The aim of the work presented here was to develop a system that can automatically identify attacks of dizziness occurring in patients suffering from positional vertigo, which occurs when sufferers move their head into certain positions. We used our novel medical device, CAVA, to record eye- and head-movement data continually for up to 30 days in patients diagnosed with a disorder called Benign Paroxysmal Positional Vertigo. Building upon our previous work, we describe a novel ensemble of five 2D Convolutional Neural Networks, using composite recognition features, including eye-movement data and three-channel accelerometer data. We achieve an F1 score of 0.63 across an 11-fold cross-fold validation experiment, demonstrating that the system can detect a few seconds of motion provoked dizziness from within over a 100 h of normal eye-movement data. We show that the system outperforms our previous 1D Neural Network approach, and that our ensemble classifier is superior to each of the individual networks it contains. We also demonstrate that our composite recognition features provide improved performance over results obtained using the individual data sources independently.