Data mining techniques on astronomical spectra data. II : Classification Analysis

Template matching is a flexible and relative straightforward technique. The classification process of template matching is to build a template database for each class, then divide the unknown data into the most similar template data (Rosenfeld & Vanderbrug, 1977). In astronomy, template matching matches spectral lines with templates and there is no training stage. So it has been widely applied in celestial object classification, redshift estimation, stellar parameters estimation and other projects (SubbaRao et al., 2002; Lupton et al., 2002; Westfall et al., 2019; Liu et al., 2015a; Zhao et al., 2012). Table 1 shows the main astronomical spectral investigations of template matching.

Template matching is often used to classify stars, galaxies and quasars and further analyze other properties of spectra. Duan et al. (2009) used spectral line matching to identify the observed spectra class and achieved a high accuracy about 92.9%, 97.9% and 98.8% for stars, galaxies and quasars, respectively. They also obtained a byproduct: high precision of redshift. Gray & Corbally (2014) used template matching for Morgan-Keenan (MK) classification and built an expert computer program imitating human classifiers. It was automatic and had comprehensible results. Wang et al. (2018) used the line intensity to classify spectra (Martins, 2018; Wang, 2019).

Template matching is also used to find peculiar objects like supernovas, M dwarfs, B stars and M giants, Double-peak emission line galaxies (Sako et al., 2018; Zhong et al., 2015b, a; Ramírez-Preciado et al., 2020; Maschmann et al., 2020). Zhong et al. (2015b) applied a template-fit method to identify and classify late-type K and M dwarfs from LAMOST. 2612 late-K and M dwarfs were identified which can help researchers to investigate the chemo-kinematics of the local Galactic disk and halo. Maschmann et al. (2020) used two Guassian functions to fit the emission lines to find double-peak candidates and finally they found 5663 double-peak emission line galaxies at z < 0.34. Meanwhile, there is an important issue for rare object identification using template matching, that is, classifiers require sufficient high-quality spectral templates. In order to obtain ample qualified rare templates, researchers tried to construct new templates (Kesseli et al., 2017; Wei et al., 2014).

Template matching has been widely used in lots of surveys. However, some spectra are of low quality, template matching can not obtain precise results on redshift estimation, stellar parameters estimation and classification (Podorvanyuk et al., 2015). Hence, for the inferior quality spectra, other machine learning algorithms like SVM based classification algorithms and artificial neural network (ANN) based classification algorithms are employed to get robust results. The other defect of template matching is that, for rare objects, we do not have enough samples to get representative template spectra. So rare objects are often misclassified.

K-Nearest Neighbor (KNN) based classification algorithms assign labels to the target based on the majority labels of its K closest objects. More in depth explanations of KNN based classification algorithms can be found in Zhang & Zhou (2007); Deng et al. (2016). The main ideas are shown in Fig. 3. They are intelligible and their time complexity is linear to the data volume. Taking these into consideration, KNN based classification algorithms have been used to classify astronomical spectra and combined with other methods to improve classification accuracy. Table 2 displays the major astronomical applications of KNN based classification algorithms.

KNN can be used for stellar classification. Brice & Andonie (2019b) used KNN and random forest (RF) for MK classification of stellar spectra. Considering high dimensional spectra data, they extracted absorption lines of spectra to reduce the time complexity. The results showed that KNN had a shorter training time but a longer testing time than RF. KNN could obtain the same accuracy as RF when using hybrid methods or oversampling balancing techniques. But for O-type stars which are few in the data sets, KNN performed poorly. This is a common phenomenon in most classification applications, that is, it is hard to get good classification results in unbalanced data sets.

For complex spectral classification tasks, it is not a good choice to only use the basic KNN based classification methods. Because from the comparison results of different classification methods, researchers found that good results were often produced by SVM or RF, rather than KNN (Xiao-Qing & Jin-Meng, 2021; Arsioli & Dedin, 2020; Pérez-Ortiz et al., 2017). To obtain better results, some improvements to KNN were also proposed, like KNN-DD to detect known outliers (Borne & Vedachalam, 2012) and ML-KNN: a lazy learning approach to multi-label learning (Zhang & Zhou, 2007). In addition, many researchers combined KNN with other methods to reduce the misclassification rate, like SVM+KNN to correct some prediction errors (Peng et al., 2013). And its classification accuracy of quasars reached 97.99%.

