Two new signal-based and one model-based fault detection methods using canonical polyadic (CP) tensor decomposition algorithms are presented, and application examples of heating systems are given for all methods. The first signal-based fault detection method uses the factor matrices of a data tensor directly, the second calculates expected values from the decomposed tensor and compares these with measured values to generate the residuals. The third fault detection method is based on multi-linear models represented by parameter tensors with elements computed by subspace parameter identification algorithms and data for different but structured operating regimes. In case of missing data or model parameters in tensor representation, an approximation method based on a special CP tensor decomposition algorithm for incomplete tensors is proposed, called the decompose-and-unfold method. As long as all relevant dynamics has been recorded, this method approximates – also from incomplete data – models for all operating regimes, which can be used for residual generation and fault detection, e.g. by parity equations.