Computational Intelligence - August 2014 - 34

to each row of the distance matrix. The
method was tested on +34,500 light curves
from early-stage periodic variable star catalogs
originated from the MACHO and OGLE [3]
surveys. The results of this process were lists of
mislabeled variables with careful explanations
of the new phenomena and reasons why they
were misclassified. Calculating the similarity matrix scales quadratically with the number of light curves in the survey. The
authors discuss this issue and provide an approximation of the
metric that reduces the computational complexity to O (N).
Also, an exact and efficient solution for distance-based outlier
detection can be found in [43], which uses a discord metric
that requires only two linear searches to find outlier light
curves as well.
A different approach for novelty detection was given in
[44], where an anomaly detection technique dubbed PCAD
(Periodic Curve Anomaly Detection) was proposed and used
to find outlier periodic variables in large astronomical databases. PCAD finds clusters using a modified k -means algorithm called phased k -means ( pk -means). This modification
is required in order to compare asynchronous time series
(arbitrary phases). By using a clustering methodology the
authors were able to find anomalies in both a global and local
sense. Local anomalies correspond to periodic variables that
lie in the frontier of a given class. Global anomalies on the
other hand differ from all the clusters. Approximately 10,000
periodic light curves from the OGLE survey were tested with
PCAD. The pre-processing of the light curves, the selection of
features, and the computation of the cross-correlations follow
the work of [42]. The cross-correlation is used as a distance
metric for the pk -means. The results obtained by the method
were then evaluated by experts and sorted as noisy light
curves, misclassified light curves and interesting outliers worthy of follow-up.
A problem with purely unsupervised methods is that prior
knowledge, when available, is not necessarily used. Semisupervised learning schemes deal with the case where labels
(supervised information) exist although not for all the available
data. Semi-supervised methods are able to find the structure of
the data distribution, learn representations and then combine
this information with what is known. Semi-supervised methods can also be used for novelty detection, with the benefit
that they may improve their discrimination by automatically
incorporating the newly extracted knowledge. The semisupervised approach is particularly interesting in the astronomical case where prior information exists, although scarce in
comparison to the bulk of available unlabeled data. In [45] a
semi-supervised scheme for classification of supernova subtypes was proposed. In this work the unlabeled supernovae
data are used to obtain optimal low-dimensional representations in an unsupervised way. A diagram of the proposed
implementation is shown in Fig. 5a. In general, features are
extracted from supernovae light curves following fixed templates. The data-driven feature extraction proposed in [45]

Astronomy has a long history of serendipitous
discovery [...] Computational intelligence and machine
learning may provide the means for facilitating the
task of novelty detection.
events does not necessarily correspond to that produced by
automated systems. Interestingly, this study establishes that there
is another problem as well: the same computational intelligence
algorithms working on different databases produced distinct
classification structures, showing that even though these databases have large numbers of examples, they have inherent biases
and may not be sufficiently large to allow the discovery of general rules. This problem has also been reported in other fields,
specifically artificial vision [39]. The work in [39] showed that
in order to produce consistent classification performances, one
could not simply use databases with hundreds of thousands of
examples, it was necessary to use close to 80 million images, far
exceeding what was traditionally considered enough by the
practitioners of the field.
Kernel Principal Component Analysis (KPCA) was used in
[40] to perform spectral clustering on light curves from the
CoRoT survey. The light curves were characterized using
three different approaches: Fourier series, autocorrelation functions, and Hidden Markov Models (HMMs). Then, dimensionality was reduced with KPCA using the Gaussian kernel.
Finally, the eigenvalues were used to find clusters of variable
stars. This novel characterization of light curves permits identifying not only periodic variable stars correctly (Fourier and
autocorrelation features), but also irregular variable stars
(HMM features).
Unsupervised learning can also be used for novelty detection, i.e., finding objects that are statistically different from
everything that is known and hence cannot be classified in one
of the existing categories. Astronomy has a long history regarding serendipitous discovery [41], i.e., to find the unexpected
(and unsought). Computational intelligence and machine
learning may provide the means for facilitating the task of novelty detection.
One may argue that the first step for novelty detection is to
define a similarity metric for astronomical time series in order
to compare time-varying astronomical objects. This is the
approach found in [42] where a methodology for outlier light
curve identification in astronomical catalogs was presented. A
similarity metric based on the correlation coefficient is computed between every pair of light curves in order to obtain a
similarity matrix. Intuitively, the outlier variable star will be dissimilar to all the other variables. Before any distance calculation, light curves are interpolated and smoothed in order to
normalize the number of points and time instants. For each
pair the lag of maximum cross-correlation is found in Fourier
space, which solves the problem of comparison between light
curves with arbitrary phases. Finally the outliers correspond to
the light curves with the lowest cross correlations with respect

IEEE ComputatIonal IntEllIgEnCE magazInE | august 2014

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