Causality in Time Series - ClopiNet

Causal analysis of fMRIv(t). In fMRI experiments the exogenous inputs v(t) mostly reflect experimental controland often have the form of a simple indicator function that can represent, for instance,the presence or absence of a visual stimulus. The vector-functions f and g can generallybe non-linear.The state-space formalism allows representation of a very large class of stochasticprocesses. Specifically, it allows representation of both so-called ‘black-box’ models,in which parameters are treated as means to adjust the fit to the data without reflectingphysically meaningful quantities, and ‘grey-box’ models, in which the adjustable parametersdo have a physical or physiological (in the case of the brain) interpretation. Aprominent example of a black-box model in econometric time-series analysis and systemsidentification is the (discrete) Vector Autoregressive Moving Average model withexogenous inputs (VARMAX model) defined as (Ljung, 1999; Reinsel, 1997):F (B)y t = G (B)v t + L(B)e t ⇔∑︀ pi=0 F iy t−i = ∑︀ sj=0G j v t− j + ∑︀ qk=0 L ke t−k(13)Here, the backshift operator B is defined, for any η t as B i η t = η t−i and F, G and Lare polynomials in the backshift operator, such that e.g. F (B) = ∑︀ pi=0 F iB i and p, sand q are the dynamic orders of the VARMAX(p,s,q) model. The minimal constraintson (13) to make it identifiable are F 0 = L 0 = I, which yields the standard VARMAXrepresentation. The VARMAX model and its various reductions (by use of only oneor two of the polynomials, e.g. VAR, VARX or VARMA models) have played a largerole in time-series prediction and WAGS influence modeling. Thus, in the context ofstate space models it is important to consider that the VARMAX model form can beequivalently formulated in a discrete linear state space form:x k+1 = Ax k + Bv k + Ke ky k = Cx k + Dv k + e k(14)In turn the discrete linear state space form can be explicitly accommodated by the continuousgeneral state-space framework in (12) when we define:f (x(t),v(t),Θ) ≃ Fx(t) +Gv(t)g(x(t),v(t),Θ) ≃ Hx(t) + Dv(t)ω(t) = ˜Kε(t)Θ = {︁ F,G, H, D, ˜K,Σ e}︁ (15)Again, the exact relations between the discrete and continuous state space parametermatrices can be derived analytically by explicit integration over time (Ljung, 1999).And, as discussed above, wherever discrete data is used to model continuous influencerelations the problems of temporal aggregation and aliasing have to be taken into account.Although analytic solutions for the discretely sampled continuous linear systemsexist, the discretization of the nonlinear stochastic model (12) does not have a uniqueglobal solution. However, physiological models of neuronal population dynamics andhemodynamics are formulated in continuous time and are mostly nonlinear while fMRIdata is inherently discrete with low sampling frequencies. Therefore, it is the discretizationof the nonlinear dynamical stochastic models that is especially relevant to causal87