By Douglas M. Hawkins, David H. Olwell
Cumulative sum (CUSUM) keep watch over charting is a necessary software for detecting and diagnosing power shifts in sequence of readings. it truly is utilized in conventional statistical method regulate (SPC) settings equivalent to production, yet is additionally powerful in settings as diversified as group of workers administration, econometrics, and standard information research. it truly is a vital software for the standard expert. This ebook covers CUSUMs from an application-oriented standpoint, whereas additionally delivering the basic theoretical underpinning. it's available to an individual with a uncomplicated statistical education, and is geared toward caliber practitioners, lecturers and scholars of caliber methodologies, and other people attracted to research of time-ordered info. The textual content is supported via a website containing CUSUM software program and information units. Douglas M. Hawkins is Chair of the dep. of utilized statistics, collage of Minnesota. he's a Fellow of the yankee Statistical organization, a Member of the overseas Statistical Institute and a Senior member of the yank Society for qc. His paintings on multivariate CUSUMs received him the Ellis R. Ott Award for the simplest paper on caliber released in 1993. He has been affiliate Editor of Technometrics and magazine of the yankee Statistical organization. David H. Olwell is affiliate Professor within the division of Mathematical Sciences on the usa army Academy. he's a member of the yankee Statistical organization, the yankee Society for qc, and the army Operations examine Society, the place his paintings on purposes of CUSUMs to coping with sexual harassment was once nominated for the 1998 Barchi prize. he's Editor of Mathematica
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5. Notice the change to linear drift from no drift. This is the basis for using the CUSUM to diagnose shifts in mean. While the process remains in control and the readings Xn follow the in-control 16 1. Introduction N(IL,0"2) distribution, the CUSUM follows a distribution centered on the horizontal axis. If the mean undergoes a step change, then the CUSUM de- velops a linear drift, and its distribution will center instead on a line whose slope 8 equals the shift in mean. The diagnosis of the CUSUM therefore consists of distinguishing the no-drift in-control behavior from the lineardrift behavior following a mean shift.
The ARL can obviously be made as large as you like by increasing k or h or both. We mentioned earlier that the choice of k follows from the choice of the shift in mean to which the CUSUM is to be tuned. The other parameter, the decision interval h, is usually fixed by deciding on the minimum tolerable in-control ARL. When X is a continuous random variable, whatever value is given to k, it is possible to find some h such that the ARL will equal any given target greater than 1. Thus for a given distribution of the process measurement X, choosing k and h implies an ARL, and choosing k and any ARL greater than 1 implies an h.
The ARLs in control and out of control are the performance measures for a CUSUM. We use the ARL from Sd". 1 35 The choice of h, the decision interval The in-control ARL of the CUSUM depends on the values of k and h. Larger values of either of these parameters leads to larger ARLs. At the extremes, if h and k are both zero, then the ARL is 1, since the first point will necessarily give a value of at least O. The ARL can obviously be made as large as you like by increasing k or h or both. We mentioned earlier that the choice of k follows from the choice of the shift in mean to which the CUSUM is to be tuned.