Feature extraction for exoplanet detection

Abstract

Detecting possible habitable planets outside of our solar system has been a growing field of study. Among several other topics, this field aims to classify stars using the transit method, i.e., using their light intensity measured over time to spot the moment when a planet follows its orbit and covers part of the star as seen by a satellite. We propose a novel approach to such classification, using an extracted set of features from individual time-series that cover three different domains: temporal, statistical, and spectral. These features are filtered based on relevant measures, and used to train and evaluate models on Kepler data. The results were compared to state-of-the-art methods evaluated on the same data set and surpass existing approaches for some data transformations. All these transformations are related to turning the time-series naïvely stationary before feature extraction. Using principal components extracted from the feature set during model training did not have a considerable impact on results. In order to better evaluate the results, a cross-validation process was performed to eliminate data set bias. During this step, the best model achieved $100\%$ recall and $98.82\%$ F1-score for the minority class. In the future, testing additional feature selection methods, as well as assessing feature importance using more explainable metrics is crucial to further understand the distinctions that separate stars with exoplanets from those without.

Appendices

Appendix A Machine learning methods

1.1 A.1 Support vector machines

SVMs, initially proposed by Boser et al. [50] and extended by Cortes and Vapnik [51], are algorithms that can be used in classification tasks that learn functions to discriminate between classes from a training set. This discrimination is achieved by constructing one or more hyper-planes in a high-dimensional space that are then used to solve the task. These hyper-planes maximally separate the data into different classes using a maximum margin, i.e., the distance between the hyper-plane and the closest data points from each class. By doing so, the algorithm finds the most robust decision boundary to properly generalize new data [52].

SVMs are particularly useful when dealing with data sets whose features have a nonlinear relationship with the response variable. This is accomplished by mapping the original feature space into a higher-dimensional space using a kernel function. This new space is then used to find a nonlinear decision boundary [53]. SVMs are also useful to handle data with high dimensionality, i.e., when the number of features is considerably larger than the number of observations [53]. Lastly, these algorithms are noticeably robust against over-fitting [51].

1.2 A.2 Random forests

Based on the work of Ho [54] and introduced by Breiman [55], RFs are an ensemble algorithm of several decision trees, each trained on a random subset of data and features. When predicting a new value, the predictions of all trees are aggregated, i.e., the class with the most votes is the predicted one [53].

The random aspect of RFs helps in preventing over-fitting, as the noise of individual trees is averaged out, improving the generalization capacity of the models and making their predictions more robust and accurate in comparison to individual decision trees [53, 55]. Furthermore, RFs also have the ability to handle high-dimensional data and provide estimates of feature importance [53]. Nonetheless, Chen and Guestrin [56] discussed that RFs can be prone to over-fitting in data sets with a high number of features, but suggested that this can be mitigated by using feature sub-sampling and bagging.

1.3 A.3 K-Nearest neighbours

KNN, proposed by Fix and Hodges [57] and later extended by Cover and Hart [58], is a nonparametric algorithm, i.e., an algorithm that does not assume the kind of the mapping function, that can be used in a supervised or unsupervised context. This algorithm is based on the assumption that similar data points have similar response values [53]. The process of finding the k nearest neighbours, with k being a hyper-parameter of this algorithm, is performed on feature space with regard to some distance measure and the outcome is the most common class among the k nearest neighbours of the predicted point [53].

1.4 A.4 Logistic regression

Introduced by Berkson [59], LR is a generalized linear model used in binary classification tasks that calculates the probability of the input being part of both classes. This probability is modelled based on the logistic/sigmoid function, i.e., $P(Y = 1 | x) = \frac{1}{1 + e^{-(\beta _0 + \sum _{i = 1}^{N} \beta _i \times x_i)}}$, where x is a vector of length N that represents the feature values of the input data and $\beta$ is a vector containing the weights for each feature as well as $\beta _0$, which represents the intercept, i.e., the log-odds of the target variable when all feature values are zero. This function maps the linear combination of features into a probability [53, 60]. Additionally, the coefficients of this model are estimated by maximizing the likelihood function of the data, which minimizes the errors between the predicted probabilities and the actual values of the target variable [60].

Appendix B OCSB test results

The results of the OCSB test are displayed on Fig. 9. Several seasonal frequencies (from 10 to 1490 with increments of 10 units) where assessed for all time-series, however, only a few stars from the majority class where detected as having a seasonal component. These range from first to third differences.

Fig. 9

Results of the OCSB test. The Y-axis consists of the count for seasonal differences needed to make a time-series stationary based on the seasonal frequency (X-axis). Red means one seasonal difference is needed, green consists of second differences and blue of third differences

Appendix C Performance analysis

Figure 10 contains several boxplots that represent the value distributions of performance metrics obtained from training and evaluating models on the original test set. The metrics were grouped by data transformation (X-axis), with each metric being represented as a facet of the plot.

Fig. 10

Minority class metric values of models trained on features extracted from time-series data. The X-axis consists of data transformations before feature extraction and the Y-axis of the values for each metric (facet)