Specific Signal Detection Charts Using Generalized Likelihood Ratios

Simulations are done to compare the performances of Generalized Likelihood Ratio (GLR) and Cuscore charts for detecting specific signals hidden in noise. Two different types of signals are considered: a sine wave and a linear trend. Two different cases are analyzed, a known signal form and parameter and a known signal form but unknown signal parameter. Both the initial performances and steady-state performances of these charts are given. Introduction Signals in the form of linear, exponential, or sinusoidal patterns are common in manufacturing processes. These patterns should be detected fast in order to reduce variations in the product quality features. Commonly used control charts are designed for detecting global forms of signals and their performance may be much worse than that of a specially designed chart. Specific signal detection charts have increased sensitivity to a single form of signal at the cost of less sensitivity to other types of signals that may be hidden in the noise. Therefore, these charts are mostly used as supplements to general statistical monitoring charts instead of replacing them. Cumulative score (Cuscore) charts were introduced to detect specific signals hidden in the noise (see, Box and Ramirez (1992), Box and o n ~ Luce (1997), and o n ~ Luce (1999)). The Cuscore statistic is based on the concept of Fisher’s efficient scores statistic. An alternative method for signal detection is to use a Generalized Likelihood Ratio (GLR) to develop a control statistic for detecting the presence of a known signal in the process data. See, for example, Lai (1995). In this paper, we compare GLR charts to Cuscore charts. Average run length simulation results are provided for the cases when the signals are a linear trend and a sine wave. Specific Signals Let Y be the monitored quality characteristic of interest with the sequential observations t t y a μ = + , ..., 1,2, , 0 = t where μ is the mean of Y , t a is the normally distributed white-noise sequence with mean zero and standard deviation σ , and t is the sequence order or time. Suppose that at unknown time , τ this statistically stable process is disturbed and that the form of this disturbing signal is known and can be represented by a function ) , , ( τ θ t f . Therefore, ( , , ) , 0,1,2, ..., t t y f t a t μ θ τ = + + = Cuscore Charts A Cuscore with handicap t k is obtained by plotting cumulative scores 1 max[{ ( ) ( , , )} ; 0] , 1,2,..., t t t t S S y k f t t θ τ − = + − = versus the sequence t , ( o n ~ Luce (1999)). A control limit, h is used to test for the presence of the hidden signal in the process, i.e. as soon as t S h ≥ an alarm is triggered for the presence of the hidden signal. o n ~ Luce (1999) suggested different Cuscore charts to be used depending on whether the signal is bounded or unbounded. For unbounded signals, a second reinitialization, which sets t = τ in ) , , ( τ θ t f when 0 t S ≤ is suggested. For bounded signals, the signal value ) , , ( τ θ t f is not reinitialized every time 0 ≤ t S . Generalized Likelihood Ratio Charts A GLR approach does not make the assumption of the known change time, but uses a maximum likelihood estimate (MLE) of the change time (see, Lai (1995)). For the normally distributed errors t a , the GLR statistic is Spring Research Conference Section on Quality & Productivity (Q&P)