# Data mining techniques on astronomical spectra data. I : Clustering Analysis

Haifeng Yang1, Chenhui Shi1, Jianghui Cai1,2, Lichan Zhou1, Yuqing Yang1,
Xujun Zhao1, Yanting He1 E-mail<EMAIL_ADDRESS>(HY);
<EMAIL_ADDRESS>(CS<EMAIL_ADDRESS>and Jing Hao1
1School of Computer Science and Technology, Taiyuan University of Science and
Technology, Taiyuan 030024, China
2School of Computer Science and Technology, North University of China, Taiyuan
030051, China

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Clustering is an effective tool for astronomical spectral analysis, to mine
clustering patterns among data. With the implementation of large sky surveys,
many clustering methods have been applied to tackle spectroscopic and
photometric data effectively and automatically. Meanwhile, the performance of
clustering methods under different data characteristics varies greatly. With
the aim of summarizing astronomical spectral clustering algorithms and laying
the foundation for further research, this work gives a review of clustering
methods applied to astronomical spectra data in three parts. First, many
clustering methods for astronomical spectra are investigated and analysed
theoretically, looking at algorithmic ideas, applications, and features.
Secondly, experiments are carried out on unified datasets constructed using
three criteria (spectra data type, spectra quality, and data volume) to
compare the performance of typical algorithms; spectra data are selected from
the Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) survey
and Sloan Digital Sky Survey (SDSS). Finally, source codes of the comparison
clustering algorithms and manuals for usage and improvement are provided on
GitHub.

###### keywords:

methods: data analysis-techniques: spectroscopic-software: data analysis

††pubyear: 2022††pagerange: Data mining techniques on astronomical spectra
data. I : Clustering Analysis–Data mining techniques on astronomical spectra
data. I : Clustering Analysis

##  1 Introduction

Astronomical spectral clustering has attracted increasing attention in
astronomy. It helps us to analyse the birth, formation and evolution of the
cosmic and astronomical objects (Fraix-Burnet et al., 2012; Chattopadhyay et
al., 2012). Nowadays, rapid developing sky survey projects have obtained a
large amount of astronomical data, which poses great challenges for effective
astronomical clustering (Luo et al., 2008). In view of the above challenges, a
wide range of clustering methods (partition-based clustering methods, density-
based clustering methods and so on) (Cai et al., 2022; Siemens & Baker, 2012;
Baker et al., 2010; Berry & Linoff, 1997; Lam & Wunsch, 2014) have been
applied to diverse astronomical tasks. However, clustering methods perform
differently on various data sources, making it difficult to evaluate the
performance and determine their applicable scenarios.

To explore the advantages of each type of clustering method, we investigate a
large number of clustering methods applied to astronomical spectra data and
analyse their applications, core ideas, merits and caveats. We then compare
the performance of classical clustering methods on unified datasets and give
objective appraisals. The datasets are constructed considering four aspects,
as follows.

(i) Different data types of some spectral feature extraction methods. Four
types of data including 1D spectra, PCA (Principal Component Analysis, a
widely used dimensionality reduction method) features of spectra, line indices
and stellar parameters are used to test different methods.
(ii) The quality of spectra. Three signal-to-noise ratios (S/Ns) are used to
analyse the robustness of clustering algorithms.
(iii) Data volume. Four sizes of spectral datasets are constructed including
8000, 20000, 40000 and 80000, however some algorithms cannot run on the size
of 80000.
(iv) Outliers detection. We know that some clustering methods can be used to
detect outliers, so datasets containing normal spectra and rare objects are
constructed to test the ability of some algorithms to detect outliers.

After each experiment, an objective analysis is given. Moreover, experimental
source codes in this paper and a brief manual about usage of the source codes
are available to readers.

This work is organized as follows: In Section 2, we briefly describe the
clustering methods from applications and theory analysis. In Section 3,
experiments on four astronomical tasks - A/F/G/K stars classification,
star/galaxy/quasar classification, subclasses of A-type star classification
and outliers detection - are carried out. Section 4 puts forward source codes
of the aforementioned experiments and a manual about the usage of the source
codes. Finally, a discussion and our future work are presented in Section 5.

##  2 Investigation of Clustering Methods on Astronomical Spectra Data

As a prevalent task, clustering has been applied in many fields, such as
imaging processing (Dorai & Jain, 1995; Forsyth & Ponce, 2011), social
networks (Ricci et al., 2011; Everton, 2012), finance security, biological
fields (Kaplan & Winther, 2013; Durbin et al., 1998), and others (Tahmasebi et
al., 2012; Fotheringham et al., 1998; Openshaw et al., 1985; Connell & Jain,
1998). It attempts to divide data into different groups according to certain
criteria. Data in the same group appears to be more similar than data in
different groups. The classical clustering process can be divided into the
following steps (Xu & Tian, 2015).

  1. (1)

Data preprocessing: extract and choose the most representative features;

  2. (2)

Clustering: design suitable clustering algorithms and train models with
unlabelled data to cluster the real data;

  3. (3)

Result analysis: explain the clustering result and evaluate the clustering
method.

Clustering methods play a vital role in astronomical spectral data analysis
(Rebbapragada et al., 2008; Balazs et al., 1996; Simpson et al., 2012; Sánchez
Almeida & Lites, 2000; Cai-Xia et al., 2020). Traditional clustering
algorithms can be divided into partition-based, density-based, hierarchical,
grid-based, and model-based. In recent years, lots of new clustering
algorithms have appeared, like fuzzy theory-based clustering algorithm,
kernel-based clustering algorithm (Couillet & Benaych-Georges, 2016), etc. The
next subsections will introduce the above typical clustering algorithms,
considering applicable scenarios, core ideas, merits, and caveats.

###  2.1 Partition-Based Clustering Algorithm

Table 1: Investigations of partition-based clustering algorithms on
astronomical spectra.

Merits | Caveats | References
---|---|---
Some improved partition-based methods classify boundary spectra effectively. K-means is simple and fast, it can deal with large dataset. |  In flux space, clustering result is sensitive to the number of clusters. K-means can not cluster arbitrary shape of groups. On classifying light curves, the classification algorithm is not robust because the data amount is small. |  Li et al. 2019, Kheirdastan & Bazarghan 2016,
Ordovás-Pascual & Sánchez Almeida 2014,
Cai et al. 2020, Sánchez Almeida et al. 2010,
Sánchez Almeida & Allende Prieto 2013,
Garcia-Dias et al. 2018, Sasdelli et al. 2016,
Hayes et al. 2020
On detecting special objects, K-means is also efficient. |  |  Chen et al. 2018a, Morales-Luis et al. 2011
Space and morphological structure can be identified by K-means. After clustering some data, lots of meaningful information can be analysed from the clustering result. |  Some distinct features in colour space which are apparent in visual examinations can not be recognized as clusters in colour space. |  Rahmani et al. 2018, Shuxin & Weimin 2017
Hogg et al. 2016, Sánchez Almeida et al. 2012,
Panos et al. 2018, Rubin & Gal-Yam 2016,
Beck et al. 2016, Chattopadhyay & Maitra 2017,
Hayes et al. 2020,Blanco-Cuaresma et al. 2015,
Mosby et al. 2015,Kiar et al. 2017

Partition-based clustering methods classify the data into different clusters
by finding optimal cluster centers and K-means is a widely used partition-
based algorithm because of its simplicity and efficiency. Astronomical
investigations based on partition-based clustering algorithm are shown in
Table 1.

Sánchez Almeida et al. (2010) used K-means to classify galaxies from SDSS DR7.
However, some spectra appeared to be between classes, so they proposed a
K-means-based method to identify marginal galaxies. Garcia-Dias et al. (2018)
analysed the application of K-means to massive Apache Point Observatory
Galactic Evolution Experiment (APOGEE) high resolution spectra. The results
showed that K-means was able to separate the bulge and halo populations and
distinguish dwarfs, subgiants, red clump (RC) stars, and red giant branch
(RGB) stars. Clustering of star, galaxy, quasar and subclasses have been
carried out extensively to analyse physical properties, chemical abundances,
boundary spectra, etc ( Chen et al. 2018a; Shuxin & Weimin 2017; Hogg et al.
2016; Kheirdastan & Bazarghan 2016; Morales-Luis et al. 2011, Sánchez Almeida
& Allende Prieto 2013). Furthermore, clustering results contain valuable
information, such as structure of Galaxy and evolutionary stages of the
Universe, which provides convenience for astronomers (Rahmani et al., 2018;
Sánchez Almeida et al., 2010; Mosby et al., 2015; Sánchez Almeida et al.,
2012; Kiar et al., 2017; Beck et al., 2016).

