# Conditional Granger causality in the frequency domain

Conditional Granger causality is a derivative of spectral Granger causality that is computed over a triplet of channels (or blocks of channels). It provides the advantage that for this triplet, it allows to differentiate between a delayed parallel drive from sources *A* to be *B* and *C* and a sequential drive from *A* to *B* to *C*.

This example illustrates the simulation and base analysis of the paper
Chen, Y., Bressler, S. L., & Ding, M. (2006). Frequency decomposition of conditional Granger causality and application to multivariate neural field potential data. *Journal of neuroscience methods, 150(2), 228-237*.

See also: the connectivity tutorial.

## Setup and simulating the data sets

First, define parameters under which samples should be simulated.

```
simcfg = [];
simcfg.ntrials = 500;
simcfg.triallength = 1;
simcfg.fsample = 200;
simcfg.nsignal = 3;
simcfg.method = 'ar';
```

We want to simulate a system with three signals. Their noise is modeled as white noise processes with zero mean and standard deviations

*σ _{1}^{2}=1*

*σ*

_{2}^{2}=0.2*σ*.

_{3}^{2}=0.3They will have the covariances ζ, η and ε. We also require a paramters μ=0.5.

```
parmeters of the model itself
mu = 0.5;
absnoise = [ 1.0 0.2 0.3 ];
```

First, we generate the sample for the case of sequential driving. We want to incorporate the system

*x(t) = ζ(t)*

*y(t) = x(t-1) + η(t)*

*z(t) = μ⋅z(t-1) + y(t-1) + ε(t)*,

which we can do like this:

```
params(i,j,k): j -> i at t=k
simcfg.params(:,:,1) = [ 0 0 0;
1.0 0 0;
0 1.0 mu];
```

Note that the matrix representation for the covariance reads from columns to row, other than the MVAR-model is read intuitively. But we still need to hand the parameters of the noise to the model:

```
paper defines stds, not cov:
simcfg.noisecov = diag(absnoise.^2);
data2 = ft_connectivitysimulation(simcfg);
```

Now create sample data for the case of differentially delayed driving,

*x(t) = ζ(t)*

*y(t) = x(t-1) + η(t)*

*z(t) = μ⋅z(t-1) + x(t-2) + ε*,

which we can write as

```
simcfg.params(:,:,1) = [ 0 0 0;
1.0 0 0;
0 0 mu];
simcfg.params(:,:,2) = [ 0 0 0;
0 0 0;
1.0 0 0];
```

We build the actual MVAR-representation…

```
data1 = ft_connectivitysimulation(simcfg);
```

… and have a first look at the data:

```
figure
plot(data.time{1}, data.trial{1})
legend(data1.label)
xlabel('time (s)')
```

Don’t be confused that we started with data2 and conclude with data1. This is just to maintain the order the systems have in the paper.

## MVAR model frequency analysis

We generate spectral representations from the MVAR representations we defined with data1 and data2. After all, we want to compute spectral Granger causality. Fast Fourier is a good starting point.

```
freq = [];
freq.freqcfg = [];
freq.freqcfg.method = 'mtmfft';
freq.freqcfg.output = 'fourier';
freq.freqcfg.tapsmofrq = 2;
freqdata1 = ft_freqanalysis(freq.freqcfg, data1);
freqdata2 = ft_freqanalysis(freq.freqcfg, data2);
```

## “Regular” Granger causality

Let first compute regular bivariate Granger causality, as this makes the difference clear to what we want.

```
grangercfg = [];
grangercfg.method = 'granger';
grangercfg.granger.conditional = 'no';
grangercfg.granger.sfmethod = 'bivariate';
gdata = [];
gdata.g1_bivar_reg = ft_connectivityanalysis(grangercfg, freqdata1);
gdata.g2_bivar_reg = ft_connectivityanalysis(grangercfg, freqdata2);
```

## Multivariate conditional Granger causality

However, we clearly want a multivariate approach. Also, we need to define channel combinations, as we now require triplets of inputs.

```
grangercfg.granger.conditional = 'yes';
grangercfg.channelcmb = {'signal001', 'signal002', 'signal003'};
grangercfg.granger.sfmethod = 'multivariate';
grangercfg.granger.conditional = 'yes';
block-wise causality
grangercfg.granger.block(1).name = freqdata1.label{1};
grangercfg.granger.block(1).label = freqdata1.label(1);
grangercfg.granger.block(2).name = freqdata1.label{2};
grangercfg.granger.block(2).label = freqdata1.label(2);
grangercfg.granger.block(3).name = freqdata1.label{3};
grangercfg.granger.block(3).label = freqdata1.label(3);
gdata.g1_multi_reg_conditional = ft_connectivityanalysis(grangercfg, freqdata1);
gdata.g2_multi_reg_conditional = ft_connectivityanalysis(grangercfg, freqdata2);
```

## Evaluation

The label combinations are 6x2 cell arrays, containing all 2-permutations
tuplets from the channels. How to interpret this?
Is the combination a, b representing F_{a→b|c}?
Let’s check this. In scenario 2, we should clearly see a higher causality
from 1→3 | 2 than in scenario 1 of the differentially delayed
drive. This corresponds to row 4 in the
gdata.g1_multi_reg_conditional.labelcmb.
So, let’s compare the labelcmb 1, 3 in both scenarios:

```
scenario1_mean = mean(gdata.g1_multi_reg_conditional.grangerspctrm(4, :));
scenario2_mean = mean(gdata.g2_multi_reg_conditional.grangerspctrm(4, :));
```