Estimating directed functional connectivity through autoregressive models and orthogonal Laguerre basis functions

Andrea Duggento^{1}, Gaetano Valenza^{2,3}, Luca Passamonti^{4,5}, Maria Guerrisi^{1}, Riccardo Barbieri^{3,6}, and Nicola Toschi^{1,7}

Estimating globally conditioned GC (GCGC) from variable Y_{t} to the variable X_{t} (Y→X) amounts to testing the null-hypothesis that knowledge of the past of Y_{t} does not improve the prediction of the future of X_{t}. To this end, in the GCGC approach two models are employed: first, the "restricted" AR model for X_{t}, which includes the past values of X_{t} itself and Z_{t}, which accounts for all other variables except Y_{t}. Second, the "unrestricted" AR model, which includes all variables X_{t}, Y_{t}, and Z_{t} [1]. We consider the restricted and unrestricted system as a MVAR system defined over the components that comes from a Volterra-Wiener expansion with Laguerre polynomials: $$x_t={\mathbf{A}}\big(\,\mathcal{L}^{(m)}(x)\oplus\mathcal{L}^{(m)}(z)\,\big)+{\varepsilon_t}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(restricted model)}$$ $$x_t={\mathbf{A'}}\big(\,\mathcal{L}^{(m)}(x)\oplus\mathcal{L}^{(m)}(y)\oplus\mathcal{L}^{(m)}(z)\,\big)+{\varepsilon_t'}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(unrestricted model)}$$ where the discrete-time Volterra-Wiener decomposition with Laguerre polynomials $$$\mathcal{L}^{(m)}(\cdot)$$$ over the discrete time signal $$${x_t}$$$ is defined as: $$\mathcal{L}^{(m)}(x) = \sum_{n=1}^N \phi_m(n) ( x_{N-n} - x_{N-n-1}) $$ The $$$m^{\text{th}}$$$-order, discrete time, orthonormal Laguerre polynomial $$$\phi_m(n)$$$ is defined as $$\phi_m(n) = \alpha^\frac{n-m}{2} (1-\alpha)^\frac{1}{2}\sum_{j=0}^m (-1)^{j}\binom{n}{j}\binom{m}{j}\alpha^{m-j}(1-\alpha)^j$$ where parameter $$$\alpha\,(0<\alpha<1)$$$ determines the rate of exponential asymptotic decline of $$$phi_m(n)$$$ [2].

The Null-hypothesis that Yt does not cause Xt., conditioned to Zt, is rejected if the $$$f_\text{ratio}$$$ of the residual sum of squares (RSS) $$f_\text{ratio}=\frac{RSS_\text{r}-RSS_\text{ur}}{RSS_\text{ur}}\frac{N_\text{obs}-2m}{m}$$ is extreme with respect of its parent distribution ($$$\chi2$$$-distribution with $$$N_\text{obs}-2m$$$ and $$$m$$$ degrees of freedom [3].

1. Seth, A.K., A.B. Barrett, and L. Barnett, Granger causality analysis in neuroscience and neuroimaging. J Neurosci, 2015. 35(8): p. 3293-7.

2. Watanabe, A. and Stark, L. Kernel method for nonlinear analysis: Identification of a biological control system. Mathematical Biosciences 27.1 (1975): 99-108.

3. Bressler, S.L. and A.K. Seth, Wiener-Granger causality: a well established methodology. Neuroimage, 2011. 58(2): p. 323-9.

4. Corbetta, M. and Gordon L.S., Control of goal-directed and stimulus-driven attention in the brain. Nature reviews neuroscience 3.3 (2002): 201-215.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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