WSEAS Transactions on Systems
Print ISSN: 1109-2777, E-ISSN: 2224-2678
Volume 24, 2025
A Novel Numerical Integration Equation Approach using the Gauss-Legendre Quadrature to Approximate the Average Run Length of Time-Series Model Running on a CUSUM Control Chart
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Abstract: Herein, we present the comprehensive validation of an approximation of the average run length (ARL) for monitoring small-to-moderate changes in the process means of long-memory fractionally integrated moving average with exogenous variables (LFIMAX) processes with different types of long memory on a cumulative sum (CUSUM) control chart based on the numerical integration equation (NIE) method. The ARL approximation using the NIE method was obtained by resolving a system of linear equations and performing integration through the partitioning and summing of the area under the curve of a function produced from the Gauss-Legendre quadrature rule. The performances of the proposed NIE and an established analytical method were compared for mean shifts of varying sizes for LFIMAX processes on a CUSUM control chart. The numerical results indicate that the proposed ARL method performed comparably with the analytical method regarding percentage accuracy Acc(%)). Moreover, a small percentage relative deviation (DEV%) is indicated, i.e., a change in magnitude of less than 0.25 can be detected rapidly in all situations. A numerical example using real-world scenario data is also provided to illustrate the practicability of the proposed method.
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Keywords: Approximated ARL, numerical integral equation, analytical method, long-memory process, LFIMAX(d, q, r) process, exponential white noise
Pages: 345-358
DOI: 10.37394/23202.2025.24.30