Genetic optimization of asteroid families’ membership

Asteroid families are groups of asteroids with a common origin, such as prior collisions or the parent body’s rotational fission. In proper [a, e, sin(i)] element domains, they are generally observed using the hierarchical clustering technique (HCMs), but the method may be ineffective in high-density regions, where it may be unable to separate near families. Previous works employed a different technique in which nine different machine learning classification algorithms were applied to the orbital distribution in proper elements of 21 known family constituents for the goal of new members’ identification. Each algorithm’s optimal hyper-parameters for every family were extensively investigated, which proved to be a time-consuming and repetitive procedure. Herein, we used a genetic algorithm-based tool to identify the most optimal machine learning algorithm for the same studied asteroid families as an alternative to the originally utilized parameter search mode. When compared to the same evaluative metrics utilized in the previous machine learning application study, the precision values of the new genetic machine learning algorithms have been consistently comparable, demonstrating that this alternative technique can be satisfactorily efficient and fast.

1 Introduction

Asteroid families are groups of minor planets that can be identified in domains of proper elements [a, e, sin(i)], with a being the asteroid’s proper semi-major axis, e the proper excentricity, and i the proper inclination. Contrary to osculating elements that change on timescales of days or less, proper elements are constant in motion and are stable on timescales of millions of years when non-gravitational forces are not considered. Proper elements can be obtained either through perturbation theory approaches or, most commonly, through numerical methods involving N-body simulations of the planets and asteroids’ orbits and Fourier analysis (Knežević and Milani, 2003). The latter elements are called synthetic proper elements. Since the 90s, the most commonly used method for detecting asteroid families has been the hierarchical clustering method (HCM, hereafter, Zappalà et al. (1995); see also Bendjoya and Zappalà (2002) for a later review). This approach has been very successful in detecting new asteroid families and is the current standard for recent surveys, like those conducted by Nesvorný et al. (2015), and for the families available at the Asteroid Families Portal (AFP, http://asteroids.matf.bg.ac.rs/fam/index.php, Radović et al. (2017)). As discussed in Milani et al. (2014), however, the large number density of asteroids in some regions of the main belt may introduce a problem not present in the smallest databases used in the 90s. Nearby asteroid families may overlap and be very close in proper element domains. As an effect, traditional HCM may fail to recognize these groups as separate. This problem, referred to as chaining by Milani et al. (2014), may be overcome if machine learning (ML) methods are employed. Carruba et al. (2020) used supervised ML algorithms to identify new members of known asteroid families, as listed at the AFP. The basic idea behind these approaches was to train ML methods to recognize patterns among the [a, e, sin(i)] distribution of already known family members, so that new asteroids with similar values of [a, e, sin(i)] could be attached to these groups without the issue of chaining. One limitation of that approach was that those algorithms were dependent on free hyper-parameters, the optimal values of which had to be found by searching over extended phase space. This was a time-consuming endeavor. Genetic algorithms, as described in Section 2.3, allow for the selection of the best-performing method and of its most optimal values of hyper-parameters automatically. The goal of this work is to verify if their use could be a safe and viable option.

2 Methods

2.1 The use of ML methods for asteroid family identification: Some background

In 2020, the AFP site reported data for asteroid families identified among a database of 631,226 numbered asteroids using the hierarchical clustering method [HCM, Bendjoya and Zappalà (2002)]; 21 asteroid families from the AFP were used to train the (ML) algorithms by dividing the samples into two parts: one for asteroids with an absolute magnitude H lower than 14, the training sample, and one for the rest, the test sample. This value of H was justified based on the work of Milani et al. (2014), which showed that asteroid families obtained for samples with H < 14 are not affected by the chaining issue and should present minimal or no overlap with other nearby dynamical families.

