deletions | additions       diff --git a/Estimating phases from signal.tex b/Estimating phases from signal.tex  index 050f4f1..66422fc 100644  --- a/Estimating phases from signal.tex  +++ b/Estimating phases from signal.tex 
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\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 trougth 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 a 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 $\sigma_{em}$.   Given this model we derived the transition and emmision probabilities needed to specify a HMM, optimized the waveform $w(\theta)$ using maximum likelihood TODO:ref and estimated the sequence of hidden states $(\theta_t,A_t,B_t)$ for each trace via the maximum or mean of its posterior distribution (Figure \ref{fig:inference_of_circadian_phase}).