KNN based classification algorithms are arguably simple and efficient machine learning algorithms. And they have been demonstrated to be competitive methods because of the high accuracy under the premise of their simplicity and rapidness (Fushiki, 2011; Sookmee et al., 2020; Guzmán et al., 2018). They use Euclidean distance to measure the similarity of data and perform better on low dimensional data. After preprocessing high dimensional spectra, KNN based classification algorithms can also be applied in astronomy, such as star/galaxy/quasar classification and classification of small radial velocity objects. However, from the investigated researches, KNN based classification algorithms mainly suffer from three disadvantages: 1) the only hyper-parameter K is difficult to determine. 2) KNN based classification algorithms are ineffective for star classification because of the misclassification between adjacent classes. 3) Unbalanced data is another challenge for KNN based classification algorithms. Recently, some algorithms like Synthetic Minority Over-Sampling Technique (SMOTE) have been employed to adjust the data volume distributions to solve the third issue.

Support vector machine (SVM) based classification algorithms are binary classifiers that learn a boundary from the training data to classify two types of data. And multiple binary SVM classifiers can be integrated into a multi-label classifier. Generally, the classification precision and robustness of SVM based classification algorithms are relatively superior to other single classifiers (non-ensemble algorithms). Table 3 shows the main astronomical researches of SVM based classification algorithms.

Spectral classification is a common astronomical task for SVM based classification algorithms (Brice & Andonie, 2019a; Barrientos et al., 2020; Tao et al., 2018; Guzmán et al., 2018; Liu et al., 2015b; Liu, 2021; Liu et al., 2018). Solarz et al. (2012) used the infrared information to separate galaxies from stars and the accuracy reached 90% for galaxies and 98% for stars. Małek et al. (2013) trained a SVM classifier to classify stars, active galactic nucleus (AGNs) and galaxies using spectroscopically confirmed sources from the VIPERS and VVDS surveys. In the stellar spectral classification, A stars and G stars can be identified easily, while it was hard to identify O, B and K stars. Because the differences in the spectral features between late B type and early A type stars or between late G and early K type stars were very weak (Liu et al., 2015b). Dong & Pan (2020) used SVM and cascaded dimensionality reduction techniques to classify spectra, which is better than principal component analysis (PCA) or t-distributed stochastic neighbor embedding (T-SNE).

In addition to classification, SVM based classification algorithms can also be used for peculiar spectra identification (Qu et al., 2020). More depth details of rare objects such as carbon stars and variable objects can be found in Gigoyan et al. (2012); Green (2013); Baran et al. (2021); Maravelias et al. (2022); Kong et al. (2018); Kou et al. (2020); Solarz et al. (2017); Qu et al. (2020). Solarz et al. (2020) detected anomalous in the mid-infrared data using one-class SVM. Among the 36 identified anomalous, 53% of them were low redshift galaxies, 33% were particular quasi-stellar objects (QSOs), 3% were galactic objects in dusty phases of their evolution and 11% were unknown objects. The main problem in this task is that the number of some types of rare samples is far smaller than normal samples. So the classification model can not identify the rare classes well. There are also many approaches to solve this problem, like data augmentation, over-sampling, etc. Liu & Zhao (2017) proposed an entropy based methods for unbalanced spectral classification. And the performance was better than using KNN and SVM directly.

SVM based classification algorithms are binary classifiers with rigorous mathematical theory. They try to find the optimal separating hyperplane to divide data into two categories (Fig. 4). For multi-label classification, One-VS-One (OVO), One-VS-All (OVA) and Directed Acyclic Graph (DAG) are the main tactics to train different classifiers. There are two important tricks of SVM based classification algorithms. One is soft margin which uses a robust partition boundary to separate two types of data and tolerates the misclassification of some abnormal data. The other one is kernel function which can map linearly inseparable data into a linearly separable high-dimensional space. However, SVM based classification algorithms adopt kernel matrix to measure the similarity of samples. So the computation time and space are two vital issues for classifiers on large amounts of data.

SVM based classification algorithms are promising classification methods that have a convincing theory and robust results. They have attracted a good deal of attention due to their high accuracy in multi-dimensional space and already have been applied to astronomical spectral classification, such as star/galaxy/quasar classification, stellar spectral classification and novelty detection. However, the time complexity of SVM based classification algorithms is exponentially related to the training size. So, it is indispensable to pre-process astronomical spectra to reduce the training time.

