With the rapid development of aeronautic and deep space exploration technologies, a large number of highresolution asteroid spectral data have been gathered, which can provide diagnostic information for identifying different categories of asteroids as well as their surface composition and mineralogical properties. However, owing to the noise of observation systems and the everchanging external observation environments, the observed asteroid spectral data always contain noise and outliers exhibiting indivisible pattern characteristics, which will bring great challenges to the precise classification of asteroids. In order to alleviate the problem and to improve the separability and classification accuracy for different kinds of asteroids, this paper presents a novel Neighboring Discriminant Component Analysis (NDCA) model for asteroid spectrum feature learning. The key motivation is to transform the asteroid spectral data from the observation space into a feature subspace wherein the negative effects of outliers and noise will be minimized while the key categoryrelated valuable knowledge in asteroid spectral data can be well explored. The effectiveness of the proposed NDCA model is verified on realworld asteroid reflectance spectra measured over the wavelength range from 0.45 to 2.45 μm, and promising classification performance has been achieved by the NDCA model in combination with different classifier models, such as the nearest neighbor (NN), support vector machine (SVM) and extreme learning machine (ELM).
Deep space exploration is the focus of space activities around the world, which aims to explore the mysteries of the universe, search for extraterrestrial life and acquire new knowledge [
The EightColor Asteroid Survey (ECAS) is the most remarkable groundbased asteroid observation survey, which gathered the spectrophotometric observations of about 600 large asteroids [
For asteroid taxonomy, Tholen et al. applied the minimal tree method by a combination with the principal component analysis (PCA) method in order to classify nearly 600 asteroid spectra from the ECAS [
Nevertheless, the question of how to automatically discover the key categoryrelated spectral characteristics/features for different kinds of asteroids remains an open problem [
Machine learning techniques have developed rapidly in recent years for spectral data processing and applications, such as the classification and target detection [
In order to define the class boundaries for asteroid classification, traditional methods always empirically determine the spectral features by relying on the presence or absence of specific features, such as the spectral curve slope, absorption wavelengths and so on, which might be intricate and less reliable. Based on the well labeled asteroid spectral dataset described in
Instead of empirically determining the spectral features via the presence or absence of specific spectral features to define asteroid class boundaries for classification, this paper presents a novel supervised Neighboring Discriminant Components Analysis (NDCA) model for discriminative asteroid spectral feature learning by simultaneously maximizing the neighboring betweenclass scatter and data variances, minimizing the neighboring withinclass scatter to alleviate the overfitting problem caused by outliers and enhancing the discrimination and generalization ability of the model.
With the neighboring discrimination learning strategy, the proposed NDCA model has stronger robustness to abnormal samples and outliers, and the generalization performance can thus be improved. In addition, the NDCA model transforms the data from the observation space into a more separable subspace, and the key categoryrelated knowledge can be well discovered and preserved for different classes of asteroids with neighboring structure preservation and label prior guidance.
The performance of the proposed NDCA model is verified on realworld asteroid dataset covering the spectral wavelengths from 0.45 to 2.45 μm by combining with different baseline classifier models, including the nearest neighbor (NN), support vector machine (SVM) and extreme learning machine (ELM). In particular, the best result is achieved by ELM, with a classification accuracy of about 95.19%.
The reminder of this paper is structured as follows.
In this paper, the observed asteroid visible and nearinfrared spectroscopy dataset is denoted as
In the process of lowdimensional feature learning, the key data knowledge and information, such as the discriminative structures, should be preserved and enhanced. Meanwhile, the noise and redundant information should be removed and suppressed. Principal component analysis (PCA) is a widely applied unsupervised statistical dimension reduction and feature learning method, which focuses on maximizing the variance of the data with significant principal components [
Unlike PCA, LDA is a supervised dimension reduction learning method and aims to maximize the separability between different classes and enhance the compactness within each class with the guidance of label information as described below [