Initialization of clusters centers and the number of clusters are two factors
that affect K-means results greatly, and many improved methods have been
proposed (Cai et al., 2020; Ordovás-Pascual & Sánchez Almeida, 2014; Li et
al., 2019). In order to optimize clustering, Cai et al. (2020), Garcia-Dias et
al. (2018) and Li et al. (2019) used new approaches to initialize cluster
centers and results suggested that clustering with better initial centers are
of high quality. Yang et al. (2022a) optimized K-means by influence space and
applied ISBFK-means on astronomical spectral data of LAMOST. Results presented
a good performance on stability and speed.

Another common application is special objects identification (e.g. rare
objects, light curves, star clusters) and retrieval tasks (Shuxin & Weimin,
2017; Panos et al., 2018; Chattopadhyay & Maitra, 2017; Sasdelli et al., 2016;
Hayes et al., 2020; Rubin & Gal-Yam, 2016; Garcia-Dias et al., 2019; Blanco-
Cuaresma et al., 2015), and the detected special objects are a significant
supplement to the astronomy category. Jin et al. (2022) used K-means
clustering to successfully divide the confirmed exoplanets from NASA dataset
into different clusters.

In astronomical applications, effective feature selection not only affects
clustering accuracy and time, but is also closely related to celestial
properties. Chen et al. (2018a) and Shuxin & Weimin (2017) adopt line indices
as spectral features to improve data quality and results indicated that they
were consistent with internal properties of stars. Optimal feature extraction
(color space, chemical pattern, etc.) can provide more information for
clustering without additional prior knowledge (Garcia-Dias et al., 2019;
Blanco-Cuaresma et al., 2015; Hogg et al., 2016; Sánchez Almeida et al., 2010;
Tammour et al., 2016).

Among partition-based clustering algorithms, K-means and K-mediods are two
typical methods; variants such as K-means++, intelligent K-means, genetic
K-means, K-modes, and kernel K-means also perform well. The goal of partition-
based clustering algorithms is that data in the same group appears more
similar than those in different groups. The pseudo-code about partition-based
clustering methods is given in Algorithm 1 and Fig. 1 shows the structure of
K-means.

Algorithm 1 Partition-Based Clustering Algorithm

1:Data; K: the number of clusters;

2:Data-Labels

3:Determine initial centers by random or other methods.

4:while not reach convergence do

5: Assign each sample to the most similar cluster center.

6: Update cluster centers with certain criterion.

7:end while

Figure 1: Structure of K-means. Yellow, purple and blue are three groups.
Triangles present cluster centers of each group.

K-means is a typical partition-based algorithm which has been widely applied
to many fields because of its advantages of simple implementation, intuitive
design and linear computational cost. It is also robust when tackling high
dimensional data and large dataset, making it an attractive tool for star
classification, multi-band photometric analysis, supernova light curve
surveys, and so on. However, the number of clusters and initial cluster
centers selection are two key factors which influence the performance of
K-means. The optimal solution to the above problems is still in the continuous
exploration stage. Most partition-based methods measure the similarity between
samples by Euclidean distance, which is not so reliable when the feature of
data is complex, so it is important to choose an appropriate similarity
measure for the data under study. Another shortcoming of partition-based
methods is that they cannot detect clusters with arbitrary shape, and only
round clusters can be recognized.

###  2.2 Density-Based Clustering Algorithm

Table 2: Investigations of density-based clustering algorithms on astronomical
spectra.

Merits | Caveats | References
---|---|---
DBSCAN can be used to classify objects on photometric and spectra data. CFSFDP is efficient on detecting outliers. | DBSCAN rarely clusters spectra directly. |  Logan & Fotopoulou 2020,
Deng & Tu 2017, Wu et al. 2020,
Tramacere et al. 2016
DBSCAN can be used to study stellar history by clustering chemical abundances. |  |  Price-Jones & Bovy 2019,
Chen et al. 2018b
DBSCAN can cluster data with arbitrary shape and size. Objects in 3D space can be detected and analysed by density-based clustering algorithms, like DBSCAN and OPTICS. |  Determination of lower spatial-density features need be improved. |  Zari et al. 2019, Carlson et al. 2013,
Dehghan & Johnston-Hollitt 2014,
Yan et al. 2020, Oliver et al. 2021,
Mahajan et al. 2018, Gao 2014,
Armstrong et al. 2015,
Sans Fuentes et al. 2017
DBSCAN can help to reduce noise. TAD algorithm can analyse the spatial-temporal density of complex or special trajectories efficiently. |  The efficiency will decrease when the dataset is too large. |  Seo et al. 2020, Yang et al. 2020

At present, density-based clustering algorithms are promising for exploring
the universe. Table 2 depicts applications of density-based methods.

Density-based clustering methods on spectra data are used to classify
celestial objects and they are more frequently used in chemical abundance,
photometric data, etc. Logan & Fotopoulou (2020) classified star/galaxy/quasar
from photometric data with hierarchical DBSCAN and the accuracy was high.
Tramacere et al. (2016) used DBSCAN and DENCLUE to classify galaxies in the
Galaxy Zoo dataset. Ordonez & Stripling (2022) combined DBSCAN and nearest
neighbor algorithm to analyse quasars; the results showed that gravitational
interaction may cause more low redshifts quasars clustering.

Another common application of density-based methods is the detection and
analysis of spatial structure of objects, like galaxy clusters, molecular
clouds and others (Carlson et al., 2013). Oliver et al. (2021) used OPTICS to
identify the structure of galaxy/halo and the low-density data points can be
identified effectively. HDBSCAN was applied to systematically find members
which are up to 50 pc from the cluster centers (Tarricq et al., 2022), and GMM
was used to fit the two dimensional distribution of cluster members in this
work; these two methods helped to find 389 open clusters. Gao (2014)
determined membership of open cluster NGC 188 and found that DBSCAN could
efficiently select possible member stars in 3D kinematic space without making
any assumptions about the distribution of cluster or field stars. Castro-
Ginard et al. (2022) used DBSCAN to identify the statistical overdensities of
stellar objects, which help to find more open clusters. The structure,
kinematics, and ages of the young stellar populations in the Orion region can
also be analysed by DBSCAN (Zari et al., 2019). Besides this, many researchers
committed to multi-space, which is a new perspective of the universe, and
software engineering also provides many useful tools for clustering analysis.

Some density-based clustering algorithms can search for outliers and cluster
centers, such as Clustering by fast search and find of density peaks (CFSFDP).
Wu et al. (2020) used CFSFDP to detect special samples in the massive low S/N
spectra.

In astronomical tasks, ’data level’ and ’algorithm level’ are two major
approaches to achieve desirable clustering performance. Deng & Tu (2017) and
Gowanlock et al. (2017) found optimal parameters and distance measurement
methods to obtain satisfactory results from ’algorithm level’. Astronomical
spectra can be represented in many forms, such as chemical abundance, line
indices, color space and other features. Efficient feature extraction is a
particularly interesting way to improve performance from ’data level’ (Price-
Jones et al., 2020; Yang et al., 2020; Price-Jones & Bovy, 2019; Gao, 2014;
Logan & Fotopoulou, 2020; Gao et al., 2015; Traven et al., 2017; Tramacere et
al., 2016). Many new density-based clustering algorithms have also been
proposed to improve the efficiency and robustness of clustering (Yang et al.,
2022b; Liang et al., 2022).

Density-based clustering algorithms group data according to region density.
High density areas are defined as classes sharing similar information. Always,
they require two input parameters: the density radius (Eps) and the minimum
number of data (MinPts) to build a cluster. DBSCAN, DENCLUE, OPTICS, CFSPDP
are classic density-based methods. In addition, some optimization approaches
have been explored to improve clustering performance, such as manhattan
distance density algorithm (MD-DBSCAN) (Deng & Tu, 2017) , normalised
euclidean distance (NED-DBSCAN) and so on. The procedure of density-based
clustering algorithm is presented in Algorithm 2.

Algorithm 2 Density-Based Clustering Algorithm

1:Data; Neighbor radius $\epsilon$; Radius density $N_{pts}$;

2:Data-Labels

3:while $X_{i}$ not none do

4: while queue not none do

5: if the number of data in $\epsilon>N_{pts}$ then

6: $X_{i}\leftarrow core\,point$

7: else

8: $X_{i}\leftarrow outlier$

9: end if

10: Assign core point and it’s neighbour points to cluster i.

11: Add neighbour points to a queue.

12: end while

13:end while

Density-based clustering algorithms have been frequently used for scientific
data analysis in various fields. It can identify irregular groups rather than
being limited to globular shapes (e.g. K-means.) and the number of clusters
need not be determined in advance. In astronomy, density-based clustering
algorithms provide crucial information for candidates identification,
substructure analysis and category supplement.

###  2.3 Model-Based Clustering Algorithm

Table 3: Investigations of model-based clustering algorithms on astronomical
spectra.

