deletions | additions       diff --git a/Hidden Markov Models.tex b/Hidden Markov Models.tex  index 6766fa2..273c7e8 100644  --- a/Hidden Markov Models.tex  +++ b/Hidden Markov Models.tex 
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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}). \ref{fig:inference_of_circadian_phase}C-D).  \subsubsection{Circadian clock}