Decision Tree (DT) based classification algorithms (Quinlan, 1996) are essential in machine learning algorithms. Their leaves represent classification results and internal nodes of branches are regarded as criteria for distinguishing objects. The graphical representation of decision tree is shown in Fig. 5. Decision tree and its variants have been applied in astronomy and many other fields (Bae, 2014; Li, 2005; Czajkowski et al., 2014; Zhao & Zhang, 2008). Table 4 shows the main astronomical investigations of decision tree based classification algorithms.

Decision tree based classification algorithms have been widely used for astronomical classification due to their good interpretability of classification results. Here are some examples of decision tree for astronomical classification. Morice-Atkinson et al. (2018) explored the classification boundaries of star and galaxy through decision tree. This visualized the classification process of the star-galaxy and helped astronomers understand the decision rules of celestial classification. Franco-Arcega et al. (2013) used parallel decision trees to classify different types of objects and evaluated the performance of classification results. Vasconcellos et al. (2011) applied 13 different decision tree algorithms to analyse the classification performance of star/galaxy, and the functional tree algorithm yielded the best results.

According to the astronomical researches using decision tree based classification algorithms, there are three tips to improve the classification performance. Firstly, effectively pre-processing the raw observational spectra will assist and speed up the classification, such as noise reduction and data compression. Secondly, extracting valid features is also important. Most familiar approaches normalize and standard spectra data by prevalent methods without additional operations (Pichara et al., 2016; Vasconcellos et al., 2011), yet these simple approaches will have high computational costs and could not improve classification accuracy effectively. So other valid features may be better to improve classification performance, such as line indices and astronomical-specific features. Thirdly, searching for appropriate methods is another vital approach to improving classification performance. Compared with other typical methods, RF performed best both on accuracy and time consuming in Xiao-Qing & Jin-Meng (2021); Brice & Andonie (2019b); Flores R. et al. (2021), etc. Alternatively, integration of decision tree and other conventional classification methods can enhance the superiority of feature selection and results interpretation respectively (Ivanov et al., 2021). However, heterogeneous data, large redshift objects, other stellar parameters regression and misclassification are still challenges for decision tree in astronomical research. These need to be solved in the future.

Iterative Dichotomiser 3 (ID3), C4.5 and Classification and Regression Trees (CART) are three widespread methods based on decision tree. ID3 adopts Information Gain (IG) as the node selection criterion for classification. While C4.5 chooses Information Gain Ratio (IGR) to alleviate the flaws of ID3 (IG: discrete data, incomplete attribution, overfitting, etc). Another upgraded method is CART which can be used for both classification and regression. It employs the Gini Index as a node selection standard instead of Information Entropy. The main advantage of decision tree based classification algorithms is interpretability of results, which is very helpful for astronomers to analyse the features of astronomical objects. And the disadvantage is that we often obtain a complex model which will be overfitting on the training data. So pruning parameters is always required to reduce overfitting.

Ensemble learning (Freund & Mason, 1999) combines multiple weak classifiers into a strong classifier to solve a task together. Generally, ensemble learning methods are sorted into bagging methods decreasing variance, boosting methods reducing deviation and stacking methods increasing prediction accuracy. Compared with the single decision tree, ensemble learning methods are more often used in astronomy.

Bagging usually trains different models with various training sets respectively and chooses one strategy to unify consequences. The principle of bagging is shown in Fig. 6. Random Forest is the most notable bagging method which consists of several unrelated decision trees. Fig. 7 describes the principle of Random Forest. RF can be applied to classification, clustering, regression and outlier detection due to its high accuracy and adaptability of high dimensional data sets.

