CUSUM

10 Key excerpts on "CUSUM"

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  eBook - ePub

  Basic Statistical Tools for Improving Quality

  • Chang W. Kang, Paul H. Kvam(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Q-sum) is short for cumulative sum, which means the values plotted on the control chart are accumulated and averaged as the process continues to produce outputs that are sampled in successive stages. In accumulating past data, each past sample has equal weight in how the control chart is constructed. The CUSUM chart is made to be sensitive to slow, insidious changes occurring in the process, unlike the Shewhart charts. For this benefit, the chart sacrifices some ability to detect a dramatic change that occurs in the process, especially if the process has been operating for a long time. While the Shewhart charts can reveal such a change in the process with a spike in the control chart, a detectable change in the plotted average can be more muted in the CUSUM chart because most of the weight in the charted value comes from past samples, when the process was presumably in control.

  The CUSUM control charts use the equally weighted information in the sequence of samples to monitor the process. At each stage, we observe how the process deviates from the target value. The CUSUM control chart uses the cumulative sum of these deviations. This enables the process manager to effectively detect a gradual shift in the process, even if it is small in magnitude.

  The CUSUM control chart is typically constructed for individual observations. When we construct the tabular CUSUM control chart, two statistics C + and C- are calculated:

  C+ = Accumulation of deviations upward from the target valueC- = Accumulation of deviations downward from target value.

  As the process manager, we can choose values for two important parameters, K and H, allowing us to construct the tabular CUSUM control charts in a way that is most beneficial to us. K is called the reference value and H is called the decision interval. We need to choose the reference value K to be one-half of the shift magnitude that we want the control chart to detect. In general, the shift is expressed in standard deviations and the CUSUM control chart is effective when the magnitude of shift is σ (one standard deviation). The value for K is 0.5 times the process standard deviation. As a rule-of-thumb, a reasonable value for H

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  eBook - PDF

  Statistical Quality Control

  A Modern Introduction

  • Douglas C. Montgomery(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  412 Part 4 ■ Other Statistical Process-Monitoring and Control Techniques 9.1 THE CUMULATIVE SUM CONTROL CHART 9.1.1 Basic Principles: The CUSUM Control Chart for Monitoring the Process Mean 9.1.2 The Tabular or Algorithmic CUSUM for Monitoring the Process Mean 9.1.3 Recommendations for CUSUM Design 9.1.4 The Standardized CUSUM 9.1.5 Improving CUSUM Responsiveness for Large Shifts 9.1.6 The Fast Initial Response or Headstart Feature 9.1.7 One-Sided CUSUMs 9.1.8 A CUSUM for Monitoring Process Variability 9.1.9 Rational Subgroups 9.1.10 CUSUMs for Other Sample Statistics 9.1.11 The V-Mask Procedure 9.1.12 The Self-Starting CUSUM 9.2 THE EXPONENTIALLY WEIGHTED MOVING AVERAGE CONTROL CHART 9.2.1 The Exponentially Weighted Moving Average Control Chart for Monitoring the Process Mean 9.2.2 Design of an EWMA Control Chart 9.2.3 Robustness of the EWMA to Non-normality 9.2.4 Rational Subgroups 9.2.5 Extensions of the EWMA 9.3 THE MOVING AVERAGE CONTROL CHART Supplemental Material for Chapter 9 S9.1 The Markov Chain Approach for Finding the ARL for CUSUM and EWMA Control Charts S9.2 Integral Equation versus Markov Chains for Finding the ARL 9 CHAPTER OUTLINE The supplemental material is on the textbook website www.wiley.com/college/montgomery. 413 Time-Weighted Control Charts 414 Chapter 9 ■ Time-Weighted Control Charts CHAPTER OVERVIEW AND LEARNING OBJECTIVES Chapters 5, 6, and 7 have concentrated on basic SPC methods. The control charts featured in these chapters are predominantly Shewhart control charts. These charts are extremely use- ful in phase I implementation of SPC, where the process is likely to be out of control and experiencing assignable causes that result in large shifts in the monitored parameters. Shewhart charts are also very useful in the diagnostic aspects of bringing an unruly process into statistical control, because the patterns on these charts often provide guidance regarding the nature of the assignable cause.

