M-estimator-based online sequential extreme learning machine for predicting chaotic time series with outliers

An M-estimator-based online sequential extreme learning machine (M-OSELM) is proposed to predict chaotic time series with outliers. The M-OSELM develops from the online sequential extreme learning machine (OSELM) algorithm and retains the same excellent sequential learning ability as OSELM, but replaces the conventional least-squares cost function with a robust M-estimator-based cost function to enhance the robustness of the model to outliers. By minimizing the M-estimator-based cost function, the possible outliers are prevented from entering the model’s output weights updating scheme. Meanwhile, in the sequential learning process of M-OSELM, a sequential parameter estimation approach based on error sliding window is introduced to estimate the threshold value of the M-estimator function for online outlier detection. Thanks to the built-in median operation and sliding window strategy, this approach is efficient to provide a stable estimator continuously without high computational costs, and then the potential outliers can be effectively detected. Simulation results show that the proposed M-OSELM has an excellent immunity to outliers and can always achieve better performance than its counterparts for prediction of chaotic time series when the training dataset contains outliers, ensuring at the same time all benefits of an online sequential approach.