Method | Merits | Caveats | References
---|---|---|---
SOM |  SOM maps high-dimension spectra into 2D graph, and similar spectra are close in graph. The ordering of spectra on the map correspond to the ordering of physical properties. SOM can be used to detect special objects. SOM method with interactive interface can greatly improve the efficiency of spectral analysis. |  SOM is not stable on clustering. On the 2D map, there is no clear clustering result. | Madhusudan et al. 2017
Rahmani et al. 2018
Ordóñez et al. 2012
Bazarghan 2011
Meusinger et al. 2017
Sasdelli et al. 2016
Fustes et al. 2013
Merényi et al. 2016
A.in der Au, et al. 2012
GMM |  GMM can detect object with a Gaussian distribution in space, like galaxy(cluster). GMM distinguish different kinds of gamma-ray bursts. |  When the data set is large, GMM will take a long time to run. | Iwasaki et al. 2019
Shin et al. 2018
Chattopadhyay & Maitra 2017
Gao 2020, Tóth et al. 2019
Shang & Oh 2012
Acuner & Ryde 2018

Investigations of model-based clustering algorithm on astronomical spectra are
shown in Table 3. Model-based methods mainly contain neural network clustering
algorithms and probability-based model clustering algorithms. Both of them
have already been developed for star, galaxy and peculiar objects clustering
(Madhusudan et al., 2017; Kuhn & Feigelson, 2017).

Self-organizing map (SOM) is a special type of neural network which maps the
data into a two-dimensional grid. It is always used to visualize the
distribution of data, and is similar to the dimensionality reduction
visualization methods like t-SNE (Van der Maaten & Hinton, 2008; Wattenberg et
al., 2016) and UMAP (McInnes et al., 2018). Bazarghan (2011) used SOM to map
158 spectra into a 13 $\times$ 13 grid and made a classification. The two-
dimensional map contains aggregated information of the data and the gaps in
the map can be regarded as the separation of different clusters. Ordóñez et
al. (2012) used SOM to distinguish different objects and divided the grid with
Fuzzy C-Means (FCM) which is a fuzzy clustering algorithms based on K-means.
Rahmani et al. (2018) trained SOM using a set of galactic templates which
covering the wavelength range from far ultraviolet to near-infrared. Compared
with other methods, the spectra of galaxies grouped together by the SOM are
more similar. And the order of sample categories on the map corresponded to
the order of physical properties. That is, in addition to representing the
type, the order of samples on the map can also represent many other
information, like physical parameters distribution. For clustering,
researchers divided the 2D map into many clusters (Merényi et al., 2016) or
mapped the data into 1 $\times$ n where n is the number of categories. For
easier use of SOM, some researchers built interactive programs to analyse
objects efficiently (A.in der Au, et al., 2012; Meusinger et al., 2017). Many
other applications like outlier analysis (Fustes et al., 2013), exploration of
the spectroscopic diversity of Type Ia supernovae (Sasdelli et al., 2016) can
also be handled or assisted by SOM.

Gaussian mixture model (GMM) is one of the most commonly used probability
models. It uses multiple multidimensional Gaussian distributions to fit data,
and each distribution represents a cluster (Reynolds, 2009). In astronomy, GMM
can be used in two ways, one is to cluster data based on features, such as 1D
spectra, chemical element abundance, etc. The other is to use Gaussian
distributions to fit targets directly, such as star clusters and molecular
clouds. Iwasaki et al. (2019) reduced X-ray spectra which are obtained from
Chandra X-ray Observatory from Tycho’s supernova remnant using a variational
autoencoder (VAE) and clustered the low-dimension features with GMM to analyse
spatial structures in Tycho’s supernova remnant. Shin et al. (2018) used an
infinite GMM to detect variability in large amounts of astronomical time
series data. GMM was also used to classify different kinds of gamma-ray bursts
(Chattopadhyay & Maitra, 2017; Tóth et al., 2019; Acuner & Ryde, 2018). Gao
(2020) used PCA+GMM to identify tidal tail around the old open cluster NGC
2506 based on Gaia-DR2. GMM can also detect and analyze objects which shaped
like Gaussian distribution. Shang & Oh (2012) used a mixture model to probe
gas motions in the intra-cluster medium and showed that the mixture parameters
can be accurately constrained by Astro-H spectra. Wagenveld et al. (2022)
proposed a probabilistic HzQ selection method which used GMM to obtain
likelihoods and used a Bayesian framework for poster probabilities of HzQ and
contaminating sources. It is useful to find more complete HzQ samples.

SOM and GMM are the most widely used model-based algorithms. Compared with
K-means, they can obtain more valuable information. The idea of SOM is to
establish a mapping between high and low dimensions. An illustration of SOM
structure (Garcia-Dias et al., 2018) is given in Fig. 2. Algorithm 3 is the
main idea of SOM.

Algorithm 3 Self-Organizing Map

1:Data; Neural units weights; Leaning rate;

2:Data-The mapping position in the 2D grid

3:while neural units weights are not stable do

4: Calculate winner neural unit with neural weights and learning rate.

5: Update the neural units weights.

6:end while

Figure 2: Structure of SOM. Three circles are data of input layer. The top
right complete layer generates two dimensional output. Weights matrix includes
weights between input layer and complete layer.

Neural network is widely used (Kiang et al., 2005) because it can learn
complex non-linear relationships. But conventional neural network model has
many parameters and consumes long time to run. Besides, it is difficult to
explain the results. In astronomical research, SOM also has the ability to
learn complex non-linear relationships. At the same time, it can visualize the
clustering results and alleviate the unexplainability of the model to a
certain extent. Numerous studies have shown that SOM can be applied to various
astronomical tasks.

GMM is a mixture of several Gaussian distributions. Its advantage is that GMM
gets rid of the limitation of measuring similarity based on Euclidean distance
and fits each dimension of data with a Gaussian distribution. Data in the same
independent Gaussian distribution belong to the same cluster (Fig. 3). It
iteratively updates the parameters of the multidimensional distributions
through EM (Expectation-Maximum) algorithm (Ng et al., 2012), and Algorithm 4
shows the procedure of GMM.

Algorithm 4 Gaussian Mixture Model

1:Data; The number of Gaussian model;

2:Data-Labels

3:Determine initial parameters of models by some methods.

4:Calculate the probability that the data belongs to each model.

5:while probability matrix not stable do

6: E-Step: calculate the probability matrix of data.

7: M-Step: update the parameters again according to probability matrix.

8:end while

Figure 3: Structure of GMM. Yellow, green and blue are three gaussian
distributions and each gaussian distribution presents one group.

Among the model-based clustering algorithms, GMM is a relatively classic
algorithm. It gives the probability of classification rather than just the
labels. But it needs enough samples to fit a accurate distribution, and needs
a long iteration time to find optimal parameters.

###  2.4 Hierarchical Clustering Algorithm

The main idea of hierarchical clustering algorithm (Karypis et al., 1999) is
to classify data into different groups with a hierarchical tree, including
hierarchical agglomerative method and hierarchical divisive method. These two
types of methods are inverse processes, that is, bottom-up and top-bottom.
Balanced Iterative Reducing and Clustering Using Hierarchies (BIRCH) (Zhang et
al., 1996), Clustering Using Representative (CURE) (Guha et al., 1998) and
Chameleon (Karypis et al., 1999) are commonly used hierarchical clustering
methods. Fielding et al. (2022) used hierarchical clustering (agglomerative
clustering) and partition-based clustering (K-means, fuzzy c-means) on
extracted features of galaxy for clustering. Results showed that hierarchical
clustering performed best. Hierarchical agglomerative clustering and
hierarchical divisive clustering are presented in Algorithm 5 and Algorithm 6
(Xu & Tian, 2015).

Algorithm 5 Hierarchical Agglomerative Clustering Algorithm

1:Data;

2:Data-Labels

3:Regard each data as an individual cluster.

4:while more than one cluster left do

5: Calculate similarity between every cluster.

6: Emerge the most similar clusters with minimal distance.

7:end while

Algorithm 6 Hierarchical Divisive Clustering Algorithm

1:Data;

2:Data-Labels

3:Suppose all data stands for one cluster.

4:while not every data is a cluster do

5: Calculate similarity between every cluster.

6: Divide the most dissimilar cluster into two clusters.

7:end while

Hierarchical clustering can find the hierarchical relationships between
classes without determining the number of clusters in advance. But it has high
computational complexity and is sensitive to outliers. Therefore, hierarchical
clustering is also usually combined with other algorithms to improve the
clustering performance. For example, researchers combined SOM and ELM with
hierarchical clustering to improve the accuracy of stellar clustering (Ordóñez
et al., 2012; Bu et al., 2016).

###  2.5 Fuzzy Theory-Based Clustering Algorithm

Clustering algorithms can be divided into two categories, namely hard
clustering and soft clustering. Fuzzy theory-based clustering algorithm is a
typical soft clustering. The basic idea of this clustering algorithm is that
data can be assigned to several clusters with various probabilities (Xu &
Tian, 2015). Fuzzy C-means (FCM), Fuzzy C-shells (FCS) and Mountain method
(MM) are classic clustering algorithms. Main processes of FCM are presented in
Algorithm 7.

