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By definition, Granger causality is a measure of linear dependence, which tests whether the variance of error for a [[linear autoregressive model]] estimation of a signal (A) can be reduced when adding a linear model estimation of a second signal (B). If this is true, signal B has a Granger causal effect on the first signal A, i.e., independent information of the past of B improves the prediction of A above and beyond the information contained in the past of A alone. The term independent is emphasized because it creates some interesting properties for GC, such as that it is invariant under rescaling of A and B, as well as the addition of a multiple of A to B. The measure of Granger Causality is nonnegative, and zero when there is no Granger causality(Geweke, 1982). | By definition, Granger causality is a measure of linear dependence, which tests whether the variance of error for a [[linear autoregressive model]] estimation of a signal (A) can be reduced when adding a linear model estimation of a second signal (B). If this is true, signal B has a Granger causal effect on the first signal A, i.e., independent information of the past of B improves the prediction of A above and beyond the information contained in the past of A alone. The term independent is emphasized because it creates some interesting properties for GC, such as that it is invariant under rescaling of A and B, as well as the addition of a multiple of A to B. The measure of Granger Causality is nonnegative, and zero when there is no Granger causality(Geweke, 1982). |
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The main advantage of Granger Causality is that it is an asymmetrical measure, in that it can dissociate between A->B versus B->A. It is important to note however that though the directionality of Granger Causality is a step closer towards measuring effective connectivity compared to symmetrical measures, it should still not be confused with “true causality”. Effective connectivity estimates the effective mechanism generating the observed data (model based approach), whereas GC is a measure of causal effect based on prediction, i.e., how well the model is improved when taking variables into account that are interacting (data-driven approach) (Barrett and Barnett, 2013). The difference with causality is best illustrated when there are more variables interacting in a system than those taken into account in the model. For example, if a variable C is causing both A and B, but with a smaller delay for B than for A, then a GC measure between A and B would show a non-zero GC for B->A, even though B is not truly causing A (Bressler and Seth, 2011). | The main advantage of Granger Causality is that it is an asymmetrical measure, in that it can dissociate between A->B versus B->A. It is important to note however that though the directionality of Granger Causality is a step closer towards measuring effective connectivity compared to symmetrical measures, it should still not be confused with “true causality”. Effective connectivity estimates the effective mechanism generating the observed data (model based approach), whereas GC is a measure of causal effect based on prediction, i.e., how well the model is improved when taking variables into account that are interacting (data-driven approach) (Barrett and Barnett, 2013). |
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{{{#!latex $\gamma$ }}} |
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The difference with causality is best illustrated when there are more variables interacting in a system than those taken into account in the model. For example, if a variable C is causing both A and B, but with a smaller delay for B than for A, then a GC measure between A and B would show a non-zero GC for B->A, even though B is not truly causing A (Bressler and Seth, 2011). | |
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== GC in Time Domain == Even though GC has been extended for nonlinear, multivariate and time-varying conditions, in this tutorial we will stick to the basic case, which is a linear, bivariate and stationary model defined in both the time and spectral domain. In the time domain this can be represented in the following way. If x represents a signal that can be modeled using a linear autoregressive model estimation (AR model) in the following two ways: |
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{{{#!latex \begin{equation} x(t) = \sum_{k=1}^{p} [A_{k}x(t-k)] +e_1 \\ x(t) = \sum_{k=1}^{p} [A_{k}x(t-k)+B_{k}y(t-k)] +e_2 \end{equation} }}} |
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Where $p$ represent the amount of past information that will be included in the prediction of the future sample, and is called the model order. In these two equations, the first models $x$ using the past (and present) of only itself whereas the second includes the past (and present) of a second signal $y$. Note that when only past measures of signals are taken into account ($k\geq 1$), the model ignores simultaneous connectivity, which makes it less susceptible to volume conduction (Cohen, 2014). Then, according to the original formulation of GC, the measure of GC is defined as follows: |
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\begin{equation} \label{eq2} F_{y \to x}=ln( \frac{var(e_1)}{var(e_2)}) \end{equation} |
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Granger causality - Mathematical Background
Granger causality (GC) is a method of functional connectivity, adapted by [[|Clive Granger]] in the 1960s, but later refined by John Geweke in the [[|form that is used today]]. Granger causality was originally formulated in economics, but has caught the attention of the neuroscience community in recent years. Before this, neuroscience traditionally relied on lesions and applying stimuli on a part of the nervous system to study it’s effect on another part. However, Granger causality made it possible to estimate the statistical influence without requiring direct intervention ?(Bressler and Seth, 2011).