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  eBook - PDF

  Statistical Quality Control for the Six Sigma Green Belt

  • Bhisham C. Gupta, H. Fred Walker(Authors)
  • 2007(Publication Date)
  • ASQ Quality Press

    (Publisher)

  Relatively speaking, the MA chart is not as effective as CUSUM and EWMA charts. The CUSUM and EWMA control charts can be used to monitor not only the process mean but also the process variance and the fraction nonconforming and nonconformities. However, our major focus of discussion of these control charts is on the process mean. An excel- lent reference for a detailed study of CUSUM control charts is Hawkins and Olwell (1998). 5.1 Basic Concepts of the CUSUM Control Chart CUSUM Control Chart versus Shewhart X – -R Control Chart The basic assumptions of a CUSUM control chart are: 1. The observations are independently and identically normally distributed with mean µ and standard deviation σ. 2. The mean µ and standard deviation σ are known. Note that in practice the parameters µ and σ are never known. Hence, we always end up estimating them. It is very important to remember that while estimating these parameters, precision is essential. Even a small error, par- ticularly in estimating µ, will have an accumulative effect, which will cause false alarms. One way to avoid such problems is to take very large samples when estimating these parameters. Phase II (Detecting Small Shifts)—SPC 103 Before we begin our formal discussion of designing the CUSUM control chart, we first use a small set of data to show how the CUSUM chart is more effective than the Shewhart control chart in detecting small shifts in the pro- cess mean. Note that the CUSUM chart we will discuss in this section has no decision interval. In other words, it is not a formal CUSUM control chart. Example 5.1 Consider a manufacturing process of auto parts. We are interested in studying a quality characteristic of the parts manufactured by the process. Let the quality characteristic when the process is under control be normally distributed with mean 20 and standard deviation 2.

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  eBook - PDF

  Statistical Quality Control

  Using MINITAB, R, JMP and Python

  • Bhisham C. Gupta(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  However, these are not the only changes that can occur. Transient special causes are also an important reality and an important source of quality problems. They cannot be ignored, and relying solely on CUSUM charts for SPC is shortsighted. Just as Shewhart charts are not particularly effective in detecting less-than-massive persistent changes in process parameter, the CUSUM charts are not particularly effective in detecting massive transient changes in process CUSUM control chart using FIR feature for the data in Table 7.6 10.0 12.5 7.5 5.0 2.5 0.0 –2.5 –5.0 Cumulative sum 1 2 3 4 5 6 7 8 9 10 Sample UCL = 5 LCL = –5 0 An estimated historical parameter is used in the calculations. Figure 7.7 Two-sided CUSUM control chart using the FIR feature for the data in Table 7.6. 216 Statistical Quality Control: Using Minitab, R, JMP, and Python parameters. In fact, the CUSUMs are meant to detect persistent but small changes in process parameters. Proper SPC requires the use of both types of controls: CUSUMs for persistent but small changes and Shewhart charts for large transient problems. Lucas (1982) introduced a combined Shewhart-CUSUM quality control scheme. The combined Shewhart-CUSUM control chart gives an out-of-control signal if the most recent sample is either outside the Shewhart control limits or beyond the CUSUM decision interval value. Lucas recom- mends using the standardized values z i for the combined chart, which is z i = x i - μ 0 σ where μ 0 is the target mean value and σ is the process standard deviation. Note that if rational sub- groups of size n > 1 are used, then z i , x i , and σ are replaced by z i , x i and σ n , respectively. In the combined Shewhart-CUSUM control chart, an out-of-control signal at time i occurs when either S + i or S - i exceeds the decision interval value h, or the z i or z i value falls outside the Shewhart control limits.