Algorithm 7 Fuzzy Theory-Based Clustering Algorithm

1:Data; Fuzzy matrix U;

2:Data-Labels

3:while fuzzy matrix U not stable do

4: Calculate cluster centers.

5: Update fuzzy matrix U.

6:end while

###  2.6 Grid-Based Clustering Algorithm

Grid-based clustering algorithms divide the space into a number of cells.
Their processing time is independent of the number of processed objects and
the processing speed is fast. Then clustering is based on the grid structure
(Saxena et al., 2017). Algorithm 8 is the main processes of them.

Algorithm 8 Grid-Based Clustering Algorithm

1:Data;

2:Data-Labels

3:Define grid cells.

4:while grid cells not stable do

5: Partition objects into proper grid cells and compute the density of each
cell.

6: Eliminate cells with density below the certain threshold.

7: Clusters can be generated in contiguous groups of dense cells.

8:end while

###  2.7 Graph-Based Clustering Algorithm

In this section, we introduce a novel approach named graph-based clustering
algorithm. Graph-based clustering operates on graphs, nodes are regarded as
data points and the edges are regarded as relationships among data points (Xu
& Tian, 2015). Spectral clustering algorithm and Affinity Propagation (AP) are
two classical graph-based clustering algorithms. Graph-based clustering
algorithms are intuitive and transform clustering into an optimization
problem.

Spectral Clustering. The core idea of spectral clustering is to treat all data
as a graph and assign different weights to each edge. It divides the whole
graph into many sub-graphs by cutting the edges with small weight, to ensure
the weights between different sub-graphs is as small as possible and the sum
of weights in the same sub-graph is as large as possible, each sub-graph is
considered to be a cluster. Duarte-Cabral et al. (2021) compiled a molecular
cloud catalogue with spectral clustering and analysed global properties of
some clouds.

Affinity Propagation (AP). The basic idea of AP algorithm is to take all data
points as potential clustering centers (called examples) and connect any two
data points to form a network (similarity matrix). Then calculate the
clustering centers of each sample through the transmission of messages
(responsibility and availability) on each side of the network. It has been
applied to star/galaxy classification, open clusters clustering and large
spectra clustering (Wang et al., 2015; Blanco-Cuaresma et al., 2015; Du et
al., 2016).

##  3 Experiment Analysis

The clustering methods mentioned in Section 2 have been successfully applied
to various astronomical data analysis. For astronomical clustering tasks,
there are many methods to choose and researchers need to select one which
works best. Due to the diversity of clustering tasks and data, it is difficult
to evaluate the advantage and disadvantage of these methods from current
literatures.

To explore the advantage of each method in various tasks, in this section, we
evaluate several commonly used clustering algorithms through building the
unified spectral datasets as experimental data, which are observed by LAMOST
survey (Luo et al., 2015).

LAMOST (The Large Sky Area Multi-Object Fiber Spectroscopic Telescope, also
called the Guo Shou Jing Telescope) is a special reflecting Schmidt telescope
with an effective aperture of 3.6 – 4.9 m and a field of view of 5°. It is
equipped with 4000 fibers, with a spectral resolution of R $\approx$ 1800 and
wavelength ranging from 3800 to 9000 $\mathop{A}\limits^{\circ}$
(http://www.lamost.org/). After seven years of surveys, LAMOST observed tens
of millions of low-resolution spectral data, which provides important data for
astronomical statistical research. The spectra we used are selected from
LAMOST DR8 which released a total of 17.23 million spectra. The number of
high-quality spectra of DR8 (S/N > 10) reaches 13.28 million and DR8 includes
a catalog of about 7.75 million groups of stellar spectral parameters.

We construct unified datasets from four aspects including feature extraction
methods for spectra, quality of spectra, dataset size and outlier detection
task, to explore advantages and disadvantages of different clustering
algorithms. In the aspect of different feature extraction methods, we organize
experiments from stellar classification, star/galaxy/quasar classification,
subclasses of A-type star classification and matching sources from LAMOST and
SDSS classification. All of these aspects need to be considered in the
practical spectral analysis, so comparative experiments are meaningful for
researchers. We compare eight clustering algorithms which are widely used in
astronomy, including K-means, K-means-dp, SOM, K-mediod, DBSCAN, CFSFDP,
Hierarchical Clustering and GMM. And the true labels we use are spectral
classes released by LAMOST.

Algorithms are evaluated by clustering accuracy. But we can not know how the
samples are divided through average accuracy, and it is not rigorous if the
clustering result is undesirable. So t-SNE algorithm and confusion matrix are
used to show the clustering errors. Data can be mapped into a 2D graph by
t-SNE, and the more similar the samples are, the closer they are in the 2D
graph. The t-SNE graphs of data before and after clustering are drawn to
observe how data is clustered. The confusion matrix can be used to determine
the exact number of false clustering samples.

###  3.1 Performance of algorithms on different spectra characteristics

Low resolution 1D spectra observed by telescope have thousands of dimensions.
And in automatic spectral analysis, astronomers always adopt some
dimensionality reduction and feature extraction techniques like PCA,
extracting spectra line indices, etc (Luo et al., 2013, 2004). Different
clustering methods will achieve different performances on these
characteristics, so it is necessary to evaluate these methods on different
spectra features. In this section, four types of data including 1D spectra,
PCA features, spectra line indices and stellar parameters are constructed to
test clustering algorithms. PCA is a feature extraction and dimension
reduction method that combines features linearly, spectral features extracted
by PCA have no physical meaning. But it is widely used in astronomy and other
fields, so we also choose PCA features to compare the clustering performance
with other features.

In this section, experiments are carried out from (1) stellar classification,
(2) star/galaxy/quasar classification, (3) classification of subclasses of
A-type star, (4) classification of matching sources from LAMOST and SDSS. Data
selections of the tasks are shown in Table 4, Table 5 and Table 6.

Table 4: Data selection of three tasks.

| 1D spectra | PCA | Line Indices | Stellar Parameters
---|---|---|---|---
Task Star | $\surd$ | $\surd$ | $\surd$ | $\surd$
Task SGQ | $\surd$ |  |  |
Task SubA | $\surd$ |  |  | $\surd$

  * 1

Task Star: Stellar Classification.

  * 2

Task SGQ: Classification of Star/Galaxy/Quasar. 1D spectra in this task refer
to original spectra and rest wavelength frame spectra.

  * 3

Task SubA: Classification of Subclass of A-type Star.

Table 5: Number of spectra of three tasks.

Task | Type | Data volume
---|---|---
Stellar Classification | A | 5000
F | 5000
G | 5000
K | 5000
Classification of Star/Galaxy/Quasar | Star | 5000
Galaxy | 5000
Quasar | 5000
Classification of Subclasses of A-type Star | A0 | 14
A1 | 6097
A2 | 2329
A3 | 514
A5 | 5551
A6 | 3383
A7 | 6046
A8 | 305
A9 | 761

Figure 4: Accuracy of Algorithms on Different Characteristics. Four bars
represent four data characteristics.

####  3.1.1 Stellar Classification of Type A/F/G/K

In the stellar classification, we use four features mentioned above. While,
the performance of data analysis using the full spectrum is inevitably
affected by reddening, and the LAMOST pipeline corrects it using polynomials
during data processing. It is difficult to obtain a consistent extinction
model, and at the same time, this paper mainly focuses on providing a
performance analysis on astronomical spectra of different techniques.
Therefore, we choose the spectra of high galactic latitude ($\mathscr{l}$ >
45°) to construct the datasets. In the preprocessing steps, we cut out the
5700$\mathop{A}\limits^{\circ}$-5900$\mathop{A}\limits^{\circ}$ wavelength
range, because LAMOST adopts relative flux calibration and there are lots of
noise at the merge between red and blue segment. The redshift of star is
small, so we use original 1D spectra released by LAMOST directly. Fig. 4 \-
Fig. 13 are the results of eight clustering algorithms on four spectra
features.

From Fig. 4, it is clear that algorithms perform differently on four spectra
characteristics. The clustering accuracies are not high and the highest ones
are only more than 80%. Among the four features, line indices work best to
separate different types of stars, followed by stellar parameters. And 1D
spectra has the same results as PCA features in most methods. But PCA, as
typical linear dimensionality-reduction method, can greatly reduce the
clustering time compared with 1D spectra. Hierarchical clustering and DBSCAN
are the worst and they group most of samples into one cluster because a small
number of outlier will be classified into other categories. GMM performs well
on various data. Compared with partition-based methods, GMM uses multiple
multi-dimensional Gaussian distributions to fit the data which is more in line
with the real characteristic distribution of the data. The main idea of
single-layer SOM is mapping data to two dimensional competitive neural units
with topology structures. In clustering, SOM maps data to 1 $\times$ N neural
units where N is the number of clusters. Its accuracy is rarely different from
K-means, but presents instability because it maps spectra to fewer data points
and needs to set the number of iteration numbers.