These families have to satisfy the criteria of being available in the Radović et al. (2017)catalog and of having at least 10 members with H < 14. The outcome of ML algorithms was measured based on metrics defined using the similarity matrix. As in the study by Carruba et al. (2019), one can define true positive (TPos) asteroids identified as a family member by both the HCM procedure method and the machine learning application, false positive (FPos) as the ones identified only by machine learning, and the false negatives (FNeg) as asteroids identified by HCM alone. Based on these quantities, f₁ or “completeness,” “precision,” and the final parameter FP metrics can be introduced. Completeness is defined as follows:

$f_1=\mathit{\text{Completeness}}=\frac{TPos}{TPos+FNeg}=\frac{TPos}{N_{Or}}.(1)$

This formula alone is not a good parameter to measure the algorithm’s performance. For example, if we have a small original family with 100 members and a retrieved family of 10,000 that includes all the 100 original members, the result of completeness would be maximum. The retrieved family is not a good approximation of the original one. To address this problem, the precision formula was defined as follows:

$Precision=\frac{TPos}{TPos+FPos}=\frac{TPos}{N_{\mathit{Retr}}},(2)$

where N_Retr= T Pos + FPos and is the number of asteroids in the retrieved family. High precision values can also be associated with a small family; for example, if we have an original family of 100 members and a retrieved one composed of 10 TPos objects, the formula outcome will be 1, but again, the retrieved family would not be a good representation of the original.

The two parameters can be combined into a single optimized measure through the final parameter formula:

$FP=\frac{1}{\sqrt{2}}\sqrt{Completenes{s}^2+Precisio{n}^2}.(3)$

2.2 Machine learning methods

Current artificial intelligence research places a strong emphasis on the machine learning field, and a variety of ML algorithm methods have been developed. Decision tree-based bagging and boosting are the most popular and widely used principles of ML algorithm codes.

Decision tree is a simple machine learning algorithm that makes decisions in the form of a tree, with branches representing decision rules and leaf nodes representing results. Extra tree uses decision trees to produce the best results by using the standalone majority vote method.

The bagging boosting principle is followed by the bagging classifier algorithm, in which the training sample is divided into multiple samples, called bootstrap, and each one is used to train an independent classifier, resulting in a variety of predictions. The final result is a weighted average of the results of each classifier.

Adaptive boosting, such as gradient methods, uses the ensemble method to combine the results of multiple standalone classifiers to arrive at the best result. However, boosting methods use the concept of the least accurate prediction getting lower weights, and the final result also follows the bagging classier idea. Adaptive boosting (AdaBoost) trains the classifier over the data points first, and the ones that were incorrectly identified are given a higher weight, influencing the outcome of the following interaction. When there are no significant errors or the number of iterations is reached, the process stops.

The gradient boosting (Gboost) algorithm applies the gradient concept to the bagging weights idea, in which the method finds under-performing estimators and attempts to fit and correct them in the next iteration by adding new estimators. When no further improvements are conceivable, the process ends.

The eXtreme gradient boosting (XGboost) technique is similar to Gboost, but it offers speed improvements such as greater multicore support, processor cache use, and the ability to control over-fitting using a more regularized model formalization.

The non-parametric supervised learning technique K-nearest neighbor (KNN) method is one of the simplest machine learning algorithms. This simple method is based on the study of the uncatalogued data’s neighbors. When the algorithm receives unseen data, it examines the classes of its closest neighbors to predict the new data category. For example, if there is an unknown color ball (data) between others with well-defined color (classes) balls, such as three reds and two blues, if we just judge by the colors of the balls, the unknown data can be identified as red due to the actual dominating class (the larger number of neighbors with the same class). This principle may be applied to any type of data and can be used to predict the class of a point by looking at its surroundings.

2.3 Genetic algorithms

Genetics algorithms, initially developed by Bremermann (1958) and popularized by Holland (1962), are a search and optimization programming technique based on Darwin’s theory of species evolution, where the strongest individual is favored and his reproduction is more likely than those of the others, forming a new generation.