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  eBook - PDF

  Quality Management Systems

  a Selective Presentation of Case-studies Showcasing Its Evolution

  • Leo D. Kounis(Author)
  • 2018(Publication Date)
  • IntechOpen

    (Publisher)

  [3] Gan FF. An optimal design of CUSUM quality control charts. Journal of Quality Technol-ogy. 1991; 23 (4):279-286 [4] Han D, Tsung F, Hu X, Wang K. CUSUM and EWMA multi-charts for detecting a range of mean shifts. Statistica Sinica. 2007; 17 (3):1139-1164 [5] Jiang W, Shu L, Apley DW. Adaptive CUSUM procedures with EWMA-based shift estimators. IIE Transactions. 2008; 40 (10):992-1003 [6] Moustakides G. Optimal stopping times for detecting changes in distributions. The Annals of Statistics. 1986; 14 (4):1379-1387 [7] Page ES. Continuous inspection schemes. Biometrika. 1954; 41 (1/2):100-115 [8] Page ES. Cumulative sum schemes using gauging. Technometrics. 1962; 4 (1):97-109 [9] Ryu JH, Wan H, Kim S. Optimal design of a CUSUM chart for a mean shift of unknown size. Journal of Quality Technology. 2010; 42 (3):311 [10] Shu J, Jiang W, Wu Z. Adaptive CUSUM procedures with Markovian Mean Estimation. Computational Statistics and Data Analysis. 2008; 52 (9):4395-4409 [11] Souza GP, Samohyl RW (2008). Monitoring forecast errors with combined CUSUM and Shewhart control charts. In: International Symposium of Forecasting. 26 [12] Sparks RS. CUSUM charts for signalling varying location shifts. Journal of Quality Tech-nology. 2000; 32 (2):157-171 [13] Sparks RS. A group of moving averages control plan for signaling varying location shifts. Quality Engineering. 2003; 15 (4):519-532 [14] Sparks RS. Shewhart dispersion charts made easy for mild to moderately autocorrelated normally distributed data. Quality Engineering. 2017 in press. http://www.tandfonline. com/doi/abs/10.1080/08982112.2017.1311415 [15] Stasinopoulos DM, Rigby RA. Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software. 2007; 23 (7):1-46 [16] Stoumbos ZG, Mittenthal J, Runger GC. Steady-state-optimal adaptive control charts based on variable sampling intervals. Stochastic Analysis and Applications. 2001; 19 (6):1025-1057 [17] WHO Regional Office for Europe.

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  eBook - PDF

  Introduction to Statistical Process Control

  • Peihua Qiu(Author)
  • 2013(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  For instance, Moustakides (1986) proved that among all control charts with a fixed ARL 0 value, the CUSUM chart with an allowance k would have the shortest ARL 1 value for detecting a persistent shift of size δ = 2 k . The SPRT connection also provides a general way to construct CUSUM charts, summarized in the box below. 144 UNIVARIATE CUSUM CHARTS General Formula for the DI Form of the CUSUM Chart Assume that the IC and OC pdf’s (or pmf’s) of the process observations are f 0 and f 1 , respectively, and { X 1 , X 2 ,... } are independent observations obtained from the process. Then, the charting statistic of the DI form of the CUSUM chart for detecting the distributional shift from f 0 to f 1 is defined as follows: for n ≥ 1, G n = max parenleftBig 0 , G n − 1 + tildewide Ψ n parenrightBig , (4.20) where G 0 = 0 and tildewide Ψ n = log ( f 1 ( X n ) / f 0 ( X n )) . Note that the description in the above box about the CUSUM chart is more gen-eral than the description in the remaining part of this subsection, in that both f 0 and f 1 are not restricted to parametric functions here, while they are assumed paramet-ric and their only difference is in the parameter values in the remaining part of this subsection. The major purpose of this generalization is that the resulting CUSUM chart (4.20) is also available to certain applications in which the process observation distributions are nonparametric, or f 0 and f 1 are parametric but their difference is not just in the parameter values. 4.3 Monitoring the Variance of a Normal Process Variability of process observations is another key characteristic that affects the qual-ity of products. In this section, we discuss monitoring of the process variability when the process is univariate and the process distribution is normal.