Classifier models, such as NN, SVM [
With the optimal output weight matrix
The remote observed asteroid spectral data usually contain noise and outliers, which will mix different categories of asteroids and make them inseparable. In addition, learning with outliers will easily cause overfitting problem, which will decrease the generalization ability of machine learning models for testing samples. Thus, the key problem is to distinguish the outliers and to select the most valuable samples for the learning of lowdimensional feature subspace and preserve the key discriminative data knowledge for different classes of asteroids.
To this end, the idea of neighboring learning is introduced to find a neighboring group of valuable samples from all the training samples as well as the samples in each class, and the outliers and noised samples are excluded in dimension reduction learning in order to enhance the generalization ability of the model. As shown in
Neighboring betweenclass scatter matrix
At the same time, the
Neighboring withinclass scatter matrix
The details for deriving Equation (13) based on Equations (10)–(12) are shown in
Then, the partial derivative of the objective function (15) with respect to
Once the above optimal projection matrix
As shown in
As previously mentioned, the smoothed asteroid spectral curves were fitted using a high order polynomial, which was furthered sampled in wavelength region from 0.45 to 2.45 μm with an increment step interval of 0.05 μm, obtaining 41 measurements for each asteroid spectrum. In order to valid the effectiveness of the proposed method, the data from different classes were firstly approximately equally divided into five groups as shown in
The performance of different dimension reduction methods under gradually increasing reduced dimension
Generally speaking, the proposed NDAC method can yield the best classification accuracies of 94.1971%, 93.6377% and 95.1895% with different classifiers.
In addition, the results show that the raw data without feature learning achieves worse classification performance among all the comparative methods. In contrast, the proposed NDCA model can achieve the highest classification accuracy by combining with different classifier models. Moreover, it should be noted that the highest accuracy can be achieved when the feature dimension is around nine. Thus, the optimal reduced dimension
Furthermore, the scatter points for the first two dimensions acquired by different methods are visualized in
Apart from the dimensionality of the derived feature subspace
The former experiments show that the proposed NDCA method can generally achieve promising and higher classification accuracy in combination with ELM. As shown in formulation (6), ELM has two key hyperparameters, i.e., the number of hidden neurons
From the above experimental results, we observe the following:
The benefits of feature learning for asteroid spectrum classification. In the experiments shown in
The advantages of the proposed NDCA model. In comparison with several representative lowdimensional feature learning methods, the proposed NDCA model can generally achieve better classification performance by combining with different classifier models. Specifically, NDCA plus NN, SVM and ELM can achieve the highest classification accuracies of 94.1971%, 93.6377% and 95.1895%, respectively. The improvements are mainly due to the following two aspects. Firstly, the NDCA model is a supervised dimension reduction method and inherits the merits of the existing methods, which can fully utilize label knowledge in order to find the key categoryrelated information of spectral data for discriminative asteroid spectral feature learning and classification. Secondly, the introduction of neighboring learning methodology can significantly reduce the side effects of outliers and noised samples in order to alleviate the overfitting problem, which will enhance the robustness of the leant lowdimensional features and finally improve the generalization ability and classification performance of the proposed model in testing.
The superiority of ELM. Three baseline classifier models, including NN, SVM and ELM, were used in the experiments. In particular, the best results are obtained by NDCA plus ELM with a classification accuracy of about 95.19%, which is generally superior to the comparing classifier models. To the best of our knowledge, this work is the first attempt to apply ELM in asteroid spectrum classification, and very competitive performance has been achieved, which can provide new application scenarios and perspectives for ELM community.
