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\subsection{Hidden Markov Models for estimating phases}  \label{sec:HMM_methods}  In order to infer the circadian phase $\theta_t$ from the observed signal, we modeled it as a diffusion-drift process, where the drift term correspond its mean angular frequency and the noise term allow the phase to deviate form its mean to explain variations in the signal.   \begin{align*}   \textrm{d}\theta_t =& \frac{2\pi}{T_1}\textrm{d}t + \sigma_{\theta}\textrm{d}W_t\\  \end{align*}  We added to the model a multiplicative amplitude term $A_t$ that allows to explain variations in the amplitude of the signal, and is modeled as an Ornstein-Uhlenbeck process $\lambda_t$ such that $A_t = \exp(\lambda_t)$.  \begin{align*}   \textrm{d}\lambda_t =& -\gamma_{\lambda} (\lambda_t-\mu_{\lambda}) \textrm{d}t + \sigma_{\lambda}\textrm{d}W_t\\  \end{align*}  Similarly an additive baseline term $B_t$ was added to take into account traces where the signal do not go back to zero. This was necessary to avoid spurious deformation of the phase around the trough of the signal. The baseline term also follows an Ornstein-Uhlenbeck process.  Finally, the model links the observed circadian signal $s_t$ to the circadian phase $\theta_t$ through a waveform $w(\theta)$, the amplitude and the baseline:  \begin{equation}  s_t = \exp(\lambda_t) w(\theta_t) + B_t +\xi,   \end{equation}  where $\xi$ is a normally distributed random variable with zero mean and variance $\sigma_{em}$. The model parameters are given in Section \ref{sec:HMM_methods}.  Given this model we derived the transition and emission probabilities needed to specify a HMM, discretized the hidden states $(\theta_t,A_t,B_t)$, optimized the waveform $w(\theta)$ using maximum likelihood (Section \ref{sec:waveform_optimization}) and estimated the sequence of hidden states for each trace via the maximum or mean of the posterior distribution, computed via the forward-backward algorithm (Figure \ref{fig:inference_of_circadian_phase}).  \subsubsection{Circadian clock}  The complete HMM for the circadian clock is defined by: