deletions | additions       diff --git a/Estimating coupling functions from phases.tex b/Estimating coupling functions from phases.tex  index 299bc04..de6721e 100644  --- a/Estimating coupling functions from phases.tex  +++ b/Estimating coupling functions from phases.tex 
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The velocities are computed by taking the finite difference of the phases $v_{\theta}(\phi,\theta) = (\theta(t+\Delta t)-\theta(t))/\Delta t$, with $\Delta t$ begin equal to $0.5h$. We then estimated the functions $F1$ and $F_2$ by doing a least square fit of a 2D Fourier serie with 10 harmonics on the collection of velocities $\{v_{\theta}\}$ and $\{v_{\phi} \}$. Before doing so we verified that the distribution of velocities at every position in the phase plane was mainly unimodal. The functions were estimated on a $40$ by $40$ grid, given by the discrete hidden states of our HMMs.  In total we selected $2753$ time traces with a minimum duration of $24h$ and at least two divisions. These gave $208'762$ velocities for each function. The data density (number of data points per discrete position in the phase plane) is shown in Supplementary  Figure \ref{fig:dataDensity}. We tested our reconstruction method on synthetic data, and confirmed that is was able to recover the directionality of the coupling as well as the signs and positions of the interaction regions (Section \ref{sec:Inferring_coupling_function_from_synthetic_data}). %An example of reconstructed functions from synthetic data is also shown in Figure \ref{fig:example_of_reconstructed_coupling_functions}.