In this paper, we present an approximation method for principal component analysis (PCA) and apply it to estimating the respiration from the overnight ECG signal. The approximation method is computationally fast with low memory requirements. We compare it to a full PCA method which is applied to segments of the ECG. Features were calculated from the two ECG derived respiration signals (EDR) and classifiers trained to detect obstructive sleep apnoea (OSA). The Extreme Learning Machine and Linear Discriminant classifier were used to classify the recordings. The data from 35 overnight ECG recordings from MIT PhysioNet Apnea-ECG training database was utilized in the paper. Apnoea detection was evaluated with leave-one-record-out cross validation. The approximated PCA method obtained the highest accuracy of 76.4% by ELM classifier at fan-out 10 and accuracy of 78.4% by LDA. While, the segmented PCA achieved lower accuracies for both classifiers, 75.9% by ELM classifier and 76.6% by LDA. We conclude that the approximation method for PCA is well suited to deriving the respiration signal from overnight ECGs.