical document, the length of the vertical line segment, that is, AB and AC, is called the judgment interval, and the straight lines BJ and CE are called judgment lines. The slopes of these judgment lines will correspond to the reference values introduced in 6.1.3 and can be measured by the number of scale units corresponding to the intervals in which each sample is plotted on the graph. The construction of the template is shown in Figure 6. Note that the lengths 5α, 10g, if necessary, can be obtained from the cumulative scale conversion or calculated using the scale factor.
The template is used by placing the geometric reference point A of the template at the most recently obtained point, or at the last 9 of a segment of particular interest.
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