Analytical and numerical methods for detection of a change in distributions

University of Technology, Sydney. Faculty of Science.This thesis aims to derive analytical approximations and numerical algorithms for analysis and design of control charts used for detection of changes in distributions. In particular, we present a new analytical approach for evaluating of characteristics of the "Exponentially Weighted Moving Average (EWMA)" procedure in the case of Gaussian and non-Gaussian distributions with light-tails. The main characteristics of a control chart are the mean of false alarm time or, Average Run Length (ARL), and the mean of delay for true alarm time or, Average Delay time (AD). ARL should be sufficiently large when the process is in-control and AD should be small when the process is out-of-control. Traditional methods for numerical evaluation ARL and AD are the Monte Carlo simulations (MC), Markov Chain Approach (MCA) and Integral Equations method (IE). These methods have the following essential drawbacks: the crude MC is very time consuming and difficult to use for finding optimal designs; MCA requires matrix inversions and, in general, is slowly convergent; IE requires intensive programming even for the case of Gaussian observations. In this thesis we develop an approach based on a combination of the martingale technique and Monte Carlo simulations. With the use of a popular symbolic/numerical software Mathematica®, this new approach allows to obtain accurate procedures for finding the optimal weights, alarm boundaries and approximations ARL and AD for the case of Gaussian observations. Further, we show that our approach can be used also for non- Gaussian distributions with light-tails and, in particular, for Poisson and Bernoulli distributions

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

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

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

N

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

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