RF is a robust classifier for spectral classification (Brice & Andonie, 2019a; Morice-Atkinson et al., 2018; Liu et al., 2019; Li et al., 2019; Bai et al., 2019; Yi et al., 2014; Hosenie et al., 2020; Biau & Scornet, 2016; Brice & Andonie, 2019b; Baqui et al., 2021). Clarke et al. (2020) trained a RF classifier on 3.1 million labelled sources from Sloan Digital Sky Survey (SDSS) and applied this model on 111 million unlabelled sources. The result showed that the classification probabilities of stars were greater than 0.9 (about 0.99). Besides, RF performed well in the process of searching for rare objects (Hou et al., 2020; Kyritsis et al., 2022). Pattnaik et al. (2021) trained a random forest classifier to determine whether a black hole or a neutron star is hosted by a Low Mass X-ray binaries (LMXBs). It is difficult to accurately classify variable stars into their respective subtypes, hence Pérez-Ortiz et al. (2017) proposed new robust feature sets and used RF to evaluate the classification performance. Akras et al. (2019) used classification tree for identifying symbiotic stars (SySts) from other H$\alpha$ emitters in photometric surveys. Guo et al. (2022) used random forest to identify white dwarfs in LAMOST DR5. Reis et al. (2018) used an unsupervised random forest to detect outliers on APO Galactic Evolution Experiment (APOGEE) stars. In addition, RF is often compared with other algorithms on classification tasks, and generally, it tends to be better than others (Arsioli & Dedin, 2020; Liu et al., 2019; Pérez-Ortiz et al., 2017).

Boosting trains models with adjusted data, that is, the weights of misclassified objects are augmented based on the former models. Fig. 8 is the principle of boosting. Gradient Boosting Decision Tree (GBDT), Adaptive boosting (Adaboost), extreme gradient boosting (XGBoost) and Light Gradient Boosting Machine (LightGBM) are prevalent boosting methods. Adaboost is a prominent boosting method that chooses single-layer decision trees as weak classifiers. In each iteration, it trains one weak classifier based on data weights generated in the last iteration. So Adaboost pays more attention on misclassified data. The other essential parameters are weights of each classifier. They are computed based on classification accuracy of every classifier. And the final results are obtained after inputting the sum of each weak classifier into a sign function. Fig. 9 is the main principle of Adaboost. GBDT (Morice-Atkinson et al., 2018; Pérez-Ortiz et al., 2017), another typical boosting algorithm, can also be regarded as an optimized version of Adaboost. GBDT chooses the residual from the previous iteration as input to train the next classifier till the residual is close to zero. Besides, GBDT can take more objective functions and train models using negative gradient, whereas Adaboost only sets data weights automatically. Fig. 10 shows the main principle of GBDT. XGBoost optimized GBDT by supporting different meta classifiers, adding regularization to limit model complexity, adapting to different data samplings and so on. Fig. 11 shows the main principle of XGBoost.

GBDT and XGBoost are two powerful ensemble classifiers (Friedman, 2001; Chen & Guestrin, 2016) and have been applied to spectral classification and rare object identification. Chao et al. (2019) used XGBoost to classify star and galaxy on dark sources of SDSS photometric data sets and the results showed that XGBoost outperformed other methods. Hu et al. (2021) searched for Cataclysmic Variables (CVs) in LAMOST-DR7 using LightGBM which is based on the ensemble tree model. They found 225 CV candidates including four new CV candidates which were verified by SIMBAD and published in catalogs. Yue et al. (2021) also identified M sub-dwarfs using XGBoost. In order to get better classification results, many new ensemble algorithms have been proposed in recent years (Chao et al., 2020; Zhao et al., 2022; Chi et al., 2022).

Stacking uses a new model to fit meta features which are obtained by multi-predictors on training sets and testing sets. And this new model will be validated with the following meta features. Fig. 12 introduces the principle of stacking.

Ensemble learning has obtained desirable results in astronomical spectral analysis. And random forest is the most frequently used ensemble method in astronomy. Because it has good generalization performance on large scale high-dimensional data sets. It is good at probabilistic prediction and is insensitive to noise. However, multi-value attribute still troubles RF. In addition, ensemble learning methods are also limited to heterogeneous data, unbalanced data and optimal parameters (the number of decision tree, weak classifiers).

Artificial Neural Network, also known as Multi Layer Perception (MLP), is a machine learning method that imitates the signal transmission mechanism in the brain. It consists of an input layer, multiple hidden layers and an output layer. The neural unit in each hidden layer tackles input data and sends results to the next fully connected layer. The output layer generates the final consequences. Fig. 13 is the principle of a neural unit. And Fig. 14 is the main principle of ANN. Particularly, Pseudo Inverse Learning (PIL) is a classic neural network. It can get globally optimal results and is faster than Back-propagation(BP) algorithm. Besides, it does not require manual tuning of parameters. So it has been used for some simple tasks. However, for complicated tasks, optimal versions of neural network are necessary. Deep learning (DL) is an essential extension of ANN, and it contains more hidden layers and complex network structures (Bergen et al., 2019). Convolutional Neural Network (CNN), Auto Decoder (AE) and Deep Belief Networks (DBN) are three chief methods of DL. Moreover, other variant versions of neural network have been proposed to adapt to different data formats, like Visual Geometry Group (VGG), Residual Networks (ResNet), Recurrent Neural Network (RNN), Generative Adversarial Networks (GAN) and others. Moreover, pre-trained models, attention blocks, transfer learning and many other tricks have been used to improve the deep learning performance effectively.

