Effect of Autocorrelation on Spc Chart Performance

Statistical Process Control (SPC) aims at quality improvements through reduction of variances. The best known tool of SPC is the control chart. Over the years the control chart has proved to be a successful practical technique for monitoring process measurements. However, its usefulness in practice is limited to those situations where it can be assumed that successive measurements are independently distributed, whereas most data sets encountered in practice exhibit some form of serial correlation. The question that is considered in this paper is what control chart methods should be used to monitor serially correlated data and how the signals on such charts should be interpreted. 1. STOCHASTIC PROCESS Classical control charts assume no correlation between successive observations of the quality characteristic. In this section we define in a more precise manner what is meant by correlation between repeatedly observed measurements of a single quality characteristic. We need to know how to estimate the serial correlation, in case it exists, and how to study its effect on classical SPC charts. To achieve these goals, the concept of a stochastic process is first necessary. A stochastic process ( ) } { I t t Y ∈ , is a family of indexed random variables, where I is called the index set. Sometimes we will refer to the mechanism generating the stochastic process simply as the process, which can be understood in double sense of the underlying stochastic process that the quality characteristic being modeled follows, or as the production process itself, which in turn generates the stochastic process (i.e. the quality characteristic). In applications in this paper, t will relate to the discrete points of time at which an observation is obtained by sampling (i.e. the index set is } { ,... 2 , 1 , 0 , 1 , 2 ..., − − = I ) and the stochastic process is said to be a discrete-time stochastic process. If { } ∞ + < < ∞ − = t t I : , the process is a continuos-time stochastic process. For discrete-time stochastic processes, it is customary to denote them as } { I t t Y ∈ , that is, a subscript is used for discrete-time indices. Discrete-time processes can be identified by describing the behavior of its tth element. Thus, we can write, for example, t t Y ε μ + = Implying that for the given process (Shewhart's model in this case) the equation holds for all discrete points in time t. In this case, the time between observation h equals Δt.