The scoring of sleep stages is an essential part of sleep studies. The main objective of this research is to provide an algorithm for the automatic classification of sleep stages using signals that may be obtained in a non-obtrusive way. After reviewing the relevant research, the authors selected a multinomial logistic regression as the basis for their approach. Several parameters were derived from movement and breathing signals, and their combinations were investigated to develop an accurate and stable algorithm. The algorithm was implemented to produce successful results: the accuracy of the recognition of Wake/NREM/REM stages is equal to 73%, with Cohen's kappa of 0.44 for the analyzed 19324 sleep epochs of 30 seconds each. This approach has the advantage of using the only movement and breathing signals, which can be recorded with less effort than heart or brainwave signals, and requiring only four derived parameters for the calculations. Therefore, the new system is a significant improvement for non-obtrusive sleep stage identification compared to existing approaches.