S3 File. Hyperparameters of prior distribution and updating rule. We explain the Bayesian linear regression for estimating phase coupling functions for cross- ...
S3 File. Hyperparameters of prior distribution and updating rule. We explain the Bayesian linear regression for estimating phase coupling functions for cross-frequency synchronization in the Materials and Methods section of the main text. Here, we describe how the posterior distribution parameters are computed. Using Eq. 12 with Eqs. 11 and 13, we can compute the posterior distribution parameters as follows: 𝝌𝒏𝒆𝒘 = 𝚺𝒊𝒏𝒆𝒘 (𝐅𝒊 𝑇 𝐝𝒊 + 𝚺𝒊𝒐𝒍𝒅 𝒊 𝚺𝒊𝒏𝒆𝒘 = (𝚺𝒊𝒐𝒍𝒅
−1
−1 𝒐𝒍𝒅 𝝌𝒊 )
+ 𝐅𝒊 𝑇 𝑭𝒊 )−1 ,
𝑇 α𝑛𝑒𝑤 = α𝑜𝑙𝑑 + , 𝑖 𝑖 2 1 𝑇 𝒐𝒍𝒅 −1 𝒐𝒍𝒅 𝑇 𝒏𝒆𝒘 −1 𝒏𝒆𝒘 β𝑛𝑒𝑤 = β𝑜𝑙𝑑 + (𝒅𝒊 𝑇 𝒅𝒊 + 𝝌𝒐𝒍𝒅 𝚺𝒊 𝝌𝒊 − 𝝌𝒏𝒆𝒘 𝚺𝒊 𝝌𝒊 ). 𝒊 𝒊 𝑖 𝑖 2 We determine the phase velocity vector and the matrices of the bases of the Fourier series at each time as follows: 𝐝𝒊 = [(𝜙𝑖 (𝑡2 ) − 𝜙𝑖 (𝑡1 ))/Δ𝑡 (𝜙𝑖 (𝑡3 ) − 𝜙𝑖 (𝑡2 ))/Δ𝑡 (𝜙𝑖 (𝑡𝑇+1 ) − 𝜙𝑖 (𝑡𝑇 ))/Δ𝑡]𝑇 , 1 𝑮𝒊,𝟏 (𝜓𝑖,1 (1)) ⋯ 𝑮𝒊,𝒊−𝟏 (𝜓𝑖,𝑖−1 (1)) 𝑮𝒊,𝒊+𝟏 (𝜓𝑖,𝑖+1 (1)) ⋯ 𝑮𝒊,𝑵 (𝜓𝑖,𝑁 (1)) 𝑭𝒊 = [ ], ⋮ 1 𝑮𝒊,𝟏 (𝜓𝑖,1 (𝑇)) ⋯ 𝑮𝒊,𝒊−𝟏 (𝜓𝑖,𝑖−1 (𝑇)) 𝑮𝒊,𝒊+𝟏 (𝜓𝑖,𝑖+1 (𝑇)) ⋯ 𝑮𝒊,𝑵 (𝜓𝑖,𝑁 (𝑇)) 𝑮𝒊𝒋 (𝜓𝑖𝑗 ) = [cos𝜓𝑖𝑗 sin𝜓𝑖𝑗 cos2𝜓𝑖𝑗 sin2𝜓𝑖𝑗 ⋯ cosM𝑖𝑗 𝜓𝑖𝑗 sinM𝑖𝑗 𝜓𝑖𝑗 ], where, 𝜓𝑖,1 (𝑡) is phase difference 𝑝𝑖 𝜙𝑗 (𝜏) − 𝑝𝑗 𝜙𝑖 (𝜏). Then, we set the hyperparameters of the prior distribution to the following values:
𝚺𝒊𝒐𝒍𝒅
𝝌𝒐𝒍𝒅 = 𝟎, α𝑜𝑙𝑑 = β𝑜𝑙𝑑 = 0, 𝜆 = 1, 𝒊 𝑖 𝑖 −1 −1 −1 −1 −1 = diag[𝜆 , 𝜆 , 𝜆 , 2𝜆 , 2𝜆 , ⋯ , 𝑀𝑖1 𝜆−1 , 𝑀𝑖1 𝜆−1 , 𝜆−1 , 𝜆−1 , 2𝜆−1 , 2𝜆−1 , ⋯ ⋯ , 𝑀𝑖2 𝜆−1 , 𝑀𝑖2 𝜆−1 , ⋯ , 𝜆−1 , 𝜆−1 , 2𝜆−1 , 2𝜆−1 , ⋯ , 𝑀𝑖𝑁 𝜆−1 , 𝑀𝑖𝑁 𝜆−1 ],
where the size of the matrix of 𝚺𝒊𝒐𝒍𝒅 is (1 + ∑𝑗≠𝑖 2𝑀𝑖𝑗 ) × (1 + ∑𝑗≠𝑖 2𝑀𝑖𝑗 ) and 𝜆 is a precision parameter. This Bayesian method was proposed and explained in a previous study [1].
1.
Ota K, Aoyagi T. Direct extraction of phase dynamics from fluctuating rhythmic data based
on a Bayesian approach. ArXiv e-prints [Internet]. 2014 May 1, 2014; 1405. Available from: http://adsabs.harvard.edu/abs/2014arXiv1405.4126O.