Convolutional Neural Network (CNN) consists of convolutional layers that extract image features, pooling layers that reduce dimensionality and fully connected layers that generate results. CNN automatically extracts features without destroying them. So it can get better accuracy and cope with high dimensional data. But its vanishing gradient problem and local optimal phenomenon still annoyed us. Fig. 15 shows the main principle of CNN.

Auto Decoder (AE) is a neural network whose input equals its output and its main idea is sparse code. It restructures the input using an encoder and a decoder. And it has been widely used for noise and dimensionality reduction to visualize data. Fig. 16 is the principle of AE.

DBN is a probabilistic generative model. Its generative model builds a joint distribution between observations and labels. DBNs consist of multiple layers of Restricted Boltzmann Machines which is a probabilistic graphical model with stochastic neural network. The output states of each neural unit are activation and deactivation.

Different examples of astrophysical research projects exploiting neural network are listed in Table 5 and summarized in the next parts.

Astronomical spectral classification is a typical task for neural network. Cabayol et al. (2019) used CNN to classify star and galaxy on low-resolution spectra from narrow-band photometry with accuracy over 98%. Jingyi et al. (2018); Astsatryan et al. (2021) used deep CNN to classify quasar and galaxy. Many new improvements of neural network emerged in recent years have been proven to be effective, like residual structures and attention mechanisms (Zou & el al., 2020). A multi-task residual neural network was applied to classify M-type star spectra. It reduced the number of parameters in spectral classification and improved the model efficiency (Lu et al., 2020). Compared to other methods, neural network always worked best on the complex data (Aghanim et al., 2015; Chen, 2021; Kerby et al., 2021; Sharma et al., 2020; Vilavicencio-Arcadia et al., 2020; Guo & Martini, 2019).

Rare object identification is another vital task of neural network (Muthukrishna et al., 2019; Jiang et al., 2020; Kou et al., 2020; Luo et al., 2008; Skoda et al., 2020; Tan et al., 2022; Zhang et al., 2022; Zheng et al., 2020; Zou et al., 2019; Margalef-Bentabol et al., 2020; Tan et al., 2022; Guo et al., 2019). Shi et al. (2014) searched for metal-poor galaxy (MPG) in large surveys and achieved an MPGs acquisition rate about 96%. Zheng & Qiu (2020) used 1D CNN to search for O stars. Muthukrishna et al. (2019); Fremling et al. (2021); Davison et al. (2022) proposed a software package that used deep learning models to classify the type, age, redshift, and host galaxy of supernova spectra. Qu et al. (2020) identified spectrum J152238.11+333136.1 from LAMOST DR5 and discussed the rare features of P-Cygni profiles.

Neural network based classification algorithms can be used to extract spectral features by different layers (i.e. hidden layers in Fig. 14, convolutional layers in Fig. 15). These layers can automatically learn rich and complex relationships between data. So neural network based algorithms can obtain high accuracy (Guo et al., 2019; Zou & el al., 2020; Liu et al., 2019; Fuqiang et al., 2014; Wang et al., 2017; Jiang et al., 2021; Jing-Min et al., 2020; Moraes et al., 2013; Portillo et al., 2020; Wang et al., 2017; Portillo et al., 2020). Furthermore, neural network could also handle input features well even without color or morphological information (Bu et al., 2019; Cabayol et al., 2019) which greatly expanded the size and formats of input data sets.

In short, neural network can learn deep features of data, which will provide subtle differences for classification. More importantly, with the introduction of tricks (i.e., residuals and attention blocks), ANN pays more attention on the valid features. In addition, ANN increases its depth to handle complex and high dimensional data. So it has been widely used in astronomy, such as star/galaxy/quasar classification, MPGs/MRGs classification, rare object identification and spectral feature selection, etc (Rastegarnia et al., 2022). Although neural network model can produce good results, it is a black box that is difficult to interpret results. Compared with decision tree, the results of neural network are difficult for astronomers to analyse the characteristics of celestial objects.

