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Analysis of sampling artifacts on the Granger causality analysis for topology extraction of neuronal dynamics.

Zhou D, Zhang Y, Xiao Y, Cai D - Front Comput Neurosci (2014)

Bottom Line: Without properly taking this issue into account, GC analysis may produce unreliable conclusions about causal influence when applied to empirical data.We use idealized linear models to illustrate possible mechanisms underlying these phenomena and to gain insight into the general spectral structures that give rise to these sampling effects.Finally, we present an approach to circumvent these sampling artifacts to obtain reliable GC values.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University Shanghai, China.

ABSTRACT
Granger causality (GC) is a powerful method for causal inference for time series. In general, the GC value is computed using discrete time series sampled from continuous-time processes with a certain sampling interval length τ, i.e., the GC value is a function of τ. Using the GC analysis for the topology extraction of the simplest integrate-and-fire neuronal network of two neurons, we discuss behaviors of the GC value as a function of τ, which exhibits (i) oscillations, often vanishing at certain finite sampling interval lengths, (ii) the GC vanishes linearly as one uses finer and finer sampling. We show that these sampling effects can occur in both linear and non-linear dynamics: the GC value may vanish in the presence of true causal influence or become non-zero in the absence of causal influence. Without properly taking this issue into account, GC analysis may produce unreliable conclusions about causal influence when applied to empirical data. These sampling artifacts on the GC value greatly complicate the reliability of causal inference using the GC analysis, in general, and the validity of topology reconstruction for networks, in particular. We use idealized linear models to illustrate possible mechanisms underlying these phenomena and to gain insight into the general spectral structures that give rise to these sampling effects. Finally, we present an approach to circumvent these sampling artifacts to obtain reliable GC values.

No MeSH data available.


Related in: MedlinePlus

Contrast between spectra and GC sampling structures for the first class. Magnitude of Sxy (/Sxy/) vs. frequency f (f = ) (black) for (A) Case 1.1. (B) Case 1.2. (C) Case 1.3. (D) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), asymptotic expressions Equation (2) in Supplementary Material (red plus) and Equation (18) (green cross) vs. sampling interval k for Case 1.1. (E) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), Equation (5) in Supplementary Material (red plus) and Equation (19) (green cross) vs. sampling interval k for Case 1.2. (F) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), Equation (9) in Supplementary Material (red plus) and Equation (21) (green cross) vs. sampling interval k for Case 1.3. Insets are corresponding log-linear plots of GC vs. sampling interval length k. The exponential decay is clearly seen in the insets. The parameters are C = 104, τd = 0.05, and β = .
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Figure 4: Contrast between spectra and GC sampling structures for the first class. Magnitude of Sxy (/Sxy/) vs. frequency f (f = ) (black) for (A) Case 1.1. (B) Case 1.2. (C) Case 1.3. (D) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), asymptotic expressions Equation (2) in Supplementary Material (red plus) and Equation (18) (green cross) vs. sampling interval k for Case 1.1. (E) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), Equation (5) in Supplementary Material (red plus) and Equation (19) (green cross) vs. sampling interval k for Case 1.2. (F) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), Equation (9) in Supplementary Material (red plus) and Equation (21) (green cross) vs. sampling interval k for Case 1.3. Insets are corresponding log-linear plots of GC vs. sampling interval length k. The exponential decay is clearly seen in the insets. The parameters are C = 104, τd = 0.05, and β = .

Mentions: In Figure 4A, which displays spectrum /Sxy/, we can observe that the magnitude of Sxy concentrates near 0 which signifies that there are no oscillations induced by the coupling b(L). In Figure 4D, which displays the GC sampling structure for this case, clearly, there are no oscillations. It can also be seen that GC obtained through the asymptotic result, [Equation (2) in Supplementary Material], agrees very well with the numerically obtained GC for all k's and the asymptotic formula Equation (18) approximates GC rather well when k is large. Therefore, we can conclude that if there are no oscillations in the coupling b(L), as signified in the peak frequency of /Sxy/ near 0 and if Sxx and Syy are constant (i.e., both are white), then F(k)y→x does not oscillate with respect to k.


