Nonparametric control chart based on change-point model
01 Jan 2009-Vol. 50, Iss: 1, pp 13-28
TL;DR: In this paper, a change-point control chart for detecting shifts in the mean of a process is developed for the case where the nominal value of the mean is unknown but some historical samples are available.
Abstract: A change-point control chart for detecting shifts in the mean of a process is developed for the case where the nominal value of the mean is unknown but some historical samples are available. This control chart is a nonparametric chart based on the Mann–Whitney statistic for a change in mean and adapted for repeated sequential use. We do not require any knowledge of the underlying distribution such as the normal assumption. Particularly, this distribution robustness could be a significant advantage in start-up or short-run situations where we usually do not have knowledge of the underlying distribution. The simulated results show that our approach has a good performance across the range of possible shifts and it can be used during the start-up stages of the process.
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TL;DR: The R package cpm is described, which provides a fast implementation of all the above change point models in both batch (Phase I) and sequential (Phase II) settings, where the sequences may contain either a single or multiple change points.
Abstract: The change point model framework introduced in Hawkins, Qiu, and Kang (2003) and Hawkins and Zamba (2005a) provides an effective and computationally efficient method for detecting multiple mean or variance change points in sequences of Gaussian random variables, when no prior information is available regarding the parameters of the distribution in the various segments. It has since been extended in various ways by Hawkins and Deng (2010), Ross, Tasoulis, and Adams (2011), Ross and Adams (2012) to allow for fully nonparametric change detection in non-Gaussian sequences, when no knowledge is available regarding even the distributional form of the sequence. Another extension comes from Ross and Adams (2011) and Ross (2014) which allows change detection in streams of Bernoulli and Exponential random variables respectively, again when the values of the parameters are unknown.
This paper describes the R package cpm, which provides a fast implementation of all the above change point models in both batch (Phase I) and sequential (Phase II) settings, where the sequences may contain either a single or multiple change points.
199 citations
TL;DR: This article reviews and synthesizes many of the important developments that pertain to the analysis of process data in Phase I and identifies the current best practices and some opportunities for future research.
Abstract: An overview and perspective is provided on the Phase I collection and analysis of data for use in process improvement and control charting.
196 citations
Cites background from "Nonparametric control chart based o..."
...Additionally, although some work has been done in developing nonparametric change-point methods (see, e.g., Zhou et al. (2009), Hawkins and Deng (2010), and Zou at al....
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TL;DR: This work considers the general problem of detecting a change in the location and/or scale parameter of a stream of random variables, and adapt several nonparametric hypothesis tests to create a streaming change detection algorithm which uses a test statistic with a null distribution independent of the data.
Abstract: The analysis of data streams requires methods which can cope with a very high volume of data points. Under the requirement that algorithms must have constant computational complexity and a fixed amount of memory, we develop a framework for detecting changes in data streams when the distributional form of the stream variables is unknown. We consider the general problem of detecting a change in the location and/or scale parameter of a stream of random variables, and adapt several nonparametric hypothesis tests to create a streaming change detection algorithm. This algorithm uses a test statistic with a null distribution independent of the data. This allows a desired rate of false alarms to be maintained for any stream even when its distribution is unknown. Our method is based on hypothesis tests which involve ranking data points, and we propose a method for calculating these ranks online in a manner which respects the constraints of data stream analysis.
177 citations
Cites methods from "Nonparametric control chart based o..."
...A sequential extension of the work of Pettitt (1979) was recently proposed by both Zhou et al. (2009) and Hawkins and Deng (2010), who used Monte Carlo simulation instead of an asymptotic argument in order to calculate exact quantiles of the test statistic....
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TL;DR: A new multivariate SPC methodology for monitoring location parameters is developed based on adapting a powerful multivariate sign test to online sequential monitoring, which results in a nonparametric counterpart of the classical multivariate EWMA (MEWMA).
