Quantifying the classification of exoplanets: in search for the right habitability metric

Abstract

What is habitability? Can we quantify it? What do we mean under the term habitable or potentially habitable planet? With estimates of the number of planets in our Galaxy alone running into billions, possibly a number greater than the number of stars, it is high time to start characterizing them, sorting them into classes/types just like stars, to better understand their formation paths, their properties and, ultimately, their ability to beget or sustain life. After all, we do have life thriving on one of these billions of planets, why not on others? Which planets are better suited for life and which ones are definitely not worth spending expensive telescope time on? We need to find sort of quick assessment score, a metric, using which we can make a list of promising planets and dedicate our efforts to them. Exoplanetary habitability is a transdisciplinary subject integrating astrophysics, astrobiology, planetary science, and even terrestrial environmental sciences. It became a challenging problem in astroinformatics, an emerging area in computational astronomy. Here, we review the existing metrics of habitability and the new classification schemes (machine learning (ML), neural networks, activation functions) of extrasolar planets, and provide an exposition of the use of computational intelligence techniques to evaluate habitability scores and to automate the process of classification of exoplanets. We examine how solving convex optimization techniques, as in computing new metrics such as Cobb–Douglas habitability score (CDHS) and constant elasticity earth similarity approach (CEESA), cross-validates ML-based classification of exoplanets. Despite the recent criticism of exoplanetary habitability ranking, we are sure that this field has to continue and evolve to use all available machinery of astroinformatics, artificial intelligence (AI) and machine learning. It might actually develop into a sort of same scale as stellar types in astronomy, to be used as a quick tool of screening exoplanets in important characteristics in search for potentially habitable planets (PHPs), or Earth-like planets, for detailed follow-up targets.

Appendices

Appendices

An introduction to machine learning (ML)

ML is a driving component of AI that enable machines to infer information from data. Analogous to the way humans learn from past experiences, ML empowers computers to make use of data and enables them to learn and improve their experiences on continuous basis. Data plays an important role in making a network learn its task and concepts. As more data is fed to machines, they become more experienced (like humans) and eventually become better at the stipulated task, without having to be programmed explicitly. When previously unseen data is fed to a machine, it makes use of information obtained from already perceived data to make decisions. In this manner, the machine learns to perform tasks by gaining insightful information from the data, and eventually evolves itself to accomplish more challenging tasks. The data which a ML algorithm uses is referred as training data. Besides this, there exists test data (unseen data), which is utilized to check predictions made by the trained algorithm. If predictions made by the algorithm are not close to the desired value, the algorithm is trained again on a new set of values till it attains the desired accuracy.

Classical ML algorithms can be categorized as supervised and unsupervised. Supervised learning makes use of labeled data whereas unsupervised learning uses unlabeled data to train the model. In either case, the algorithm tries to search for a pattern in the data, a pattern that can be used later for identification of unknown (test) data. Polynomial regression, Random Forest, Linear Regression, Logistic Regression, Decision Tree, KNN and Naive Bayes are all supervised learning algorithms, while K-Means Clustering, Apriori, Hierarchical Clustering, Principal Component Analysis and Singular Value Decomposition fall under unsupervised learning. A slightly different and less popular category is the Reinforcement Learning. Reinforcement Learning makes use of the three components—agents, environment and actions—to achieve the defined target. An agent is a decision maker which gets to understand behavior of the environment and takes necessary actions to achieve its goal. Rewards are computed for every chosen action. Strategically, the agent interacts with its environment and chooses action that generates maximum rewards. With trial and error, the agent discovers data, generate possible actions, computes reward for each action and keep record of actions that reaches to the target.

1.1 Working of neural networks

A robust ML tool, Neural Network, capable to perform computationally challenging tasks on complex datasets, is inspired from the working of the human brain. The network is made up of processing neurons that are functionally similar to neurons existing in brains. These neurons are arranged in layers across the network, from one end to another and, between two layers, neurons are connected with synaptic weights. A synaptic weight describes the strength of the connection between neurons. This structure enables the processing units to perform high-speed parallel computing on input data. Neurons are equipped with a special function called activation function AF, which squashes the neuron input and brings it to a specific range before propagating it to other layers. There are three types of AFs: binary step function, linear and non-linear AFs. Binary and linear AFs are not suitable to be used in back-propagation (will be explored later) owing to the fact that the derivative of the function becomes constant. On the other hand, non-linear AF allows the network to perform complex mapping of input and output and thus facilitates modelling complex data to solve problems related to the Computer Vision, Artificial Intelligence, Deep Learning and Natural Language Processing.

Essentially, neural networks work on the principle of back-propagation algorithm which runs in two phases to achieve classification on given data set. Figure 6 shows a simple NN, consisting of one hidden layer with 5 hidden units.

Fig. 6

A sample neural network

The network weights are initialized with random values. To attain the right set of network weights, a training set is chosen as input such that the network tries to identify patterns from the training values to make correct future predictions. The first phase involves computation of inputs of neurons of first layer. AF converts these inputs into a definite range and propagates it to neurons of connected layer. The process goes on till the output is received at output layer. In the second phase, error (cost) associated with training sample is computed and error gradients are propagated back to the network in the form of weight updates. The process is repeated for multiple training samples in numerous iterations or epochs, till the expected output is received at the output neurons.