Analysis of sampling artifacts on the Granger causality analysis for topology extraction of neuronal dynamics.

Zhou D, Zhang Y, Xiao Y, Cai D - Front Comput Neurosci (2014)

Contrast between spectra and GC sampling structures for the first class. Magnitude of Sxy (/Sxy/) vs. frequency f (f = ) (black) for (A) Case 1.1. (B) Case 1.2. (C) Case 1.3. (D) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), asymptotic expressions Equation (2) in Supplementary Material (red plus) and Equation (18) (green cross) vs. sampling interval k for Case 1.1. (E) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), Equation (5) in Supplementary Material (red plus) and Equation (19) (green cross) vs. sampling interval k for Case 1.2. (F) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), Equation (9) in Supplementary Material (red plus) and Equation (21) (green cross) vs. sampling interval k for Case 1.3. Insets are corresponding log-linear plots of GC vs. sampling interval length k. The exponential decay is clearly seen in the insets. The parameters are C = 104, τd = 0.05, and β = .
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4115622&req=5

Figure 4: Contrast between spectra and GC sampling structures for the first class. Magnitude of Sxy (/Sxy/) vs. frequency f (f = ) (black) for (A) Case 1.1. (B) Case 1.2. (C) Case 1.3. (D) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), asymptotic expressions Equation (2) in Supplementary Material (red plus) and Equation (18) (green cross) vs. sampling interval k for Case 1.1. (E) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), Equation (5) in Supplementary Material (red plus) and Equation (19) (green cross) vs. sampling interval k for Case 1.2. (F) Comparison of F(k)y → x obtained through numerical solution Equation (15) (blue), Equation (9) in Supplementary Material (red plus) and Equation (21) (green cross) vs. sampling interval k for Case 1.3. Insets are corresponding log-linear plots of GC vs. sampling interval length k. The exponential decay is clearly seen in the insets. The parameters are C = 104, τd = 0.05, and β = .
Mentions: In Figure 4A, which displays spectrum /Sxy/, we can observe that the magnitude of Sxy concentrates near 0 which signifies that there are no oscillations induced by the coupling b(L). In Figure 4D, which displays the GC sampling structure for this case, clearly, there are no oscillations. It can also be seen that GC obtained through the asymptotic result, [Equation (2) in Supplementary Material], agrees very well with the numerically obtained GC for all k's and the asymptotic formula Equation (18) approximates GC rather well when k is large. Therefore, we can conclude that if there are no oscillations in the coupling b(L), as signified in the peak frequency of /Sxy/ near 0 and if Sxx and Syy are constant (i.e., both are white), then F(k)y→x does not oscillate with respect to k.

Bottom Line: Without properly taking this issue into account, GC analysis may produce unreliable conclusions about causal influence when applied to empirical data.We use idealized linear models to illustrate possible mechanisms underlying these phenomena and to gain insight into the general spectral structures that give rise to these sampling effects.Finally, we present an approach to circumvent these sampling artifacts to obtain reliable GC values.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University Shanghai, China.

ABSTRACT
Granger causality (GC) is a powerful method for causal inference for time series. In general, the GC value is computed using discrete time series sampled from continuous-time processes with a certain sampling interval length τ, i.e., the GC value is a function of τ. Using the GC analysis for the topology extraction of the simplest integrate-and-fire neuronal network of two neurons, we discuss behaviors of the GC value as a function of τ, which exhibits (i) oscillations, often vanishing at certain finite sampling interval lengths, (ii) the GC vanishes linearly as one uses finer and finer sampling. We show that these sampling effects can occur in both linear and non-linear dynamics: the GC value may vanish in the presence of true causal influence or become non-zero in the absence of causal influence. Without properly taking this issue into account, GC analysis may produce unreliable conclusions about causal influence when applied to empirical data. These sampling artifacts on the GC value greatly complicate the reliability of causal inference using the GC analysis, in general, and the validity of topology reconstruction for networks, in particular. We use idealized linear models to illustrate possible mechanisms underlying these phenomena and to gain insight into the general spectral structures that give rise to these sampling effects. Finally, we present an approach to circumvent these sampling artifacts to obtain reliable GC values.

No MeSH data available.


Related in: MedlinePlus