Figure 5: t-SNE distribution of true labels for A/F/G/K stars on different
data characteristics. From Left to right is 1D spectra, PCA, Line Indices, and
Stellar Parameters. Different colors represents various types of stars: Blue-A
stars, Yellow-F stars, Gray-G stars, Purple-K stars. Figure 6: t-SNE
distribution of eight clustering algorithms results of A/F/G/K stars on 1D
spectra. Each subgraph represents the results of one clustering algorithm on
the 1D spectra. Different colors in each subgraph represent different classes
in the clustering results. The same color in different subplots is not
necessarily the same class. Figure 7: t-SNE distribution of eight clustering
algorithms results of A/F/G/K stars on PCA. Each subgraph represents the
results of one clustering algorithm on the PCA. Different colors in each
subgraph represent different classes in the clustering results. The same color
in different subplots is not necessarily the same class. Figure 8: t-SNE
distribution of eight clustering algorithms results of A/F/G/K stars on line
indices. Each subgraph represents the results of one clustering algorithm on
the line indices. Different colors in each subgraph represent different
classes in the clustering results. The same color in different subplots is not
necessarily the same class. Figure 9: t-SNE distribution of eight clustering
algorithms results of A/F/G/K stars on stellar parameters. Each subgraph
represents the results of one clustering algorithm on the stellar parameters.
Different colors in each subgraph represent different classes in the
clustering results. The same color in different subplots is not necessarily
the same class.

In order to intuitively understand how data is divided in the clustering,
t-SNE algorithm is used to draw the two-dimensional distribution of datasets
before and after clustering (Fig. 5, Fig. 6 \- Fig. 9). Different colors are
used to distinguish different types. But the same color in different graphs
after clustering does not represent the same type of stars, because clustering
algorithms just divide data into some groups based on the similarity. In the
four plots on true labels (Fig. 5), there are overlaps between adjacent types,
especially on F stars and G stars, G stars and K stars. t-SNE map of PCA
features is basically the same as that of 1D spectra. However, there is a
circle of outliers around 1D spectra which does not exist in PCA features. The
outliers are some incomplete spectra due to observational conditions and this
shows that PCA can reduce the impact of unusual values on the data. The
distribution of stars with the same types is more concentrated in the graph of
the line indices. In the stellar parameters, the graph shows small
aggregations.

From Fig. 6 and Fig. 7, GMM can tackle complicated data distributions compared
with partition-based algorithms. The reason is that, partition-based
clustering algorithms use Euclidean distance to measure the similarity between
samples which makes it easy to identify clusters with spherical distribution,
but they can not consider some special dimensional features. GMM uses Gaussian
distribution to fit features in each dimension, so it has better performance
on 1D spectra and PCA features. While on the dataset of line indices (Fig. 8),
which has a more concentrated distribution of each type of star, partition-
based methods, GMM and SOM have the same clustering results. DBSCAN and
hierarchical algorithms always divide data into one cluster. CFSFDP algorithm
sometimes gets results similar to K-means and sometimes gets poor results.
Because it divides samples into a cluster according to the local density of
samples without considering the global distribution.

Figure 10: Confusion matrix of eight algorithms on 1D spectra. Predicted
label: color and digit in each cell are the consistent probability between
predicted label and true label. Color is in direct proportion to the figure:
bigger numbers and deeper color. Figure 11: Confusion matrix of eight
algorithms on PCA. Predicted label: color and digit in each cell are the
consistent probability between predicted label and true label. Color is in
direct proportion to the figure: bigger numbers and deeper color. Figure 12:
Confusion matrix of eight algorithms on line indices. Predicted label: color
and digit in each cell are the consistent probability between predicted label
and true label. Color is in direct proportion to the figure: bigger numbers
and deeper color. Figure 13: Confusion matrix of eight algorithms on stellar
parameters. Predicted label: color and digit in each cell are the consistent
probability between predicted label and true label. Color is in direct
proportion to the figure: bigger numbers and deeper color.

Meanwhile, we show confusion matrix of each kind of algorithms to observe
misclassified samples straightforward (Fig. 10 \- Fig. 13). Through
experiments, sometimes it is more appropriate to divide four types of spectra
into five clusters, for that a small number of samples will be classified as
one cluster, and we name them "outliers" in the confusion matrix.

Compared with other methods, performance of partition-based algorithms and GMM
are stable and good because they obtain the optimal results from the global
perspective of data. Misclassified phenomenon always exists in the adjacent
clusters and the clustering accuracy of A stars is highest.

####  3.1.2 Classification of Star/Galaxy/Quasar

Classification of Star/Galaxy/Quasar is an essential part in astronomy.
Galaxies and quasars always have large redshift and the positions of line
features are shifted variously. So we deredshift the spectra of galaxies and
quasars in the dataset. There are many excellent methods to deredshift, but
our purpose is to compare clustering algorithms, so we use the redshift values
released by LAMOST directly. To have more shared wavelengths of star, galaxy
and quasar, we choose galaxies and quasars with redshifts less than 0.3.
Meanwhile, we also construct a dataset on original spectra without
deredshifting to compare the performance, their results are shown in Fig. 14
\- Fig. 19.

From Fig. 15, we can know that the distribution of quasars on rest wavelength
frame has less overlaps with stars and galaxies, so the clustering accuracy of
quasars is higher than original spectra (Fig. 18, Fig. 19). But the average
accuracy of rest wavelength frame spectra is not better than original spectra
(Fig. 14). Because galaxies and stars are not separated well. The distribution
of stars and galaxies are not spherical (or circular), so partition-based
methods cannot classify them, but GMM can fit the distribution of each type
very well. Density-based methods (DBSCAN) can distinguish stars from others
while cannot separate galaxies and quasars. Because most stars only have
emission lines, and galaxies and quasars both have absorption lines. SOM is
also a partition-based method when it is used for clustering, but our
experiments find that its clustering results are not stable.

Figure 14: Accuracy of eight algorithms for star/galaxy/quasar on original
spectra and rest wavelength frame spectra. Two bars represent tow spectra
characteristics. Figure 15: t-SNE distribution of star/galaxy/quasar on
original spectra and rest wavelength frame spectra. Left is the original
spectra, right is the rest wavelength frame spectra. Red: Star, Green: Galaxy,
Purple: QSO. Figure 16: t-SNE distribution of the results of algorithms on
original spectra. Each subgraph represents the results of one clustering
algorithm on the original spectra. Different colors in each subgraph represent
different classes in the clustering results. The same color in different
subplots is not necessarily the same class. Figure 17: t-SNE distribution of
the results of algorithms on rest wavelength frame spectra. Each subgraph
represents the results of one clustering algorithm on the rest wavelength
frame spectra. Different colors in each subgraph represent different classes
in the clustering results. The same color in different subplots is not
necessarily the same class. Figure 18: Confusion matrix of eight algorithms on
original spectra. Predicted label: color and digit in each cell are the
consistent probability between predicted label and true label. Color is in
direct proportion to the figure: bigger numbers and deeper color. Figure 19:
Confusion matrix of eight algorithms on rest wavelength frame spectra.
Predicted label: color and digit in each cell are the consistent probability
between predicted label and true label. Color is in direct proportion to the
figure: bigger numbers and deeper color.

####  3.1.3 Classification of Subclasses of A-type Star

1D spectra and stellar parameters are used to cluster subclasses of A-type
star. Considering that there are many subclasses of A-type star, confusion
matrices are not plotted in this section, the t-SNE maps and clustering
results are shown in Fig. 20 \- Fig. 22.

Figure 20: t-SNE distribution of subclasses of A-star with true labels. Left
to right is 1D spectra and stellar parameters. Different colors represent
subclasses of A-star. Figure 21: t-SNE distribution of eight algorithms on 1D
spectra of subclass of A stars. Each subgraph represents the results of one
clustering algorithm on the 1D spectra. Different colors in each subgraph
represent different classes in the clustering results. The same color in
different subplots is not necessarily the same class. Figure 22: t-SNE
distribution of eight algorithms on stellar parameters of subclass of A stars.
Each subgraph represents the results of one clustering algorithm on the
stellar parameters. Different colors in each subgraph represent different
classes in the clustering results. The same color in different subplots is not
necessarily the same class.

t-SNE map (Fig. 20) shows that there are significant overlaps between
different subclasses on 1D spectra. On stellar parameters, there are obvious
gaps between the subclasses with large differences, but the overlap between
the subclasses with small differences is also serious. And none of the
algorithms can separate 1D spectra of subclasses well, even GMM which performs
better on other clustering tasks. On stellar parameters, their performances
are a little better. Stellar parameters with significant difference can be
separated, but very similar ones cannot be separated well. Subclasses with
small number are always divided into that subclasses with large number and
cannot be separated out individually. So it is not a good idea to classify
subclasses of stars by clustering methods.