By definition, Granger causality is a measure of linear dependence, which tests whether the variance of error for a ?linear autoregressive model estimation of a signal (A) can be reduced when adding a linear model estimation of a second signal (B). If this is true, signal B has a Granger causal effect on the first signal A, i.e., independent information of the past of B improves the prediction of A above and beyond the information contained in the past of A alone. The term independent is emphasized because it creates some interesting properties for GC, such as that it is invariant under rescaling of A and B, as well as the addition of a multiple of A to B. The measure of Granger Causality is nonnegative, and zero when there is no Granger causality(Geweke, 1982).
The main advantage of Granger Causality is that it is an asymmetrical measure, in that it can dissociate between A->B versus B->A. It is important to note however that though the directionality of Granger Causality is a step closer towards measuring effective connectivity compared to symmetrical measures, it should still not be confused with “true causality”. Effective connectivity estimates the effective mechanism generating the observed data (model based approach), whereas GC is a measure of causal effect based on prediction, i.e., how well the model is improved when taking variables into account that are interacting (data-driven approach) (Barrett and Barnett, 2013).
The difference with causality is best illustrated when there are more variables interacting in a system than those taken into account in the model. For example, if a variable C is causing both A and B, but with a smaller delay for B than for A, then a GC measure between A and B would show a non-zero GC for B->A, even though B is not truly causing A (Bressler and Seth, 2011).
IMAGE
GC in Time Domain
Even though GC has been extended for nonlinear, multivariate and time-varying conditions, in this tutorial we will stick to the basic case, which is a linear, bivariate and stationary model defined in both the time and spectral domain. In the time domain this can be represented in the following way. If x represents a signal that can be modeled using a linear autoregressive model estimation (AR model) in the following two ways:
Where $p$ represent the amount of past information that will be included in the prediction of the future sample, and is called the model order. In these two equations, the first models $x$ using the past (and present) of only itself whereas the second includes the past (and present) of a second signal $y$. Note that when only past measures of signals are taken into account ($k\geq 1$), the model ignores simultaneous connectivity, which makes it less susceptible to volume conduction (Cohen, 2014). Then, according to the original formulation of GC, the measure of GC is defined as follows:
\begin{equation} \label{eq2} F_{y \to x}=ln( \frac{var(e_1)}{var(e_2)}) \end{equation}
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Temporal resolution: the high time resolution offered by MEG/EEG and intracranial EEG allows for a very powerful application of GC and also offers the important advantage of spectral analysis.
Stationarity: the GC methods described so far are all based on AR models, and therefore assume stationarity of the signal (constant auto-correlation over time). However, neuroscience data, especially task-based data such as event-related potentials are mostly nonstationary. There are two possible approaches to solve this problem. The first is to apply methods such as differencing, filtering, and smoothing to make the data stationary (see a recommendation for time domain GC). Dynamical changes in the connectivity profile cannot be detected with the first approach. The second approach is to turn to versions of GC that have been adapted for nonstationary data, either by using a non-parametric estimation of GC or through measures of time-varying GC, which estimate dynamic parameters with adaptive or short-time window methods (Bressler and Seth, 2011).
Number of variables: Granger causality is very time-consuming in the multivariate case for many variables (O(m^2) where m represents the number of variables). Since each connection pair results in two values, there will also be a large number of statistical comparisons that need to be controlled for. When performing GC in the spectral domain, this number increases even more as statistical tests have to be performed per frequency. Therefore, it is usually recommended to select a limited number of ROIs or electrodes based on some hypothesis found in previous literature, or on some initial processing with a more simple and less computationally heavy measure of connectivity.
Pre-processing: The influence of pre-processing steps such as filtering and smoothing on GC estimates is a crucial issue. Studies have generally suggested to limit filtering only for artifact removal or to improve the stationarity of the data but cautioned against band-pass filtering to isolate causal influence within a specific frequency band (Barnett and Seth, 2011).
Volume Conduction: Granger causality can be performed both in the scalp domain or in the source domain. Though spectral domain GC generally does not incorporate present values of the signals in the model, it is still not immune from spurious connectivity measures due to volume conduction (for a discussion see (Steen et al., 2016)). Therefore, it is recommended to reduce the problem of signal mixing using additional processing steps such as performing source localization and doing connectivity in the source domain.
Data length: because of the extent of parameters that need to be estimated, the number of data points should be sufficient for a good fit of the model. This is especially true for windowing approaches, where data is cut into smaller epochs. A rule of thumb is that the number of estimated parameters should be at least (~10) several times smaller than the number of data points.