A new statistical training algorithm for a single multiplicative neuron model artificial neural network

The single multiplicative neuron model has been frequently used by researchers in recent years, as it does not have a complex structure and does not include the hidden layer unit number problem, unlike many feed-forward artificial neural network models. The model of single multiplicative neuron model artificial neural networks does not have statistical assumptions just like in many artificial neural network models. The random error term is not used in the mathematical model of single multiplicative neuron model artificial neural networks. This situation is not acceptable considering that artificial neural networks work with random samples. Based on this idea, for the first time, by including a random error term in the single multiplicative neuron model artificial neural network model, mathematical equations of likelihood functions are given for normal, cauchy, logistic, gumbel, and laplace distributions. A new statistical training algorithm is proposed to obtain optimal weights and bias values of the network. In the new training algorithm, particle swarm optimization proposed by Kennedy and Eberhart (in: Proceedings of IEEE international conference on neural networks (ICNN '95). IEEE, pp 1942–1948, 1995) is used in maximizing likelihood functions. In the performance evaluation of the proposed method, Nasdaq and S&P500 time series in different years are analyzed and the analysis results are compared with many artificial neural network models in the literature. Finally, it is concluded that the proposed method produces very successful forecasting results.