Randomly weighted neural networks in statistical modeling

This chapter reviews Randomly Weighted Neural Networks (RWNNs), with particular emphasis on Extreme Learning Machines (ELM) and Random Vector Functional Link (RVFL) networks. These models belong to a class of feedforward neural networks in which the hidden-layer parameters are randomly generated and kept fixed, while the output weights are estimated analytically through regularized least-squares procedures. From a statistical standpoint, RWNNs are not introduced as a new modeling paradigm, but rather as fast and flexible training strategies for classical neural network models. Their functional form coincides with that of standard nonparametric regression models based on linear combinations of nonlinear basis functions, and they can be interpreted as sieve estimators or random-basis approximations. The chapter first reviews neural networks as statistical models for supervised learning, with primary emphasis on nonlinear regression. On this foundation, the subsequent sections extend the same framework to classification and nonlinear time series forecasting. The discussion then turns to ELM and RVFL models, presenting their formal specification and examining their estimation, regularization, approximation properties, and numerical stability. Extensions to deep architectures and multi-output settings are also considered, with particular attention to the roles of randomization and skip connections. The chapter further investigates RWNNs for time series forecasting within nonlinear autoregressive frameworks, discussing recursive, direct, and multi-output forecasting strategies. Finally, ensemble methods based on randomization, bagging, and boosting are reviewed, showing how RWNNs can be combined effectively to enhance robustness and predictive accuracy. Overall, the chapter positions RWNNs as computationally efficient, statistically grounded tools that bridge classical nonparametric modeling and modern machine learning, making them especially attractive for tabular data, time-series applications, and scenarios requiring fast training, stability, and interpretability.