Biology Reference
In-Depth Information
investigate modeling and aberration detection approaches, taking such
trend and seasonality into account.
12.3 Statistical Framework for Aberration Detection
Denote by {
y
t
, t
= 1,2,…} the univariate time series to monitor. In this chap-
ter,
y
t
will always be a discrete univariate random variable, but continuous
and multivariate versions are just as conceivable. The aim of aberration
detection is to on-line detect an important change in the process occurring
at an unknown time
t
. This could, for example, be a change in the pro-
cess parameters resulting in a change in level or variation of the process.
Using terminology from statistical process control, the process can thus
be in one of two states at each time point
t
:
in-control
, that is,
s
< τ, or
out-
of-control
(that is,
s
≥ τ). The binary 0/1 indicator
x
(
t
) will denote the true
but unknown state of the process at time
t
, assuming that
x
(
t
) = 1 means
out-of-control.
At time s ≥ 1, where a decision about the state of
x
(
s
) is to be made, the
available process information is
y
s
= {
y
t
;
t
≤
s
}. A detection method is now a
rule that predicts the unknown state of
x
(
s
) based on
y
s
. This is done by com-
puting a summary
r
(
y
s
) based on
y
s
, which is then compared to a threshold
value
g
and, consequently,
ˆ
()
xs
=
Ir
(( )
y
> ,
g
)
s
where
I
(×) is an indicator function, that is, the function returns 1 if
r
(
y
s
) >
g
and zero otherwise. The time of the first out-of-control alarm is then a ran-
dom variable
T
=
min{
s
≥ : > .
1
r
()
y
g
}
(12.1)
A
s
After the change to the out-of-control state at time τ, the decision rule should
as quickly as possible sound an alarm. However, it might take a number
of observations after τ before enough evidence has been collected to do so.
Two important target variables for evaluating the performance of a detection
method are the
in-control run-length
T
A
|τ
= ∝, that is, the number of epochs
before the first wrong alarm, and the
out-of-control run-length
T
A
|τ
= 1, that is,
the number of epochs to detect an already occurred change. Various sum-
maries such as expectation or median can be computed of these run-length
variables. Specifically, the expectation of the in-control run-length E(
T
A
|τ
=
∞)—known as the
average in-control run-length
or
P
(
T
A
Y
t
a
|τ = ∞)—the prob-
ability to get a false alarm within the first
t
a
epochs of the monitoring—is
