Statistics for Environmental Engineers

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The Cuscore Statistic


Consider a statistical model in which the yt are observations, в is some unknown parameter, and the xt’s are known independent variables. This can be written in the form:


yt = f(*t, в) + at t = 1,2,…,n


Assume that when в is the true value of the unknown parameter, the resulting at’s are a sequence of


2 2


independently, identically, normally distributed random variables with mean zero and variance aa = a . The series of at’s is called a white noise sequence. The model is a way of reducing data to white noise:


at = yt f (xt, в)


The Cusum chart is based on the simplest possible model, yt = n+at. As long as the process is in control (varying randomly about the mean), subtracting the mean reduces the series of yt to a series of white noise. The cumulative sum of the white noise series is the Cusum statistic and this is plotted on the Cusum chart. In a more general way, the Cusum is a Cuscore that relates how the residuals change with respect to changes in the mean (the parameter n).


Box and Ramirez (1992) defined the Cuscore associated with the parameter value в = в0 as:


Q    at о dto


where dt0 = -_t    is a discrepancy vector related to how the residuals change with respect to changes


дв в=в0


in в. The subscripts (0) indicate the reference condition for the in-control process. When the process is in control, the parameter has the value в0 and the process model generates residuals at. If the process shifts out of control, в ф в0 and the residuals are inflated as described by1:

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