There are two main difficulties in clustering subclasses of A-type star. One
is that the spectra of some subclasses are very similar, the other is the
number of different subclasses varies greatly. This will cause the classifier
to make wrong classification, even the pipline of LAMOST which distinguishes
subclasses by template matching. Compared to 1D spectra, stellar parameters
are more reliable to classify subclasses. The poor classification results of
the experiment is not only due to the clustering algorithms but also possibly
because the errors exist in the input labels, so clustering accuracies in this
task do not make much sense. But the experiment could provide a clear view of
distributions of subclasses of A-type star and help researchers to study the
principle of clustering on them.

####  3.1.4 Classification on Matching Sources of LAMOST and SDSS

There are lots of overlapping sources of LAMOST and SDSS, we construct two
datasets from them to compare clustering performances. Spectra of SDSS are
selected from DR16. The task is stellar classification and the datasets are
shown in Table 6. The clustering results are shown in Fig. 23 \- Fig. 28 .

From the t-SNE maps of LAMOST and SDSS (Fig. 24), it can be seen that K-type
star overlap less with other three types. However, A-type and F-type, F-type
and G-type have more overlaps. The map of SDSS is clearer than that of LAMOST,
it has fewer noise points, mainly because there is always some noise in the
splicing at the red and blue segments of LAMOST spectra.

In the process of clustering, we set 5 cluster centers for 4 classes of stars
so that clustering algorithms can separate F and G type stellar spectra, but
meanwhile, K-type star may be divided into two categories.

The clustering accuracy of SDSS is slightly better than that of LAMOST (Fig.
23). This is mainly due to the calibration quality, compared to SDSS, the
fiber-to-fiber sensitivity variations in LAMOST sometimes leads to wrong
overall calibration which will affect the classification of spectra. And the
mismatch between red/blue segments in some spectra also introduces problems.
In the clustering results, hierarchical clustering and DBSCAN also cluster
most spectra into one class. One thing worth noticing is that GMM can classify
G-type stars very well both in LAMOST and SDSS even they have parts of spectra
similar to F-type stars. SOM and CFSFDP cannot separate F-type and G-type
stars well on LAMOST spectra, but can separate them well on SDSS, this shows
that the degree of spectra discernment of SDSS is higher than LAMOST.

Table 6: Datasets of LAMOST and SDSS

| Survey |  |  | Type |  |  | Data volume |
---|---|---|---|---|---|---|---|---
| LAMOST |  |  | A |  |  | 5824 |
F |  |  | 5380 |
G |  |  | 4151 |
K |  |  | 6240 |
| SDSS |  |  | A |  |  | 5797 |
F |  |  | 5355 |
G |  |  | 4144 |
K |  |  | 6229 |

Figure 23: Accuracy of eight algorithms on LAMOST and SDSS. Two bars represent
spectra from LAMOST and SDSS. Figure 24: t-SNE distribution of true labels
from LAMOST and SDSS. Left to right is LAMOST and SDSS. Different colors
represent A, F, G, K stars. Figure 25: t-SNE distribution of eight algorithms
on LAMOST spectra. Each subgraph represents the results of one clustering
algorithm on the spectra from LAMOST. Different colors in each subgraph
represent different classes in the clustering results. The same color in
different subplots is not necessarily the same class. Figure 26: t-SNE
distribution of eight algorithms on SDSS spectra. Each subgraph represents the
results of one clustering algorithm on spectra from SDSS. Different colors in
each subgraph represent different classes in the clustering results. The same
color in different subplots is not necessarily the same class. Figure 27:
Confusion matrix of eight algorithms on LAMOST spectra. Predicted label: color
and digit in each cell are the consistent probability between predicted label
and true label. Color is in direct proportion to the figure: bigger numbers
and deeper color. Figure 28: Confusion matrix of eight algorithms on SDSS
spectra. Predicted label: color and digit in each cell are the consistent
probability between predicted label and true label. Color is in direct
proportion to the figure: bigger numbers and deeper color.

###  3.2 Performance of algorithms on different quality of spectra

The quality of spectra observed by telescope will be affected by the
observation conditions, and the signal-to-noise ratio of the spectra will be
low if the conditions are not good. Noise in the spectra will increase the
difficulty of spectral analysis, so it is important to find out clustering
methods which are insensitive to noise. In this section, three spectral
datasets of different S/Ns are constructed to study the robustness of
clustering algorithms.

Table 7: Three datasets with different S/Ns

| S/N of g-band | S/N of i-band | Size | Data
---|---|---|---|---
Dataset H | >30 | >30 | 20000 | PCA
Dataset M | 10-30 | 10-30 | 20000 | PCA
Dataset L | <10 | <10 | 20000 | PCA

  * 1

H, M and L represent high S/N, medium S/N and low S/N.

  * 2

There are equal numbers of four types of stars (A,F,G,K) in each dataset.

Three different S/Ns include high (g band > 30 and i band > 30), middle (g
band:10-30 or i band:10-30) and low (g band < 10 and i band < 10). The task of
this comparative experiment is to cluster stellar spectra because the number
of galaxy and quasar spectra with S/N > 30 is very small. Spectra with S/N >
10 are considered qualified, so we also add them to the accuracy chart. Table
7 shows the configuration of datasets used in this section.

Figure 29: Accuracy of eight algorithms on spectra of different S/Ns. Four
bars represent four S/Ns.

Fig. 29 shows the average accuracy of eight algorithms on different S/Ns
datasets and the accuracy is proportional to the signal-to-noise ratio. As
before, GMM still has the highest accuracy. Fig. 30 is the t-SNE distribution
of datasets and Fig. 31 \- Fig. 36 are the clustering results and their
confusion matrices.

Figure 30: t-SNE distribution of true labels of A/F/G/K stars on different
S/Ns. Left to right: >30, 10-30, <10\. Four colors represent A, F, G, K stars.
Figure 31: t-SNE distribution of eight algorithms on high S/N of spectra. Each
subgraph represents the results of one clustering algorithm on the high S/N of
spectra. Different colors in each subgraph represent different classes in the
clustering results. The same color in different subplots is not necessarily
the same class. Figure 32: t-SNE distribution of eight algorithms on medium
S/N of spectra. Each subgraph represents the results of one clustering
algorithm on the medium S/N of spectra. Different colors in each subgraph
represent different classes in the clustering results. The same color in
different subplots is not necessarily the same class. Figure 33: t-SNE
distribution of eight algorithms on low S/N of spectra. Each subgraph
represents the results of one clustering algorithm on the low S/N of spectra.
Different colors in each subgraph represent different classes in the
clustering results. The same color in different subplots is not necessarily
the same class.

Data distributions vary in different S/Ns. There seems to be less overlaps in
S/N $>30$, but it is serious in S/N $<10$, and S/N:10-30 is between the above
two. Results of partition-based algorithms also tend to be circle. GMM
performs very well on the data with S/N $>10$ and the clustering accuracy of A
stars reaches 92.5%. But there is a phenomenon that misclassification always
occurs in backward types but rarely happens in the forward types. For example,
in the results of GMM on high S/N, 27 % of G-type stars are classified as
K-type, but only 0.5 % of K-type stars were classified as G-type and 19.8 % of
K-type stars are grouped separately. Compared with K-means, GMM is more
susceptible to signal-to-noise ratio, although GMM has good performance.

Since there is excessive noise in the spectra with S/N < 10, clustering
algorithms perform poorly, and the results are of little use for the spectral
classification tasks. So for the low S/N spectra, other effective methods
should be used for analysis.

Figure 34: Confusion matrix of eight algorithms on high S/N of spectra.
Predicted label: color and digit in each cell are the consistent probability
between predicted label and true label. Color is in direct proportion to the
figure: bigger numbers and deeper color. Figure 35: Confusion matrix of eight
algorithms on medium S/N of spectra. Predicted label: color and digit in each
cell are the consistent probability between predicted label and true label.
Color is in direct proportion to the figure: bigger numbers and deeper color.
Figure 36: Confusion matrix of eight algorithms on low S/N of spectra.
Predicted label: color and digit in each cell are the consistent probability
between predicted label and true label. Color is in direct proportion to the
figure: bigger numbers and deeper color. Figure 37: Confusion matrix of eight
algorithms on S/N $>$10\. Predicted label: color and digit in each cell are
the consistent probability between predicted label and true label. Color is in
direct proportion to the figure: bigger numbers and deeper color.

###  3.3 Performance of algorithms on different data volumes

Table 8: Four datasets with different volumes of spectra.

| Size | S/N | Data
---|---|---|---
Dataset s | 8000 | >10 | PCA
Dataset m | 20000 | >10 | PCA
Dataset l | 40000 | >10 | PCA
Dataset e | 80000 | >10 | PCA

In machine learning tasks, the size of dataset is always a factor that affects
the results. When the amount of data is small, overfitting will occur. But if
the size is too large, some algorithms with high time complexity will take a
long time to run and some algorithms may not be able to run. So, this section
performs eight clustering algorithms on spectral datasets size of 8000, 20000,
40000 and 80000 to analyse their effectiveness and efficiency. The clustering
task is also stellar classification and the number of four types of stars in
the dataset is the same. Table 8 shows the configuration of four datasets.

