Page 243 - Special Topic Session (STS) - Volume 1

STS426 Asis K.C.
P256, P1024 are peak flux measured in 64,256 and 1024 ms bin, respectively;
T50, T90 are time within which 50% and 90% of the flux arrive. Unit of fluence
is given in ergs per square centimeter (ergs cm−2), unit of peak flux is count
per square centimeter per second (cm−2 s−1) and unit of time is second (s).
First, observations on each variable are standardized because the ranges of
the variables vary largely. Then, for a particular choice of kernel, KPCA is
performed on them. They extracted nonlinear features using the significant
kernel principal components. Then using them as study variables, k-means
clustering method (Hartigan–Wong clustering algorithm (Hartigan et al. 1979))
is performed on them in which the number of clusters is determined with the
help of gap statistic (Tibshirani et al. (2001)).
Kernel Principal Components are supposed to carry less information and
more noise with increasing order and after a certain order they fail to give any
relevant information regarding the data under study. So, they started choosing
first few KPCs and number of chosen KPCs is increased as long as their
performance gets better in terms of an accuracy measure. In this context, they
used one accuracy measure as the Dunn index (Dunn (1974)), which indicates
the internal validation of a performed clustering, taking values between 0 and
∞ with greater value indicating better clustering.

3. Results and interpretation
First, k-means clustering method is applied to the standardized variables
of GRB data set in which gap statistic gives no clustering in GRBs, i.e., raw GRB
data set fails to reveal the inherent clustering nature in GRBs. Then the same
method is applied to the principal components, extracted from the GRB data
through principal component analysis. Linear features (first two PCs),
explaining more than 80% variation in data, results in one group of GRBs. Thus
linear information on data can't expose the natural groups present in GRBs. A
new choice of Kernel successfully reveals the inherent clustering nature in
GRBs, by extracting the relevant nonlinear information from raw data in terms
of kernel principal components. We see the first two KPCs, extracted through
kernel (10) with p < 1 and for every choice of s considered, are enough to
describe the data. In Chattopadhyay et al. (2007), k-means clustering approach
is directly applied to differently chosen study variables and 1594 GRBs are
clustered in three groups of sizes 622, 423, and 549, respectively with 4.08 %
1-NN classification error rate. While their clustering of 1972 GRBs based on
the first two kernel principal components, extracted by proposed kernel with
p = 1/2 and s = σ1, group those 1594 GRBs in three clusters of sizes 827, 438,
and 329, respectively with 0.2% 1-NN classification error rate.
Here they not only reduced the burden of the data, but also extracted the
inherent information from the data, on which simple clustering method reveals
the natural groups in GRBs. We propose a new possible way, kernel principal

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