AKS comment: This page contains reviews and my responses as of June 20, 2007.
This is a nice contribution, but there is definitely room to improve it. The description by Granger is obviously good, and so it the section after that (" Mathematical formulation").
The section about the spectral approach has fairly heavy notation, and a number of derivations that are used to obtain a certain result. But why is that result important or useful? Could one have obtained a similar result in the time domain? What advantages, if any, are obtained by going in the frequency domain? If this information is not given, this section is hardly useful (it just turns the reader off because of the math, with no gain).
AKS response: I have qualified the maths here with the phrase 'for completeness' and referred to the Application in Neuroscience section, which explains why frequency decompositions are particularly useful. I also give an intuitive description of what spectral G-causality reveals, in terms of power. By definition, one cannot get frequency information by remaining in the time domain.
One feels that there are also other important issues that have not been touched on. What are the possible confounds in the computation of G-causality? What are its pros and cons compared to other techniques?
AKS response: Substantial new material addresses this point. There is a discussion of conditional G-causality which describes potential confounds due to indirect interactions or differently delayed inputs. Pros and cons w.r.t other techniques are now addressed more explicitly, and a new Concluding Remarks section summarizes these points.
In the context of neuroscience, for example, one of the key issues is whether it can be used to infer connectivity, or at least functional connectivity. Are there examples that can be given, at least of toy systems, which might illustrate the pros and cons of this technique?
AKS response: A new paragraph within Application in Neuroscience addresses this point.
Overall, the concept of Granger Causality is clearly, and timely, described. But I do have a few suggests that hopefully would add some improvements:
1. The math description seems too heavy, which would scare away people who are not comfortable with math.
AKS response: See response to first review.
2. Kaminski et al. 2001 showed the equivalence between Granger Causality and DTF (the directed transfer function). This point is not reflected in the current version.
AKS response: Fixed.
3. Limitations and extensions is a nice section. Multivariate Granger Causality (currently bivariate) would deserve some attention.
AKS response: The new discussion of conditional G-causality serves this purpose.
I do not want to make any changes to the article but I am not sure if the section on the use of spectral methods is correct as I can reverse the direction of time and get the same spectrum.
AKS response: I have checked through the material as closely as I can and do not see any error. The analysis is based closely on Brovelli et al. (2004) and Kaminski et al. (2001) but has roots in the original work of Geweke. I have also checked with other workers in the field who use the spectral version regularly, they didn't seem to find a problem. But if there is a problem I will be very pleased to address it in a future revision, of course.
some comments were sent directly to the author.
User 5 (User:): Transfer entropy comment
The article states that Transfer Entropy "... is currently restricted to bivariate situations". I guess more accurate would be "it is currently restricted to bivariate situations, although multivariate extensions have been proposed (Lizier et al., 2011)."
The reference is Lizier, J.T., Heinzle, J., Horstmann, A., Haynes, J.-D. & Prokopenko, M., 2011 Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity, Journal of Computational Neuroscience, 30, 85-107.
AKS response: As of Mar 2011, the article is definitely in need of a substantial revision, which I will work on. Regarding transfer entropy specifically, I and colleagues Barnett and Barrett have now established a formal equivalence which holds in the multivariate and conditional cases (Phys Rev Letters 2009). I will be interested to read the Lizier ref in this context - it looks very interesting!
User 6: Typo in section on Spectral Granger causality?
"\Sigma is the covariance matrix of the residuals E(t)"
Shouldn't that be "\Sigma is the spectral matrix of the residuals E(t)" ? Regards SKM.
User 7: Another Typo?
In this passage: "It is possible to formulate statistical tests for which I now designate as G-causality" I suggest "which" should be "what".