Figure 38: Accuracy of algorithms on different data volumes. Four bars
represent four data volumes.

Fig. 39 shows the t-SNE distribution of 8000, 20000, 40000, 80000 of data.
Some algorithms do not have the accuracy of size 80000 because they need a lot
of memory space and 64GB is not enough for them to run. Fig. 40 \- Fig. 46 are
the clustering results and confusion matrices and Fig. 38 shows their average
accuracies.

Results show that data volume makes a little influence on accuracy of most
algorithms in our experiments, and it can also be explained by t-SNE. CFSFDP
performs worse with data volume increasing. Because it identifies cluster
centers by neighborhood density of adjacent data rather than an iterative
method. So data volume has a large impact on CFSFDP.

Figure 39: t-SNE of eight algorithms on data volumes of 8000, 20000, 40000 and
80000. Four colors represent A, F, G, K stars. Figure 40: t-SNE of eight
algorithms on data volume of 8000. Each subgraph represents the results of one
clustering algorithm on the data volume of 8000. Different colors in each
subgraph represent different classes in the clustering results. The same color
in different subplots is not necessarily the same class. Figure 41: t-SNE of
eight algorithms on data volume of 20000. Each subgraph represents the results
of one clustering algorithm on data volume of 20000. Different colors in each
subgraph represent different classes in the clustering results. The same color
in different subplots is not necessarily the same class. Figure 42: t-SNE of
eight algorithms on data volume of 40000. Each subgraph represents the results
of one clustering algorithm on data volume of 40000. Different colors in each
subgraph represent different classes in the clustering results. The same color
in different subplots is not necessarily the same class. Figure 43: Confusion
matrix of eight algorithms on data volume of 8000. Predicted label: color and
digit in each cell are the consistent probability between predicted label and
true label. Color is in direct proportion to the figure: bigger numbers and
deeper color. Figure 44: Confusion matrix of eight algorithms on data volume
of 20000. Predicted label: color and digit in each cell are the consistent
probability between predicted label and true label. Color is in direct
proportion to the figure: bigger numbers and deeper color. Figure 45:
Confusion matrix of eight algorithms on data volume of 40000. Predicted label:
color and digit in each cell are the consistent probability between predicted
label and true label. Color is in direct proportion to the figure: bigger
numbers and deeper color. Figure 46: K-means, GMM, SOM on data volume of
80000. Predicted label: color and digit in each cell are the consistent
probability between predicted label and true label. Color is in direct
proportion to the figure: bigger numbers and deeper color.

Although data volume is an important factor affecting the results for
supervised classification tasks, it is not the case for clustering. And most
clustering algorithms are not sensitive to data volume. Because clustering is
to find the optimal division results on all data and it does not need a large
amount of data to improve generalization. But some algorithms cannot run when
the size of data is too large, because they need to calculate the distance
matrix which requires a large memory space.

###  3.4 Performance of algorithms on outliers detection

In the big data era, it is crucial to find outliers. Although outliers
sometimes have bad effect on the information we need, they also could provide
some useful knowledge to many fields, e.g. credit card fraud, e-commerce
crime, medical diagnosis, etc. And rare objects detection is also an
attractive research in astronomy. In addition to typical outlier methods,
clustering is also an effective tool to detect outliers.

In this section, we choose three types of special spectra to verify the
ability of clustering methods to find these outliers, (a) physical outliers:
carbon star; (b) overlapping sources: double star; (c) artefacts outliers: bad
merging of red and blue segments. And we use two types of normal spectra, one
containing only stars and the other one containing stars, galaxies and
quasars. Put outliers into these normal spectra and observe the effect of
detecting them by different algorithms. We also compare the ability of
detecting outliers (carbon stars) from datasets on different spectra features,
different qualities of spectra and different data volumes. Their data
introductions are shown in Table 9 and Table 10.

In this task, K-means, GMM, CFSFDP, DBSCAN, Local Outier Factor (LOF) are
selected to detect outliers. LOF is a typical outlier detection algorithm, so
it is also included in our comparison. Various clustering algorithms have
different ways to find outliers. In the experiments, K-means regards samples
far away from most cluster centers as outliers. Samples with small
probabilities of every Gaussian distribution can be outliers in GMM. Data in
left-top corner of $\rho$-$\delta$, core graph in CFSFDP, are outliers. DBSCAN
can detect all outliers with small densities and group normal samples into
some clusters. In LOF we calculate and rank outlier factor of each sample,
Samples with large outlier factors are selected as outliers.

Table 9: Datasets of outlier detection on two types of normal objects

|  | Normal Spectra |  | Outliers
---|---|---|---|---
|  |  |  | Carbon stars:100
Dataset outlier 1 |  | A:F:G:K=2500:2500:2500:2500 |  | Double stars:100
|  |  |  | Artefacts:100
|  |  |  | Carbon stars:200
Dataset outlier 2 |  | Star:Galaxy:Quasar=10000:6500:3300 |  | Double stars:200
|  |  |  | Artefacts:200

Table 10: Nine datasets with three criteria to detect carbon stars.

| Dataset | Abnormal spectra size | Rare spectra size | S/N | Data type
---|---|---|---|---|---
Different Spectral Characteristics | Dataset ds | 20000 | 100 | >10 | 1D Spectra
Dataset dp | 20000 | 100 | >10 | PCA
Dataset dl | 20000 | 100 | >10 | Line Indices
Different S/Ns | Dataset rh | 20000 | 100 | >30 | PCA
Dataset rm | 20000 | 100 | 10-30 | PCA
Dataset rl | 20000 | 100 | <10 | PCA
Different Data Volumes | Dataset vs | 8000 | 40 | >10 | PCA
Dataset vm | 20000 | 100 | >10 | PCA
Dataset vl | 40000 | 200 | >10 | PCA

We evaluate their performances by ROC curve (receiver operating characteristic
curve, or sensitivity curve), which is a widely used index to measure the
ability of outliers detection. The horizontal axis of the curve is false
positive rate, it is the ratio of normal samples in detected targets to all
normal samples. Vertical axis is true positive rate which is the ratio of true
outliers in detected targets to all true outliers. The closer the curve is to
the upper left corner of the graph, indicates the method has a stronger
ability to detect outliers.

From the ROC curves of results (Fig. 47 and Fig. 48), we can know that
compared with the classic outlier algorithm LOF, clustering algorithms also
perform well in outlier detection tasks. It is easier to detect outliers in
stellar spectra than in all spectra (including stars, galaxies, and quasars).
The reason is that spectra of quasars are also unusual and very few in number,
so they are often detected as outliers too. The performance of algorithms on
detecting spectra with bad-merge in red and blue segment is better than carbon
star and double star only except GMM. Some badly merged spectra have a
wavelength range without flux values, and the Euclidean distance between them
and normal spectra is far, so they can be detected easily. GMM uses several
Gaussian distributions to fit data and does not measure the similarity between
spectra, so its ability to detect badly merged spectra is poor.

Figure 47: ROC curves of detecting three types of outliers from stars.
Different colors represents five algorithms. X axis: outlier detection FPR, Y
axis: PR. Curves near left-top are prefer to find outliers.  Figure 48: ROC
curves of detecting three types of outliers from all spectra (stars, galaxies
and quasars). Different colors represents five algorithms. X axis: outlier
detection FPR, Y axis: PR. Curves near left-top are prefer to find outliers.
Figure 49: ROC curves of detecting carbon stars from stars on different
features. Different colors represents five algorithms. X axis: outlier
detection FPR, Y axis: PR. Curves near left-top are prefer to find outliers.

Fig. 49 reveals desirable results of five algorithms on finding outliers on
different spectra features. Algorithms perform worse on 1D spectra than on PCA
features, because normal 1D spectra sometimes contain few incomplete spectra
which are usually regarded as outliers. In Fig. 49, clustering algorithms can
get more satisfactory consequences than LOF algorithm. It indicates that
clustering methods also have good ability to outliers detection. Another
interesting thing is that though DBSCAN has difficulty on clustering spectra,
it is able to detect outliers. K-means and GMM select spectra far away from
cluster centers as outliers and they both have good performances. In our
experiments, we find that it is not a good idea to regard the samples in
clusters with small number in the clustering results as outliers. By
clustering methods, clusters or cluster centers are found, and samples
farthest from the cluster centers are more likely to be outliers.

Figure 50: ROC curves of detecting carbon stars from stars on different S/Ns.
Different colors represents five algorithms. X axis: outlier detection FPR, Y
axis: PR. Curves near left-top are prefer to find outliers.

Fig. 50 reveals the effect of S/N on outlier detection, and the quality of
spectra has a great impact on the results. In the high S/N spectra, each
algorithm can find outliers well. But in the low S/N spectra, algorithms
basically have no ability to find outliers. K-means and DBSCAN are inferior to
others on S/N > 30\. While K-means is superior among all algorithms on S/N <
10 and S/N:10-30 that shows a little robustness.