Abstract: Nonparametric control charts are useful in statistical process control (SPC) when there is a lack of or limited knowledge about the underlying process distribution, especially when the process measurement is multivariate. This article develops a new multivariate SPC methodology for monitoring location parameters. It is based on adapting a powerful multivariate sign test to online sequential monitoring. The weighted version of the sign test is used to formulate the charting statistic by incorporating the exponentially weighted moving average control (EWMA) scheme, which results in a nonparametric counterpart of the classical multivariate EWMA (MEWMA). It is affine-invariant and has a strictly distribution-free property over a broad class of population models. That is, the in-control (IC) run length distribution can attain (or is always very close to) the nominal one when using the same control limit designed for a multivariate normal distribution. Moreover, when the process distribution comes from the elli...
153 citations
Cites background from "Nonparametric control chart based o..."
...The analogous phenomenon for univariate nonparametric charts has been mentioned in the literature, for example, by Hackel and Ledolter (1991) and Zhou et al. (2009)....
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TL;DR: This work presents two distribution-free charts that can detect arbitrary changes to the process distribution during Phase II monitoring, formed by integrating the omnibus Kolmogorov–Smirnov and Cramer—von-Mises tests into the widely researched change-point model framework.
Abstract: The authors present two distribution-free charts that can detect arbitrary changes to the process distribution during Phase II monitoring.
152 citations
Cites background or methods from "Nonparametric control chart based o..."
...This is something that did not receive adequate attention in Zou and Tsung (2010), where only increases were considered....
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...…extending this methodology for use in Phase II, and CPMs have been proposed for parametric monitoring of a Gaussian mean (Hawkins et al. (2003)), Gaussian standard deviation (Hawkins and Zamba (2005)), and distribution-free monitoring of the location (Hawkins and Deng (2010); Zhou et al. (2009))....
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References
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TL;DR: In this paper, the authors show that the limit distribution is normal if n, n$ go to infinity in any arbitrary manner, where n = m = 8 and n = n = 8.
Abstract: Let $x$ and $y$ be two random variables with continuous cumulative distribution functions $f$ and $g$. A statistic $U$ depending on the relative ranks of the $x$'s and $y$'s is proposed for testing the hypothesis $f = g$. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis $f = g$ the probability of obtaining a given $U$ in a sample of $n x's$ and $m y's$ is the solution of a certain recurrence relation involving $n$ and $m$. Using this recurrence relation tables have been computed giving the probability of $U$ for samples up to $n = m = 8$. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if $m, n$ go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives $f(x) > g(x)$ for every $x$.
11,055 citations
Book•
01 Jan 1985
TL;DR: In this article, the authors present a survey of statistical process control and capability analysis techniques for improving the quality of a business process in the modern business environment, using a variety of techniques.
Abstract: Quality Improvement in the Modern Business Environment.STAISTICAL METHODS USEFUL IN QUALITY IMPROVEMENT.Modeling Process Quality.Inferences About Process Quality.BASIC METHODS OF STATISTICAL PROCESS CONTROL AND CAPABILITY ANALYSIS.Methods and Philosophy of Statistical Process Control.Control Charts for Variables.Control Charts for Attributes.Process and Measurement Systems System Capability Analysis.OTHER STATISTICAL PROCESS MONITORING AND CONTROL TECHNIQUES.Cumulative Sum and Exponentially Weighted Moving Average Control Charts.Other Univariate SPC Techniques.Multivariate Process Monitoring and Control.Engineering Process Control and SPC.PROCESS DESIGN AND IMPROVEMENT WITH DESIGNED EXPERIMENTS.Factorial and Fractional Factorial Designs for Process Design and Improvement.Process Optimization with Designed Experiments.ACCEPTANCE SAMPLING.Lot--by--Lot Acceptance Sampling for Attributes.Other Acceptance Sampling Techniques.Appendix.Bibliography.Answers to Selected Exercises.Index.
7,312 citations
Book•
01 Dec 1971
TL;DR: Theoretical Bases for Calculating the ARE Examples of the Calculations of Efficacy and ARE Analysis of Count Data.