1.2 Activation functions (AF)

It must be admitted that the machine classification is a non-trivial task as the classes are non-linearly separated from each other. There is a large number of linear and non-linear AFs currently in use in Neural Networks. These mathematical functions are tied-up to every neuron in the network to facilitate decision making while solving a complex problem at hand. The function output allow neurons to determine whether it should be activated or not. Moreover, the function also normalizes output values of neurons and brings it to either [0, 1] or $[-1,1]$ range. In this section, we explore the non-linear AFs as they speed up solving complex problems by stacking of multiple hidden layers thus forming deep layered architectures.

1.2.1 Sigmoid function

Also known as logistic function, Sigmoid function [77] looks like a S-shaped curve which maps real values between 0 and 1. The mathematical notation of the function is given as

$$\begin{aligned} S(x) = \frac{1}{1 + \mathrm {e}^{-x}}\,. \end{aligned}$$

(A.1)

Sigmoid suffers from vanishing gradient problem as it is evident that for very large and very small values of x, the gradient of the function becomes zero.

1.2.2 ReLU

Rectified linear unit (ReLU) is a function that returns zero for negative and identity for positive inputs. Mathematically, the function looks like the following:

$$\begin{aligned} S(x) = \mathrm{max}(0,x)\,. \end{aligned}$$

(A.2)

Computation of gradient is an crucial step in back propagation and unlike Sigmoid, ReLU provides a constant gradient for positive inputs. But visibly, ReLU’s gradient and also the output from neurons is zero for negative values.

1.2.3 A-ReLU

Approximate ReLu (A-ReLU) is another variant of ReLU that is defined by

$$\begin{aligned} S\left( x \right) = \left\{ \begin{matrix} 0 &{} x\le 0\\ kx^{n} &{} x> 0 \end{matrix}\right. . \end{aligned}$$

(A.3)

The function, unlike ReLU, is differentiable at $x=0$ and, hence, it can be shown that its derivative is continuous.

1.2.4 Swish

This AF, proposed by the Google Brain team [78], works better than ReLU in various Deep Learning architectures on the most challenging data. The smooth, non-monotonic function is mathematically represented as

$$\begin{aligned} S(x) = \frac{x}{1 + \mathrm {e}^{-x}}\,. \end{aligned}$$

(A.4)

Experimentally, it was observed that Swish and ReLU perform at par till 40 layers in Deep Learning architectures. Beyond 40 layers, Swish outperforms ReLU, which is particularly seen on few datasets while training deep layered architectures. However, on shallow networks this might not be true.

1.3 SBAF—classification of exoplanets for cross-validating metrics

AFs such as Sigmoid suffer from local oscillation and flat mean-square-error (MSE) problems. This means that over successive iterations, the error between the target and predicted labels may not minimize due to the oscillating local minima. Saha et al. [70] mitigated this problem by proposing a new AF, SBAF⁵,

$$\begin{aligned} y = \frac{1}{1 + kx^{\alpha }(1-x)^{1-\alpha }} \,, \end{aligned}$$

(A.5)

where where k and $\alpha$ are the parameters of the function, and the derivative is computed as

$$\begin{aligned} \frac{dy}{dx} = -y^{2} \cdot \frac{\alpha -x}{x(1-x)} \cdot \frac{1-y}{y} \frac{y(1-y)}{x(1-x)}\cdot (x-\alpha )\,. \end{aligned}$$

(A.6)

SBAF is a non-monotonic function that does not exhibit vanishing gradient problem during training. The characteristics of the function indicate the presence of local minima and maxima not present in other popularly used AFs. It is also shown that unlike the Sigmoid, this function does not have a saddle point [71]. It is also argued that AFs need not be monotonic at all when analyzed over the entire domain. The following properties of SBAF help to achieve a near-perfect classification of exoplanets, matching habitability labels with Earth similarity:

• Does not admit a saddle point;

• Has local minima at $x=\alpha$;

• Satisfies Universal Approximation theorem;

• Does not admit flat MSE;

• Does not suffer from vanishing gradient problem.

An evaluation of SBAF is incomplete without stating its importance in the context of other AFs described above. It is also observed that the classification performance of classical ML algorithms (Support Vector Machine, Random Forest, Naive Bayes, etc.) drops abysmally when applied on the restricted-feature data.

However, pure empiricism does not determine the existence of an AF, its state-of-the-art (SOTA) performance notwithstanding. AFs such as Swish and Mish claim SOTA results on complex neural network architectures. But, are these even AFs? An AF has to be discriminatory in the sense that it should be able to approximate any nonlinear function describing the data over a neural network [71]. There is no discussion in the literature on this property known as the Universal Approximation [71]. Does this mean that we dilute an AF to a level of just being continuous and smooth? In other words, can any function be an AF? The answer is an emphatic negative, but the Deep Learning literature is laden with such empirical observations. Additionally, an AF needs to be smooth everywhere (no singularity), so that the first derivative can be used in back propagation in tandem with the Lipschitz continuity [72] for fast acceleration [73]. RELU suffers from this singularity problem. Sigmoid is not devoid of problems either. It flattens out for most values in the input range and is, therefore, only suited for inputs normalized to [0, 1]. It has its second derivative zero, which explains the difficulty of Sigmoid overcoming local minima/maxima. Another important fact that SBAF helped establish in the significant work of Saha et al. [71] is that we no longer need AFs to be monotone. This was a common wisdom in the Deep Learning community but based purely on empirical observation.