deletions | additions       diff --git a/Estimating phases from signal.tex b/Estimating phases from signal.tex  index b5c2f08..21149be 100644  --- a/Estimating phases from signal.tex  +++ b/Estimating phases from signal.tex 
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\end{equation}  where $\xi$ is a normally distributed random variable. The model for the cell cycle is similar but links the nucleus area to the cell cycle phase. In order to distinguish between amplitude or baseline changes and phase deformations, we imposed that the amplitude and the baseline can only change on long timescale (about a day). More precisely we modeled the phase as a diffusion-drift process and the amplitude and background as Ornstein-Uhlenbeck processes.  Using that model we calibrated HMM's capable of inferring the two phases from the signals (Methods \ref{sec:HMM_methods}). Once the estimated phases are known for a given cell trace, the trajectory can be plotted in the $(\theta, \phi)$ plane and the velocity in both direction can be estimated by computing the discrete derivative of the trajectory (Figure \ref{fig:inference_of_circadian_phase}B). \ref{fig:inference_of_circadian_phase}B) \cite{Rosenblum2001}.  Given a large collection of such traces it is possible to estimate the velocity on the whole $(\theta, \phi)$ plane\cite{Rosenblum2001}  and thus to reconstruct the dynamics of the system. Inferred phase in the data are shown in Figure \ref{fig:inference_of_circadian_phase}C-D.