The central goal is to design a model that makes useful classifications for the different orbit types.
This is a classification project, since the variable to be predicted is categorical: AMO, APO, ATE. These orbit types are asteroid orbits, so the model will classify orbits according to the above asteroid orbit classes.
The first step is to prepare the data set, which is the source of information for the classification problem. For that, we need to configure the following concepts:
The variables are:
The above target variables are asteroid orbits.
AMO is referring to Amor asteroids: near-Earth asteroids. The orbital perihelion of these objects is close to, but greater than, the orbital aphelion of Earth (a > 1.0 AU and 1.017 AU < q < 1.3 AU).
APO is referring to Apollo asteroids: near-Earth asteroid orbits which cross the Earth's orbit (a > 1.0 AU and q < 1.017 AU).
ATE refers to Aten asteroids: a dynamical group of asteroids whose orbits bring them into proximity with Earth. By definition, Atens are Earth-crossing asteroids (a<1.0 AU and Q>0.983 AU).
These differents asteroid orbits are shown in the following image:
Note that neural networks work with numbers. In this regard, the categorical variable "class" is transformed into three numerical variables as follows:
The instances are divided into training, selection, and testing subsets. They represent 60% (1034), 20% (344), and 20% (344) of the original instances, respectively, and are split at random.
We can calculate the distributions of all variables. The next figure is the pie chart for the orbit types.
As we can see, most of the samples are APO orbits.
Finally, the inputs-targets correlations might indicate to us what factors most influence.
Here, the most correlated variables with the classification are a, q, and Q, the semi-major axis, perihelion distance, and aphelion distance of the orbit, respectively. Also are not many correlated variables like M, mean anomaly, or H, absolute V-magnitude.
The second step is to choose a neural network. For classification problems, it is usually composed by:
The scaling layer contains the statistics on the inputs calculated from the data file and the method for scaling the input variables. Here the minimum and maximum method has been set. Nevertheless, the mean and standard deviation method would produce very similar results.
The number of perceptron layers is 1. This perceptron layer has 11 inputs and 3 neurons.
The neural network has three outputs since the target variable contains 3 classes (AMO, APO, ATE).
The next figure is a graphical representation of this classification neural network:
Here, the yellow circles represent scaling neurons, the blue circles represent perceptron neurons, and the red circles represent probabilistic neurons. The number of inputs is 11, and the number of outputs is 3.
The fourth step is to set the training strategy, which is composed of:
The following chart shows how the training and selection errors decrease with the epochs during the training process.
The final values are training error = 0.0469 NSE (blue), and selection error = 0.189 NSE (orange).
Order selection algorithms train several network architectures with a different number of neurons and select that with the smallest selection error.
The incremental order method starts with a small number of neurons and increases the complexity at each iteration.
The purpose of the testing analysis is to validate the generalization performance of the model. Here we compare the neural network outputs to the corresponding targets in the testing instances of the data set.
In the confusion matrix, the rows represent the targets (or real values) and the columns the corresponding outputs (or predictive values). The diagonal cells show the correctly classified cases, and the off-diagonal cells show the misclassified cases.
|Predicted APO||Predicted ATE||Predicted AMO|
|Real APO||283 (84.0%)||0 (0.0%)||1 (0.3%)|
|Real ATE||1 (0.3%)||30 (8.7%)||0 (0.0%)|
|Real AMO||7 (2.0%)||0 (0.0%)||16 (4.7%)|
As we can see, the number of instances that the model can correctly predict is 335 (97.4%), while it misclassifies is 9 (2.6%). This shows that our predictive model has a great classification accuracy.
The neural network is now ready to predict outputs for inputs that it has never seen. This process is called model deployment.
To classify a given orbit, we calculate the neural network outputs from the differents variables. For instance:
The neural network would classify the orbit as Apollo asteroid orbit for this particular case since it has the highest probability.
