Average Run Lengths in Cumulative Chart Quality Control Schemes
TL;DR: In this paper, average run-lengths are evaluated for V-mask quality control schemes based on cumulative deviation charts when the observations are Normally distributed and either independent or members of a certain serially correlated class.
Abstract: Average run-lengths are evaluated for V-mask quality control schemes based on cumulative deviation charts when the observations are Normally distributed and either independent or members of a certain serially correlated class.
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TL;DR: A unified framework for the design and the performance analysis of the algorithms for solving change detection problems and links with the analytical redundancy approach to fault detection in linear systems are established.
Abstract: This book is downloadable from http://www.irisa.fr/sisthem/kniga/. Many monitoring problems can be stated as the problem of detecting a change in the parameters of a static or dynamic stochastic system. The main goal of this book is to describe a unified framework for the design and the performance analysis of the algorithms for solving these change detection problems. Also the book contains the key mathematical background necessary for this purpose. Finally links with the analytical redundancy approach to fault detection in linear systems are established. We call abrupt change any change in the parameters of the system that occurs either instantaneously or at least very fast with respect to the sampling period of the measurements. Abrupt changes by no means refer to changes with large magnitude; on the contrary, in most applications the main problem is to detect small changes. Moreover, in some applications, the early warning of small - and not necessarily fast - changes is of crucial interest in order to avoid the economic or even catastrophic consequences that can result from an accumulation of such small changes. For example, small faults arising in the sensors of a navigation system can result, through the underlying integration, in serious errors in the estimated position of the plane. Another example is the early warning of small deviations from the normal operating conditions of an industrial process. The early detection of slight changes in the state of the process allows to plan in a more adequate manner the periods during which the process should be inspected and possibly repaired, and thus to reduce the exploitation costs.
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23 Jan 1998
TL;DR: In this article, the authors present a CUSUM chart for a normal mean and compare it to the Shewhart Xbar chart for the same purpose, showing the effect of the change in the normal distribution.
Abstract: 1 Introduction.- 1.1 Common-cause and special-cause variability.- 1.2 Transient and persistent special causes.- 1.3 The Shewhart and CUSUM charts.- 1.4 Basis for the CUSUM chart for a normal mean.- 1.4.1 Statistical properties of the CUSUM.- 1.5 Out-of-control distribution of the CUSUM.- 1.6 Testing for a shift -the V mask.- 1.7 Estimation following a signal.- 1.8 Using individual readings or rational groups.- 1.9 The decision interval form of the CUSUM.- 1.9.1 Example.- 1.10 Summary.- 1.11 Further reading.- 2 CUSUM design.- 2.1 The choice of k and h.- 2.1.1 Reference value k - "tuning" for a specific shift.- 2.2 Runs, run length, and average run length.- 2.2.1 The choice of h, the decision interval.- 2.2.2 Calculating the k, h, ARL relationship.- 2.2.3 A closer look at the choice of in-control ARL.- 2.2.4 Designing a CUSUM of Xbar.- 2.3 The Shewhart Xbar chart as CUSUM.- 2.4 Summary.- 2.5 Further reading.- 3 More about normal data.- 3.1 In-control ARLs.- 3.2 Out-of-control ARLs.- 3.2.1 Model.- 3.2.2 The ARL following a shift in mean.- 3.3.3 ARL sensitivity to choice of K.- 3.2.4 Out-of-control states and two-sided CUSUMs.- 3.3 FIR CUSUMs: zero start and steady state start.- 3.3.1 Introduction.- 3.3.2 Out-of-control ARL of the FIR CUSUM.- 3.3.3 ARL of two-sided FIR CUSUMS.- 3.3.4 Initial and steady-state ARL.- 3.4 Controlling for the mean within a range.- 3.4.1 Example.- 3.5 The impact of variance shifts.- 3.5.1 Individual data -the approximate normal transform.- 3.5.2 Rational groups-variance CUSUMs.- 3.6 Combined Shewhart and CUSUM charts.- 3.6.1 Example.- 3.7 Effect of model departures.- 3.7.1 Nonnormality.- 3.7.2 Independence.- 3.8 Weighted CUSUMs.- 3.8.1 Example.- 3.9 Summary.- 3.10 Further reading.- 4 Other continuous distributions.- 4.1 The gamma family and normal variances.- 4.1.1 Background.- 4.1.2 Normal variances.- 4.1.3 Design of the CUSUM for scale.- 4.1.4 Example: Sugar bags.- 4.1.5 Shift in the gamma shape parameter ?.- 4.1.6 Example - shift in ss.- 4.2 The inverse Gaussian family.- 4.2.1 Background.- 4.2.2 Shift in mean.- 4.2.3 Shift in scale parameter.- 4.3 Example from General Motors.- 4.3.1 CUSUM chart for location.- 4.3.2 CUSUM chart for ?.- 4.3.3 Remarks.- 4.4 Comments.- 4.5 Further reading.- 5 Discrete data.- 5.1 Types of discrete data.- 5.1.1 Binomial data.- 5.1.2 Count data.- 5.2 The graininess of the ARL function.- 5.3 The Poisson distribution and count data.- 5.3.1 Useful properties of the Poisson distribution.- 5.4 The Poisson and CUSUMs.- 5.4.1 Design for an upward shift.- 5.4.2 Downward shift.- 5.4.3 ARLs.- 5.4.4 Example.- 5.4.5 The effect of departures from Poisson.- 5.4.6 Checking conformity to the Poisson model.- 5.5 Weighted Poisson CUSUMs.- 5.6 The binomial distribution.- 5.6.1 Background.- 5.6.2 Examples.- 5.6.3 The choice of m.- 5.7 Weighted binomial CUSUMs.