Graphics Reference
In-Depth Information
he covariance matrix Σ
SPD
CDU/CSU
GRUENE
FDP
LINKE
−
−
SPD
.
−
.
−
.
−
.
.
−
−
CDU/CSU
−
.
.
.
.
−
.
−
−
GRUENE
−
.
.
.
.
−
.
−
−
FDP
−
.
.
.
.
−
.
−
−
LINKE
.
−
.
−
.
−
.
.
is proportional to the correlation matrix
SPD
CDU/CSU
GRUENE
FDP
LINKE
SPD
.
−
.
−
.
−
.
.
CDU/CSU
−
.
.
.
.
−
.
GRUENE
−
.
.
.
.
−
.
FDP
−
.
.
.
.
−
.
LINKE
.
−
.
−
.
−
.
.
Numerical results for the remaining four components are omitted for brevity and
simplicity. To visualize the results, we again use a principal component projection of
the data into two dimensions. Principal component analysis returns a linear projec-
tion.Let
bethecorrespondingprojectionmatrix.Alineartransformationofamul-
tivariate Gaussian distribution is another multivariate Gaussian, with parameters
A
μ
k
=A
μ
k
(
.
)
Σ
k
T
=A
Σ
k
A
(
.
)
Figure
.
shows the five cluster components projected onto the first two prin-
cipal components. he inner solid ellipse for each cluster gives the
% confidence
region of the corresponding Gaussian distribution, the dashed outer ellipse the
%
confidence region. At first sight, the main difference between this segmentation and
the PAM result from Sect.
.
is that the eastern states have been split into two clus-
ters (instead of one), and the western states into three (instead of four). here also
seems to be huge overlap within these two groups of clusters, which may make one
wonder why the BIC favors a five-cluster solution over a
-cluster solution (east vs.
west). Obviously a more thorough investigation of the result is necessary.
Fitting a mixture model to the data does not directly partition the data into dis-
joint groups, but it does provide probability values that indicate how likely it is that
a given point belongs to a segment - the so-called posterior probability that obser-
vation
(
x, y
)
belongs to class j:
π
j
f
j
(
y
x, θ
j
)
P
(
j
x, y
)=
.
(
.
)
k
π
k
f
k
(
y
x, θ
k
)
Histograms or rootograms of the posterior class probabilities can be used to visually
assess the cluster structure (Tantrum et al.,
). Rootograms are very similar to
