Digital Signal Processing Reference
In-Depth Information
10
9
8
7
6
5
4
3
10
−
1
10
0
10
1
10
2
Linearity Parameter (K)
FIGURE 5.6
Values of the myriad as a function of
K
for the following data sets: (solid) original data set
=
0
,
1
,
3
,
6
,
7
,
8
,
9; (dash-dot) original set plus an additional observation at 20; (dotted) additional
observation at 100; (dashed) additional observations at 800,
−
500, and 700.
To illustrate the above, it is instructive to look at the behavior of the sample
myriad shown in Figure 5.6. The solid line shows the values of the myriad as
a function of
K
for the data set
{
}
.Itcan be observed that, as
K
increases, the myriad tends asymptotically to the sample average. On the
other hand, as
K
is decreased, the myriad favors the value 7, which indicates
the location of the cluster formed by the samples 6
,
7
,
8
,
9. This is a typical
behavior of the myriad for small
K
:ittends to favor values where samples
are more likely to occur or cluster. The term
myriad
has been coined as a result
of this characteristic.
The dotted line shows how the sample myriad is affected by an additional
observation of value 100. For large values of
K
, the myriad is very sensitive
to this new observation. In contrast, for small
K
, the variability of the data is
assumed to be small, and the new observation is considered an outlier, not
influencing significantly the value of the myriad.
More interestingly, if the additional observations are the very large data 800,
0
,
1
,
3
,
6
,
7
,
8
,
9
−
500, 700 (dashed curve), the myriad is practically unchanged for moderate
values of
K
(
K
10). This behavior exhibits a very desirable outlier rejection
property, not found, for example, in median-type estimators.
<
5.3.3
Scale-Invariant Operation
Unlike the sample mean or median, the operation of the sample myriad is not
scale invariant; i.e., for fixed values of the linearity parameter, its behavior can
vary depending on the units of the data. This is formalized in the following
property stated without proof (see Reference 25).
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