Future work discussion. First, future work will consider employing feature selection methods in order to study the asteroid spectral characteristics. Distinct from feature learning/extraction methods, which adopts the idea of data transformation, feature/band selection methods use the idea of selection and aim to automatically select a small subset of representative spectral bands in order to remove spectral redundancy while simultaneously preserving the significant spectral knowledge. Since the feature selection is performed in the original observation space, the specific selected bands have clearer physical meanings with better interpretability. As a result, feature/band selection is an important technique for spectral dimensionality reduction and has room for further improvement. Second, the visualization in
This paper has introduced a novel supervised NDCA learning model for asteroid spectral feature learning and classification. The key idea is to distinguish the outliers and noised samples in order to alleviate the overfitting problem and to find the significant categoryrelated features such that the classification performance can be improved. The goals are technically achieved by simultaneously maximizing the neighboring betweenclass scatter, minimizing the withinclass scatter and preserving the neighboring principal components. Experimental results on reflectance spectrum characteristics measured across the spectral wavelengths ranging from 0.45 to 2.45 μm show the effectiveness of the proposed model by combining with different baseline classifier models, including NN, SVM and ELM, and the highest classification accuracy is achieved using the ELM classifier, which also verifies the superiority of ELM for multiclass classification problem.
All the authors made significant contributions to the study. T.G. and X.P.L. conceived and designed the global structure and methodology of the manuscript; T.G. analyzed the data and wrote the manuscript. Y.X.Z. and K.Y. provided some valuable advice and proofread the manuscript. All authors have read and agreed to the published version of the manuscript.
This work is supported by The Science and Technology Development Fund, Macau SAR (No. 0073/2019/A2). Tan Guo is also funded by The Macao Young Scholars Program under Grant AM2020008, The Natural Science Foundation of Chongqing under Grant cstc2020jcyjmsxmX0636, The Key Scientific and Technological Innovation Project for “ChengduChongqing Double City Economic Circle” under grant KJCXZD2020025, The National Key Research and Development Program of China under Grant2019YFB2102001 and The 2019 Outstanding Chinese and Foreign Youth Exchange Program of China Association of Science and Technology (CAST).
Not applicable.
Not applicable.
The data presented in this study are available upon request from the author.
The authors would like to thank Francesca E. DeMeo from MIT for providing the asteroid spectral dataset.
The authors declare no conflict of interest.
Equation (13) shows the model formulation for the proposed NDCA method and contains three key components, i.e.,
Deriving
Indicate
Since
It can be easily observed that
Thus, Equation (A3) can be rewritten as below.
Furthermore, the trace of Equation (A5) is used for the optimization of subspace projection matrix
Following the above derivations from Equations (A1)–(A6), the component
Deriving
Signify
Equation (A8) can be transformed into the following form.
Furthermore, the trace of Equation (A10) is used for optimization as described below.
In this way, the component
Equation (11) via the derivations from Equations (A7)–(A11).
Deriving
Denote
Substitute
It can be observed that
Similarly, the trace of Equation (A15) is used for the optimization of subspace projection matrix
According to the above derivations from Equation (A12)–(A16), the component
Overview of the asteroid feature learning and classification scheme.
Illustration of the proposed Neighboring Discriminant Component Analysis (NDCA) model.
The spectral preprocessing for (1)
The spectral preprocessing for (2957)
The spectral preprocessing for (1807)
Fivefold cross verification scheme for asteroid spectral data.
The performance of different dimension reduction methods under different reduced dimensions using NN as the classifier.
The performance of different dimension reduction methods under different reduced dimensions using SVM as the classifier.
The performance of different dimension reduction methods under different reduced dimensions using ELM as the classifier.
Visualization for the scatter points of the first two components acquired by different methods. By comparison, the proposed NDCA model shows better withinclass compactness and betweenclass separation characteristics.
NN plus NDCA performance under different combinations of parameters. (
SVM plus NDCA performance under different combinations of parameters. (
ELM plus NDCA performance under different combinations of parameters. (
Classification performance variations of NDCA plus ELM under different settings of
Description of the asteroid spectral datasets for 371 asteroids with 24 classes.
Class  
# samples  6  4  13  3  1  10  18  16 
Class  ‘K’  ‘L’  ‘O’  ‘Q’  ‘R’  ‘S’  ‘Sa’  ‘Sq’ 
# samples  16  22  1  8  1  144  2  29 
Class  ‘Sr’  ‘Sv’  ‘T’  ‘V’  ‘X’  ‘Xc’  ‘Xe’  ‘Xk’ 
# samples  22  2  4  17  4  3  7  18 
Important notations used in this paper.
Notation  Meaning  Notation  Meaning 