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  Statistics in Research and Development

  • R. Caulcutt(Author)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  I hope you are not disappointed with the failure of the adapted CUSUM technique to find dramatic changes in the variability of yield and impurity. This lack of success could be regarded as welcome news. Any significant change in variability would invalidate the earlier CUSUM analyses we did on mean yield and mean impurity, for they are based on the assumption that the process variability is constant. You may recall that we encountered a similar assumption with the two-sample Mest, when it was required that the two populations should be equally variable. We will meet a similar assumption when we use regression analysis in Chapter 10. In general, any techniques which focus on differences in mean will also take account of variability and will require that magnitude of this variability be constant. 9.6 T H E USE O F C U S U M S IN S T A T I S T I C A L P R O C E S S C O N T R O L In Chapter 8 we used average run length curves to compare statistical process control (SPC) charts. We examined in some detail the conven-tional Shewhart chart, the moving average chart and the exponentially weighed moving average (EWMA) chart. We used the average run length curves to show how quickly each chart would detect changes of different sizes. Amongst the average run length curves I included one for the CUSUM control chart. You may recall that it showed how powerful the CUSUM chart is, when faced with relatively small changes. I promised that the CUSUM technique would be introduced in Chapter 9, and so it has been. However, we have so far discussed only the post-mortem application of CUSUMs. Let us now turn to the use of CUSUMs in SPC. SPC operates in real time, unlike the CUSUM post-mortem analysis. When using the latter we have all our data to hand before we start the analysis. With SPC, however, the data arrive in small quantities at regular intervals and we wish to make a decision about the process as each point is added to the chart.

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  eBook - PDF

  Statistical Process Control For Quality Improvement- Hardcover Version

  • J. Koronacki, J.R. Thompson(Authors)
  • 2001(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  We note that (4.23) indicates a procedure somewhat similar to that of a control chart for the mean. A major difference is that, at any stage j , we base our test on ¯ ¯ x j rather than on ¯ x j . Clearly such a cumula-tive procedure (CUSUM) is not so much oriented to detecting “Pareto glitches,” but rather to discovering a fundamental and persistent change in the mean. Now let us return to the case of two levels for µ 0 and µ 1 . How shall we adapt such a test to process control? We might take the upper limit for acceptable µ and call it µ 1 . Then we could take the lower acceptable limit for µ and call it µ 0 . Typically, but not always, the target value is CUSUM test for shift of the mean 133 the midpoint µ ∗ = µ 0 + µ 1 2 . (4.24) Then the analogue of the acceptance interval in the control chart ap-proach becomes the region of uncertainty ln( k 0 ) < N µ 1 − µ 0 σ 2 /n [ ¯ ¯ x N − µ 0 + µ 1 2 ] < ln( k 1 ) . (4.25) Suppose that we observe the first N observations to be equal to µ ∗ . Then our statistic becomes R 1 = N µ 1 − µ 0 σ 2 /n [ µ 0 + µ 1 2 − µ 0 + µ 1 2 ] = 0 . (4.26) Then, suppose the next observation is µ ∗ + δ . The statistic becomes R 1 = ( N + 1) µ 1 − µ 0 σ 2 /n [ µ 0 + µ 1 2 + δ N + 1 − µ 0 + µ 1 2 ] (4.27) = µ 1 − µ 0 σ 2 /n δ. We note that this statistic is independent of N . So, in one sense, the statistic does not have the highly undesirable property that a run of 100 lots, each precisely on target, can mask a 101’st bad lot. On the other hand, suppose that the 101’st lot gives ¯ x 101 = µ ∗ + δ , and the 102’nd lot gives ¯ x 102 = µ ∗ − δ . Then, the pooled test will have returned our statistic to a neutral R 1 = 0. If we believe that Pareto glitches both below and above the mean can occur, we note that the sequential test allows (undesirably) for the cancellation of positive departures from the standard by negative ones. We note an additional problem with the kind of sequential test de-veloped above.