Abstract: Introduction and Fundamentals Introduction Fundamental Statistical Concepts Order Statistics, Quantiles, and Coverages Introduction Quantile Function Empirical Distribution Function Statistical Properties of Order Statistics Probability-Integral Transformation Joint Distribution of Order Statistics Distributions of the Median and Range Exact Moments of Order Statistics Large-Sample Approximations to the Moments of Order Statistics Asymptotic Distribution of Order Statistics Tolerance Limits for Distributions and Coverages Tests of Randomness Introduction Tests Based on the Total Number of Runs Tests Based on the Length of the Longest Run Runs Up and Down A Test Based on Ranks Tests of Goodness of Fit Introduction The Chi-Square Goodness-of-Fit Test The Kolmogorov-Smirnov One-Sample Statistic Applications of the Kolmogorov-Smirnov One-Sample Statistics Lilliefors's Test for Normality Lilliefors's Test for the Exponential Distribution Anderson-Darling Test Visual Analysis of Goodness of Fit One-Sample and Paired-Sample Procedures Introduction Confidence Interval for a Population Quantile Hypothesis Testing for a Population Quantile The Sign Test and Confidence Interval for the Median Rank-Order Statistics Treatment of Ties in Rank Tests The Wilcoxon Signed-Rank Test and Confidence Interval The General Two-Sample Problem Introduction The Wald-Wolfowitz Runs Test The Kolmogorov-Smirnov Two-Sample Test The Median Test The Control Median Test The Mann-Whitney U Test and Confidence Interval Linear Rank Statistics and the General Two-Sample Problem Introduction Definition of Linear Rank Statistics Distribution Properties of Linear Rank Statistics Usefulness in Inference Linear Rank Tests for the Location Problem Introduction The Wilcoxon Rank-Sum Test and Confidence Interval Other Location Tests Linear Rank Tests for the Scale Problem Introduction The Mood Test The Freund-Ansari-Bradley-David-Barton Tests The Siegel-Tukey Test The Klotz Normal-Scores Test The Percentile Modified Rank Tests for Scale The Sukhatme Test Confidence-Interval Procedures Other Tests for the Scale Problem Applications Tests of the Equality of k Independent Samples Introduction Extension of the Median Test Extension of the Control Median Test The Kruskal-Wallis One-Way ANOVA Test and Multiple Comparisons Other Rank-Test Statistics Tests against Ordered Alternatives Comparisons with a Control Measures of Association for Bivariate Samples Introduction: Definition of Measures of Association in a Bivariate Population Kendall's Tau Coefficient Spearman's Coefficient of Rank Correlation The Relations between R and T E(R), tau, and rho Another Measure of Association Applications Measures of Association in Multiple Classifications Introduction Friedman's Two-Way Analysis of Variance by Ranks in a k x n Table and Multiple Comparisons Page's Test for Ordered Alternatives The Coefficient of Concordance for k Sets of Rankings of n Objects The Coefficient of Concordance for k Sets of Incomplete Rankings Kendall's Tau Coefficient for Partial Correlation Asymptotic Relative Efficiency Introduction Theoretical Bases for Calculating the ARE Examples of the Calculations of Efficacy and ARE Analysis of Count Data Introduction Contingency Tables Some Special Results for k x 2 Contingency Tables Fisher's Exact Test McNemar's Test Analysis of Multinomial Data Summary Appendix of Tables Answers to Problems References Index A Summary and Problems appear at the end of each chapter.
2,988 citations
TL;DR: In this paper, nonparametric techniques are introduced for the change point problem and exact and approximate results are obtained for testing the null hypothesis of no change for zero-one observations, Binomial observations, and continuous observations.
Abstract: Non‐parametric techniques are introduced for the change‐point problem. Exact and approximate results are obtained for testing the null hypothesis of no change. The methods are illustrated by the analysis of three sets of data illustrating the techniques for zero–one observations, Binomial observations and continuous observations. Some comparisons are made with methods based on cusums.
2,671 citations
Journal Article•
TL;DR: The recognition that an EWMA control scheme can be represented as a Markov chain allows its properties to be evaluated more easily and completely than has previously been done.
1,624 citations