Assuming that features are independent, Gaussian naive Bayes based classification algorithms simplify the Bayesian algorithm. They prefer to deal with features in a Gaussian distribution and the maximum posterior probability is the final results. Eq 1 is the objective of Gaussian Naive Bayes based classification algorithms and Eq 2 is Gaussian probabilities. Table 6 represents astronomical studies of Gaussian naive Bayes based classification algorithms.

where

${\mathop{\delta}\nolimits_{y}}$ is variance of ${\mathop{x}\nolimits_{i}}$ (${i}$=1, 2, ……, n) and ${\mathop{\mu}\nolimits_{y}}$ is average of ${\mathop{x}\nolimits_{i}}$ in Eq 2.

Guassian Naive Bayes based classification algorithms are good at dealing with continuous small data generated from Gaussian distribution. Under the assumption of reliable and sufficient prior spectral information, they could identify rare objects from a large number of spectra data, such as carbon stars (Wallerstein & Knapp, 1998; Lloyd Evans, 2010; Arsioli & Dedin, 2020; Pruzhinskaya et al., 2019; Hoyle et al., 2015). And they were good at reducing noise of stellar spectra, which increased classification accuracy (Kang et al., 2021).

Bayesian Logistic Regression (LR) based classification algorithms obtain posterior probability distributions from linear regression models. And we can get classification results through the sigmoid function. The main researches of LR based classification algorithms are shown in Table 6. Fig. 17 is the principle of Bayesian Logistic Regression based classification algorithms.

LR based classification algorithms can be used for quick regression. However, they can not get desirable accuracy due to underfitting, bipartition data and linear data in small feature spaces. In astronomy, logistic regression based classification algorithms were often combined with other techniques to predict physical parameters and classify celestial objects (Pérez-Galarce et al., 2021; Tao et al., 2018; Luo et al., 2008).

Partial Least Squares Discriminant Analysis (PLS-DA) belongs to the discriminant analysis of multi-variance data analysis techniques and can be used for classification and discrimination. It handles data in the same cluster rather than data in different clusters. Data in the same group varies widely. And data volumes between groups differ a lot. It extracts principle components of the independent variable X and the controlled variable Y, and finds the relationship between principle components in a two high-dimensional space. Table 6 displays the main astronomical researches of CRC-PLS based classification algorithms.

CRC is a novel machine learning algorithm that represents a query by a linear integral of training samples. And CRC classifies the above queries based on the representation (Daniel et al., 2011). It has the ability to handle unbalanced, non-linear and multi-label data.

CRC-PLS reaps the merits of PLS regression and CRC. So it can classify the high-dimensional spectra data (Song et al., 2018).

Ranking based positive-unlabelled(PU) learning algorithms have been frequently used in astrophysical object retrieval. Graph based ranking methods successfully identify carbon stars from massive astronomical spectra data, such as manifold algorithm and efficient manifold algorithm (Si et al., 2015), Locally linear embedding. The bipartite ranking is another typical method to improve ranking performance and it has been introduced to search for carbon stars (Du et al., 2016). Alternatively, bagging is a popular method to obtain better performance by integrating different classifiers. The idea of bagging has been well applied in rare object retrieval wonderfully (Li et al., 2018; Du et al., 2016).

The core idea of ranking based classification methods is to learn a ranking based model which usually ranks data sets by predefined evaluation methods. They have two goals: 1) positive samples are ranked ahead of negative samples. 2) the scores of related samples tend to be similar. Many optimal ranking methods have emerged to improve classification performance and reduce time consumption, such as efficient manifold algorithms and bagging TopPush. And these methods have already discovered carbon stars from extensive spectra data which is a significant supplement to the catalogs of carbon stars (Table 6 ).

In the experimental design, we construct several groups of data sets using the spectra data from LAMOST (Luo et al., 2015) and SDSS.

LAMOST (The Large Sky Area Multi-Object Fiber Spectroscopic Telescope, also known as Guo Shou Jing Telescope) is a special reflective Schmidt telescope with an effective aperture of 3.6–4.9 meters and a field of view of 5$\degree$. It is equipped with 4000 fibers, a spectra resolution of R $\approx$ 1800 and a wavelength ranging from 3800 to 9000 $\mathop{A}\limits^{\circ}$ (http://www.lamost.org/public/?locale=en). Its scientific goal is to make a 20000 square degrees spectroscopic survey (DEC: -10$\degree$ $\sim$ +90$\degree$). After seven years of surveying, LAMOST has observed tens of millions of low-resolution spectra data, providing important data for astronomical statistical research.

