ELM parameter estimation in view of maximum likelihood

Extreme Learning Machine (ELM) can be considered a probabilistic model, wherein the output is a random variable characterized by its mean and variance, both crucial for assessing and enhancing ELM's generalization ability. Despite its importance, limited research has been conducted on quantitative variance estimation and the underlying factors that influence connecting weight regularization. In this paper, we establish a connection between the variance and weights of ELM for regression problems, and propose maximum likelihood-based estimations for model parameters. We introduce a novel activation function and algorithms to generate hidden layer outputs and predicted outputs that adhere or approximate a normal distribution, as validated through experiments. Additionally, by leveraging the established relationship, we derive estimators for weights, variance, and confidence intervals of predicted values using Maximum Likelihood Estimation (MLE). The variance estimator enables a quantitative evaluation of generalization, and the link between variance and weights offers a theoretical basis for ELM regularization techniques.