Precise Average Run Length of an Exponentially Weighted Moving Average Control Chart for Time Series Model

This study aims to develop a precise formula for calculating the average run length in the context of an exponentially weighted moving average (EWMA) control chart, specifically in the presence of a seasonal autoregressive with exogenous variable (SARX(P,r)(L)) model. The research also introduces a novel method for estimating the average run length using numerical integral equations, facilitating a comparison between the outcomes derived from the formula and those obtained through the numerical integral equation method. Additionally, control charts are applied to real-world data across diverse domains. The explicit formula is evaluated based on the absolute percentage difference and CPU time. The results show that the average run length calculated using the proposed method precisely corresponds to the findings from the numerical integral equation method. In addition, it's important to mention that the explicit formulas demonstrated a significant improvement in computational efficiency, requiring much fewer computations than the NIE approach.