Page 236 - Special Topic Session (STS) - Volume 1

STS426 Tanuka C.
the mixture distributions along with Bayesian Feature Selection to extract
the directions. For an extensive discussion with mathematical details and
implementations to both real and simulated datasets please refer to ?.
The mclust software has a function MclustDR() which implements the
method discussed in the paper mentioned above. The function, MclustDR()
also has an argument lambda which is basically a tuning parameter in the
range [0, 1] as described in ?. This argument can be tuned to recover the
directions that mostly separate the estimated clusters. The package is
maintained by the author himself and thus provides the best possible way
to implement the discussed method.
We use this method in our present work just to see how distinct our
clusters are and thus helping us to validate the obtained results.

4. Analyses and Interpretations of the Data
The dataset that we have prepared has many desirable characteristics of
our nearby galaxies. We proceed to the analysis part with the following steps:
(i) We first perform Gaussian-Mixture-Model-Based Clustering (GMMBC) on
the selected number of variables. This step selects the number of optimal
clusters with the memberships in each of the clusters/groups.
(ii) Next, we perform the Bayesian LASSO within the selected clusters with all
of the variables that we have used for prediction purpose. This step will
create a variable selection within the clusters created from the previous
step in order to find values.
(iii) Next, we perform full Bayesian Regression on these clusters/groups with
the selected variables from the previous step.
(iv) After all of the above, we come to the interpretation part.
Figure 1 shows the plot of Bayesian Information Criterion (BIC) against the
number of mixture components required, i.e., the number of groups. We will
choose that number of mixture components (the number of clusters) to be
optimal for which the BIC is maximum. The BIC criterion (Figure 1) for selecting
the optimal number of clusters was giving rise to 6 clusters. After performing
Bayesian LASSO, we observed that Group 4 & 5 had the same set of variables
selected and after merging these two clusters gave the same set of selected
variables from the LASSO. Also, GMMBC with 6 clusters and GMMBC with 5
clusters only had a difference of 94.1 (1.2%) in BIC. Thus, noting these points
and for better interpretability of the Galaxy Clusters we continue our work with
5 clusters (viz. G1 - G5) each of them having unique set of variables explaining
their characteristics.
The methods described in Section 3.4 are performed in our data
accordingly. The two different directions describing the maximum amount of
separation or uniqueness between the clusters/groups are extracted from the
data set using the methods described in the paper ?.

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