One‐sided runs‐rules schemes to monitor autocorrelated time series data using a first‐order autoregressive model with skip sampling strategies

Abstract

In some industrial or health-related processes, it makes more practical sense to monitor either an increase only or a decrease only in the quality characteristic of interest. Consequently, in this paper, we propose four one-sided Shewhart X charts supplemented with runs-rules to monitor the mean of autocorrelated normally distributed samples using a stationary first-order autoregressive model. To counteract the negative effect of autocorrelation, we implement a sampling strategy which involves sampling of non-neighboring observations to form rational subgroups. The Markov chain technique is used to derive zero-state and steady-state closed-form expressions of the specific shift performance metric, ie, average run-length (ARL). Moreover, we compute the expected ARL metric which evaluates each monitoring scheme based on all specified range of possible values of the shift parameter, or more specifically, from a global point of view and thus gives a different perspective from the specific shift ARL metric. We observed that the steady-state improved w-of-w and the improved 2-of-(H + 1) schemes yield a better overall performance than their corresponding basic counterparts for all different levels of autocorrelation. A real-life example is provided to illustrate the implementation of the monitoring schemes proposed here.

On monitoring the process mean of autocorrelated observations with measurement errors using the w‐of‐w runs‐rules scheme

“…It is worth mentioning that Equations (16) to (19), or similar versions, have been done in earlier and recent publications for the i.i.d. and autocorrelated observations scenarios using one-and two-sided schemes; see Page, 38 Champ, 39 Acosta-Mejia, 40 Lim and Cho, 42 Shongwe et al, 26,44 and Shongwe and Malela-Majika. 45 A number of authors have shown that if a monitoring scheme is designed based on one specific shift size δ, it would perform poorly when it is considerably different from the assumed one; see Reynolds and Lou 50 and Ryu et al 51 This makes the ARL inefficient in assessing the overall performance of a monitoring scheme.…”

“…Some of these models are known as autoregressive (AR), moving average (MA), autoregressive moving average (ARMA), autoregressive integrated moving average (ARIMA), etc; see Box et al for more thorough discussion on these. In this paper, we only consider the well‐known first‐order AR model (ie, AR(1)) as a starting point (other models will be discussed in upcoming articles) and because according to Alwan and Radson, this is the most commonly used time series model in SPM applications; see also Wardell et al, Runger and Willemain, Claro et al, Kazemzadeh et al, Costa and Machado, Chang and Wu, Keramatpour et al, Franco et al, Hu and Sun, Osei‐Aning et al, Garza‐Venegas et al, Shongwe et al, Ahmad et al, etc, for additional indication that AR(1) is the most used model in SPM due to its simplicity as compared with other stationary time series processes.…”

“…observations are done in Acosta‐Mejia, Antzoulakos and Rakitzis, Lim and Cho, Khilare and Shirke, Shongwe et al, etc. For autocorrelated observations, w‐of‐w runs‐rules schemes to monitor the process mean using one‐ and two‐sided schemes are discussed in Shongwe et al and Shongwe and Malela‐Majika, respectively. However, none of these w‐of‐w runs‐rules studies have accounted for (a) effect of measurement error only and (b) combined effect of measurement errors and autocorrelation.…”

On autoregressive model selection for the exponentially weighted moving average control chart of residuals in monitoring the mean of autocorrelated processes

“…The first one is to sample from the observation data stream less frequently. [5][6][7][8] It seems to be an easy solution; however, it has the disadvantage of making a considerable amount of available data wasted. For example, if we sample every tenth observation, 90% of the information is discarded.…”

Exponentially weighted moving average chart with a likelihood ratio test for monitoring autocorrelated processes

“…Polunchenko and Raghavan provided a comparative study of the cumulative sum (CUSUM) chart and the Shiryaev‐Roberts procedure for the nonasymptotic setting with correlated observations. Shongwe et al presented four one‐sided Shewhart‐ charts with runs‐rules to monitoring the mean of autocorrelated normally distributed samples under an AR(1) model. Shongwe and Malela‐Majika proposed two Shewhart‐ charts with w‐of‐w runs‐rules based on the skipping sampling strategy for an AR(1) model.…”