Development of EWMA Control Chart for Detecting Changes in AR(p) with Quadratic Trend Model

Abstract

This study is intended to propose a formula for the Average Run Length (ARL) of the Exponentially Weighted Moving Average (EWMA) control chart when the observed data follow an autoregressive model of order p with quadratic trend. This research emphasizes the fundamental importance of developing precise ARL computation techniques with optimal processing efficiency, as ARL remains the predominant criterion for control chart performance evaluation. The derivation of the explicit ARL formula employs Fredholm’s integral equation methodology, with solution uniqueness assured through the application of Banach’s Fixed Point Theorem. Performance validation involves comparative analysis against approximate ARL values obtained via Numerical Integral Equation (NIE) approaches, specifically utilizing the Midpoint rule technique. The efficiency of the explicit formula of ARL is evaluated using two criteria: absolute percentage difference and CPU Time. The empirical results confirm that the ARL values derived from the explicit formula closely approximate those obtained via numerical integral equation methods, exhibiting an absolute percentage difference of less than 0.001%. Computationally, the proposed explicit formula achieves processing times of approximately 0.001 seconds, while the Midpoint rule method takes 2-3 seconds. In conclusion, the results demonstrate that the proposed explicit ARL formulas for EWMA charts provide accuracy comparable to the NIE method while significantly reducing computational time. This confirms the efficiency and practice applicability to the explicit formulas for monitoring real-world data, such as pneumonia cases at Siriraj Hospital.