Digital Signal Processing Reference
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produces a higher product of distances. As
K
is reduced, the myriad searches
clusters as shown in Figure 5.3b. If
K
is made large, all distances become close
and it can be shown that the myriad tends to the sample mean.
5.3.2
The Tuning of
K
The linear and mode properties indicate the behavior of the myriad estimator
for large and small values of
K
.Fromapractical point of view, it is important
to determine whether a given value of
K
is large (or small) enough for the
linear (or mode) property to hold approximately. With this in mind, it is
instructive to look at the myriad as the maximum likelihood location estimator
generated by a Cauchy distribution with dispersion
K
(geometrically,
K
is
equivalent to half the interquartile range). Given a fixed set of samples, the
maximum likelihood (ML) method locates the generating distribution in a
position where the probability of the specific sample set to occur is maximum.
When
K
is large, the generating distribution is highly dispersed, and its
density function looks flat (see the density function corresponding to
K
2
in
Figure 5.4). If
K
is large enough,
all
the samples can be accommodated in-
side the interquartile range of the distribution, and the ML estimator visual-
izes them as “well behaved” (no outliers.) In this case, a desirable estimator
would be the sample average, in complete agreement with the linear property.
From this consideration, it should be clear that a fair approximation to the
K
k = 10k
2 1
K
^
k1
^
k2
β
β
K1
K2
2k
1
2K
2K
2
FIGURE 5.4
The role of the linearity parameter when the myriad is viewed as an ML estimator. When
K
is
large, the generating density function is spread and the data are visualized as “well behaved”
(the optimal estimator is the sample average). For small values of
K
, the generating density
becomes highly localized, and the data are visualized as very impulsive (the optimal estimator
is a cluster locator).
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