Subspace projection matrix 

Label matrix 

Highdimensional dataset with dimension 
Data points with index 


Lowerdimensional features of X with dimension 
N  Number of datapoints 

Number of classes in 
N 
Number of data points in 

Balance parameter in ELM model  Lowerdimensional features for 
Description of asteroid spectral datasets used in the experiments.
Class  Total  
# Samples  6  45  16  16  22  8  199  17  32  361 
Experimental data partition of 5folds.
Class  Fold 1  Fold 2  Fold 3  Fold 4  Fold 5  Total 

‘A’  1  1  1  1  2  6 
‘C’  9  9  9  9  9  45 
‘D’  3  3  3  4  3  16 
‘K’  4  3  3  3  3  16 
‘L’  4  4  4  5  5  22 
‘Q’  2  2  1  2  1  8 
‘S’  40  40  40  40  39  199 
‘V’  3  4  4  3  3  17 
‘X’  6  6  7  6  7  32 
# samples  72  72  72  73  72  361 
Experiment settings with different fold partitions.
Experiments  Training Dataset  Testing Dataset 

Exp. 1  fold 1, fold 2, fold 3 and fold 4 ( 
fold 5 ( 
Exp. 2  fold 1, fold 2, fold 3 and fold 5 ( 
fold 4 ( 
Exp. 3  fold 1, fold 2, fold 4 and fold 5 ( 
fold 3 ( 
Exp. 4  fold 1, fold 3, fold 4 and fold 5 ( 
fold 2 ( 
Exp. 5  fold 2, fold 3, fold 4 and fold 5 ( 
fold 1 ( 
Classification accuracy (%) of different dimension reduction algorithms using NN as the classifier.
Methods  Exp. 1  Exp. 2  Exp. 3  Exp. 4  Exp. 5  Average 

Raw  94.4444  84.9315  87.5000  93.0556  86.1111  89.2085 
PCA  94.4444  84.9315  87.5000  93.0556  86.1111  89.2085 
LDA  95.8333  90.4110  88.8889  97.2222  88.8889  92.2489 
LPP  90.2778  87.6712  90.2778  91.6667  88.8889  89.7565 
LPDP  95.8333 

91.6667  97.2222  88.8889 

NDCA 

89.0411 




Classification accuracy (%) of different dimension reduction algorithms using SVM as the classifier.
Methods  Exp. 1  Exp. 2  Exp. 3  Exp. 4  Exp. 5  Average 

Raw  94.4444  86.3014  93.0556  93.0556  91.6667  91.7047 
PCA  94.4444  89.0411  93.0556  93.0556  91.6667  92.2527 
LDA  94.4444  90.4110  88.8889  94.4444  91.6667  91.9711 
LPP 

86.3014  90.2778 


92.8158 
LPDP  94.4444  90.4110  93.0556  94.4444  91.6667 

NDCA  94.4444 



93.0556 

Classification accuracy (%) of different dimension reduction algorithms using ELM as the classifier.
Methods  Exp. 1  Exp. 2  Exp. 3  Exp. 4  Exp. 5  Average 

Raw  94.8611  89.3151  92.0833  95.6944  91.5278  92.6963 
PCA  95.0000  90.4110  92.0833  96.5278  92.3611  93.2766 
LDA  95.4167 

91.8056  97.2222  93.4722  94.4874 
LPP  95.8333  90.4110  93.0556 

94.4444  94.4711 
LPDP  95.9722 

92.6389  97.2222  93.3333 

NDCA 

91.7808 

97.2222 


Performance improvement between different pairs methods by using different classifiers.
Classifiers  Comparison Pairs  

<Ours, Raw>  <Ours, PCA>  <Ours, LDA>  <Ours, LPP>  <Ours, LPDP>  
NN  ↑ 4.9886%  ↑ 4.9886%  ↑ 1.9482%  ↑ 4.4406%  ↑ 1.3927% 
SVM  ↑ 1.9330%  ↑ 1.3850%  ↑ 1.6666%  ↑ 0.8219%  ↑ 0.8333% 
ELM  ↑ 2.4932%  ↑ 1.9129%  ↑ 0.7021%  ↑ 0.7184%  ↑ 0.4521% 