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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

The GC sampling structure as sampling interval length tends to zero. Fx→y (red), Fy→x (cyan) obtained from voltage time series and Fx→y (red dash), Fy→x (cyan dash) obtained from spike train time series with sampling interval length τ. Note that, by the asymptotic distribution theory of GC (Geweke, 1982), the estimator of a directed GC has a bias , where p is the regression order and n is the length of the discrete time series. We have used the Bayesian information criterion (Schwarz, 1978) to determine the regression order p and have subtracted this type of biases in the figure (see Appendix C in Supplementary Material for details). The time series are generated by the I&F network whose topology is shown in Figure 1A with parameters ν = 1 ms−1, λ = 0.0177, s = 0.02.
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Figure 3: The GC sampling structure as sampling interval length tends to zero. Fx→y (red), Fy→x (cyan) obtained from voltage time series and Fx→y (red dash), Fy→x (cyan dash) obtained from spike train time series with sampling interval length τ. Note that, by the asymptotic distribution theory of GC (Geweke, 1982), the estimator of a directed GC has a bias , where p is the regression order and n is the length of the discrete time series. We have used the Bayesian information criterion (Schwarz, 1978) to determine the regression order p and have subtracted this type of biases in the figure (see Appendix C in Supplementary Material for details). The time series are generated by the I&F network whose topology is shown in Figure 1A with parameters ν = 1 ms−1, λ = 0.0177, s = 0.02.

Mentions: As demonstrated through the oscillation phenomenon of GC above, clearly, we cannot choose sampling interval length arbitrarily in the application of GC analysis. Then, an important issue arises: what should be the criteria of choosing a correct sampling interval length to obtain discrete time series in measurement for reliable GC analysis. For a natural discrete-time dynamics, we can simply use their intrinsic intervals, e.g., for discrete time series generated by an autoregressive model, and we can then obtain reliable GC values for causal interpretations because the entire information is incorporated in the causal analysis. However, most physical quantities are continuous in time, one does not have particular intrinsic intervals for sampling. In order to obtain reliable GC values, one possible scenario, similar to the discrete case, is that it is always better if one uses more finely sampled time series because apparently more information is incorporated for causality determination. To examine this scenario, we study the convergence property of the GC sampling structure as the sampling interval length τ tends to 0. The corresponding numerical results are shown in Figure 3 for a two-neuron I&F network.


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)

The GC sampling structure as sampling interval length tends to zero. Fx→y (red), Fy→x (cyan) obtained from voltage time series and Fx→y (red dash), Fy→x (cyan dash) obtained from spike train time series with sampling interval length τ. Note that, by the asymptotic distribution theory of GC (Geweke, 1982), the estimator of a directed GC has a bias , where p is the regression order and n is the length of the discrete time series. We have used the Bayesian information criterion (Schwarz, 1978) to determine the regression order p and have subtracted this type of biases in the figure (see Appendix C in Supplementary Material for details). The time series are generated by the I&F network whose topology is shown in Figure 1A with parameters ν = 1 ms−1, λ = 0.0177, s = 0.02.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 3: The GC sampling structure as sampling interval length tends to zero. Fx→y (red), Fy→x (cyan) obtained from voltage time series and Fx→y (red dash), Fy→x (cyan dash) obtained from spike train time series with sampling interval length τ. Note that, by the asymptotic distribution theory of GC (Geweke, 1982), the estimator of a directed GC has a bias , where p is the regression order and n is the length of the discrete time series. We have used the Bayesian information criterion (Schwarz, 1978) to determine the regression order p and have subtracted this type of biases in the figure (see Appendix C in Supplementary Material for details). The time series are generated by the I&F network whose topology is shown in Figure 1A with parameters ν = 1 ms−1, λ = 0.0177, s = 0.02.
Mentions: As demonstrated through the oscillation phenomenon of GC above, clearly, we cannot choose sampling interval length arbitrarily in the application of GC analysis. Then, an important issue arises: what should be the criteria of choosing a correct sampling interval length to obtain discrete time series in measurement for reliable GC analysis. For a natural discrete-time dynamics, we can simply use their intrinsic intervals, e.g., for discrete time series generated by an autoregressive model, and we can then obtain reliable GC values for causal interpretations because the entire information is incorporated in the causal analysis. However, most physical quantities are continuous in time, one does not have particular intrinsic intervals for sampling. In order to obtain reliable GC values, one possible scenario, similar to the discrete case, is that it is always better if one uses more finely sampled time series because apparently more information is incorporated for causality determination. To examine this scenario, we study the convergence property of the GC sampling structure as the sampling interval length τ tends to 0. The corresponding numerical results are shown in Figure 3 for a two-neuron I&F network.

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