Multivariate CUSUM and EWMA Control Charts for Skewed Populations Using Weighted Standard Deviations

Abstract

This article proposes a heuristic method of constructing multivariate cumulative sum and exponentially weighted moving average control charts for skewed populations based on the weighted standard deviation method which adjusts the variancecovariance matrix of quality characteristics and approximates the probability density function using several multivariate normal distributions. These control charts, however, reduce to the conventional control charts when the underlying distribution is symmetric. In-control and out-of-control average run lengths of the proposed control charts are compared with those of the conventional control charts for multivariate lognormal and Weibull distributions. Simulation results show that considerable improvements over the standard method can be achieved when the underlying distribution is skewed.

A synthetic multivariate exponentially weighted moving average scheme for monitoring of bivariate Gamma distributed processes

“…This leads the MCUSUM scheme more sensitive to the small-to-moderate shifts. It is worth noting that there were some research works about the MCUSUM type scheme for monitoring skewed distributed data, for instance, Chang 23 and Xie et al 3 Hence, a MCUSUM type scheme is employed here for monitoring the individual BGD observations. According to Crosier, 7 the charting statistic 𝐒 𝑡 of the MCUSUM scheme can be defined as…”

“…Apart from the nonparametric methods, various data transformation techniques have also been studied in literature to construct multivariate parametric control charts for skewed distributions. See, for example, Xie et al, 1,3 Chang 23 , Khan et al, 24 and Marchant et al 25 The data transformation techniques need special attention as such transformations may lead to some loss of useful information. See, Xie et al, 3 for more details.…”

A study on the effects of a skewed distribution on the EWMA and MA charts

“…One common case is that the process distribution is skewed [3][4]. For example, the distributions of measurements from filling processes, chemical processes, semiconductor processes, cutting tool wear processes are usually skewed [5][6]. When the underlying process distribution is skewed, the performance of a control chart is substantially affected as it will produce higher incidence of false alarms or Type-I error risk.…”

A multivariate CUSUM control chart for monitoring Gumbel's bivariate exponential data

“…[13][14][15][16] Although attractive, these nonparametric control charts are difficult to apply in practice due to the expensive computations required for their implementation. On the other hand, with the help of some data transformations, a multivariate nonnormal distribution can be approximately transformed into a multivariate normal distribution or into the combination of different normal distributions, for example, using the double square-root method (see Kittlitz 17 ) or the weighted standard deviation method (see Chang 18,19 ). Because data transformation may lead to some loss of useful information, it should be carefully considered as an appropriate method for process monitoring.…”

“…It is known that there are many similarities between the MEWMA and the MCUSUM charts (see the literature 19,27 ). Both of them use the information from historical observations, which makes these two charts more sensitive to detect small shifts in the process.…”