Spectral Dynamic Causal Modeling

Implementation of spectral dynamic mode decomposition (DCM) in Julia. Code partially translated from the DCM implementation in MATLAB as part of SPM12 (https://www.fil.ion.ucl.ac.uk/spm/software/spm12/)

Define notational correspondence between SPM and this code:

the following two precision matrices will not be updated by the code, they belong to the assumed prior distribution p (fixed, but what if it isn't the ground truth?) ipC = Πθ_pr # precision matrix of prior of parameters p(θ) ihC = Πλ_pr # precision matrix of prior of hyperparameters p(λ)

Variational distribution parameters: pE, Ep = μθ_pr, μθ # prior expectation of parameters (q(θ)) pC, Cp = θΣ, Σθ # prior covariance of parameters (q(θ)) hE, Eh = μλ_pr, μλ # prior expectation of hyperparameters (q(λ)) hC, Ch = λΣ, Σλ # prior covariance of hyperparameters (q(λ))

Σ, iΣ # data covariance matrix (likelihood), and its inverse (precision of likelihood - use Π only for those precisions that don't change) Q # components of iΣ; definition: iΣ = sum(exp(λ)*Q)