deletions | additions       diff --git a/Hidden Markov Models.tex b/Hidden Markov Models.tex  index 273c7e8..2144511 100644  --- a/Hidden Markov Models.tex  +++ b/Hidden Markov Models.tex 
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\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*} \begin{align}  \textrm{d}\theta_t =& \frac{2\pi}{T_1}\textrm{d}t + \sigma_{\theta}\textrm{d}W_t\\  \end{align*} \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*} \begin{align}  \textrm{d}\lambda_t =& -\gamma_{\lambda} (\lambda_t-\mu_{\lambda}) \textrm{d}t + \sigma_{\lambda}\textrm{d}W_t\\  \end{align*} \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. 
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The HMM for inferring the cell cycle phase $\phi$ from the two Fucci signals ($s^R_t$ and $s^G_t$, for Green and Red) is the same than the one described in the previous section, except that the two signals are related to the phase trough two waveforms and a shared amplitude:  \begin{align*} \begin{align}  s^R_t =& \exp(\alpha_t) w^R(\phi_t) +\xi\\  s^G_t =& \exp(\alpha_t) w^G(\phi_t) +\xi\\  \end{align*} \end{align}  We assumed independence and wrote the emission probability as the product of the two terms: $P(s^R_t,s^G_t|\phi_t,\alpha_t)=P(s^R_t|\phi_t,\alpha_t)P(s^G_t|\phi_t,\alpha_t)$.