Periodograms are used as a key significance assessment and visualisation tool to display the significant periodicities in unevenly sampled time series. We introduce a framework of periodograms, called "Agatha", to disentangle periodic signals from correlated noise and to solve the 2-dimensional model selection problem: signal dimension and noise model dimension. These periodograms are calculated by applying likelihood maximization and marginalization and combined in a self-consistent way. We compare Agatha with other periodograms for the detection of Keplerian signals in synthetic radial velocity data produced for the Radial Velocity Challenge as well as in radial velocity datasets of several Sun-like stars. In our tests we find Agatha is able to recover signals to the adopted detection limit of the radial velocity challenge. Applied to real radial velocity, we use Agatha to confirm previous analysis of CoRoT-7 and to find two new planet candidates with minimum masses of 15.1 $M_\oplus$ and 7.08 $M_\oplus$ orbiting HD177565 and HD41248, with periods of 44.5 d and 13.4 d, respectively. We find that Agatha outperforms other periodograms in terms of removing correlated noise and assessing the significances of signals with more robust metrics. Moreover, it can be used to select the optimal noise model and to test the consistency of signals in time. Agatha is intended to be flexible enough to be applied to time series analyses in other astronomical and scientific disciplines. Agatha is available at http://www.agatha.herts.ac.uk.