Analytical and numerical solutions of average run length integral equations for an EWMA control chart over a long memory SARFIMA process

The efficiency of a process, especially a long memory seasonal autoregressive fractional integral moving average (SARFIMA) process, has commonly been measured through the quality control chart. In this paper, a generalized long memory SARFIMA process of the exponentially weighted moving average (EWMA) control chart is carried out and shown. Also, analytical and numerical average run length (ARL) were designed to measure the efficiency of the EWMA control. Existence and uniqueness by the fixed point theory are proven for the analytical ARL. Error and convergence of numerical integration equations are also given for the numerical ARL. The findings indicated that the analytical ARL was evaluated more quickly and accurately than the numerical ARL. As a result, the analytical ARL is an alternative for measuring the efficiency of an EWMA control chart over a long memory SARFIMA process

Solving the Linear Integral Equations Based on Radial Basis Function Interpolation

The radial basis function (RBF) method, especially the multiquadric (MQ) function, was introduced in solving linear integral equations. The procedure of MQ method includes that the unknown function was firstly expressed in linear combination forms of RBFs, then the integral equation was transformed into collocation matrix of RBFs, and finally, solving the matrix equation and an approximation solution was obtained. Because of the superior interpolation performance of MQ, the method can acquire higher precision with fewer nodes and low computations which takes obvious advantages over thin plate splines (TPS) method. In implementation, two types of integration schemes as the Gauss quadrature formula and regional split technique were put forward. Numerical results showed that the MQ solution can achieve accuracy of 1E-5. So, the MQ method is suitable and promising for integral equations