The mathematical expression of the trained neural network is listed below.
scaled_a = a*(1+1)/(17.81870079-(0.6369649768))-0.6369649768*(1+1)/(17.81870079-0.6369649768)-1; scaled_e = e*(1+1)/(0.9560419917-(0.02542470023))-0.02542470023*(1+1)/(0.9560419917-0.02542470023)-1; scaled_i = i*(1+1)/(75.41239929-(0.1460839957))-0.1460839957*(1+1)/(75.41239929-0.1460839957)-1; scaled_w = w*(1+1)/(359.6629944-(0.5218380094))-0.5218380094*(1+1)/(359.6629944-0.5218380094)-1; scaled_Node = Node*(1+1)/(359.855011-(0.1360419989))-0.1360419989*(1+1)/(359.855011-0.1360419989)-1; scaled_M = M*(1+1)/(359.8250122-(0.05216519907))-0.05216519907*(1+1)/(359.8250122-0.05216519907)-1; scaled_q = q*(1+1)/(1.060099959-(0.09279999882))-0.09279999882*(1+1)/(1.060099959-0.09279999882)-1; scaled_Q = Q*(1+1)/(34.68000031-(0.9900000095))-0.9900000095*(1+1)/(34.68000031-0.9900000095)-1; scaled_P = P*(1+1)/(75.22000122-(0.5099999905))-0.5099999905*(1+1)/(75.22000122-0.5099999905)-1; scaled_H = H*(1+1)/(22-(14.10000038))-14.10000038*(1+1)/(22-14.10000038)-1; scaled_MOID = MOID*(1+1)/(0.04998699948-(9.999999747e-06))-9.999999747e-06*(1+1)/(0.04998699948-9.999999747e-06)-1; perceptron_layer_0_output_0 = tanh[ 0.998564 + (scaled_a*0.483063)+ (scaled_e*4.23015)+ (scaled_i*0.033193)+ (scaled_w*0.00368636)+ (scaled_Node*-0.0750109)+ (scaled_M*0.0156837)+ (scaled_q*4.51185)+ (scaled_Q*0.387602)+ (scaled_P*-0.48082)+ (scaled_H*-0.124104)+ (scaled_MOID*0.0301184) ]; perceptron_layer_0_output_1 = tanh[ 0.378967 + (scaled_a*0.0679598)+ (scaled_e*2.29236)+ (scaled_i*-0.259866)+ (scaled_w*-0.415081)+ (scaled_Node*-0.00389391)+ (scaled_M*-0.0223087)+ (scaled_q*-1.3599)+ (scaled_Q*0.12361)+ (scaled_P*-0.168758)+ (scaled_H*0.596372)+ (scaled_MOID*-0.376615) ]; perceptron_layer_0_output_2 = tanh[ -2.16292 + (scaled_a*1.04587)+ (scaled_e*0.149125)+ (scaled_i*0.107164)+ (scaled_w*-0.0590388)+ (scaled_Node*0.0606977)+ (scaled_M*-0.0259366)+ (scaled_q*5.8545)+ (scaled_Q*0.931787)+ (scaled_P*1.49335)+ (scaled_H*0.115587)+ (scaled_MOID*0.0850156) ]; probabilistic_layer_combinations_0 = -0.922149 +3.8602*perceptron_layer_0_output_0 +1.09178*perceptron_layer_0_output_1 -3.9471*perceptron_layer_0_output_2 probabilistic_layer_combinations_1 = 1.00764 -4.69034*perceptron_layer_0_output_0 +1.0006*perceptron_layer_0_output_1 -1.231*perceptron_layer_0_output_2 probabilistic_layer_combinations_2 = -0.0237251 +0.837826*perceptron_layer_0_output_0 -2.09368*perceptron_layer_0_output_1 +5.1967*perceptron_layer_0_output_2 sum_ = exp(probabilistic_layer_combinations_0 + exp(probabilistic_layer_combinations_1 + exp(probabilistic_layer_combinations_2; APO = exp(probabilistic_layer_combinations_0)/sum_; ATE = exp(probabilistic_layer_combinations_1)/sum_; AMO = exp(probabilistic_layer_combinations_2)/sum_;
We have just built a predictive model to determine the possible asteroid orbit type.