An integral equation approach for analysis of control charts

University of Technology, Sydney. Faculty of Science.This thesis is concerned with the use of Statistical Process Control (SPC) charts for detection of change-point in distributions in quality control and surveillance problems. We derive explicit analytical formulas and develop numerical algorithms for evaluating important characteristics of "Exponentially Weighted Moving Average" (EWMA) control charts for a range of distributions. The most popular characteristics of a control chart are Average Run Length (ARL) - the mean of observations/times that are taken before a system is signalled to be out-of-control when it is actually still in-control, and Average Delay (AD) time - the mean of delay of true alarm times before a system that is actually out-of-control is signalled to be out-of- control. An important property required of ARL is that it should be sufficiently large when the process is in-control to reduce a number of false alarms. On the other side, if the process is actually out-of-control then its AD should be as small as possible. Traditional methods that are used for evaluating chart characteristics include Markov Chain Approach (MCA), Integral Equation (IE) and Monte Carlo simulation (MC) methods. Some crucial features of the methods are as follows: the MCA requires many matrix inversions and there is no theoretical proof of convergence of the method; the IE is most advanced method and it was used before only for Gaussian distribution; the MC is very time consuming. In this thesis, we find explicit formulas for ARL and AD of EWMA in the case when observations are exponentially distributed. These explicit formulas can be applied to some other distributions, e.g. the Pareto distribution. The numerical results obtained from our explicit formulas are compared with results obtained from the Monte Carlo simulation (MC) and Markov Chain Approach (MCA). We also compare the performance of the EWMA procedure with charts obtained with the CUSUM and Shiryayev-Roberts procedures. The technique that we use to derive the formulas for an exponential distribution cannot be used to derive formulas for gamma and Wei bull distributions. However, we have developed a different method for evaluating the ARL and AD for the case of gamma and Weibull distributions . This method is based on a numerical solution of Integral Equations based on Gauss-Legendre integration rules to approximate the integrals. Numerical results for these distributions are compared with results from other approaches

On EWMA procedure for AR(1) observations with exponential white noise

In this paper, we use Fredholm second kind integral equations method to solve the corresponding Average Run Length (ARL), when the observations of a random process are serially-correlated. We derive explicit expressions for the ARL of an EWMA control chart, or its corresponding AR(1) process, when the observations follow an exponential distribution white noise. The analytical expressions derived, are easy to implement in any computer packages, and as a consequence, it reduces considerably the computational time comparable with the traditional numerical methods used to solve integral equations. © 2012 Academic Publications, Ltd

Fitting pareto distribution with hyperexponential to evaluate the arl for cusum chart

Explicit formulas for the Average Run Length (ARL) of Cumulative Sum (CUSUM) chart are very complicated in regarding the analytical derivation when observations are Long-tailed distributions. The objective of this paper is to fitting Pareto distribution with the hyperexponential distribution to evaluate ARL of CUSUM procedure. The numerical results obtained from analytical solution for the ARL and from numerical approximations are derived and we compared the result with integral equations approach. © 2012 Academic Publications, Ltd

Explicit analytical solutions for the average run length of CUSUM and EWMA charts

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Analytical Method of Average Run Length for Trend Exponential AR(1) Processes in EWMA Procedure

The Exponentially Weighted Moving Average (EWMA) procedure are used for monitoring and detecting small shifts in the process mean which performs quicker than the Shewhart control chart. Usually, the common assumption of the Statistical Process Control (SPC) is the observations are independent and identically distributed (IID). In practice, however, the observed data are from industry and finance is serially correlated with trend. In this paper, we extend to use CUSUM procedure to compare with EWMA procedure. The performance of latter is superior to the former when the magnitudes of shift are small to moderate. It is shown that EWMA procedure performs better than the CUSUM procedure for the case of trend exponential AR(1) processes

Analysis of Average Run Length for CUSUM Procedure with Negative Exponential Data

The Average Run Length (ARL) is a performance measure that is frequently used in control charts. Cumulative Sum (CUSUM) is a popular procedure in quality control as it is a sensitive detector of small shifts in values of distribution parameters. In this paper, we use an integral equation approach to derive explicit formulas for the ARL (the first passage times) for CUSUM when observations are negative exponential distributed. Simulations are carried out to compare the performance of the explicit formulas with that of numerical approximations. The computational time for the explicit formulas is found to be approximately 10 seconds, which is much less than the computational time required for numerical approximations