A major drawback of nonlinear dimensionality reduction (DR) techniques is their inability to preserve some authentic information from the source domain, leading to projections that are often hard to interpret when it comes to observing anything other than the topological structure of the data. In this paper, we propose a nonlinear DR approach enforcing projection constraints resulting from an a priori knowledge about the structure of the data in multivariate periodical time series. We then propose several ways of exploiting this constrained projection to extract user-relevant information, such as the nominal behavior of a periodical dynamical system or the deviant behaviors which may occur at different time scales. The techniques are demonstrated on both a synthetic dataset composed of simulated multivariate data exhibiting a periodical behavior, and a real dataset corresponding to six months of sensor data acquisitions and recordings inside experimental buildings