MOST ACCURATE NON-LINEAR APPROXIMATION OF STANDARD NORMAL DISTRIBUTION INTEGRAL BASED ON ARTIFICIAL NEURAL NETWORKS
Keywords:
Artificial neural network, standard normal distribution, approximation, cumulative distribution function, non-linear modelAbstract
Approximating the cumulative distribution function values of a standard normaldistribution with the highest accuracies still remains a challenging task. For this purpose,the non-linear prediction formulas based on artificial neural networks are applicable tothe non-linear nature of a standard normal distribution integral. In this study, a datasetconsisting of almost real integral values of a standard normal distribution was preparedranging from -5 to 10 by increments of 0.01. The dataset was used to train 16 artificialneural networks each of which was repeated 100 times to reach the best performanceamong them by considering the number of neurons, including 1, 2, 3, 5, 15, 25, 35, and 45.The test dataset was constructed ranging from -10 to 10 by increments of 0.001 withoutincluding the training dataset. Two different types of ANN models were considered inwhich their transfer functions of the hidden layers were hyperbolic tangent and those ofthe output layers were either hyperbolic tangent or linear (purelin) . Three evaluationmetrics, the mean squared error (MSE), absolute error (AE), and relative error (RE)were used to compare the results of the proposed models and another 7 accurateliterature approximation formulas. The results of the predicted points against theiralmost real values were illustrated and their measurement metric values were calculatedand compared with those of the 7 literature formulas. The highest accuracies with 8 to 9digits of accuracy were achieved by the 2 proposed equations based on ANN models using.
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