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  Six Sigma

  Advanced Tools for Black Belts and Master Black Belts

  • Loon Ching Tang, Thong Ngee Goh, Hong See Yam, Timothy Yoap(Authors)
  • 2007(Publication Date)
  • Wiley

    (Publisher)

  In the example below, the change in mean of the forecast errors was detected in period 6 when S 6 , 2 exceeded L 2 . Recognizing that forecast monitoring is done sequentially, one can easily see that Ta-ble 25.1 contains unnecessary calculations. The only important information for control Table 25.1 Example of the BCUSUM method. 19 Period Forecast error S 1 S 2 S 3 S 4 S 5 S 6 1 − 10 − 10 2 20 20 10 3 15 15 35 25 4 5 5 20 40 30 5 − 25 − 25 − 20 − 5 15 5 6 − 25 − 25 − 50 − 45 − 30 − 10 − 20 Control limits: L 1 to L 6 ± 30 ± 40 ± 50 ± 60 ± 70 ± 80 384 CUSUM and Backward CUSUM for Autocorrelated Observations purposes is that contained in the last row. The BCUSUMs in rows 1--5 are of no use at all unless we are trying to detect change not on an on-line or real-time basis. For example, it is easy to see that when there is a BCUSUM value in row 4 that exceeds the specified control limits, one does not need the ensuing observations (i.e., observations 5 and 6) to conclude that there has been a change in mean of the forecast errors. At period 4, the forecaster already knows that there has been a change or intervention affecting the system. Thus, if one needs to implement a BCUSUM considering only the last few forecast errors, one need only calculate the following BCUSUM: S i = i j = 1 e n − j + 1 , i = 1 , 2 ,..., m , where m represents the number of latest forecast errors being monitored. One drawback in basing a monitoring scheme only on the latest few m observations is that small process shifts will be difficult to detect. In detecting a change in the mean of a normal random variable, for example, the ARL of a two-sided CUSUM chart ( h = 4 . 8 , k = 0 . 5 , ARL 0 = 390) for a 0.25 σ mean shift is approximately 120. Monitoring only the latest 6--10 observations will not provide enough sensitivity to detect such a change. This ARL figure may not have any practical importance in forecasting since forecasters are usually dealing with few samples.

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  eBook - PDF

  Introduction to Statistical Quality Control

  • Douglas C. Montgomery(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  How long would it take to recover the cost of this new equipment? 10.28. You have been hired by a health care organization to improve the performance in their major surgical center. Management of the organization wants to use control charts to monitor surgical errors. What are the issues that you would need to consider in under- taking this project? For example, are all surgical errors of the same severity? Should they be analyzed by procedure and/or by surgeon? What risk factors could possibly be considered? CHAPTER 10 OTHER UNIVARIATE STATISTICAL PROCESS-MONITORING AND CONTROL TECHNIQUES CHAPTER OVERVIEW AND LEARNING OBJECTIVES The widespread successful use of the basic SPC methods described in Part 3 and the CUSUM and EWMA control charts in the previous chapter have led to the development of many new tech- niques and procedures over the past 20 years. This chapter is an overview of some of the more useful recent developments. We begin with a discussion of SPC methods for short production runs and concentrate on how conventional control charts can be modified for this situation. Although there are other techniques that can be applied to the short-run scenario, this approach seems to be most widely used in practice. We then discuss modified and acceptance control charts. These techniques find some application in situations where process capability is high, such as the Six Sigma manufacturing environment. Multiple-stream processes are encountered in many indus- tries. An example is container filling on a multiple-head machine. We present the group control chart (a classical method for the multiple-stream process) and another procedure based on control charts for monitoring the specific types of assignable causes associated with these systems. We also discuss techniques for monitoring processes with autocorrelated data, a topic of considerable importance in the chemical and process industries.

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