In terms of the genetic algorithm, each individual in the population is a solution, and every individual’s collection of hyper-parameters (genes) determines this solution’s characteristics when we have a problem to solve. A genetic algorithm’s functionality can be divided into five sections: population, fitness function, selection, crossover, and mutation. The fitness function assigns a fitness score to each solution, indicating the individual’s capacity to compete with the others. The fittest solutions are chosen during the selection phase, and their characteristics are passed down to the next generation. Each pair of parents is mated, and a crossover point is picked at random from the available genes. The offspring is formed by exchanging the available genes until the stipulated crossover point is reached, and then the newly formed individual is included in the population. The crossover is defined by this phase. At the mutation stage, newly produced offspring may be subjected to a low-probability random mutation, altering their genetic structure. The algorithm determines if the new population is considerably different from the previous generation at the termination phase and if the genetic algorithm supplied a set of solutions to our problem. If not, a new circle begins.

TPOT (Le et al., 2020; Olson et al., 2016a; Olson et al., 2016b) is a Python-based automated machine learning tool that employs genetic algorithms to identify the most appropriate machine learning pipeline for a particular task. The user must provide simple inputs after manually cleaning the raw user data: generations (number of genetic algorithm training iterations), population size (number of individuals retained in the population each generation), cross-validation—cv (used to evaluate each pipeline using a simple parameter K, which corresponds to the number of groups into which the data sample is divided), and the random state (random number generator seed for reproducibility). The tool then automatically produces the machine learning pipeline’s best model.

2.4 Procedure

The main goal of the genetic algorithm is to find a minimum in the fitness function. TPOT may not always find a global minimum of this function. This means that, occasionally, the machine learning pipeline produced by TPOT may not be the best-performing one. In this work, we estimate the efficiency of using genetic algorithms compared to other methods of optimization of hyper-parameters like those used in Carruba et al. (2020).

The main criteria for a family to be included in this research are that 1) it has to be listed in the Radović et al. (2017) catalog and 2) it has to have at least 10 members with H < 14 and be located in the main belt regions (Carruba et al., 2013); 21 asteroid families satisfy our selection criteria on the Asteroid Families Portal (AFP). For these families, we first selected the region where the group is located (Table 1). We then label asteroids in the region with 1 if the asteroid belongs to a given family of interest, and 0 otherwise. TPOT is then used to train on this set of elements, yielding the optimum machine learning pipeline for this family.

TABLE 1

TABLE 1. Table of the main belt orbital regions. We report the region name, where H.I. indicates for highly inclined, the a_min value and the Jupiter mean-motion resonance name associated with that value, a_max value and its related Jupiter mean-motion resonance name, and the minimum value of sin(i_min).

3 Results

The main objective of the research is to investigate the use of genetic algorithms to optimize machine learning predictions of new asteroid family members as an improvement to the work carried out by Carruba et al. (2020). The machine learning algorithm parameters generated by TPOT are listed in Table 2 for extra tree and random forest classifier methods. The other families’ algorithm parameters (XGBoost, gradient boosting classifier, and K neighbors classifier) are available upon request. Table 3 shows the new metrics values, as well as the best TPOT, generated machine learning algorithm for each family¹.

TABLE 2

TABLE 2. Extratree and random forest classifier machine learning parameters as generated by the TPOT tool, with the number of estimators fixed at 100. The columns report the asteroid families IDs, the values of bootstrap, criterion, max_features, min_sample_leaf, and min_sample_split.

TABLE 3

TABLE 3. Table reports the metrics of the 21 asteroid families from the Asteroid Families Portal using provided genetic machine learning algorithm by TPOT. There is a representation of the metrics f₁ (completeness), precision, the FP coefficients, and the best estimator of each family.