The Sloan Digital Sky Survey (SDSS) is an international collaboration of scientists to build the most detailed Three-Dimensional Imagery of the Universe. It uses a wide-field telescope with a diameter of 2.5 meters and a field of view of 3$\degree$. The photometric system is matched with five filters in u, g, r, i and z bands to photograph celestial objects. It covers 7500 square degrees of the sky around the South Galactic Pole and records data on nearly 2 million celestial objects.

Experimental data are selected from LAMOST DR8 and SDSS DR16. The LAMOST DR8 data sets include a total of 17.23 million released spectra. The number of high-quality spectra of DR8 (that is, the S/N > 10) reaches 13.28 million and DR8 includes a catalog of about 7.75 million groups of stellar spectral parameters. The SDSS DR16 covers more than one-third of the sky and contains about 5,789,200 total spectra and 4,846,156 useful spectra. And DR16 contains new optical and infrared spectra, including the first infrared spectra observed by Las Campanas Observatory in Chile.

We select and preprocess the spectra from four aspects. These are shown in Table 7.

(1) Data release. We select spectra from LAMOST DR8 and SDSS DR16.

(2) Extinction problem. In order to decrease the influence of reddening on classification performance, 1D spectra in data sets are selected from LAMOST (45$\degree$< l) (Yang et al., 2022a).

(3) Flux calibration. LAMOST uses relative flux calibration. We cut off the overlapping region (5700Å$<\lambda<$ 5900Å) known to have calibration issues to minimize their effect on our classification.

(4) Redshift. For star/galaxy/quasar classification, we convert original spectra into the rest-frame wavelengths by applying the redshift values from LAMOST and compare the performance of classification on the rest wavelength frame spectra and original spectra. Because the radial velocity of stellar spectra are small under the current resolution of LAMOST, which has little influence on classification results. Spectra for stellar classification are left in the observed-frame wavelengths.

We determine three classification tasks among multiple astronomical researches, including A/F/G/K stars classification, star/galaxy/quasar classification and rare object identification. Rare objects includes carbon stars (Wallerstein & Knapp, 1998; Gigoyan et al., 2012; Lloyd Evans, 2010), double stars, artefacts: bad merging of red and blue segments (A common phenomenon that occurs in the spectra of LAMOST).

We design six groups of data sets for the above tasks. Data sets 1 - data sets 3 are constructed for A/F/G/K stars classifications. They are divided by data characteristics, S/Ns and data volumes, and each data sets contain three or four sub-datasets. Data sets 4 are used to evaluate the classification performance of star/galaxy/quasar on original spectra and rest wavelength frame spectra. Data set 5 is used to identify rare objects: carbon stars, double stars and artefacts. And the classifier is trained on 200 rare objects and 19900 other non-rare objects. Non-rare objects include 10000 normal stars, 6500 galaxies and 3400 quasars. We analyse the results of rare object identification by accuracy, precision, recall and F1 score. Spectra of the first five groups of data sets are selected from LAMOST. Because the sources of LAMOST have considerable overlaps with SDSS, we construct the matching data sets (data sets 6 in Table 8) from SDSS and LAMOST to compare the classification performances on them. The analyses of experimental results on data sets 6 are elucidated in Section 3.2.1.

The composition of testing sets in all data sets is the same as their training sets. The ratio of training sets and testing sets for data sets 1, data sets 2, data sets 3, data sets 4 and data sets 6 is 8:2 and the ratio of training sets and testing sets for data set 5 is 1:1. Details of data sets are shown in Table 8.

In this section, nine basic methods including K-Nearest Neighbour, Support Vector Machine, Decision Tree, Random Forest, Gradient Boosting Decision Tree, Logistic Regression, Pseudo Inverse Learning and Convolutional Neural Network are tested on astronomical spectra data and we fairly evaluate the classification performance.

Our experiments use grid search (Syarif et al., 2016) to identify the optimal parameters of each algorithm. And we take the average accuracy of 5-fold cross validation (Fushiki, 2011) as the final accuracy to avoid the influence of sample selection.