Abstract
Manufacturing for a multitude of continuous processing applications in the era of automation and ‘Industry 4.0’ is focused on rapid throughput while producing products of acceptable quality that meet customer specifications. Monitoring the stability or statistical control of key process parameters using data acquired from online sensors is fundamental to successful automation in manufacturing applications. This study addresses the significant problem of positive autocorrelation in data collected from online sensors, which may impair assessment of statistical control. Sensor data collected at short time intervals typically have significant autocorrelation, and traditional statistical process control (SPC) techniques cannot be deployed. There is a plethora of literature on techniques for SPC in the presence of positive autocorrelation. This paper contributes to this area of study by investigating the performance of ‘Copula’ based control charts by assessing the average run length (ARL) when the subsequent observations are correlated and follow the AR(1) model. The conditional distribution of yt given yt−1 is used in deriving the control chart limits for three different categories of Copulas: Gaussian, Clayton, and Farlie-Gumbel-Morgenstern Copulas. Preliminary results suggest that the overall performance of the Clayton Copula and Farlie-Gumbel-Morgenstern Copula is better compared to other Archimedean Copulas. The Clayton Copula is the more robust with respect to changes in the process standard deviation as the correlation coefficient increases.
Highlights
The origin of the time series goes back to the 1930s, in the context of the ARMA (Auto Regressive Moving Average) models that were developed by Herman Wold [1] for the stationary time series
In addition to forecasting, these time series models can be used in quality control, especially in the construction of the popular EWMA (Exponentially Weighted Moving Average) control charts, under the assumption that the averages follow the AR(1) model [5,6,7,8,9,10,11]
We perform a simulation study in order to compare the performance of the Gaussian Copula against the Clayton Copula and the FGM Copula on the basis of the average run length (ARL0), under the assumption that the null hypothesis H0 is true
Summary
The origin of the time series goes back to the 1930s, in the context of the ARMA (Auto Regressive Moving Average) models that were developed by Herman Wold [1] for the stationary time series. Smoothing techniques were used in time series [3,4] These models use numerically iterative procedures to estimate the parameters involved in the time series models. In addition to forecasting, these time series models can be used in quality control, especially in the construction of the popular EWMA (Exponentially Weighted Moving Average) control charts, under the assumption that the averages follow the AR(1) (or First Order Auto Regressive) model [5,6,7,8,9,10,11]. We use Copula models to construct control charts for a stationary process under the assumption that the process follows the AR(1) series. Our interest is in finding a suitable Copula model for approximating the joint distribution among these correlated time series observations
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