- 5.7.1 Example.- 5.8 Other discrete distributions.- 5.9 Summary.- 5.10 Further reading.- 6 Theoretical foundations of the CUSUM.- 6.1 General theory.- 6.1.1 Relation of the SPRT to the CUSUM.- 6.1.2 Optimality properties.- 6.2 The general exponential family.- 6.2.1 Derivation of the CUSUM for a normal mean shift.- 6.2.2 The gamma family and normal variance.- 6.2.3 Relation to normal variances.- 6.2.4 The Poisson family.- 6.2.5 The binomial family.- 6.2.6 The negative binomial family.- 6.2.7 The inverse Gaussian family.- 6.2.8 The Weibull distribution.- 6.2.9 Distributions outside the exponential family.- 6.3 The Markov property of CUSUMs.- 6.4 Getting the ARL.- 6.4.1 The renewal equations.- 6.4.2 The Markov chain approach.- 6.4.3 Simulation using variance reduction techniques.- 6.5 Summary.- 6.6 Further reading.- 7 Calibration and short runs.- 7.1 The self-starting approach.- 7.2 The self-starting CUSUM for a normal mean.- 7.2.1 Special features of self-starting charts.- 7.3 Self-starting CUSUMs for gamma data.- 7.3.1 Background.- 7.3.2 The scheme.- 7.3.3 Example.- 7.3.4 Normal data - control of mean and variance.- 7.3.5 Comments.- 7.4 Discrete data.- 7.4.1 The Poisson distribution.- 7.4.2 The binomial distribution.- 7.4.3 Updating the targets.- 7.5 Summary.- 7.6 Further reading.- 8 Multivariate data.- 8.1 Outline of the multivariate normal.- 8.2 Shewhart charting-Hotelling's T2.- 8.3 CUSUM charting - various approaches.- 8.3.1 Collections of unvariate CUSUMs.- 8.4 Regression adjustment.- 8.4.1 Example.- 8.4.2 SPC use of regression-adjusted variables.- 8.4.3 Example - monitoring a carbide plant.- 8.5 Choice of regression adjustment.- 8.6 The use of several regression-adjusted variables.- 8.6.1 Example.- 8.7 The multivariate exponentially weighted moving average.- 8.8 Summary.- 8.9 Further reading.- 9 Special topics.- 9.1 Robust CUSUMs.- 9.2 Recursive residuals in regression.- 9.2.1 Definition and properties.- 9.2.2 Example.- 9.3 Autocorrelated data.- 9.3.1 Example.- 9.4 Summary.- 9.5 Further reading.- 9.5.1 Time series.- 9.5.2 Score methods.- 9.5.3 Robustification.- 9.5.4 Recursive residuals.- 10 Software.- 10.1 Programs and templates.- 10.2 Data files.- References.
653 citations
TL;DR: A unified theory of sequential changepoint detection is introduced which leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on-line implementation and yet are nearly optimal under several performance criteria.
Abstract: After a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing, we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection. This theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on-line implementation and yet are nearly optimal under several performance criteria.
563 citations
TL;DR: In this article, the authors unify and extend previously published characterizations of moving average, geometric moving average and cumulative sum control chart procedures, and present comparable characterisations of two procedures based on tests described but not evaluated in earlier papers.
Abstract: This paper unifies and extends previously published characterizations of moving average, geometric moving average, and cumulative sum control chart procedures. It presents comparable characterizations of two procedures based on tests described but not evaluated in earlier papers. One of these procedures is based on a test devised by Girshick and Rubin that is optimal under a particular set of idealized conditions. The other procedure is based on run sum tests, which are generalizations of the type of run test that counts the number of consecutive points that exceed a limit, the generalization taking into account the extent that points in such a run exceed the limit.
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TL;DR: In this article, the authors consider the problem of controlling the quality of the output of a continuous manufacturing process where quality will normally be maintained at an acceptable level for long periods of time, but where a change can take place which will result in the process manufacturing material to a quality level which is unacceptable.
Abstract: We consider here the problem of controlling the quality of the output of a continuous manufacturing process where quality will normally be maintained at an acceptable level for long periods of time, but where a change can take place which will result in the process manufacturing material to a quality level which is unacceptable. Our first attempts in this field were to take standard sequential and multiple sampling schemes derived for batch inspection problems and to use theae schemes retrospectively, the first sample being the current sample the second sample being that taken immediately previous to the current sample and so oni. An example of such a scheme is quoted by Barnard (1954). Page (1954) introduced the concept of cumulative sum charts based on Wald sequential schemes and since this original paper we have adopted this principle to give a wide range of inispection schemes with known properties. Barnard (1959) has recently drawn attention again to the cumulative sum chart and has prepared certain empirical methods for deciding when changes have occurred and for estimating the magnitude of these. We describe, in this paper, the methods which we have developed for our own use and which we have now been applying in practice since 1955. The method is entirely systematic, the parameters of a scheme with required properties can be determined easily, and in our experience the technique is very simple to apply either in graphical or tabular form.
221 citations