Carruba et al. (2020) used machine learning parameter algorithms applied to an old database of numbered asteroids at the Asteroid Families Portal, making a direct percentual comparison (Eq. 4) to current results, which are based on an updated asteroid database, complicated. Previously, HCM was applied to a database of 631,226 numbered asteroids. There were 793,310 registered objects in May 2022. Changes between the FP’s (Eq. (3)) coefficient metrics obtained in Carruba et al. (2020) and the current ones are displayed in Table 4.

$R=\frac{\left(F{P}_{2022}\right)-\left(F{P}_{2020}\right)}{\left(F{P}_{2022}\right)},(4)$

where (FP ₂₀₂₀) refers to the final parameter metric of Carruba et al. (2020) machine learning applications and (FP ₂₀₂₂) is linked to the current work metrics.

TABLE 4

TABLE 4. Table with FP coefficient metric comparison results of Carruba et al. (2020) and the current work. We report the asteroid families’ names, the value of the result metric FP of 2020, the current results of the FP, and a percentual comparison R (see Eq. 4).

This problem can be handled by reapplying the Carruba et al. (2020) machine learning algorithms to the updated database of the 21 asteroid families investigated. Table 5 shows the new percentual difference R.

TABLE 5

TABLE 5. Table shows the Fp’s metrics percentual comparison R between Carruba et al. (2020) machine learning algorithms applied to an updated database (displayed in the second column) and the current machine learning algorithms generated by genetic algorithms (column FP 2022).

Figures 1 and 2 show the distribution of the original and retrieved family members in the cases of the Eunomia and Maria families. Those families were chosen based on the metric comparison: 15 Eunomia had the worst value of R (-0.124), while Maria displayed the best result, with R = 0.141. In all cases, the algorithms were able to retrieve asteroids in the right orbital regions of the asteroid families, with modest gains for the case of genetic algorithms applied to the Maria family.

FIGURE 1

FIGURE 1. Proper (a, e) and (a,sin(i)) projections of the 15 Eunomia family. Results with Carruba et al. (2020) machine learning algorithms applied to an updated database of Asteroid Families Portal site are shown in the left panels, while the current genetic algorithm applied to the same family database appears on the right. The original family members are represented by the black dots, while the retrieved members are represented by the red points.

FIGURE 2

FIGURE 2. Same as in Figure 1 but for the 170 Maria family.

The frequency of R distribution in the 21 studied asteroid families can be seen in Figure 3.

FIGURE 3

FIGURE 3. Histogram of R (Eq. 4). The graphics display the percentual frequency of R on the 21 studied asteroid families. Data distribution moments: mean = 0.0086, standard deviation = 0.063, skewness = 0.268, and kurtosis = 0.485.

4 Conclusion

According to the histogram data (Figure 3), the R distribution is slightly positively skewed (to the right, where the genetic algorithms perform better) and has more values in the tails than a normal distribution. A small standard deviation shows that the sample values are condensed close to the mean, indicating a homogeneous sample.

The R fluctuation demonstrates a connection between old machine learning codes and current genetic algorithm implementations. Although there is a minor improvement in some families’ precision and a small drop in others using the genetic method, the overall picture shows that the new application is satisfactory, demonstrating that using the genetic algorithm to solve this problem is safe.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

ML and VC contributed to the conception and design of the study. VC and ML organized the database. ML and VC performed the statistical analysis. ML and VC wrote the first draft of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

Funding

We are grateful to the Brazilian National Research Council, CNPq, for supporting the work of ML (grant PIBIC 2021/2698) and VC (grant 304168/2021-1).

Acknowledgments

This research has made use of the Asteroid Families Portal maintained at the Department of Astronomy/University of Belgrade. We are grateful to Prof. Rita C. Domingos and Safwan Aljbaae for useful suggestions and comments.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

¹In this work, we will refer to an asteroid family with the identification and name of its lowest numbered member, that is, 15 Eunomia, for example. Please note that the correct way to do that would be by using parenthesis around the identification number, that is, (15) Eunomia. For the sake of simplicity, here, we will prefer to use our shortest nomenclature.

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