deletions | additions       diff --git a/Hidden Markov Models.tex b/Hidden Markov Models.tex  index 81936f3..8ccc39e 100644  --- a/Hidden Markov Models.tex  +++ b/Hidden Markov Models.tex 
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The complete HMM for the circadian clock is defined by:  \begin{equation} \label{eqn:circ_HMM0} \label{eqn:circ_HMM1}  \textrm{d}\theta_t =& \frac{2\pi}{T_1}\textrm{d}t + \sigma_{\theta}\textrm{d}W_t  \end{equation}  %  \begin{equation} \label{eqn:circ_HMM2}  \textrm{d}\lambda_t =& -\gamma_{\lambda} (\lambda_t-\mu_{\lambda}) \textrm{d}t + \sigma_{\lambda}\textrm{d}W_t   \end{equation}  %  \begin{equation} \label{eqn:circ_HMM3}  \textrm{d}B_t =& -\gamma_{B} (B_t-\mu_{B}) \textrm{d}t + \sigma_{B}\textrm{d}W_t   \end{equation}  %  \begin{equation} \label{eqn:circ_HMM4}  s_t =& \exp(\lambda_t) w(\theta_t) + B_t +\xi  \end{equation}  \begin{align}   \textrm{d}\theta_t %\begin{align}   %\textrm{d}\theta_t  =& \frac{2\pi}{T_1}\textrm{d}t + \sigma_{\theta}\textrm{d}W_t \label{eqn:circ_HMM1}\\ \textrm{d}\lambda_t %\textrm{d}\lambda_t  =& -\gamma_{\lambda} (\lambda_t-\mu_{\lambda}) \textrm{d}t + \sigma_{\lambda}\textrm{d}W_t \label{eqn:circ_HMM2} \\ \textrm{d}B_t %\textrm{d}B_t  =& -\gamma_{B} (B_t-\mu_{B}) \textrm{d}t + \sigma_{B}\textrm{d}W_t\\ s_t %s_t  =& \exp(\lambda_t) w(\theta_t) + B_t +\xi\\  \end{align} +\xi \\  %\end{align}  For the phase $\theta$ we used a mean period $T_1$ of $24$h and a phase diffusion coefficient $\sigma_{\theta}$ of $0.15$ $\textrm{rad} \; h^{-1/2}$. For the amplitude $\lambda$ we used a timescale $\gamma_{\lambda}^{-1}$ of $30$h a zero mean value $\mu_{\lambda}$ and a diffusion coefficient of $0.07$. For the baseline $B$ we used a timescale of $30$h, a zero mean and a diffusion coefficient of $0.022$. These parameters were chosen such that the amplitude and the baseline smoothly follow the maximums and the minimums of the signal, without explaining variations in the shape of the signal, which we aim to capture in the phase.  Finally the variance of the noise $\xi$ was set to $0.1$.