Peer review of: Partial cross mapping eliminates indirect causal influences (2020)



Reviewers' comments: 
 
Reviewer #3 (Remarks to the Author): 
 
The authors have substantially improved the manuscript. However, there are several remaining points 
that need to be addressed before this referee can recommend the paper for publication. 
 
1.Universality. The authors have made a great effort to indicate the originality of the approach they 
introduce. However, it seems that they have omitted discussion on some of the previous works on the 
problem of direct-indirect interactions, which is quite a well-studied topic. For example, works like 
i. Schelter, Winterhalder, Hellwig, Guschlbauer, Lücking, and Timmer, Direct or indirect? Graphical 
models for neural oscillators, Journal of Physiology-Paris(2006), 99 (1), 37-46 
ii. Nawrath, Romano, Thiel, Kiss, Wickramasinghe, Timmer, Kurths, and Schelter, Distinguishing Direct 
from Indirect Interactions in Oscillatory Networks with Multiple Time Scales, Phys Rev Lett (2010), 
104, 038701, 10.1103/PhysRevLett.104.038701 
are quite relevant to the work presented in the manuscript, and the authors should carefully consider 
and discuss the novelty of their approach in comparison to these earlier ideas. 
 
2. Comparison with the DBI. In general, it is easy to agree that the proposed PCM and DBI are 
complementary. However, the comparison and the arguments in Table S5 are rather 'cherry-picking' 
arguments on the side of PCM. Looking at the table in detail - 
i. 'Not require, a priori, knowledge of model equations' - this is true in general. The DBI does not 
require specific knowledge for oscillators (as the model is based on Fourier series), yet, in general, it is 
a model-based method. 
ii. 'Infer coupling functions using function basis set' - indeed; yet, not only coupling functions but also 
other dynamical mechanisms too. In this respect the PCM is a 'functional connectivity' while DBI is a 
'effective connectivity' measure [see e.g. Park& Friston (2013) Science, 342(6158), 1238411] 
iii. 'Distinguish direct couplings from indirect ones', true. Though DBI is not designed for this problem, 
this is the complementary aspect. 
iv. Here one very important aspect is left out - the DBI is a stochastic measure and can infer the noise 
dynamics, while PCM is not. 
v. 'Infer structures with slowly-switched durations' - this is not true for PCM in general. The slowly-
switched duration is a relative term (how slow? in respect of what?), in this respect. During the 
swiching from moderate to fast the PCM will not work by definition, as it is based on the Takens-Mane 
theorem. DBI will perform quite satisfactorily in such cases by definition (see e.g. fig5 in Duggento et 
al, PRE 86(6), 061126, 2012). 
So, either delete this argument all-together, or have two rows/arguments where for slow (but define 
what slow means) both PCM and DBI are fine, and one for moderate and fast where only DBI is fine. 
 
3. The writing style. There is some degree of aggression in the writing style - especially in the main 
document (though not in the response letter, at all), and the authors may consider editing it to 
smooth it down.