Electroencephalography(EEG), helps to analyze the neuronal activity of a human brain in the form of electrical signals with high temporal resolution in the millisecond range. In the case of e-health cares, the internet is the medium for remote patients.To extract clean clinical information from EEG signals, it is essential to remove unwanted signals, called as artifacts, that are due to different causes including at the time of acquisition. In this piece of work, the authors considered the EEG signal contaminated with ocular artifacts. The clean EEG as well as artifactual EEG are taken from the openly available Mendeley database for verifying the performance of the proposed algorithm.Being the artifactual signal is non-linear and non-stationary the Deep Wavelet Vector Functional Link Network(WVFLN) model is used in this case. The Machine Learning approach has taken a leading role in every field of current research and WVFLN is one of them. For the proof of adaptive nature, the model is designed with EEG as a reference and artifactual EEG as input.To vary the weight and reduce the error, an exponentially weighted Recursive Least Square(RLS) algorithm is used to update the output layer weight of the novel WVFLN model. The Wavelet coefficients are considered in this model with a radial basis function to satisfy the required signal experimentation. It is found that the result is excellent in terms of Mean Square Error(MSE), Root Mean Square Error(RMSE).The MSE and RMSE are found as 0.18μV2 and 0.4242 μV respectively.Also, the proposed method is compared with the earlier methods to show its efficacy.