Manuel Cabral Morais

On the performance of combined EWMA schemes for μ and σ : A Markovian approach

Changes in the process mean (μ) or in the process standard deviation (σ) ought to be regarded as an indication that a production process is out of control. This paper considers the problem of the joint monitoring of these two parameters — when the quality characteristic follows a normal distribution —, using a combined Exponentially Weighted Moving Average (CEWMA) scheme. Three performance measures of this joint control scheme are investigated under shifts in the process mean or inflations of the process standard deviation, and under the adoption of head starts: the average run length, the run length percentage points and the probability of a misleading signal. Approximations to these three performance indicators will be obtained considering a two-dimensional Markov chain. The independence between the horizontal and vertical transitions of this approximating two-dimensional Markov chain plays an important role in providing simple expressions to those performance measures which avoid the computation of a probability transition matrix with unusual dimensions. A numerical comparison between these three performance measures and the corresponding ones of the matched combined Shewhart (CShewhart) scheme will be also presented, leading to the conclusion that the substituition of this combined scheme by the CEWMA scheme can improve the joint monitoring of the process mean and standard deviation.

Misleading Signals in Simultaneous Residual Schemes for the Mean and Variance of a Stationary Process

The performance assessment of simultaneous surveillance schemes for the process mean (μ) and variance (σ2) requires a special performance measure, in addition to the average run length. It refers to two events which can be likely to happen when such schemes are at use: the individual chart for μ triggers a signal before the one for σ2, even though the process mean is on-target and the variance is off-target; the constituent chart for σ2 triggers a signal before the one for μ, although the variance is in-control and the process mean is out-of-control. These are called misleading signals since they correspond to a misinterpretation of a mean (variance) change as a shift in the process variance (mean) and can lead the quality control operator or engineer to a misdiagnosis of assignable causes and to deploy incorrect actions to bring the process back to target. This article discusses the impact of autocorrelation on the probability of misleading signals of simultaneous Shewhart and EWMA residual schemes for the mean and variance of a stationary process.

Two stochastic properties of one‐sided exponentially weighted moving average control charts∗

This paper establishes two stochastic monotonicity results concerning the run length of an upper one‐sided Exponentially Weighted Moving Average (EWMA) control chart, based on the logarithm of the sample variance, for monitoring a process standard deviation. These properties cast interesting light on the control chart performance, and their extension to other one‐sided EWMA and CUSUM control charts is straightforward.

Misleading Signals in Joint Schemes for μ and σ

The joint monitoring of the process mean and variance can be achieved by running what is termed a joint scheme. The process is deemed out-of-control whenever a signal is observed on either individual chart of a joint scheme. Thus, the two following events are likely to happen: a signal is triggered by the chart for the mean although it is on-target and the standard deviation is off-target; the mean is out-of-control and the variance is in-control, however, a signal is given by the chart for the standard deviation.

SOME STOCHASTIC PROPERTIES OF UPPER ONE-SIDED AND EWMA CHARTS FOR μ IN THE PRESENCE OF SHIFTS IN σ

In this paper we shall discuss certain stochastic properties of the run length of upper one-sided andEWMA control charts for the process mean when the quality characteristic is normally distributed. We look at the performance of these two charts in the presence of shifts in the process mean and the standard deviation, and at their ability to detect these latter shifts and to give misleading signals.

On ARL-Unbiased Control Charts

Manufacturing processes are usually monitored by making use of control charts for variables or attributes. Controlling both increases and decreases in a parameter, by using a control statistic with an asymmetrical distribution, frequently leads to an ARL-biased chart, in the sense that some out-of-control average run length (ARL) values are larger than the in-control ARL, i.e., it takes longer to detect some shifts in the parameter than to trigger a false alarm. In this paper, we are going to: explore what Pignatiello et al. (4th Industrial Engineering Research Conference, 1995) and Acosta-Mejia et al. (J Qual Technol 32:89–102, 2000) aptly called an ARL-unbiased chart; provide instructive illustrations of ARL-(un)biased charts of the Shewhart-, exponentially weighted moving average (EWMA)-, and cumulative sum (CUSUM)-type; relate ARL-unbiased Shewhart charts with the notions of unbiased and uniformly most powerful unbiased (UMPU) tests; briefly discuss the design of EWMA charts not based on ARL(-unbiasedness).

Assessing the Impact of Autocorrelation in Misleading Signals in Simultaneous Residual Schemes for the Process Mean and Variance: A Stochastic Ordering Approach

Misleading signals (MS) correspond to the misinterpretation of a shift in the process mean (variance) as a shift in the process variance (mean). MS occur when: The individual chart for the mean triggers a signal before the one for the variance, even though the process mean is on-target and the variance is off-target; The individual chart for the variance triggers a signal before the one for the mean, although the variance is in-control and the process mean is out-of-control.

An ARL-unbiased c-chart

In statistical process control (SPC), it is usual to assume that counts have a Poisson distribution. The non-negative, discrete, and asymmetrical character of a control statistic with such a distribution and the value of its target mean may prevent the quality control practitioner to deal with a c-chart with a pre-specified in-control average run length (ARL) or the ability to control not only increases but also decreases in the mean of those counts in a timely fashion. Furthermore, the c-charts proposed in the SPC literature tend to be ARL-biased, in the sense that some out-of-control ARL values are larger than the in-control ARL. In this paper, we explore the notions of randomized and uniformly most powerful unbiased tests to eliminate the bias of the ARL function of the c-chart. Copyright

Misleading Signals in Simultaneous Schemes for the Mean Vector and Covariance Matrix of a Bivariate Process

In a bivariate setting, misleading signals (MS) correspond to valid alarms which lead to the misinterpretation of a shift in the mean vector (resp. covariance matrix) as a shift in the covariance matrix (resp. mean vector). While dealing with bivariate output and two univariate control statistics (one for each parameter), MS occur when: The individual chart for the mean vector triggers a signal before the one for the covariance matrix, although the mean vector is on-target and the covariance matrix is off-target. The individual chart for the variance triggers a signal before the one for the mean, despite the fact that the covariance matrix is in-control and the mean vector is out-of-control.

EWMA Charts for Multivariate Output: Some Stochastic Ordering Results

We investigate the impact of sustained shifts in the covariance matrix on the run length (RL) of EWMA charts for detecting shifts in the process mean vector of multivariate normal i.i.d. output. We prove that some changes in the covariance matrix can cause an undesirable stochastic decrease in the detection speed of specific shifts in the process mean vector, and this should not be tolerated by practitioners. The illustration of this and other RL-related stochastic ordering results is based on extensive Monte Carlo simulations. This article can be thought as a multivariate extension of the results of Morais and Pacheco (2001) concerning the stochastic behavior of the RL of upper one-sided EWMA schemes for the process mean of i.i.d. output.