Figure 51: ROC curves of detecting carbon stars from stars on different data
volumes. Different colors represents five algorithms. X axis: outlier
detection FPR, Y axis: PR. Curves near left-top are prefer to find outliers.

In the experiment of data volumes (Fig. 51), the proportion of normal spectra
and outliers is 1000:5. When the size of dataset is large, clustering
algorithms are better than LOF, since there are more abnormal spectra in large
dataset, and LOF will treat them as outliers.

Figure 52: Mapping 400 spectra of A, F, G, K stars to a 20 $\times$ 20 grid
with SOM. The darker color means more spectra mapped in this cell and the
blank area means no spectra mapped here. Spectra separated by blank space are
very different on shape.

Rare objects make significant sense to research in astronomy. To search some
kinds of rare objects, we can use supervised classification methods. However,
it is difficult to obtain a robust model due to the small number of rare
samples. In this situation, clustering methods are good choices to deal with
this task. And SOM is also a commonly used method, which can map similar
spectra to the same or close locations on two-dimensional competing layer with
topology structure. Candidates of rare objects can be adjacent spectra of
known rare spectra in the map.

In Fig. 52, 400 spectra including A, F, G and K stars are mapped on a complete
layer of 20 $\times$ 20\. Deeper background means more spectra are mapped on
this position. Spectra with similar shapes are closer in the map and spectra
separated by blank are more different than adjacent spectra. In this graph, A
stars, F stars, G stars and K stars are located in different positions
respectively. And there are gaps between anomalous spectra and normal spectra,
so we also can use SOM to detect outliers.

Fig. LABEL:fig:SOM_rare shows the idea of searching carbon stars by SOM. In
the figure, grey points represent normal objects while red points are known
carbon stars and purple points are unknown carbon stars. We can recognize
adjacent data of red points as rare object candidates and then do further
validation manually.

##  4 Source Code and Manual

Table 11: Source codes of algorithms used in this paper.

Code Type |  |  | Methods |
---|---|---|---|---
Spectra processing |  |  | PCA |
|  | Extract line indices |
Clustering algorithms |  |  | K-means |
|  | K-mediod |
|  | K-means_dp |
|  | GMM |
|  | SOM |
|  | CFSFDP |
|  | Hierarchical clustering |
|  | DBSCAN |
Others |  |  | t-SNE |

Source codes of clustering algorithms used in this paper is provided
(https://github.com/shichenhui/SpectraClustering) and Table 11 presents main
codes list. The specific steps and precautions of code usage are also given in
the above link. Clustering results of some algorithms will be influenced by
the parameters so we adjust parameters and find optimal parameters.

The code is written in Python which is widely used for machine learning and
data analysis. The dependent packages of the codes are: numpy (Harris et al.,
2020), sklearn, matplotlib (Hunter, 2007), pandas, scipy. Each algorithm is
organized in the following steps: 1) load training datasets; 2) plot the t-SNE
distribution of the data; 3) cluster data; 4) plot t-SNE distribution of
results; 5) evaluate clustering results.

These codes load data from *.csv files which store tabular data in text and a
row of data is a spectrum. Users need to convert their spectra data to such a
format or modify the data loading mode. Some of algorithms are implemented
directly through the sklearn package. It is efficient to adopt best algorithms
for astronomical spectral analysis.

The number of clusters K is the main parameter of partition-based methods and
it has a great influence on the clustering results. Sometimes we can set the
number of clusters K to be more than the actual number, because some rare
objects or outliers will be classified as clusters. In our experiment, one or
two more than actual number of clusters can be good results. If we have no
idea of the number of clusters, t-SNE or UMAP method could be used to have a
rough idea of how the data is aggregated. Euclidean distance is used in
K-means to measure the similarity between samples and it can be replaced by
other similarity measures, like Manhattan distance, distance in manifold space
and others. This should be selected according to the characteristics of the
data. K-mediod chooses a sample as cluster center which has the smallest sum
of distances from all other samples in the cluster instead of an average
point. This can reduce the impact of a small number of outliers on the cluster
center. But the time complexity is higher than K-means. As for the selection
of initial cluster centers, CFSFDP method is a good choice.

Neighborhood radius and the number of min-samples in a cluster are two main
parameters in DBSCAN. Generally, it requires multiple tests to obtain optimal
values. To obtain good clustering results, DBSCAN has two requirements for
data, one is the degree of sample aggregation of different clusters cannot be
very different, otherwise suitable parameter neighborhood radius will not be
found. The other is that there needs to be obvious separation between
clusters, so that the clusters are not grouped together. DBSCAN performs bad
on spectra because these two conditions are not satisfied. However, in the
tasks of analysing spatial structure, DBSCAN can get good results.

SOM needs to set the number of iterations, usually hundreds of times to
achieve convergence results. It can map data into 1D or 2D space. 1D space is
convenient for analyzing the physical properties and the input data must be a
small number of relevant features. But for clustering, the results in 1D space
are less stable. 2D space can effectively get the data distribution for
clustering or outliers detection. The shape of grids mapped with SOM could be
rectangular or hexagonal, and hexagonal grids can represent more detailed
distribution information. An important parameter in GMM is covariance-type
which describes the type of covariance, four types could be chosen,
"spherical", "tied", "diag" and "full". "full" is the most flexible type
meaning that each cluster can have its own arbitrary shape. But it may not
give the best results in real applications. The choice of covariance-type
requires experience or testing to obtain optimal results. Our experimental
results found that the "tied" and "diag" were better than "spherical" and
"full".

Agglomerative and divisive are two models of hierarchical clustering. In our
experiment, hierarchical clustering could not work well but it can be combined
with other algorithms. CFSFDP calculates two values ($\delta$ and $\rho$) for
each sample to select cluster centers and outliers. And we can use them to
cluster or detect outliers with decision graph method.

##  5 Discussion

Clustering methods used in recent astronomical spectroscopic studies are
investigated in this paper and they are briefly introduced. Considering the
different data used in respective research, it is difficult to find the
strength of each method. So we construct some unified spectral datasets to
analyse their advantages under different conditions. Some clustering methods
can also be used to find outliers, so datasets also include normal spectra and
rare spectra to test their performance on searching for outliers.

Through the experiments, we found that GMM performs better than others on 1D
spectra and PCA features. On stellar spectra line indices, GMM performs as
well as partition-based methods. Spectra line indices can extract stellar
spectra features effectively and the clustering results of many methods on
line indices are better than 1D spectra. Density-based algorithms and
hierarchical clustering perform poorly on spectra related datasets, although
they have many advantages on benchmark datasets. The reason is that, in
spectra dataset, there is no clear separation between different types of
spectra and the density distribution of clusters may be different. So it is
impossible to find appropriate parameters for efficient clustering.

Although for supervised classification algorithms, overfitting can be reduced
when the dataset is large. But for clustering, the amount of data has little
influence on the clustering results. And some algorithms can not run on the
large datasets. GMM works well on spectra, but its running time on large
datasets is much higher than other methods. K-means is still a good choice if
researchers want to make a fast clustering of the data.

The experiments also show that clustering methods are very effective to find
abnormal spectra. Multiple cluster centers need to be found first, then the
samples far from the cluster centers can be regarded as outliers and these
methods are very robust. When researchers want to observe the distribution of
spectra data, dimensionality reduction and visualization methods are very
intuitive, such as t-SNE, UMAP and SOM. And SOM is also widely used in
astronomy to find special spectra.

The purpose of this paper is to provide an analysis of the advantages and
disadvantages of different algorithmic ideas. The methods we use are all basic
algorithms and the data preprocessing is also general steps. Improved
algorithms and better preprocessing will be more effective to improve
clustering performance, such as extinction processing, feature extraction,
etc.

## Acknowledgements

We would like to thank the reviewer, Igor Chilingarian, for his valuable
comments and suggestions to improve the paper.

The Guo Shou Jing Telescope (the Large Sky Area Multi-Object Fiber
Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built
by the Chinese Academy of Sciences. Funding for the project has been provided
by the National Development and Reform Commission. LAMOST is operated and
managed by National Astronomical Observatories, Chinese Academy of Sciences.

The work is supported by the National Natural Science Foundation of China
(Grant Nos. U1931209), Key Research and Development Projects of Shanxi
Province (Grant Nos. 201903D121116), and the central government guides local
science and technology development funds (Grant No. 20201070). Fundamental
Research Program of Shanxi Province(Grant No. 20210302123223,
202103021224275).

## Data Availability

Experimental data used for this study is obtained from The Guo Shou Jing
Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope,
LAMOST) Data Release 8 (http://www.lamost.org/lmusers/) and Sloan Digital Sky
Survey (SDSS) Data Release 16 (https://www.sdss.org/dr16/). Codes used in this
paper is also available online at
(https://www.github.com/shichenhui/SpectraClustering).

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