Description length guided nonlinear unified Granger causality analysis

Abstract

Most Granger causality analysis (GCA) methods still remain a two-stage scheme guided by different mathematical theories; both can actually be viewed as the same generalized model selection issues. Adhering to Occam's razor, we present a unified GCA (uGCA) based on the minimum description length principle. In this research, considering the common existence of nonlinearity in functional brain networks, we incorporated the nonlinear modeling procedure into the proposed uGCA method, in which an approximate representation of Taylor's expansion was adopted. Through synthetic data experiments, we revealed that nonlinear uGCA was obviously superior to its linear representation and the conventional GCA. Meanwhile, the nonlinear characteristics of high-order terms and cross-terms would be successively drowned out as noise levels increased. Then, in real fMRI data involving mental arithmetic tasks, we further illustrated that these nonlinear characteristics in fMRI data may indeed be drowned out at a high noise level, and hence a linear causal analysis procedure may be sufficient. Next, involving autism spectrum disorder patients data, compared with conventional GCA, the network property of causal connections obtained by uGCA methods appeared to be more consistent with clinical symptoms.

Keywords: Description length; Functional MRI; Nonlinear modeling; Unified Granger causality analysis.

Figure 1.

The obtained causal connections. Data length was 300, simulation number was 1,000. ‘Low’ denoted a low noise level (variance = 0.2), the noise variance in ‘Mid’ (‘high’) was 0.6 (1.2).

Figure 2.

The comparison between linear data and nonlinear data with the high-order term (cross-term).

Figure 3.

The relationships of simulation data sets in the six-node network.

Figure 4.

The obtained causal connections. The L denotes data length, its noise variance variance = 0.4.

Figure 5.

Mental arithmetic of CSA control state under the two stimuli. The activation regions were processed by SPM12; the control state meant that the sample was in rest state. (A) CSA control state under visual stimulus. (B) CSA control state under auditory stimulus (p < 0.0001, uncorrected).

Figure 6.

Mental arithmetic networks under different stimuli obtained by nonlinear uGCA and nonlinear conventional GCA.

Figure 7.

The obtained network similarity under the same stimulus between using nonlinear modeling and linear modeling.

Figure 8.

Causal connection matrices of social brain network identified by nonlinear uGCA and nonlinear conventional GCA.

Figure 9.

The out-degree and in-degree of subnetwork in social brain network within nonlinear uGCA and nonlinear conventional GCA. The four subnetworks are limbic, high-level, visual-sensory, and intermediate network.

Figure 10.

The small-world property of social brain networks obtained by nonlinear uGCA and nonlinear conventional GCA.