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At this point, readers who are not interested in time series analysis, or who have not studied this
subject, may wish to skip to the Comments section. The next chapter brings us back to a comfortable topic.
The model for Figure 13.2 is an integrated moving average (IMA) time series model in its simplest form:
y t – y t−1 = a t – θa t−1

2
where a t is a random normal independent variable with mean zero and constant variance σ . This is the
model for which the exponentially weighted moving average (EWMA) with smoothing constant θ is the
minimum mean square error forecast of y t from origin t – 1.
An IMA model describes a time series that is nonstationary; that is, it has no fixed mean level, except
as an approximation over a short time interval. An IMA time series drifts. Many real processes do drift
and the IMA can be a suitable model in these cases.
The disturbance model is y t − y t−1 = a t − (θ + δ )a t−1 , where δ is the change in the parameter. Tiao et al.
(2000) has designed a Cuscore chart to detect this change.
The discrepancy vector is:

∂ –
d t = – ------- = ------------
a ˆ t
a t
θ ∂ 1 θ–
is the exponentially weighted moving average of previous a t ’s with smoothing constant θ. This
where a ˆ t
is upgraded using:
a ˆ t = ( 1 – θ)a t−1 + θa ˆ t−1

The Cuscore statistic is:
1
Q = ------------ ∑ a t0 a ˆ t0
1 θ–
If θ 0 is the true value of θ, the a t0 ’s would be independent and the exponentially weighted average a ˆ 10
of previous a t0 ’s would not be correlated with a t0 .
The first-order autoregressive model (AR1) is another common time series model. This model describes
a stationary time series, that is, a series that has a fixed mean level:
z t = φz t−1 + a t
where z t is the deviation y t − T from the target value. We wish to monitor for a possible change in the
parameter from its normal value φ 0 to some new value φ 0 + δ. The null model is a t0 = z t − φ 0 z t−1 and
the discrepancy model is a t0 = z t − (φ 0 + δ )z t −1 . Box and Luceno (1997) explain how this is done. We
lack the space to provide more detailed explanation and example calculations. Details and applications
are discussed in Box and Luceno (1997) and Tiao et al. (2000).

Comments

Control charts have developed well beyond the classical design of Shewhart. Automated data collection
provides information that does not always fit a Shewhart chart. Individual observations are charted, and
often there is no central limit property to rely on, so the reference distribution of the plotted values is
not always the normal distribution. Autocorrelation and process drift are common, and data transforma-
tions are needed. Also, statistics other than the mean value are charted. Disturbances other than a simple
shift in mean are important: a change in slope signals wear of a machine or tool, a sinusoidal disturbance
signals vibration of a compressor shaft, a bump signals a possible harmful shock to the system, and so on.
The Cusum chart shown in Chapter 12 was one powerful development. It serves the same purpose as
a Shewhart chart but has greater sensitivity. It will detect a smaller change than a Shewhart chart can
detect. It gains this sensitivity by plotting the cumulative change from the target level. The accumulation
of information about deviations provides useful knowledge about a process.
© 2002 By CRC Press LLC